N2.qti

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<questestinterop xmlns="http://www.imsglobal.org/xsd/ims_qtiasiv1p2" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:schemaLocation="http://www.imsglobal.org/xsd/ims_qtiasiv1p2 http://www.imsglobal.org/xsd/ims_qtiasiv1p2p1.xsd">
  <objectbank ident="N2">
    <qtimetadata>
      <qtimetadatafield><fieldlabel>bank_title</fieldlabel><fieldentry>Differential Equations -- N2</fieldentry></qtimetadatafield>
    </qtimetadata>
  <item ident="N2-2998" title="N2 | Euler's method for approximating IVP solutions | ver. 2998"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.2 )" alt="x( 1.2 )" title="x( 1.2 )" data-latex="x( 1.2 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.2 )" alt="y( 1.2 )" title="y( 1.2 )" data-latex="y( 1.2 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -2 \, x^{2} y^{2} - t y^{2} + 3 \hspace{2em} x( 1 )= 2" alt="x'= -2 \, x^{2} y^{2} - t y^{2} + 3 \hspace{2em} x( 1 )= 2" title="x'= -2 \, x^{2} y^{2} - t y^{2} + 3 \hspace{2em} x( 1 )= 2" data-latex="x'= -2 \, x^{2} y^{2} - t y^{2} + 3 \hspace{2em} x( 1 )= 2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= t^{2} x - 2 \, t^{2} y - 1 \hspace{2em} y( 1 )= 1" alt="y'= t^{2} x - 2 \, t^{2} y - 1 \hspace{2em} y( 1 )= 1" title="y'= t^{2} x - 2 \, t^{2} y - 1 \hspace{2em} y( 1 )= 1" data-latex="y'= t^{2} x - 2 \, t^{2} y - 1 \hspace{2em} y( 1 )= 1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.2%20)" alt="x( 1.2 )" title="x( 1.2 )" data-latex="x( 1.2 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.2%20)" alt="y( 1.2 )" title="y( 1.2 )" data-latex="y( 1.2 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-2%20%5C,%20x%5E%7B2%7D%20y%5E%7B2%7D%20-%20t%20y%5E%7B2%7D%20+%203%20%5Chspace%7B2em%7D%20x(%201%20)=%202" alt="x'= -2 \, x^{2} y^{2} - t y^{2} + 3 \hspace{2em} x( 1 )= 2" title="x'= -2 \, x^{2} y^{2} - t y^{2} + 3 \hspace{2em} x( 1 )= 2" data-latex="x'= -2 \, x^{2} y^{2} - t y^{2} + 3 \hspace{2em} x( 1 )= 2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20t%5E%7B2%7D%20x%20-%202%20%5C,%20t%5E%7B2%7D%20y%20-%201%20%5Chspace%7B2em%7D%20y(%201%20)=%201" alt="y'= t^{2} x - 2 \, t^{2} y - 1 \hspace{2em} y( 1 )= 1" title="y'= t^{2} x - 2 \, t^{2} y - 1 \hspace{2em} y( 1 )= 1" data-latex="y'= t^{2} x - 2 \, t^{2} y - 1 \hspace{2em} y( 1 )= 1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.1 )\approx 1.40" alt="x( 1.1 )\approx 1.40" title="x( 1.1 )\approx 1.40" data-latex="x( 1.1 )\approx 1.40"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.1 )\approx 0.900" alt="y( 1.1 )\approx 0.900" title="y( 1.1 )\approx 0.900" data-latex="y( 1.1 )\approx 0.900"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.2 )\approx 1.29" alt="x( 1.2 )\approx 1.29" title="x( 1.2 )\approx 1.29" data-latex="x( 1.2 )\approx 1.29"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.2 )\approx 0.752" alt="y( 1.2 )\approx 0.752" title="y( 1.2 )\approx 0.752" data-latex="y( 1.2 )\approx 0.752"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.1%20)%5Capprox%201.40" alt="x( 1.1 )\approx 1.40" title="x( 1.1 )\approx 1.40" data-latex="x( 1.1 )\approx 1.40"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.1%20)%5Capprox%200.900" alt="y( 1.1 )\approx 0.900" title="y( 1.1 )\approx 0.900" data-latex="y( 1.1 )\approx 0.900"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.2%20)%5Capprox%201.29" alt="x( 1.2 )\approx 1.29" title="x( 1.2 )\approx 1.29" data-latex="x( 1.2 )\approx 1.29"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.2%20)%5Capprox%200.752" alt="y( 1.2 )\approx 0.752" title="y( 1.2 )\approx 0.752" data-latex="y( 1.2 )\approx 0.752"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-2652" title="N2 | Euler's method for approximating IVP solutions | ver. 2652"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.80 )" alt="x( -0.80 )" title="x( -0.80 )" data-latex="x( -0.80 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.80 )" alt="y( -0.80 )" title="y( -0.80 )" data-latex="y( -0.80 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= t^{2} x^{2} + 4 \, t^{2} y^{2} + 1 \hspace{2em} x( -1 )= 0" alt="x'= t^{2} x^{2} + 4 \, t^{2} y^{2} + 1 \hspace{2em} x( -1 )= 0" title="x'= t^{2} x^{2} + 4 \, t^{2} y^{2} + 1 \hspace{2em} x( -1 )= 0" data-latex="x'= t^{2} x^{2} + 4 \, t^{2} y^{2} + 1 \hspace{2em} x( -1 )= 0"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 2 \, x y^{2} + 4 \, t y - 3 \hspace{2em} y( -1 )= -1" alt="y'= 2 \, x y^{2} + 4 \, t y - 3 \hspace{2em} y( -1 )= -1" title="y'= 2 \, x y^{2} + 4 \, t y - 3 \hspace{2em} y( -1 )= -1" data-latex="y'= 2 \, x y^{2} + 4 \, t y - 3 \hspace{2em} y( -1 )= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.80%20)" alt="x( -0.80 )" title="x( -0.80 )" data-latex="x( -0.80 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.80%20)" alt="y( -0.80 )" title="y( -0.80 )" data-latex="y( -0.80 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20t%5E%7B2%7D%20x%5E%7B2%7D%20+%204%20%5C,%20t%5E%7B2%7D%20y%5E%7B2%7D%20+%201%20%5Chspace%7B2em%7D%20x(%20-1%20)=%200" alt="x'= t^{2} x^{2} + 4 \, t^{2} y^{2} + 1 \hspace{2em} x( -1 )= 0" title="x'= t^{2} x^{2} + 4 \, t^{2} y^{2} + 1 \hspace{2em} x( -1 )= 0" data-latex="x'= t^{2} x^{2} + 4 \, t^{2} y^{2} + 1 \hspace{2em} x( -1 )= 0"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%202%20%5C,%20x%20y%5E%7B2%7D%20+%204%20%5C,%20t%20y%20-%203%20%5Chspace%7B2em%7D%20y(%20-1%20)=%20-1" alt="y'= 2 \, x y^{2} + 4 \, t y - 3 \hspace{2em} y( -1 )= -1" title="y'= 2 \, x y^{2} + 4 \, t y - 3 \hspace{2em} y( -1 )= -1" data-latex="y'= 2 \, x y^{2} + 4 \, t y - 3 \hspace{2em} y( -1 )= -1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.90 )\approx 0.500" alt="x( -0.90 )\approx 0.500" title="x( -0.90 )\approx 0.500" data-latex="x( -0.90 )\approx 0.500"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.90 )\approx -0.900" alt="y( -0.90 )\approx -0.900" title="y( -0.90 )\approx -0.900" data-latex="y( -0.90 )\approx -0.900"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.80 )\approx 0.883" alt="x( -0.80 )\approx 0.883" title="x( -0.80 )\approx 0.883" data-latex="x( -0.80 )\approx 0.883"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.80 )\approx -0.795" alt="y( -0.80 )\approx -0.795" title="y( -0.80 )\approx -0.795" data-latex="y( -0.80 )\approx -0.795"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.90%20)%5Capprox%200.500" alt="x( -0.90 )\approx 0.500" title="x( -0.90 )\approx 0.500" data-latex="x( -0.90 )\approx 0.500"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.90%20)%5Capprox%20-0.900" alt="y( -0.90 )\approx -0.900" title="y( -0.90 )\approx -0.900" data-latex="y( -0.90 )\approx -0.900"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.80%20)%5Capprox%200.883" alt="x( -0.80 )\approx 0.883" title="x( -0.80 )\approx 0.883" data-latex="x( -0.80 )\approx 0.883"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.80%20)%5Capprox%20-0.795" alt="y( -0.80 )\approx -0.795" title="y( -0.80 )\approx -0.795" data-latex="y( -0.80 )\approx -0.795"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-7373" title="N2 | Euler's method for approximating IVP solutions | ver. 7373"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.20 )" alt="x( 0.20 )" title="x( 0.20 )" data-latex="x( 0.20 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.20 )" alt="y( 0.20 )" title="y( 0.20 )" data-latex="y( 0.20 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -2 \, t^{2} x^{2} - 2 \, t y^{2} + 3 \hspace{2em} x( 0 )= -2" alt="x'= -2 \, t^{2} x^{2} - 2 \, t y^{2} + 3 \hspace{2em} x( 0 )= -2" title="x'= -2 \, t^{2} x^{2} - 2 \, t y^{2} + 3 \hspace{2em} x( 0 )= -2" data-latex="x'= -2 \, t^{2} x^{2} - 2 \, t y^{2} + 3 \hspace{2em} x( 0 )= -2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -3 \, x^{2} y^{2} - 2 \, t^{2} x - 3 \hspace{2em} y( 0 )= -2" alt="y'= -3 \, x^{2} y^{2} - 2 \, t^{2} x - 3 \hspace{2em} y( 0 )= -2" title="y'= -3 \, x^{2} y^{2} - 2 \, t^{2} x - 3 \hspace{2em} y( 0 )= -2" data-latex="y'= -3 \, x^{2} y^{2} - 2 \, t^{2} x - 3 \hspace{2em} y( 0 )= -2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.20%20)" alt="x( 0.20 )" title="x( 0.20 )" data-latex="x( 0.20 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.20%20)" alt="y( 0.20 )" title="y( 0.20 )" data-latex="y( 0.20 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-2%20%5C,%20t%5E%7B2%7D%20x%5E%7B2%7D%20-%202%20%5C,%20t%20y%5E%7B2%7D%20+%203%20%5Chspace%7B2em%7D%20x(%200%20)=%20-2" alt="x'= -2 \, t^{2} x^{2} - 2 \, t y^{2} + 3 \hspace{2em} x( 0 )= -2" title="x'= -2 \, t^{2} x^{2} - 2 \, t y^{2} + 3 \hspace{2em} x( 0 )= -2" data-latex="x'= -2 \, t^{2} x^{2} - 2 \, t y^{2} + 3 \hspace{2em} x( 0 )= -2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-3%20%5C,%20x%5E%7B2%7D%20y%5E%7B2%7D%20-%202%20%5C,%20t%5E%7B2%7D%20x%20-%203%20%5Chspace%7B2em%7D%20y(%200%20)=%20-2" alt="y'= -3 \, x^{2} y^{2} - 2 \, t^{2} x - 3 \hspace{2em} y( 0 )= -2" title="y'= -3 \, x^{2} y^{2} - 2 \, t^{2} x - 3 \hspace{2em} y( 0 )= -2" data-latex="y'= -3 \, x^{2} y^{2} - 2 \, t^{2} x - 3 \hspace{2em} y( 0 )= -2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.10 )\approx -1.70" alt="x( 0.10 )\approx -1.70" title="x( 0.10 )\approx -1.70" data-latex="x( 0.10 )\approx -1.70"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.10 )\approx -7.10" alt="y( 0.10 )\approx -7.10" title="y( 0.10 )\approx -7.10" data-latex="y( 0.10 )\approx -7.10"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.20 )\approx -2.41" alt="x( 0.20 )\approx -2.41" title="x( 0.20 )\approx -2.41" data-latex="x( 0.20 )\approx -2.41"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.20 )\approx -51.1" alt="y( 0.20 )\approx -51.1" title="y( 0.20 )\approx -51.1" data-latex="y( 0.20 )\approx -51.1"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.10%20)%5Capprox%20-1.70" alt="x( 0.10 )\approx -1.70" title="x( 0.10 )\approx -1.70" data-latex="x( 0.10 )\approx -1.70"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.10%20)%5Capprox%20-7.10" alt="y( 0.10 )\approx -7.10" title="y( 0.10 )\approx -7.10" data-latex="y( 0.10 )\approx -7.10"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.20%20)%5Capprox%20-2.41" alt="x( 0.20 )\approx -2.41" title="x( 0.20 )\approx -2.41" data-latex="x( 0.20 )\approx -2.41"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.20%20)%5Capprox%20-51.1" alt="y( 0.20 )\approx -51.1" title="y( 0.20 )\approx -51.1" data-latex="y( 0.20 )\approx -51.1"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-0204" title="N2 | Euler's method for approximating IVP solutions | ver. 0204"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.20 )" alt="x( 0.20 )" title="x( 0.20 )" data-latex="x( 0.20 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.20 )" alt="y( 0.20 )" title="y( 0.20 )" data-latex="y( 0.20 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -3 \, t x^{2} - t y + 1 \hspace{2em} x( 0 )= 0" alt="x'= -3 \, t x^{2} - t y + 1 \hspace{2em} x( 0 )= 0" title="x'= -3 \, t x^{2} - t y + 1 \hspace{2em} x( 0 )= 0" data-latex="x'= -3 \, t x^{2} - t y + 1 \hspace{2em} x( 0 )= 0"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 2 \, x y^{2} - 3 \, t y - 3 \hspace{2em} y( 0 )= -2" alt="y'= 2 \, x y^{2} - 3 \, t y - 3 \hspace{2em} y( 0 )= -2" title="y'= 2 \, x y^{2} - 3 \, t y - 3 \hspace{2em} y( 0 )= -2" data-latex="y'= 2 \, x y^{2} - 3 \, t y - 3 \hspace{2em} y( 0 )= -2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.20%20)" alt="x( 0.20 )" title="x( 0.20 )" data-latex="x( 0.20 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.20%20)" alt="y( 0.20 )" title="y( 0.20 )" data-latex="y( 0.20 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-3%20%5C,%20t%20x%5E%7B2%7D%20-%20t%20y%20+%201%20%5Chspace%7B2em%7D%20x(%200%20)=%200" alt="x'= -3 \, t x^{2} - t y + 1 \hspace{2em} x( 0 )= 0" title="x'= -3 \, t x^{2} - t y + 1 \hspace{2em} x( 0 )= 0" data-latex="x'= -3 \, t x^{2} - t y + 1 \hspace{2em} x( 0 )= 0"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%202%20%5C,%20x%20y%5E%7B2%7D%20-%203%20%5C,%20t%20y%20-%203%20%5Chspace%7B2em%7D%20y(%200%20)=%20-2" alt="y'= 2 \, x y^{2} - 3 \, t y - 3 \hspace{2em} y( 0 )= -2" title="y'= 2 \, x y^{2} - 3 \, t y - 3 \hspace{2em} y( 0 )= -2" data-latex="y'= 2 \, x y^{2} - 3 \, t y - 3 \hspace{2em} y( 0 )= -2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.10 )\approx 0.100" alt="x( 0.10 )\approx 0.100" title="x( 0.10 )\approx 0.100" data-latex="x( 0.10 )\approx 0.100"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.10 )\approx -2.30" alt="y( 0.10 )\approx -2.30" title="y( 0.10 )\approx -2.30" data-latex="y( 0.10 )\approx -2.30"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.20 )\approx 0.223" alt="x( 0.20 )\approx 0.223" title="x( 0.20 )\approx 0.223" data-latex="x( 0.20 )\approx 0.223"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.20 )\approx -2.43" alt="y( 0.20 )\approx -2.43" title="y( 0.20 )\approx -2.43" data-latex="y( 0.20 )\approx -2.43"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.10%20)%5Capprox%200.100" alt="x( 0.10 )\approx 0.100" title="x( 0.10 )\approx 0.100" data-latex="x( 0.10 )\approx 0.100"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.10%20)%5Capprox%20-2.30" alt="y( 0.10 )\approx -2.30" title="y( 0.10 )\approx -2.30" data-latex="y( 0.10 )\approx -2.30"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.20%20)%5Capprox%200.223" alt="x( 0.20 )\approx 0.223" title="x( 0.20 )\approx 0.223" data-latex="x( 0.20 )\approx 0.223"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.20%20)%5Capprox%20-2.43" alt="y( 0.20 )\approx -2.43" title="y( 0.20 )\approx -2.43" data-latex="y( 0.20 )\approx -2.43"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-9935" title="N2 | Euler's method for approximating IVP solutions | ver. 9935"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.80 )" alt="x( -0.80 )" title="x( -0.80 )" data-latex="x( -0.80 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.80 )" alt="y( -0.80 )" title="y( -0.80 )" data-latex="y( -0.80 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -4 \, t^{2} x + 4 \, x y^{2} - 3 \hspace{2em} x( -1 )= -2" alt="x'= -4 \, t^{2} x + 4 \, x y^{2} - 3 \hspace{2em} x( -1 )= -2" title="x'= -4 \, t^{2} x + 4 \, x y^{2} - 3 \hspace{2em} x( -1 )= -2" data-latex="x'= -4 \, t^{2} x + 4 \, x y^{2} - 3 \hspace{2em} x( -1 )= -2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -t^{2} x^{2} + 3 \, x^{2} y^{2} - 3 \hspace{2em} y( -1 )= -2" alt="y'= -t^{2} x^{2} + 3 \, x^{2} y^{2} - 3 \hspace{2em} y( -1 )= -2" title="y'= -t^{2} x^{2} + 3 \, x^{2} y^{2} - 3 \hspace{2em} y( -1 )= -2" data-latex="y'= -t^{2} x^{2} + 3 \, x^{2} y^{2} - 3 \hspace{2em} y( -1 )= -2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.80%20)" alt="x( -0.80 )" title="x( -0.80 )" data-latex="x( -0.80 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.80%20)" alt="y( -0.80 )" title="y( -0.80 )" data-latex="y( -0.80 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-4%20%5C,%20t%5E%7B2%7D%20x%20+%204%20%5C,%20x%20y%5E%7B2%7D%20-%203%20%5Chspace%7B2em%7D%20x(%20-1%20)=%20-2" alt="x'= -4 \, t^{2} x + 4 \, x y^{2} - 3 \hspace{2em} x( -1 )= -2" title="x'= -4 \, t^{2} x + 4 \, x y^{2} - 3 \hspace{2em} x( -1 )= -2" data-latex="x'= -4 \, t^{2} x + 4 \, x y^{2} - 3 \hspace{2em} x( -1 )= -2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-t%5E%7B2%7D%20x%5E%7B2%7D%20+%203%20%5C,%20x%5E%7B2%7D%20y%5E%7B2%7D%20-%203%20%5Chspace%7B2em%7D%20y(%20-1%20)=%20-2" alt="y'= -t^{2} x^{2} + 3 \, x^{2} y^{2} - 3 \hspace{2em} y( -1 )= -2" title="y'= -t^{2} x^{2} + 3 \, x^{2} y^{2} - 3 \hspace{2em} y( -1 )= -2" data-latex="y'= -t^{2} x^{2} + 3 \, x^{2} y^{2} - 3 \hspace{2em} y( -1 )= -2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.90 )\approx -4.70" alt="x( -0.90 )\approx -4.70" title="x( -0.90 )\approx -4.70" data-latex="x( -0.90 )\approx -4.70"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.90 )\approx 2.10" alt="y( -0.90 )\approx 2.10" title="y( -0.90 )\approx 2.10" data-latex="y( -0.90 )\approx 2.10"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.80 )\approx -11.8" alt="x( -0.80 )\approx -11.8" title="x( -0.80 )\approx -11.8" data-latex="x( -0.80 )\approx -11.8"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.80 )\approx 29.3" alt="y( -0.80 )\approx 29.3" title="y( -0.80 )\approx 29.3" data-latex="y( -0.80 )\approx 29.3"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.90%20)%5Capprox%20-4.70" alt="x( -0.90 )\approx -4.70" title="x( -0.90 )\approx -4.70" data-latex="x( -0.90 )\approx -4.70"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.90%20)%5Capprox%202.10" alt="y( -0.90 )\approx 2.10" title="y( -0.90 )\approx 2.10" data-latex="y( -0.90 )\approx 2.10"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.80%20)%5Capprox%20-11.8" alt="x( -0.80 )\approx -11.8" title="x( -0.80 )\approx -11.8" data-latex="x( -0.80 )\approx -11.8"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.80%20)%5Capprox%2029.3" alt="y( -0.80 )\approx 29.3" title="y( -0.80 )\approx 29.3" data-latex="y( -0.80 )\approx 29.3"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-9631" title="N2 | Euler's method for approximating IVP solutions | ver. 9631"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.2 )" alt="x( 1.2 )" title="x( 1.2 )" data-latex="x( 1.2 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.2 )" alt="y( 1.2 )" title="y( 1.2 )" data-latex="y( 1.2 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 4 \, t^{2} x^{2} + 2 \, x y^{2} - 3 \hspace{2em} x( 1 )= 1" alt="x'= 4 \, t^{2} x^{2} + 2 \, x y^{2} - 3 \hspace{2em} x( 1 )= 1" title="x'= 4 \, t^{2} x^{2} + 2 \, x y^{2} - 3 \hspace{2em} x( 1 )= 1" data-latex="x'= 4 \, t^{2} x^{2} + 2 \, x y^{2} - 3 \hspace{2em} x( 1 )= 1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -t^{2} y - 4 \, x y - 1 \hspace{2em} y( 1 )= -1" alt="y'= -t^{2} y - 4 \, x y - 1 \hspace{2em} y( 1 )= -1" title="y'= -t^{2} y - 4 \, x y - 1 \hspace{2em} y( 1 )= -1" data-latex="y'= -t^{2} y - 4 \, x y - 1 \hspace{2em} y( 1 )= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.2%20)" alt="x( 1.2 )" title="x( 1.2 )" data-latex="x( 1.2 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.2%20)" alt="y( 1.2 )" title="y( 1.2 )" data-latex="y( 1.2 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%204%20%5C,%20t%5E%7B2%7D%20x%5E%7B2%7D%20+%202%20%5C,%20x%20y%5E%7B2%7D%20-%203%20%5Chspace%7B2em%7D%20x(%201%20)=%201" alt="x'= 4 \, t^{2} x^{2} + 2 \, x y^{2} - 3 \hspace{2em} x( 1 )= 1" title="x'= 4 \, t^{2} x^{2} + 2 \, x y^{2} - 3 \hspace{2em} x( 1 )= 1" data-latex="x'= 4 \, t^{2} x^{2} + 2 \, x y^{2} - 3 \hspace{2em} x( 1 )= 1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-t%5E%7B2%7D%20y%20-%204%20%5C,%20x%20y%20-%201%20%5Chspace%7B2em%7D%20y(%201%20)=%20-1" alt="y'= -t^{2} y - 4 \, x y - 1 \hspace{2em} y( 1 )= -1" title="y'= -t^{2} y - 4 \, x y - 1 \hspace{2em} y( 1 )= -1" data-latex="y'= -t^{2} y - 4 \, x y - 1 \hspace{2em} y( 1 )= -1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.1 )\approx 1.30" alt="x( 1.1 )\approx 1.30" title="x( 1.1 )\approx 1.30" data-latex="x( 1.1 )\approx 1.30"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.1 )\approx -0.600" alt="y( 1.1 )\approx -0.600" title="y( 1.1 )\approx -0.600" data-latex="y( 1.1 )\approx -0.600"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.2 )\approx 1.91" alt="x( 1.2 )\approx 1.91" title="x( 1.2 )\approx 1.91" data-latex="x( 1.2 )\approx 1.91"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.2 )\approx -0.315" alt="y( 1.2 )\approx -0.315" title="y( 1.2 )\approx -0.315" data-latex="y( 1.2 )\approx -0.315"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.1%20)%5Capprox%201.30" alt="x( 1.1 )\approx 1.30" title="x( 1.1 )\approx 1.30" data-latex="x( 1.1 )\approx 1.30"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.1%20)%5Capprox%20-0.600" alt="y( 1.1 )\approx -0.600" title="y( 1.1 )\approx -0.600" data-latex="y( 1.1 )\approx -0.600"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.2%20)%5Capprox%201.91" alt="x( 1.2 )\approx 1.91" title="x( 1.2 )\approx 1.91" data-latex="x( 1.2 )\approx 1.91"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.2%20)%5Capprox%20-0.315" alt="y( 1.2 )\approx -0.315" title="y( 1.2 )\approx -0.315" data-latex="y( 1.2 )\approx -0.315"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-6130" title="N2 | Euler's method for approximating IVP solutions | ver. 6130"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.80 )" alt="x( -0.80 )" title="x( -0.80 )" data-latex="x( -0.80 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.80 )" alt="y( -0.80 )" title="y( -0.80 )" data-latex="y( -0.80 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -2 \, t^{2} x - 2 \, t^{2} y - 2 \hspace{2em} x( -1 )= 2" alt="x'= -2 \, t^{2} x - 2 \, t^{2} y - 2 \hspace{2em} x( -1 )= 2" title="x'= -2 \, t^{2} x - 2 \, t^{2} y - 2 \hspace{2em} x( -1 )= 2" data-latex="x'= -2 \, t^{2} x - 2 \, t^{2} y - 2 \hspace{2em} x( -1 )= 2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -t^{2} x^{2} - x y^{2} \hspace{2em} y( -1 )= 0" alt="y'= -t^{2} x^{2} - x y^{2} \hspace{2em} y( -1 )= 0" title="y'= -t^{2} x^{2} - x y^{2} \hspace{2em} y( -1 )= 0" data-latex="y'= -t^{2} x^{2} - x y^{2} \hspace{2em} y( -1 )= 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.80%20)" alt="x( -0.80 )" title="x( -0.80 )" data-latex="x( -0.80 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.80%20)" alt="y( -0.80 )" title="y( -0.80 )" data-latex="y( -0.80 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-2%20%5C,%20t%5E%7B2%7D%20x%20-%202%20%5C,%20t%5E%7B2%7D%20y%20-%202%20%5Chspace%7B2em%7D%20x(%20-1%20)=%202" alt="x'= -2 \, t^{2} x - 2 \, t^{2} y - 2 \hspace{2em} x( -1 )= 2" title="x'= -2 \, t^{2} x - 2 \, t^{2} y - 2 \hspace{2em} x( -1 )= 2" data-latex="x'= -2 \, t^{2} x - 2 \, t^{2} y - 2 \hspace{2em} x( -1 )= 2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-t%5E%7B2%7D%20x%5E%7B2%7D%20-%20x%20y%5E%7B2%7D%20%5Chspace%7B2em%7D%20y(%20-1%20)=%200" alt="y'= -t^{2} x^{2} - x y^{2} \hspace{2em} y( -1 )= 0" title="y'= -t^{2} x^{2} - x y^{2} \hspace{2em} y( -1 )= 0" data-latex="y'= -t^{2} x^{2} - x y^{2} \hspace{2em} y( -1 )= 0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.90 )\approx 1.40" alt="x( -0.90 )\approx 1.40" title="x( -0.90 )\approx 1.40" data-latex="x( -0.90 )\approx 1.40"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.90 )\approx -0.400" alt="y( -0.90 )\approx -0.400" title="y( -0.90 )\approx -0.400" data-latex="y( -0.90 )\approx -0.400"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.80 )\approx 1.04" alt="x( -0.80 )\approx 1.04" title="x( -0.80 )\approx 1.04" data-latex="x( -0.80 )\approx 1.04"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.80 )\approx -0.581" alt="y( -0.80 )\approx -0.581" title="y( -0.80 )\approx -0.581" data-latex="y( -0.80 )\approx -0.581"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.90%20)%5Capprox%201.40" alt="x( -0.90 )\approx 1.40" title="x( -0.90 )\approx 1.40" data-latex="x( -0.90 )\approx 1.40"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.90%20)%5Capprox%20-0.400" alt="y( -0.90 )\approx -0.400" title="y( -0.90 )\approx -0.400" data-latex="y( -0.90 )\approx -0.400"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.80%20)%5Capprox%201.04" alt="x( -0.80 )\approx 1.04" title="x( -0.80 )\approx 1.04" data-latex="x( -0.80 )\approx 1.04"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.80%20)%5Capprox%20-0.581" alt="y( -0.80 )\approx -0.581" title="y( -0.80 )\approx -0.581" data-latex="y( -0.80 )\approx -0.581"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-7227" title="N2 | Euler's method for approximating IVP solutions | ver. 7227"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.2 )" alt="x( 1.2 )" title="x( 1.2 )" data-latex="x( 1.2 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.2 )" alt="y( 1.2 )" title="y( 1.2 )" data-latex="y( 1.2 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -2 \, t^{2} y + 4 \, t x - 3 \hspace{2em} x( 1 )= 2" alt="x'= -2 \, t^{2} y + 4 \, t x - 3 \hspace{2em} x( 1 )= 2" title="x'= -2 \, t^{2} y + 4 \, t x - 3 \hspace{2em} x( 1 )= 2" data-latex="x'= -2 \, t^{2} y + 4 \, t x - 3 \hspace{2em} x( 1 )= 2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -2 \, t x^{2} + t y + 3 \hspace{2em} y( 1 )= 0" alt="y'= -2 \, t x^{2} + t y + 3 \hspace{2em} y( 1 )= 0" title="y'= -2 \, t x^{2} + t y + 3 \hspace{2em} y( 1 )= 0" data-latex="y'= -2 \, t x^{2} + t y + 3 \hspace{2em} y( 1 )= 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.2%20)" alt="x( 1.2 )" title="x( 1.2 )" data-latex="x( 1.2 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.2%20)" alt="y( 1.2 )" title="y( 1.2 )" data-latex="y( 1.2 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-2%20%5C,%20t%5E%7B2%7D%20y%20+%204%20%5C,%20t%20x%20-%203%20%5Chspace%7B2em%7D%20x(%201%20)=%202" alt="x'= -2 \, t^{2} y + 4 \, t x - 3 \hspace{2em} x( 1 )= 2" title="x'= -2 \, t^{2} y + 4 \, t x - 3 \hspace{2em} x( 1 )= 2" data-latex="x'= -2 \, t^{2} y + 4 \, t x - 3 \hspace{2em} x( 1 )= 2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-2%20%5C,%20t%20x%5E%7B2%7D%20+%20t%20y%20+%203%20%5Chspace%7B2em%7D%20y(%201%20)=%200" alt="y'= -2 \, t x^{2} + t y + 3 \hspace{2em} y( 1 )= 0" title="y'= -2 \, t x^{2} + t y + 3 \hspace{2em} y( 1 )= 0" data-latex="y'= -2 \, t x^{2} + t y + 3 \hspace{2em} y( 1 )= 0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.1 )\approx 2.50" alt="x( 1.1 )\approx 2.50" title="x( 1.1 )\approx 2.50" data-latex="x( 1.1 )\approx 2.50"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.1 )\approx -0.500" alt="y( 1.1 )\approx -0.500" title="y( 1.1 )\approx -0.500" data-latex="y( 1.1 )\approx -0.500"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.2 )\approx 3.42" alt="x( 1.2 )\approx 3.42" title="x( 1.2 )\approx 3.42" data-latex="x( 1.2 )\approx 3.42"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.2 )\approx -1.63" alt="y( 1.2 )\approx -1.63" title="y( 1.2 )\approx -1.63" data-latex="y( 1.2 )\approx -1.63"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.1%20)%5Capprox%202.50" alt="x( 1.1 )\approx 2.50" title="x( 1.1 )\approx 2.50" data-latex="x( 1.1 )\approx 2.50"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.1%20)%5Capprox%20-0.500" alt="y( 1.1 )\approx -0.500" title="y( 1.1 )\approx -0.500" data-latex="y( 1.1 )\approx -0.500"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.2%20)%5Capprox%203.42" alt="x( 1.2 )\approx 3.42" title="x( 1.2 )\approx 3.42" data-latex="x( 1.2 )\approx 3.42"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.2%20)%5Capprox%20-1.63" alt="y( 1.2 )\approx -1.63" title="y( 1.2 )\approx -1.63" data-latex="y( 1.2 )\approx -1.63"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-3803" title="N2 | Euler's method for approximating IVP solutions | ver. 3803"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.2 )" alt="x( 1.2 )" title="x( 1.2 )" data-latex="x( 1.2 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.2 )" alt="y( 1.2 )" title="y( 1.2 )" data-latex="y( 1.2 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -4 \, t x^{2} + 4 \, t y^{2} - 2 \hspace{2em} x( 1 )= -1" alt="x'= -4 \, t x^{2} + 4 \, t y^{2} - 2 \hspace{2em} x( 1 )= -1" title="x'= -4 \, t x^{2} + 4 \, t y^{2} - 2 \hspace{2em} x( 1 )= -1" data-latex="x'= -4 \, t x^{2} + 4 \, t y^{2} - 2 \hspace{2em} x( 1 )= -1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= t^{2} y^{2} + 3 \, t x \hspace{2em} y( 1 )= 2" alt="y'= t^{2} y^{2} + 3 \, t x \hspace{2em} y( 1 )= 2" title="y'= t^{2} y^{2} + 3 \, t x \hspace{2em} y( 1 )= 2" data-latex="y'= t^{2} y^{2} + 3 \, t x \hspace{2em} y( 1 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.2%20)" alt="x( 1.2 )" title="x( 1.2 )" data-latex="x( 1.2 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.2%20)" alt="y( 1.2 )" title="y( 1.2 )" data-latex="y( 1.2 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-4%20%5C,%20t%20x%5E%7B2%7D%20+%204%20%5C,%20t%20y%5E%7B2%7D%20-%202%20%5Chspace%7B2em%7D%20x(%201%20)=%20-1" alt="x'= -4 \, t x^{2} + 4 \, t y^{2} - 2 \hspace{2em} x( 1 )= -1" title="x'= -4 \, t x^{2} + 4 \, t y^{2} - 2 \hspace{2em} x( 1 )= -1" data-latex="x'= -4 \, t x^{2} + 4 \, t y^{2} - 2 \hspace{2em} x( 1 )= -1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20t%5E%7B2%7D%20y%5E%7B2%7D%20+%203%20%5C,%20t%20x%20%5Chspace%7B2em%7D%20y(%201%20)=%202" alt="y'= t^{2} y^{2} + 3 \, t x \hspace{2em} y( 1 )= 2" title="y'= t^{2} y^{2} + 3 \, t x \hspace{2em} y( 1 )= 2" data-latex="y'= t^{2} y^{2} + 3 \, t x \hspace{2em} y( 1 )= 2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.1 )\approx 0.000" alt="x( 1.1 )\approx 0.000" title="x( 1.1 )\approx 0.000" data-latex="x( 1.1 )\approx 0.000"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.1 )\approx 2.10" alt="y( 1.1 )\approx 2.10" title="y( 1.1 )\approx 2.10" data-latex="y( 1.1 )\approx 2.10"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.2 )\approx 1.74" alt="x( 1.2 )\approx 1.74" title="x( 1.2 )\approx 1.74" data-latex="x( 1.2 )\approx 1.74"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.2 )\approx 2.63" alt="y( 1.2 )\approx 2.63" title="y( 1.2 )\approx 2.63" data-latex="y( 1.2 )\approx 2.63"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.1%20)%5Capprox%200.000" alt="x( 1.1 )\approx 0.000" title="x( 1.1 )\approx 0.000" data-latex="x( 1.1 )\approx 0.000"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.1%20)%5Capprox%202.10" alt="y( 1.1 )\approx 2.10" title="y( 1.1 )\approx 2.10" data-latex="y( 1.1 )\approx 2.10"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.2%20)%5Capprox%201.74" alt="x( 1.2 )\approx 1.74" title="x( 1.2 )\approx 1.74" data-latex="x( 1.2 )\approx 1.74"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.2%20)%5Capprox%202.63" alt="y( 1.2 )\approx 2.63" title="y( 1.2 )\approx 2.63" data-latex="y( 1.2 )\approx 2.63"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-1188" title="N2 | Euler's method for approximating IVP solutions | ver. 1188"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.2 )" alt="x( 1.2 )" title="x( 1.2 )" data-latex="x( 1.2 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.2 )" alt="y( 1.2 )" title="y( 1.2 )" data-latex="y( 1.2 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 3 \, t^{2} x^{2} + 3 \, x y \hspace{2em} x( 1 )= -1" alt="x'= 3 \, t^{2} x^{2} + 3 \, x y \hspace{2em} x( 1 )= -1" title="x'= 3 \, t^{2} x^{2} + 3 \, x y \hspace{2em} x( 1 )= -1" data-latex="x'= 3 \, t^{2} x^{2} + 3 \, x y \hspace{2em} x( 1 )= -1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -t^{2} y^{2} + 3 \, x^{2} y - 1 \hspace{2em} y( 1 )= 0" alt="y'= -t^{2} y^{2} + 3 \, x^{2} y - 1 \hspace{2em} y( 1 )= 0" title="y'= -t^{2} y^{2} + 3 \, x^{2} y - 1 \hspace{2em} y( 1 )= 0" data-latex="y'= -t^{2} y^{2} + 3 \, x^{2} y - 1 \hspace{2em} y( 1 )= 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.2%20)" alt="x( 1.2 )" title="x( 1.2 )" data-latex="x( 1.2 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.2%20)" alt="y( 1.2 )" title="y( 1.2 )" data-latex="y( 1.2 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%203%20%5C,%20t%5E%7B2%7D%20x%5E%7B2%7D%20+%203%20%5C,%20x%20y%20%5Chspace%7B2em%7D%20x(%201%20)=%20-1" alt="x'= 3 \, t^{2} x^{2} + 3 \, x y \hspace{2em} x( 1 )= -1" title="x'= 3 \, t^{2} x^{2} + 3 \, x y \hspace{2em} x( 1 )= -1" data-latex="x'= 3 \, t^{2} x^{2} + 3 \, x y \hspace{2em} x( 1 )= -1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-t%5E%7B2%7D%20y%5E%7B2%7D%20+%203%20%5C,%20x%5E%7B2%7D%20y%20-%201%20%5Chspace%7B2em%7D%20y(%201%20)=%200" alt="y'= -t^{2} y^{2} + 3 \, x^{2} y - 1 \hspace{2em} y( 1 )= 0" title="y'= -t^{2} y^{2} + 3 \, x^{2} y - 1 \hspace{2em} y( 1 )= 0" data-latex="y'= -t^{2} y^{2} + 3 \, x^{2} y - 1 \hspace{2em} y( 1 )= 0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.1 )\approx -0.700" alt="x( 1.1 )\approx -0.700" title="x( 1.1 )\approx -0.700" data-latex="x( 1.1 )\approx -0.700"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.1 )\approx -0.100" alt="y( 1.1 )\approx -0.100" title="y( 1.1 )\approx -0.100" data-latex="y( 1.1 )\approx -0.100"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.2 )\approx -0.502" alt="x( 1.2 )\approx -0.502" title="x( 1.2 )\approx -0.502" data-latex="x( 1.2 )\approx -0.502"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.2 )\approx -0.216" alt="y( 1.2 )\approx -0.216" title="y( 1.2 )\approx -0.216" data-latex="y( 1.2 )\approx -0.216"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.1%20)%5Capprox%20-0.700" alt="x( 1.1 )\approx -0.700" title="x( 1.1 )\approx -0.700" data-latex="x( 1.1 )\approx -0.700"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.1%20)%5Capprox%20-0.100" alt="y( 1.1 )\approx -0.100" title="y( 1.1 )\approx -0.100" data-latex="y( 1.1 )\approx -0.100"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.2%20)%5Capprox%20-0.502" alt="x( 1.2 )\approx -0.502" title="x( 1.2 )\approx -0.502" data-latex="x( 1.2 )\approx -0.502"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.2%20)%5Capprox%20-0.216" alt="y( 1.2 )\approx -0.216" title="y( 1.2 )\approx -0.216" data-latex="y( 1.2 )\approx -0.216"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-4689" title="N2 | Euler's method for approximating IVP solutions | ver. 4689"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.8 )" alt="x( -1.8 )" title="x( -1.8 )" data-latex="x( -1.8 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.8 )" alt="y( -1.8 )" title="y( -1.8 )" data-latex="y( -1.8 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= t^{2} x - 3 \, t^{2} y + 3 \hspace{2em} x( -2 )= 1" alt="x'= t^{2} x - 3 \, t^{2} y + 3 \hspace{2em} x( -2 )= 1" title="x'= t^{2} x - 3 \, t^{2} y + 3 \hspace{2em} x( -2 )= 1" data-latex="x'= t^{2} x - 3 \, t^{2} y + 3 \hspace{2em} x( -2 )= 1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -x y^{2} + 2 \, t x - 2 \hspace{2em} y( -2 )= -1" alt="y'= -x y^{2} + 2 \, t x - 2 \hspace{2em} y( -2 )= -1" title="y'= -x y^{2} + 2 \, t x - 2 \hspace{2em} y( -2 )= -1" data-latex="y'= -x y^{2} + 2 \, t x - 2 \hspace{2em} y( -2 )= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.8%20)" alt="x( -1.8 )" title="x( -1.8 )" data-latex="x( -1.8 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.8%20)" alt="y( -1.8 )" title="y( -1.8 )" data-latex="y( -1.8 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20t%5E%7B2%7D%20x%20-%203%20%5C,%20t%5E%7B2%7D%20y%20+%203%20%5Chspace%7B2em%7D%20x(%20-2%20)=%201" alt="x'= t^{2} x - 3 \, t^{2} y + 3 \hspace{2em} x( -2 )= 1" title="x'= t^{2} x - 3 \, t^{2} y + 3 \hspace{2em} x( -2 )= 1" data-latex="x'= t^{2} x - 3 \, t^{2} y + 3 \hspace{2em} x( -2 )= 1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-x%20y%5E%7B2%7D%20+%202%20%5C,%20t%20x%20-%202%20%5Chspace%7B2em%7D%20y(%20-2%20)=%20-1" alt="y'= -x y^{2} + 2 \, t x - 2 \hspace{2em} y( -2 )= -1" title="y'= -x y^{2} + 2 \, t x - 2 \hspace{2em} y( -2 )= -1" data-latex="y'= -x y^{2} + 2 \, t x - 2 \hspace{2em} y( -2 )= -1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.9 )\approx 2.90" alt="x( -1.9 )\approx 2.90" title="x( -1.9 )\approx 2.90" data-latex="x( -1.9 )\approx 2.90"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.9 )\approx -1.70" alt="y( -1.9 )\approx -1.70" title="y( -1.9 )\approx -1.70" data-latex="y( -1.9 )\approx -1.70"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.8 )\approx 6.09" alt="x( -1.8 )\approx 6.09" title="x( -1.8 )\approx 6.09" data-latex="x( -1.8 )\approx 6.09"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.8 )\approx -3.84" alt="y( -1.8 )\approx -3.84" title="y( -1.8 )\approx -3.84" data-latex="y( -1.8 )\approx -3.84"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.9%20)%5Capprox%202.90" alt="x( -1.9 )\approx 2.90" title="x( -1.9 )\approx 2.90" data-latex="x( -1.9 )\approx 2.90"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.9%20)%5Capprox%20-1.70" alt="y( -1.9 )\approx -1.70" title="y( -1.9 )\approx -1.70" data-latex="y( -1.9 )\approx -1.70"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.8%20)%5Capprox%206.09" alt="x( -1.8 )\approx 6.09" title="x( -1.8 )\approx 6.09" data-latex="x( -1.8 )\approx 6.09"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.8%20)%5Capprox%20-3.84" alt="y( -1.8 )\approx -3.84" title="y( -1.8 )\approx -3.84" data-latex="y( -1.8 )\approx -3.84"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-4511" title="N2 | Euler's method for approximating IVP solutions | ver. 4511"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.20 )" alt="x( 0.20 )" title="x( 0.20 )" data-latex="x( 0.20 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.20 )" alt="y( 0.20 )" title="y( 0.20 )" data-latex="y( 0.20 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 2 \, x^{2} y + 4 \, t x - 1 \hspace{2em} x( 0 )= 0" alt="x'= 2 \, x^{2} y + 4 \, t x - 1 \hspace{2em} x( 0 )= 0" title="x'= 2 \, x^{2} y + 4 \, t x - 1 \hspace{2em} x( 0 )= 0" data-latex="x'= 2 \, x^{2} y + 4 \, t x - 1 \hspace{2em} x( 0 )= 0"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -x^{2} y^{2} + 2 \, t y^{2} - 3 \hspace{2em} y( 0 )= 0" alt="y'= -x^{2} y^{2} + 2 \, t y^{2} - 3 \hspace{2em} y( 0 )= 0" title="y'= -x^{2} y^{2} + 2 \, t y^{2} - 3 \hspace{2em} y( 0 )= 0" data-latex="y'= -x^{2} y^{2} + 2 \, t y^{2} - 3 \hspace{2em} y( 0 )= 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.20%20)" alt="x( 0.20 )" title="x( 0.20 )" data-latex="x( 0.20 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.20%20)" alt="y( 0.20 )" title="y( 0.20 )" data-latex="y( 0.20 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%202%20%5C,%20x%5E%7B2%7D%20y%20+%204%20%5C,%20t%20x%20-%201%20%5Chspace%7B2em%7D%20x(%200%20)=%200" alt="x'= 2 \, x^{2} y + 4 \, t x - 1 \hspace{2em} x( 0 )= 0" title="x'= 2 \, x^{2} y + 4 \, t x - 1 \hspace{2em} x( 0 )= 0" data-latex="x'= 2 \, x^{2} y + 4 \, t x - 1 \hspace{2em} x( 0 )= 0"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-x%5E%7B2%7D%20y%5E%7B2%7D%20+%202%20%5C,%20t%20y%5E%7B2%7D%20-%203%20%5Chspace%7B2em%7D%20y(%200%20)=%200" alt="y'= -x^{2} y^{2} + 2 \, t y^{2} - 3 \hspace{2em} y( 0 )= 0" title="y'= -x^{2} y^{2} + 2 \, t y^{2} - 3 \hspace{2em} y( 0 )= 0" data-latex="y'= -x^{2} y^{2} + 2 \, t y^{2} - 3 \hspace{2em} y( 0 )= 0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.10 )\approx -0.100" alt="x( 0.10 )\approx -0.100" title="x( 0.10 )\approx -0.100" data-latex="x( 0.10 )\approx -0.100"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.10 )\approx -0.300" alt="y( 0.10 )\approx -0.300" title="y( 0.10 )\approx -0.300" data-latex="y( 0.10 )\approx -0.300"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.20 )\approx -0.205" alt="x( 0.20 )\approx -0.205" title="x( 0.20 )\approx -0.205" data-latex="x( 0.20 )\approx -0.205"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.20 )\approx -0.598" alt="y( 0.20 )\approx -0.598" title="y( 0.20 )\approx -0.598" data-latex="y( 0.20 )\approx -0.598"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.10%20)%5Capprox%20-0.100" alt="x( 0.10 )\approx -0.100" title="x( 0.10 )\approx -0.100" data-latex="x( 0.10 )\approx -0.100"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.10%20)%5Capprox%20-0.300" alt="y( 0.10 )\approx -0.300" title="y( 0.10 )\approx -0.300" data-latex="y( 0.10 )\approx -0.300"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.20%20)%5Capprox%20-0.205" alt="x( 0.20 )\approx -0.205" title="x( 0.20 )\approx -0.205" data-latex="x( 0.20 )\approx -0.205"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.20%20)%5Capprox%20-0.598" alt="y( 0.20 )\approx -0.598" title="y( 0.20 )\approx -0.598" data-latex="y( 0.20 )\approx -0.598"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-3777" title="N2 | Euler's method for approximating IVP solutions | ver. 3777"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.8 )" alt="x( -1.8 )" title="x( -1.8 )" data-latex="x( -1.8 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.8 )" alt="y( -1.8 )" title="y( -1.8 )" data-latex="y( -1.8 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -3 \, t y + 2 \, x y \hspace{2em} x( -2 )= 2" alt="x'= -3 \, t y + 2 \, x y \hspace{2em} x( -2 )= 2" title="x'= -3 \, t y + 2 \, x y \hspace{2em} x( -2 )= 2" data-latex="x'= -3 \, t y + 2 \, x y \hspace{2em} x( -2 )= 2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 4 \, t x - 4 \, t y \hspace{2em} y( -2 )= 1" alt="y'= 4 \, t x - 4 \, t y \hspace{2em} y( -2 )= 1" title="y'= 4 \, t x - 4 \, t y \hspace{2em} y( -2 )= 1" data-latex="y'= 4 \, t x - 4 \, t y \hspace{2em} y( -2 )= 1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.8%20)" alt="x( -1.8 )" title="x( -1.8 )" data-latex="x( -1.8 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.8%20)" alt="y( -1.8 )" title="y( -1.8 )" data-latex="y( -1.8 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-3%20%5C,%20t%20y%20+%202%20%5C,%20x%20y%20%5Chspace%7B2em%7D%20x(%20-2%20)=%202" alt="x'= -3 \, t y + 2 \, x y \hspace{2em} x( -2 )= 2" title="x'= -3 \, t y + 2 \, x y \hspace{2em} x( -2 )= 2" data-latex="x'= -3 \, t y + 2 \, x y \hspace{2em} x( -2 )= 2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%204%20%5C,%20t%20x%20-%204%20%5C,%20t%20y%20%5Chspace%7B2em%7D%20y(%20-2%20)=%201" alt="y'= 4 \, t x - 4 \, t y \hspace{2em} y( -2 )= 1" title="y'= 4 \, t x - 4 \, t y \hspace{2em} y( -2 )= 1" data-latex="y'= 4 \, t x - 4 \, t y \hspace{2em} y( -2 )= 1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.9 )\approx 3.00" alt="x( -1.9 )\approx 3.00" title="x( -1.9 )\approx 3.00" data-latex="x( -1.9 )\approx 3.00"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.9 )\approx 0.200" alt="y( -1.9 )\approx 0.200" title="y( -1.9 )\approx 0.200" data-latex="y( -1.9 )\approx 0.200"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.8 )\approx 3.23" alt="x( -1.8 )\approx 3.23" title="x( -1.8 )\approx 3.23" data-latex="x( -1.8 )\approx 3.23"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.8 )\approx -1.93" alt="y( -1.8 )\approx -1.93" title="y( -1.8 )\approx -1.93" data-latex="y( -1.8 )\approx -1.93"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.9%20)%5Capprox%203.00" alt="x( -1.9 )\approx 3.00" title="x( -1.9 )\approx 3.00" data-latex="x( -1.9 )\approx 3.00"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.9%20)%5Capprox%200.200" alt="y( -1.9 )\approx 0.200" title="y( -1.9 )\approx 0.200" data-latex="y( -1.9 )\approx 0.200"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.8%20)%5Capprox%203.23" alt="x( -1.8 )\approx 3.23" title="x( -1.8 )\approx 3.23" data-latex="x( -1.8 )\approx 3.23"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.8%20)%5Capprox%20-1.93" alt="y( -1.8 )\approx -1.93" title="y( -1.8 )\approx -1.93" data-latex="y( -1.8 )\approx -1.93"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-1068" title="N2 | Euler's method for approximating IVP solutions | ver. 1068"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.80 )" alt="x( -0.80 )" title="x( -0.80 )" data-latex="x( -0.80 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.80 )" alt="y( -0.80 )" title="y( -0.80 )" data-latex="y( -0.80 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 4 \, x^{2} y + 2 \, t x + 3 \hspace{2em} x( -1 )= 0" alt="x'= 4 \, x^{2} y + 2 \, t x + 3 \hspace{2em} x( -1 )= 0" title="x'= 4 \, x^{2} y + 2 \, t x + 3 \hspace{2em} x( -1 )= 0" data-latex="x'= 4 \, x^{2} y + 2 \, t x + 3 \hspace{2em} x( -1 )= 0"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= t y^{2} + 3 \, x y - 3 \hspace{2em} y( -1 )= 2" alt="y'= t y^{2} + 3 \, x y - 3 \hspace{2em} y( -1 )= 2" title="y'= t y^{2} + 3 \, x y - 3 \hspace{2em} y( -1 )= 2" data-latex="y'= t y^{2} + 3 \, x y - 3 \hspace{2em} y( -1 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.80%20)" alt="x( -0.80 )" title="x( -0.80 )" data-latex="x( -0.80 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.80%20)" alt="y( -0.80 )" title="y( -0.80 )" data-latex="y( -0.80 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%204%20%5C,%20x%5E%7B2%7D%20y%20+%202%20%5C,%20t%20x%20+%203%20%5Chspace%7B2em%7D%20x(%20-1%20)=%200" alt="x'= 4 \, x^{2} y + 2 \, t x + 3 \hspace{2em} x( -1 )= 0" title="x'= 4 \, x^{2} y + 2 \, t x + 3 \hspace{2em} x( -1 )= 0" data-latex="x'= 4 \, x^{2} y + 2 \, t x + 3 \hspace{2em} x( -1 )= 0"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20t%20y%5E%7B2%7D%20+%203%20%5C,%20x%20y%20-%203%20%5Chspace%7B2em%7D%20y(%20-1%20)=%202" alt="y'= t y^{2} + 3 \, x y - 3 \hspace{2em} y( -1 )= 2" title="y'= t y^{2} + 3 \, x y - 3 \hspace{2em} y( -1 )= 2" data-latex="y'= t y^{2} + 3 \, x y - 3 \hspace{2em} y( -1 )= 2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.90 )\approx 0.300" alt="x( -0.90 )\approx 0.300" title="x( -0.90 )\approx 0.300" data-latex="x( -0.90 )\approx 0.300"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.90 )\approx 1.30" alt="y( -0.90 )\approx 1.30" title="y( -0.90 )\approx 1.30" data-latex="y( -0.90 )\approx 1.30"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.80 )\approx 0.593" alt="x( -0.80 )\approx 0.593" title="x( -0.80 )\approx 0.593" data-latex="x( -0.80 )\approx 0.593"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.80 )\approx 0.966" alt="y( -0.80 )\approx 0.966" title="y( -0.80 )\approx 0.966" data-latex="y( -0.80 )\approx 0.966"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.90%20)%5Capprox%200.300" alt="x( -0.90 )\approx 0.300" title="x( -0.90 )\approx 0.300" data-latex="x( -0.90 )\approx 0.300"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.90%20)%5Capprox%201.30" alt="y( -0.90 )\approx 1.30" title="y( -0.90 )\approx 1.30" data-latex="y( -0.90 )\approx 1.30"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.80%20)%5Capprox%200.593" alt="x( -0.80 )\approx 0.593" title="x( -0.80 )\approx 0.593" data-latex="x( -0.80 )\approx 0.593"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.80%20)%5Capprox%200.966" alt="y( -0.80 )\approx 0.966" title="y( -0.80 )\approx 0.966" data-latex="y( -0.80 )\approx 0.966"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-7662" title="N2 | Euler's method for approximating IVP solutions | ver. 7662"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.2 )" alt="x( 1.2 )" title="x( 1.2 )" data-latex="x( 1.2 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.2 )" alt="y( 1.2 )" title="y( 1.2 )" data-latex="y( 1.2 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -3 \, t^{2} x - 4 \, x y + 1 \hspace{2em} x( 1 )= 1" alt="x'= -3 \, t^{2} x - 4 \, x y + 1 \hspace{2em} x( 1 )= 1" title="x'= -3 \, t^{2} x - 4 \, x y + 1 \hspace{2em} x( 1 )= 1" data-latex="x'= -3 \, t^{2} x - 4 \, x y + 1 \hspace{2em} x( 1 )= 1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -2 \, t y^{2} + 4 \, t x - 3 \hspace{2em} y( 1 )= -2" alt="y'= -2 \, t y^{2} + 4 \, t x - 3 \hspace{2em} y( 1 )= -2" title="y'= -2 \, t y^{2} + 4 \, t x - 3 \hspace{2em} y( 1 )= -2" data-latex="y'= -2 \, t y^{2} + 4 \, t x - 3 \hspace{2em} y( 1 )= -2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.2%20)" alt="x( 1.2 )" title="x( 1.2 )" data-latex="x( 1.2 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.2%20)" alt="y( 1.2 )" title="y( 1.2 )" data-latex="y( 1.2 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-3%20%5C,%20t%5E%7B2%7D%20x%20-%204%20%5C,%20x%20y%20+%201%20%5Chspace%7B2em%7D%20x(%201%20)=%201" alt="x'= -3 \, t^{2} x - 4 \, x y + 1 \hspace{2em} x( 1 )= 1" title="x'= -3 \, t^{2} x - 4 \, x y + 1 \hspace{2em} x( 1 )= 1" data-latex="x'= -3 \, t^{2} x - 4 \, x y + 1 \hspace{2em} x( 1 )= 1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-2%20%5C,%20t%20y%5E%7B2%7D%20+%204%20%5C,%20t%20x%20-%203%20%5Chspace%7B2em%7D%20y(%201%20)=%20-2" alt="y'= -2 \, t y^{2} + 4 \, t x - 3 \hspace{2em} y( 1 )= -2" title="y'= -2 \, t y^{2} + 4 \, t x - 3 \hspace{2em} y( 1 )= -2" data-latex="y'= -2 \, t y^{2} + 4 \, t x - 3 \hspace{2em} y( 1 )= -2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.1 )\approx 1.60" alt="x( 1.1 )\approx 1.60" title="x( 1.1 )\approx 1.60" data-latex="x( 1.1 )\approx 1.60"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.1 )\approx -2.70" alt="y( 1.1 )\approx -2.70" title="y( 1.1 )\approx -2.70" data-latex="y( 1.1 )\approx -2.70"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.2 )\approx 2.85" alt="x( 1.2 )\approx 2.85" title="x( 1.2 )\approx 2.85" data-latex="x( 1.2 )\approx 2.85"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.2 )\approx -3.90" alt="y( 1.2 )\approx -3.90" title="y( 1.2 )\approx -3.90" data-latex="y( 1.2 )\approx -3.90"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.1%20)%5Capprox%201.60" alt="x( 1.1 )\approx 1.60" title="x( 1.1 )\approx 1.60" data-latex="x( 1.1 )\approx 1.60"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.1%20)%5Capprox%20-2.70" alt="y( 1.1 )\approx -2.70" title="y( 1.1 )\approx -2.70" data-latex="y( 1.1 )\approx -2.70"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.2%20)%5Capprox%202.85" alt="x( 1.2 )\approx 2.85" title="x( 1.2 )\approx 2.85" data-latex="x( 1.2 )\approx 2.85"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.2%20)%5Capprox%20-3.90" alt="y( 1.2 )\approx -3.90" title="y( 1.2 )\approx -3.90" data-latex="y( 1.2 )\approx -3.90"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-3860" title="N2 | Euler's method for approximating IVP solutions | ver. 3860"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.20 )" alt="x( 0.20 )" title="x( 0.20 )" data-latex="x( 0.20 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.20 )" alt="y( 0.20 )" title="y( 0.20 )" data-latex="y( 0.20 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -2 \, t^{2} y^{2} + 3 \, x y + 2 \hspace{2em} x( 0 )= -2" alt="x'= -2 \, t^{2} y^{2} + 3 \, x y + 2 \hspace{2em} x( 0 )= -2" title="x'= -2 \, t^{2} y^{2} + 3 \, x y + 2 \hspace{2em} x( 0 )= -2" data-latex="x'= -2 \, t^{2} y^{2} + 3 \, x y + 2 \hspace{2em} x( 0 )= -2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= x y^{2} + 2 \, t y \hspace{2em} y( 0 )= 2" alt="y'= x y^{2} + 2 \, t y \hspace{2em} y( 0 )= 2" title="y'= x y^{2} + 2 \, t y \hspace{2em} y( 0 )= 2" data-latex="y'= x y^{2} + 2 \, t y \hspace{2em} y( 0 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.20%20)" alt="x( 0.20 )" title="x( 0.20 )" data-latex="x( 0.20 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.20%20)" alt="y( 0.20 )" title="y( 0.20 )" data-latex="y( 0.20 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-2%20%5C,%20t%5E%7B2%7D%20y%5E%7B2%7D%20+%203%20%5C,%20x%20y%20+%202%20%5Chspace%7B2em%7D%20x(%200%20)=%20-2" alt="x'= -2 \, t^{2} y^{2} + 3 \, x y + 2 \hspace{2em} x( 0 )= -2" title="x'= -2 \, t^{2} y^{2} + 3 \, x y + 2 \hspace{2em} x( 0 )= -2" data-latex="x'= -2 \, t^{2} y^{2} + 3 \, x y + 2 \hspace{2em} x( 0 )= -2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20x%20y%5E%7B2%7D%20+%202%20%5C,%20t%20y%20%5Chspace%7B2em%7D%20y(%200%20)=%202" alt="y'= x y^{2} + 2 \, t y \hspace{2em} y( 0 )= 2" title="y'= x y^{2} + 2 \, t y \hspace{2em} y( 0 )= 2" data-latex="y'= x y^{2} + 2 \, t y \hspace{2em} y( 0 )= 2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.10 )\approx -3.00" alt="x( 0.10 )\approx -3.00" title="x( 0.10 )\approx -3.00" data-latex="x( 0.10 )\approx -3.00"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.10 )\approx 1.20" alt="y( 0.10 )\approx 1.20" title="y( 0.10 )\approx 1.20" data-latex="y( 0.10 )\approx 1.20"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.20 )\approx -3.88" alt="x( 0.20 )\approx -3.88" title="x( 0.20 )\approx -3.88" data-latex="x( 0.20 )\approx -3.88"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.20 )\approx 0.793" alt="y( 0.20 )\approx 0.793" title="y( 0.20 )\approx 0.793" data-latex="y( 0.20 )\approx 0.793"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.10%20)%5Capprox%20-3.00" alt="x( 0.10 )\approx -3.00" title="x( 0.10 )\approx -3.00" data-latex="x( 0.10 )\approx -3.00"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.10%20)%5Capprox%201.20" alt="y( 0.10 )\approx 1.20" title="y( 0.10 )\approx 1.20" data-latex="y( 0.10 )\approx 1.20"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.20%20)%5Capprox%20-3.88" alt="x( 0.20 )\approx -3.88" title="x( 0.20 )\approx -3.88" data-latex="x( 0.20 )\approx -3.88"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.20%20)%5Capprox%200.793" alt="y( 0.20 )\approx 0.793" title="y( 0.20 )\approx 0.793" data-latex="y( 0.20 )\approx 0.793"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-1055" title="N2 | Euler's method for approximating IVP solutions | ver. 1055"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.2 )" alt="x( 1.2 )" title="x( 1.2 )" data-latex="x( 1.2 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.2 )" alt="y( 1.2 )" title="y( 1.2 )" data-latex="y( 1.2 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 2 \, x^{2} y^{2} + 4 \, t^{2} y - 2 \hspace{2em} x( 1 )= 0" alt="x'= 2 \, x^{2} y^{2} + 4 \, t^{2} y - 2 \hspace{2em} x( 1 )= 0" title="x'= 2 \, x^{2} y^{2} + 4 \, t^{2} y - 2 \hspace{2em} x( 1 )= 0" data-latex="x'= 2 \, x^{2} y^{2} + 4 \, t^{2} y - 2 \hspace{2em} x( 1 )= 0"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -3 \, t^{2} x - t y^{2} - 1 \hspace{2em} y( 1 )= -2" alt="y'= -3 \, t^{2} x - t y^{2} - 1 \hspace{2em} y( 1 )= -2" title="y'= -3 \, t^{2} x - t y^{2} - 1 \hspace{2em} y( 1 )= -2" data-latex="y'= -3 \, t^{2} x - t y^{2} - 1 \hspace{2em} y( 1 )= -2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.2%20)" alt="x( 1.2 )" title="x( 1.2 )" data-latex="x( 1.2 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.2%20)" alt="y( 1.2 )" title="y( 1.2 )" data-latex="y( 1.2 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%202%20%5C,%20x%5E%7B2%7D%20y%5E%7B2%7D%20+%204%20%5C,%20t%5E%7B2%7D%20y%20-%202%20%5Chspace%7B2em%7D%20x(%201%20)=%200" alt="x'= 2 \, x^{2} y^{2} + 4 \, t^{2} y - 2 \hspace{2em} x( 1 )= 0" title="x'= 2 \, x^{2} y^{2} + 4 \, t^{2} y - 2 \hspace{2em} x( 1 )= 0" data-latex="x'= 2 \, x^{2} y^{2} + 4 \, t^{2} y - 2 \hspace{2em} x( 1 )= 0"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-3%20%5C,%20t%5E%7B2%7D%20x%20-%20t%20y%5E%7B2%7D%20-%201%20%5Chspace%7B2em%7D%20y(%201%20)=%20-2" alt="y'= -3 \, t^{2} x - t y^{2} - 1 \hspace{2em} y( 1 )= -2" title="y'= -3 \, t^{2} x - t y^{2} - 1 \hspace{2em} y( 1 )= -2" data-latex="y'= -3 \, t^{2} x - t y^{2} - 1 \hspace{2em} y( 1 )= -2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.1 )\approx -1.00" alt="x( 1.1 )\approx -1.00" title="x( 1.1 )\approx -1.00" data-latex="x( 1.1 )\approx -1.00"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.1 )\approx -2.50" alt="y( 1.1 )\approx -2.50" title="y( 1.1 )\approx -2.50" data-latex="y( 1.1 )\approx -2.50"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.2 )\approx -1.16" alt="x( 1.2 )\approx -1.16" title="x( 1.2 )\approx -1.16" data-latex="x( 1.2 )\approx -1.16"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.2 )\approx -2.93" alt="y( 1.2 )\approx -2.93" title="y( 1.2 )\approx -2.93" data-latex="y( 1.2 )\approx -2.93"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.1%20)%5Capprox%20-1.00" alt="x( 1.1 )\approx -1.00" title="x( 1.1 )\approx -1.00" data-latex="x( 1.1 )\approx -1.00"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.1%20)%5Capprox%20-2.50" alt="y( 1.1 )\approx -2.50" title="y( 1.1 )\approx -2.50" data-latex="y( 1.1 )\approx -2.50"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.2%20)%5Capprox%20-1.16" alt="x( 1.2 )\approx -1.16" title="x( 1.2 )\approx -1.16" data-latex="x( 1.2 )\approx -1.16"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.2%20)%5Capprox%20-2.93" alt="y( 1.2 )\approx -2.93" title="y( 1.2 )\approx -2.93" data-latex="y( 1.2 )\approx -2.93"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-4139" title="N2 | Euler's method for approximating IVP solutions | ver. 4139"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.8 )" alt="x( -1.8 )" title="x( -1.8 )" data-latex="x( -1.8 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.8 )" alt="y( -1.8 )" title="y( -1.8 )" data-latex="y( -1.8 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 4 \, t^{2} y^{2} - 3 \, x^{2} y \hspace{2em} x( -2 )= 0" alt="x'= 4 \, t^{2} y^{2} - 3 \, x^{2} y \hspace{2em} x( -2 )= 0" title="x'= 4 \, t^{2} y^{2} - 3 \, x^{2} y \hspace{2em} x( -2 )= 0" data-latex="x'= 4 \, t^{2} y^{2} - 3 \, x^{2} y \hspace{2em} x( -2 )= 0"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -3 \, t x^{2} - 2 \, t^{2} y + 3 \hspace{2em} y( -2 )= 2" alt="y'= -3 \, t x^{2} - 2 \, t^{2} y + 3 \hspace{2em} y( -2 )= 2" title="y'= -3 \, t x^{2} - 2 \, t^{2} y + 3 \hspace{2em} y( -2 )= 2" data-latex="y'= -3 \, t x^{2} - 2 \, t^{2} y + 3 \hspace{2em} y( -2 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.8%20)" alt="x( -1.8 )" title="x( -1.8 )" data-latex="x( -1.8 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.8%20)" alt="y( -1.8 )" title="y( -1.8 )" data-latex="y( -1.8 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%204%20%5C,%20t%5E%7B2%7D%20y%5E%7B2%7D%20-%203%20%5C,%20x%5E%7B2%7D%20y%20%5Chspace%7B2em%7D%20x(%20-2%20)=%200" alt="x'= 4 \, t^{2} y^{2} - 3 \, x^{2} y \hspace{2em} x( -2 )= 0" title="x'= 4 \, t^{2} y^{2} - 3 \, x^{2} y \hspace{2em} x( -2 )= 0" data-latex="x'= 4 \, t^{2} y^{2} - 3 \, x^{2} y \hspace{2em} x( -2 )= 0"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-3%20%5C,%20t%20x%5E%7B2%7D%20-%202%20%5C,%20t%5E%7B2%7D%20y%20+%203%20%5Chspace%7B2em%7D%20y(%20-2%20)=%202" alt="y'= -3 \, t x^{2} - 2 \, t^{2} y + 3 \hspace{2em} y( -2 )= 2" title="y'= -3 \, t x^{2} - 2 \, t^{2} y + 3 \hspace{2em} y( -2 )= 2" data-latex="y'= -3 \, t x^{2} - 2 \, t^{2} y + 3 \hspace{2em} y( -2 )= 2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.9 )\approx 6.40" alt="x( -1.9 )\approx 6.40" title="x( -1.9 )\approx 6.40" data-latex="x( -1.9 )\approx 6.40"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.9 )\approx 0.701" alt="y( -1.9 )\approx 0.701" title="y( -1.9 )\approx 0.701" data-latex="y( -1.9 )\approx 0.701"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.8 )\approx -1.50" alt="x( -1.8 )\approx -1.50" title="x( -1.8 )\approx -1.50" data-latex="x( -1.8 )\approx -1.50"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.8 )\approx 23.8" alt="y( -1.8 )\approx 23.8" title="y( -1.8 )\approx 23.8" data-latex="y( -1.8 )\approx 23.8"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.9%20)%5Capprox%206.40" alt="x( -1.9 )\approx 6.40" title="x( -1.9 )\approx 6.40" data-latex="x( -1.9 )\approx 6.40"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.9%20)%5Capprox%200.701" alt="y( -1.9 )\approx 0.701" title="y( -1.9 )\approx 0.701" data-latex="y( -1.9 )\approx 0.701"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.8%20)%5Capprox%20-1.50" alt="x( -1.8 )\approx -1.50" title="x( -1.8 )\approx -1.50" data-latex="x( -1.8 )\approx -1.50"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.8%20)%5Capprox%2023.8" alt="y( -1.8 )\approx 23.8" title="y( -1.8 )\approx 23.8" data-latex="y( -1.8 )\approx 23.8"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-4090" title="N2 | Euler's method for approximating IVP solutions | ver. 4090"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.8 )" alt="x( -1.8 )" title="x( -1.8 )" data-latex="x( -1.8 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.8 )" alt="y( -1.8 )" title="y( -1.8 )" data-latex="y( -1.8 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -x^{2} y + 3 \, t y - 1 \hspace{2em} x( -2 )= -1" alt="x'= -x^{2} y + 3 \, t y - 1 \hspace{2em} x( -2 )= -1" title="x'= -x^{2} y + 3 \, t y - 1 \hspace{2em} x( -2 )= -1" data-latex="x'= -x^{2} y + 3 \, t y - 1 \hspace{2em} x( -2 )= -1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -3 \, x^{2} y^{2} + 3 \, t y + 3 \hspace{2em} y( -2 )= -2" alt="y'= -3 \, x^{2} y^{2} + 3 \, t y + 3 \hspace{2em} y( -2 )= -2" title="y'= -3 \, x^{2} y^{2} + 3 \, t y + 3 \hspace{2em} y( -2 )= -2" data-latex="y'= -3 \, x^{2} y^{2} + 3 \, t y + 3 \hspace{2em} y( -2 )= -2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.8%20)" alt="x( -1.8 )" title="x( -1.8 )" data-latex="x( -1.8 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.8%20)" alt="y( -1.8 )" title="y( -1.8 )" data-latex="y( -1.8 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-x%5E%7B2%7D%20y%20+%203%20%5C,%20t%20y%20-%201%20%5Chspace%7B2em%7D%20x(%20-2%20)=%20-1" alt="x'= -x^{2} y + 3 \, t y - 1 \hspace{2em} x( -2 )= -1" title="x'= -x^{2} y + 3 \, t y - 1 \hspace{2em} x( -2 )= -1" data-latex="x'= -x^{2} y + 3 \, t y - 1 \hspace{2em} x( -2 )= -1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-3%20%5C,%20x%5E%7B2%7D%20y%5E%7B2%7D%20+%203%20%5C,%20t%20y%20+%203%20%5Chspace%7B2em%7D%20y(%20-2%20)=%20-2" alt="y'= -3 \, x^{2} y^{2} + 3 \, t y + 3 \hspace{2em} y( -2 )= -2" title="y'= -3 \, x^{2} y^{2} + 3 \, t y + 3 \hspace{2em} y( -2 )= -2" data-latex="y'= -3 \, x^{2} y^{2} + 3 \, t y + 3 \hspace{2em} y( -2 )= -2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.9 )\approx 0.299" alt="x( -1.9 )\approx 0.299" title="x( -1.9 )\approx 0.299" data-latex="x( -1.9 )\approx 0.299"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.9 )\approx -1.70" alt="y( -1.9 )\approx -1.70" title="y( -1.9 )\approx -1.70" data-latex="y( -1.9 )\approx -1.70"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.8 )\approx 1.18" alt="x( -1.8 )\approx 1.18" title="x( -1.8 )\approx 1.18" data-latex="x( -1.8 )\approx 1.18"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.8 )\approx -0.510" alt="y( -1.8 )\approx -0.510" title="y( -1.8 )\approx -0.510" data-latex="y( -1.8 )\approx -0.510"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.9%20)%5Capprox%200.299" alt="x( -1.9 )\approx 0.299" title="x( -1.9 )\approx 0.299" data-latex="x( -1.9 )\approx 0.299"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.9%20)%5Capprox%20-1.70" alt="y( -1.9 )\approx -1.70" title="y( -1.9 )\approx -1.70" data-latex="y( -1.9 )\approx -1.70"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.8%20)%5Capprox%201.18" alt="x( -1.8 )\approx 1.18" title="x( -1.8 )\approx 1.18" data-latex="x( -1.8 )\approx 1.18"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.8%20)%5Capprox%20-0.510" alt="y( -1.8 )\approx -0.510" title="y( -1.8 )\approx -0.510" data-latex="y( -1.8 )\approx -0.510"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-5684" title="N2 | Euler's method for approximating IVP solutions | ver. 5684"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.20 )" alt="x( 0.20 )" title="x( 0.20 )" data-latex="x( 0.20 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.20 )" alt="y( 0.20 )" title="y( 0.20 )" data-latex="y( 0.20 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= t^{2} y^{2} + 2 \, t x \hspace{2em} x( 0 )= 1" alt="x'= t^{2} y^{2} + 2 \, t x \hspace{2em} x( 0 )= 1" title="x'= t^{2} y^{2} + 2 \, t x \hspace{2em} x( 0 )= 1" data-latex="x'= t^{2} y^{2} + 2 \, t x \hspace{2em} x( 0 )= 1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 2 \, x^{2} y^{2} + 3 \, t x^{2} + 1 \hspace{2em} y( 0 )= 0" alt="y'= 2 \, x^{2} y^{2} + 3 \, t x^{2} + 1 \hspace{2em} y( 0 )= 0" title="y'= 2 \, x^{2} y^{2} + 3 \, t x^{2} + 1 \hspace{2em} y( 0 )= 0" data-latex="y'= 2 \, x^{2} y^{2} + 3 \, t x^{2} + 1 \hspace{2em} y( 0 )= 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.20%20)" alt="x( 0.20 )" title="x( 0.20 )" data-latex="x( 0.20 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.20%20)" alt="y( 0.20 )" title="y( 0.20 )" data-latex="y( 0.20 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20t%5E%7B2%7D%20y%5E%7B2%7D%20+%202%20%5C,%20t%20x%20%5Chspace%7B2em%7D%20x(%200%20)=%201" alt="x'= t^{2} y^{2} + 2 \, t x \hspace{2em} x( 0 )= 1" title="x'= t^{2} y^{2} + 2 \, t x \hspace{2em} x( 0 )= 1" data-latex="x'= t^{2} y^{2} + 2 \, t x \hspace{2em} x( 0 )= 1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%202%20%5C,%20x%5E%7B2%7D%20y%5E%7B2%7D%20+%203%20%5C,%20t%20x%5E%7B2%7D%20+%201%20%5Chspace%7B2em%7D%20y(%200%20)=%200" alt="y'= 2 \, x^{2} y^{2} + 3 \, t x^{2} + 1 \hspace{2em} y( 0 )= 0" title="y'= 2 \, x^{2} y^{2} + 3 \, t x^{2} + 1 \hspace{2em} y( 0 )= 0" data-latex="y'= 2 \, x^{2} y^{2} + 3 \, t x^{2} + 1 \hspace{2em} y( 0 )= 0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.10 )\approx 1.00" alt="x( 0.10 )\approx 1.00" title="x( 0.10 )\approx 1.00" data-latex="x( 0.10 )\approx 1.00"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.10 )\approx 0.100" alt="y( 0.10 )\approx 0.100" title="y( 0.10 )\approx 0.100" data-latex="y( 0.10 )\approx 0.100"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.20 )\approx 1.02" alt="x( 0.20 )\approx 1.02" title="x( 0.20 )\approx 1.02" data-latex="x( 0.20 )\approx 1.02"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.20 )\approx 0.232" alt="y( 0.20 )\approx 0.232" title="y( 0.20 )\approx 0.232" data-latex="y( 0.20 )\approx 0.232"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.10%20)%5Capprox%201.00" alt="x( 0.10 )\approx 1.00" title="x( 0.10 )\approx 1.00" data-latex="x( 0.10 )\approx 1.00"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.10%20)%5Capprox%200.100" alt="y( 0.10 )\approx 0.100" title="y( 0.10 )\approx 0.100" data-latex="y( 0.10 )\approx 0.100"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.20%20)%5Capprox%201.02" alt="x( 0.20 )\approx 1.02" title="x( 0.20 )\approx 1.02" data-latex="x( 0.20 )\approx 1.02"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.20%20)%5Capprox%200.232" alt="y( 0.20 )\approx 0.232" title="y( 0.20 )\approx 0.232" data-latex="y( 0.20 )\approx 0.232"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-9274" title="N2 | Euler's method for approximating IVP solutions | ver. 9274"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.2 )" alt="x( 1.2 )" title="x( 1.2 )" data-latex="x( 1.2 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.2 )" alt="y( 1.2 )" title="y( 1.2 )" data-latex="y( 1.2 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 3 \, x^{2} y^{2} - 4 \, t x \hspace{2em} x( 1 )= -2" alt="x'= 3 \, x^{2} y^{2} - 4 \, t x \hspace{2em} x( 1 )= -2" title="x'= 3 \, x^{2} y^{2} - 4 \, t x \hspace{2em} x( 1 )= -2" data-latex="x'= 3 \, x^{2} y^{2} - 4 \, t x \hspace{2em} x( 1 )= -2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 3 \, t^{2} x^{2} + 3 \, x y^{2} + 3 \hspace{2em} y( 1 )= -2" alt="y'= 3 \, t^{2} x^{2} + 3 \, x y^{2} + 3 \hspace{2em} y( 1 )= -2" title="y'= 3 \, t^{2} x^{2} + 3 \, x y^{2} + 3 \hspace{2em} y( 1 )= -2" data-latex="y'= 3 \, t^{2} x^{2} + 3 \, x y^{2} + 3 \hspace{2em} y( 1 )= -2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.2%20)" alt="x( 1.2 )" title="x( 1.2 )" data-latex="x( 1.2 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.2%20)" alt="y( 1.2 )" title="y( 1.2 )" data-latex="y( 1.2 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%203%20%5C,%20x%5E%7B2%7D%20y%5E%7B2%7D%20-%204%20%5C,%20t%20x%20%5Chspace%7B2em%7D%20x(%201%20)=%20-2" alt="x'= 3 \, x^{2} y^{2} - 4 \, t x \hspace{2em} x( 1 )= -2" title="x'= 3 \, x^{2} y^{2} - 4 \, t x \hspace{2em} x( 1 )= -2" data-latex="x'= 3 \, x^{2} y^{2} - 4 \, t x \hspace{2em} x( 1 )= -2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%203%20%5C,%20t%5E%7B2%7D%20x%5E%7B2%7D%20+%203%20%5C,%20x%20y%5E%7B2%7D%20+%203%20%5Chspace%7B2em%7D%20y(%201%20)=%20-2" alt="y'= 3 \, t^{2} x^{2} + 3 \, x y^{2} + 3 \hspace{2em} y( 1 )= -2" title="y'= 3 \, t^{2} x^{2} + 3 \, x y^{2} + 3 \hspace{2em} y( 1 )= -2" data-latex="y'= 3 \, t^{2} x^{2} + 3 \, x y^{2} + 3 \hspace{2em} y( 1 )= -2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.1 )\approx 3.60" alt="x( 1.1 )\approx 3.60" title="x( 1.1 )\approx 3.60" data-latex="x( 1.1 )\approx 3.60"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.1 )\approx -2.90" alt="y( 1.1 )\approx -2.90" title="y( 1.1 )\approx -2.90" data-latex="y( 1.1 )\approx -2.90"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.2 )\approx 34.7" alt="x( 1.2 )\approx 34.7" title="x( 1.2 )\approx 34.7" data-latex="x( 1.2 )\approx 34.7"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.2 )\approx 11.2" alt="y( 1.2 )\approx 11.2" title="y( 1.2 )\approx 11.2" data-latex="y( 1.2 )\approx 11.2"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.1%20)%5Capprox%203.60" alt="x( 1.1 )\approx 3.60" title="x( 1.1 )\approx 3.60" data-latex="x( 1.1 )\approx 3.60"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.1%20)%5Capprox%20-2.90" alt="y( 1.1 )\approx -2.90" title="y( 1.1 )\approx -2.90" data-latex="y( 1.1 )\approx -2.90"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.2%20)%5Capprox%2034.7" alt="x( 1.2 )\approx 34.7" title="x( 1.2 )\approx 34.7" data-latex="x( 1.2 )\approx 34.7"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.2%20)%5Capprox%2011.2" alt="y( 1.2 )\approx 11.2" title="y( 1.2 )\approx 11.2" data-latex="y( 1.2 )\approx 11.2"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-9378" title="N2 | Euler's method for approximating IVP solutions | ver. 9378"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.20 )" alt="x( 0.20 )" title="x( 0.20 )" data-latex="x( 0.20 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.20 )" alt="y( 0.20 )" title="y( 0.20 )" data-latex="y( 0.20 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= t^{2} y^{2} - t^{2} x - 1 \hspace{2em} x( 0 )= 0" alt="x'= t^{2} y^{2} - t^{2} x - 1 \hspace{2em} x( 0 )= 0" title="x'= t^{2} y^{2} - t^{2} x - 1 \hspace{2em} x( 0 )= 0" data-latex="x'= t^{2} y^{2} - t^{2} x - 1 \hspace{2em} x( 0 )= 0"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 2 \, t x^{2} - 2 \, t y + 1 \hspace{2em} y( 0 )= 1" alt="y'= 2 \, t x^{2} - 2 \, t y + 1 \hspace{2em} y( 0 )= 1" title="y'= 2 \, t x^{2} - 2 \, t y + 1 \hspace{2em} y( 0 )= 1" data-latex="y'= 2 \, t x^{2} - 2 \, t y + 1 \hspace{2em} y( 0 )= 1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.20%20)" alt="x( 0.20 )" title="x( 0.20 )" data-latex="x( 0.20 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.20%20)" alt="y( 0.20 )" title="y( 0.20 )" data-latex="y( 0.20 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20t%5E%7B2%7D%20y%5E%7B2%7D%20-%20t%5E%7B2%7D%20x%20-%201%20%5Chspace%7B2em%7D%20x(%200%20)=%200" alt="x'= t^{2} y^{2} - t^{2} x - 1 \hspace{2em} x( 0 )= 0" title="x'= t^{2} y^{2} - t^{2} x - 1 \hspace{2em} x( 0 )= 0" data-latex="x'= t^{2} y^{2} - t^{2} x - 1 \hspace{2em} x( 0 )= 0"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%202%20%5C,%20t%20x%5E%7B2%7D%20-%202%20%5C,%20t%20y%20+%201%20%5Chspace%7B2em%7D%20y(%200%20)=%201" alt="y'= 2 \, t x^{2} - 2 \, t y + 1 \hspace{2em} y( 0 )= 1" title="y'= 2 \, t x^{2} - 2 \, t y + 1 \hspace{2em} y( 0 )= 1" data-latex="y'= 2 \, t x^{2} - 2 \, t y + 1 \hspace{2em} y( 0 )= 1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.10 )\approx -0.100" alt="x( 0.10 )\approx -0.100" title="x( 0.10 )\approx -0.100" data-latex="x( 0.10 )\approx -0.100"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.10 )\approx 1.10" alt="y( 0.10 )\approx 1.10" title="y( 0.10 )\approx 1.10" data-latex="y( 0.10 )\approx 1.10"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.20 )\approx -0.199" alt="x( 0.20 )\approx -0.199" title="x( 0.20 )\approx -0.199" data-latex="x( 0.20 )\approx -0.199"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.20 )\approx 1.18" alt="y( 0.20 )\approx 1.18" title="y( 0.20 )\approx 1.18" data-latex="y( 0.20 )\approx 1.18"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.10%20)%5Capprox%20-0.100" alt="x( 0.10 )\approx -0.100" title="x( 0.10 )\approx -0.100" data-latex="x( 0.10 )\approx -0.100"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.10%20)%5Capprox%201.10" alt="y( 0.10 )\approx 1.10" title="y( 0.10 )\approx 1.10" data-latex="y( 0.10 )\approx 1.10"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.20%20)%5Capprox%20-0.199" alt="x( 0.20 )\approx -0.199" title="x( 0.20 )\approx -0.199" data-latex="x( 0.20 )\approx -0.199"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.20%20)%5Capprox%201.18" alt="y( 0.20 )\approx 1.18" title="y( 0.20 )\approx 1.18" data-latex="y( 0.20 )\approx 1.18"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-5085" title="N2 | Euler's method for approximating IVP solutions | ver. 5085"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.80 )" alt="x( -0.80 )" title="x( -0.80 )" data-latex="x( -0.80 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.80 )" alt="y( -0.80 )" title="y( -0.80 )" data-latex="y( -0.80 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -2 \, t x^{2} + 2 \, t^{2} y + 3 \hspace{2em} x( -1 )= -2" alt="x'= -2 \, t x^{2} + 2 \, t^{2} y + 3 \hspace{2em} x( -1 )= -2" title="x'= -2 \, t x^{2} + 2 \, t^{2} y + 3 \hspace{2em} x( -1 )= -2" data-latex="x'= -2 \, t x^{2} + 2 \, t^{2} y + 3 \hspace{2em} x( -1 )= -2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -3 \, t y + x y - 3 \hspace{2em} y( -1 )= 0" alt="y'= -3 \, t y + x y - 3 \hspace{2em} y( -1 )= 0" title="y'= -3 \, t y + x y - 3 \hspace{2em} y( -1 )= 0" data-latex="y'= -3 \, t y + x y - 3 \hspace{2em} y( -1 )= 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.80%20)" alt="x( -0.80 )" title="x( -0.80 )" data-latex="x( -0.80 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.80%20)" alt="y( -0.80 )" title="y( -0.80 )" data-latex="y( -0.80 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-2%20%5C,%20t%20x%5E%7B2%7D%20+%202%20%5C,%20t%5E%7B2%7D%20y%20+%203%20%5Chspace%7B2em%7D%20x(%20-1%20)=%20-2" alt="x'= -2 \, t x^{2} + 2 \, t^{2} y + 3 \hspace{2em} x( -1 )= -2" title="x'= -2 \, t x^{2} + 2 \, t^{2} y + 3 \hspace{2em} x( -1 )= -2" data-latex="x'= -2 \, t x^{2} + 2 \, t^{2} y + 3 \hspace{2em} x( -1 )= -2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-3%20%5C,%20t%20y%20+%20x%20y%20-%203%20%5Chspace%7B2em%7D%20y(%20-1%20)=%200" alt="y'= -3 \, t y + x y - 3 \hspace{2em} y( -1 )= 0" title="y'= -3 \, t y + x y - 3 \hspace{2em} y( -1 )= 0" data-latex="y'= -3 \, t y + x y - 3 \hspace{2em} y( -1 )= 0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.90 )\approx -0.900" alt="x( -0.90 )\approx -0.900" title="x( -0.90 )\approx -0.900" data-latex="x( -0.90 )\approx -0.900"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.90 )\approx -0.300" alt="y( -0.90 )\approx -0.300" title="y( -0.90 )\approx -0.300" data-latex="y( -0.90 )\approx -0.300"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.80 )\approx -0.503" alt="x( -0.80 )\approx -0.503" title="x( -0.80 )\approx -0.503" data-latex="x( -0.80 )\approx -0.503"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.80 )\approx -0.654" alt="y( -0.80 )\approx -0.654" title="y( -0.80 )\approx -0.654" data-latex="y( -0.80 )\approx -0.654"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.90%20)%5Capprox%20-0.900" alt="x( -0.90 )\approx -0.900" title="x( -0.90 )\approx -0.900" data-latex="x( -0.90 )\approx -0.900"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.90%20)%5Capprox%20-0.300" alt="y( -0.90 )\approx -0.300" title="y( -0.90 )\approx -0.300" data-latex="y( -0.90 )\approx -0.300"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.80%20)%5Capprox%20-0.503" alt="x( -0.80 )\approx -0.503" title="x( -0.80 )\approx -0.503" data-latex="x( -0.80 )\approx -0.503"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.80%20)%5Capprox%20-0.654" alt="y( -0.80 )\approx -0.654" title="y( -0.80 )\approx -0.654" data-latex="y( -0.80 )\approx -0.654"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-5899" title="N2 | Euler's method for approximating IVP solutions | ver. 5899"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.80 )" alt="x( -0.80 )" title="x( -0.80 )" data-latex="x( -0.80 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.80 )" alt="y( -0.80 )" title="y( -0.80 )" data-latex="y( -0.80 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -2 \, t^{2} y^{2} + 2 \, t x^{2} - 3 \hspace{2em} x( -1 )= -2" alt="x'= -2 \, t^{2} y^{2} + 2 \, t x^{2} - 3 \hspace{2em} x( -1 )= -2" title="x'= -2 \, t^{2} y^{2} + 2 \, t x^{2} - 3 \hspace{2em} x( -1 )= -2" data-latex="x'= -2 \, t^{2} y^{2} + 2 \, t x^{2} - 3 \hspace{2em} x( -1 )= -2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -x^{2} y^{2} + t x^{2} - 2 \hspace{2em} y( -1 )= -1" alt="y'= -x^{2} y^{2} + t x^{2} - 2 \hspace{2em} y( -1 )= -1" title="y'= -x^{2} y^{2} + t x^{2} - 2 \hspace{2em} y( -1 )= -1" data-latex="y'= -x^{2} y^{2} + t x^{2} - 2 \hspace{2em} y( -1 )= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.80%20)" alt="x( -0.80 )" title="x( -0.80 )" data-latex="x( -0.80 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.80%20)" alt="y( -0.80 )" title="y( -0.80 )" data-latex="y( -0.80 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-2%20%5C,%20t%5E%7B2%7D%20y%5E%7B2%7D%20+%202%20%5C,%20t%20x%5E%7B2%7D%20-%203%20%5Chspace%7B2em%7D%20x(%20-1%20)=%20-2" alt="x'= -2 \, t^{2} y^{2} + 2 \, t x^{2} - 3 \hspace{2em} x( -1 )= -2" title="x'= -2 \, t^{2} y^{2} + 2 \, t x^{2} - 3 \hspace{2em} x( -1 )= -2" data-latex="x'= -2 \, t^{2} y^{2} + 2 \, t x^{2} - 3 \hspace{2em} x( -1 )= -2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-x%5E%7B2%7D%20y%5E%7B2%7D%20+%20t%20x%5E%7B2%7D%20-%202%20%5Chspace%7B2em%7D%20y(%20-1%20)=%20-1" alt="y'= -x^{2} y^{2} + t x^{2} - 2 \hspace{2em} y( -1 )= -1" title="y'= -x^{2} y^{2} + t x^{2} - 2 \hspace{2em} y( -1 )= -1" data-latex="y'= -x^{2} y^{2} + t x^{2} - 2 \hspace{2em} y( -1 )= -1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.90 )\approx -3.30" alt="x( -0.90 )\approx -3.30" title="x( -0.90 )\approx -3.30" data-latex="x( -0.90 )\approx -3.30"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.90 )\approx -2.00" alt="y( -0.90 )\approx -2.00" title="y( -0.90 )\approx -2.00" data-latex="y( -0.90 )\approx -2.00"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.80 )\approx -6.20" alt="x( -0.80 )\approx -6.20" title="x( -0.80 )\approx -6.20" data-latex="x( -0.80 )\approx -6.20"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.80 )\approx -7.53" alt="y( -0.80 )\approx -7.53" title="y( -0.80 )\approx -7.53" data-latex="y( -0.80 )\approx -7.53"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.90%20)%5Capprox%20-3.30" alt="x( -0.90 )\approx -3.30" title="x( -0.90 )\approx -3.30" data-latex="x( -0.90 )\approx -3.30"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.90%20)%5Capprox%20-2.00" alt="y( -0.90 )\approx -2.00" title="y( -0.90 )\approx -2.00" data-latex="y( -0.90 )\approx -2.00"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.80%20)%5Capprox%20-6.20" alt="x( -0.80 )\approx -6.20" title="x( -0.80 )\approx -6.20" data-latex="x( -0.80 )\approx -6.20"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.80%20)%5Capprox%20-7.53" alt="y( -0.80 )\approx -7.53" title="y( -0.80 )\approx -7.53" data-latex="y( -0.80 )\approx -7.53"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-4107" title="N2 | Euler's method for approximating IVP solutions | ver. 4107"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.80 )" alt="x( -0.80 )" title="x( -0.80 )" data-latex="x( -0.80 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.80 )" alt="y( -0.80 )" title="y( -0.80 )" data-latex="y( -0.80 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 3 \, t x + x y - 2 \hspace{2em} x( -1 )= 2" alt="x'= 3 \, t x + x y - 2 \hspace{2em} x( -1 )= 2" title="x'= 3 \, t x + x y - 2 \hspace{2em} x( -1 )= 2" data-latex="x'= 3 \, t x + x y - 2 \hspace{2em} x( -1 )= 2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -2 \, x^{2} y^{2} - 3 \, t y^{2} - 3 \hspace{2em} y( -1 )= -2" alt="y'= -2 \, x^{2} y^{2} - 3 \, t y^{2} - 3 \hspace{2em} y( -1 )= -2" title="y'= -2 \, x^{2} y^{2} - 3 \, t y^{2} - 3 \hspace{2em} y( -1 )= -2" data-latex="y'= -2 \, x^{2} y^{2} - 3 \, t y^{2} - 3 \hspace{2em} y( -1 )= -2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.80%20)" alt="x( -0.80 )" title="x( -0.80 )" data-latex="x( -0.80 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.80%20)" alt="y( -0.80 )" title="y( -0.80 )" data-latex="y( -0.80 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%203%20%5C,%20t%20x%20+%20x%20y%20-%202%20%5Chspace%7B2em%7D%20x(%20-1%20)=%202" alt="x'= 3 \, t x + x y - 2 \hspace{2em} x( -1 )= 2" title="x'= 3 \, t x + x y - 2 \hspace{2em} x( -1 )= 2" data-latex="x'= 3 \, t x + x y - 2 \hspace{2em} x( -1 )= 2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-2%20%5C,%20x%5E%7B2%7D%20y%5E%7B2%7D%20-%203%20%5C,%20t%20y%5E%7B2%7D%20-%203%20%5Chspace%7B2em%7D%20y(%20-1%20)=%20-2" alt="y'= -2 \, x^{2} y^{2} - 3 \, t y^{2} - 3 \hspace{2em} y( -1 )= -2" title="y'= -2 \, x^{2} y^{2} - 3 \, t y^{2} - 3 \hspace{2em} y( -1 )= -2" data-latex="y'= -2 \, x^{2} y^{2} - 3 \, t y^{2} - 3 \hspace{2em} y( -1 )= -2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.90 )\approx 0.801" alt="x( -0.90 )\approx 0.801" title="x( -0.90 )\approx 0.801" data-latex="x( -0.90 )\approx 0.801"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.90 )\approx -4.30" alt="y( -0.90 )\approx -4.30" title="y( -0.90 )\approx -4.30" data-latex="y( -0.90 )\approx -4.30"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.80 )\approx 0.0410" alt="x( -0.80 )\approx 0.0410" title="x( -0.80 )\approx 0.0410" data-latex="x( -0.80 )\approx 0.0410"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.80 )\approx -1.98" alt="y( -0.80 )\approx -1.98" title="y( -0.80 )\approx -1.98" data-latex="y( -0.80 )\approx -1.98"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.90%20)%5Capprox%200.801" alt="x( -0.90 )\approx 0.801" title="x( -0.90 )\approx 0.801" data-latex="x( -0.90 )\approx 0.801"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.90%20)%5Capprox%20-4.30" alt="y( -0.90 )\approx -4.30" title="y( -0.90 )\approx -4.30" data-latex="y( -0.90 )\approx -4.30"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.80%20)%5Capprox%200.0410" alt="x( -0.80 )\approx 0.0410" title="x( -0.80 )\approx 0.0410" data-latex="x( -0.80 )\approx 0.0410"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.80%20)%5Capprox%20-1.98" alt="y( -0.80 )\approx -1.98" title="y( -0.80 )\approx -1.98" data-latex="y( -0.80 )\approx -1.98"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-5651" title="N2 | Euler's method for approximating IVP solutions | ver. 5651"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 2.2 )" alt="x( 2.2 )" title="x( 2.2 )" data-latex="x( 2.2 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 2.2 )" alt="y( 2.2 )" title="y( 2.2 )" data-latex="y( 2.2 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -3 \, x^{2} y^{2} + t^{2} x + 1 \hspace{2em} x( 2 )= 2" alt="x'= -3 \, x^{2} y^{2} + t^{2} x + 1 \hspace{2em} x( 2 )= 2" title="x'= -3 \, x^{2} y^{2} + t^{2} x + 1 \hspace{2em} x( 2 )= 2" data-latex="x'= -3 \, x^{2} y^{2} + t^{2} x + 1 \hspace{2em} x( 2 )= 2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 3 \, x^{2} y + 4 \, t y^{2} + 3 \hspace{2em} y( 2 )= 2" alt="y'= 3 \, x^{2} y + 4 \, t y^{2} + 3 \hspace{2em} y( 2 )= 2" title="y'= 3 \, x^{2} y + 4 \, t y^{2} + 3 \hspace{2em} y( 2 )= 2" data-latex="y'= 3 \, x^{2} y + 4 \, t y^{2} + 3 \hspace{2em} y( 2 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%202.2%20)" alt="x( 2.2 )" title="x( 2.2 )" data-latex="x( 2.2 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%202.2%20)" alt="y( 2.2 )" title="y( 2.2 )" data-latex="y( 2.2 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-3%20%5C,%20x%5E%7B2%7D%20y%5E%7B2%7D%20+%20t%5E%7B2%7D%20x%20+%201%20%5Chspace%7B2em%7D%20x(%202%20)=%202" alt="x'= -3 \, x^{2} y^{2} + t^{2} x + 1 \hspace{2em} x( 2 )= 2" title="x'= -3 \, x^{2} y^{2} + t^{2} x + 1 \hspace{2em} x( 2 )= 2" data-latex="x'= -3 \, x^{2} y^{2} + t^{2} x + 1 \hspace{2em} x( 2 )= 2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%203%20%5C,%20x%5E%7B2%7D%20y%20+%204%20%5C,%20t%20y%5E%7B2%7D%20+%203%20%5Chspace%7B2em%7D%20y(%202%20)=%202" alt="y'= 3 \, x^{2} y + 4 \, t y^{2} + 3 \hspace{2em} y( 2 )= 2" title="y'= 3 \, x^{2} y + 4 \, t y^{2} + 3 \hspace{2em} y( 2 )= 2" data-latex="y'= 3 \, x^{2} y + 4 \, t y^{2} + 3 \hspace{2em} y( 2 )= 2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 2.1 )\approx -1.90" alt="x( 2.1 )\approx -1.90" title="x( 2.1 )\approx -1.90" data-latex="x( 2.1 )\approx -1.90"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 2.1 )\approx 7.90" alt="y( 2.1 )\approx 7.90" title="y( 2.1 )\approx 7.90" data-latex="y( 2.1 )\approx 7.90"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 2.2 )\approx -70.0" alt="x( 2.2 )\approx -70.0" title="x( 2.2 )\approx -70.0" data-latex="x( 2.2 )\approx -70.0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 2.2 )\approx 69.1" alt="y( 2.2 )\approx 69.1" title="y( 2.2 )\approx 69.1" data-latex="y( 2.2 )\approx 69.1"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%202.1%20)%5Capprox%20-1.90" alt="x( 2.1 )\approx -1.90" title="x( 2.1 )\approx -1.90" data-latex="x( 2.1 )\approx -1.90"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%202.1%20)%5Capprox%207.90" alt="y( 2.1 )\approx 7.90" title="y( 2.1 )\approx 7.90" data-latex="y( 2.1 )\approx 7.90"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%202.2%20)%5Capprox%20-70.0" alt="x( 2.2 )\approx -70.0" title="x( 2.2 )\approx -70.0" data-latex="x( 2.2 )\approx -70.0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%202.2%20)%5Capprox%2069.1" alt="y( 2.2 )\approx 69.1" title="y( 2.2 )\approx 69.1" data-latex="y( 2.2 )\approx 69.1"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-1522" title="N2 | Euler's method for approximating IVP solutions | ver. 1522"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.2 )" alt="x( 1.2 )" title="x( 1.2 )" data-latex="x( 1.2 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.2 )" alt="y( 1.2 )" title="y( 1.2 )" data-latex="y( 1.2 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -4 \, t^{2} y + 2 \, x y^{2} - 2 \hspace{2em} x( 1 )= 1" alt="x'= -4 \, t^{2} y + 2 \, x y^{2} - 2 \hspace{2em} x( 1 )= 1" title="x'= -4 \, t^{2} y + 2 \, x y^{2} - 2 \hspace{2em} x( 1 )= 1" data-latex="x'= -4 \, t^{2} y + 2 \, x y^{2} - 2 \hspace{2em} x( 1 )= 1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -2 \, t^{2} x + 3 \, x y - 3 \hspace{2em} y( 1 )= -1" alt="y'= -2 \, t^{2} x + 3 \, x y - 3 \hspace{2em} y( 1 )= -1" title="y'= -2 \, t^{2} x + 3 \, x y - 3 \hspace{2em} y( 1 )= -1" data-latex="y'= -2 \, t^{2} x + 3 \, x y - 3 \hspace{2em} y( 1 )= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.2%20)" alt="x( 1.2 )" title="x( 1.2 )" data-latex="x( 1.2 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.2%20)" alt="y( 1.2 )" title="y( 1.2 )" data-latex="y( 1.2 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-4%20%5C,%20t%5E%7B2%7D%20y%20+%202%20%5C,%20x%20y%5E%7B2%7D%20-%202%20%5Chspace%7B2em%7D%20x(%201%20)=%201" alt="x'= -4 \, t^{2} y + 2 \, x y^{2} - 2 \hspace{2em} x( 1 )= 1" title="x'= -4 \, t^{2} y + 2 \, x y^{2} - 2 \hspace{2em} x( 1 )= 1" data-latex="x'= -4 \, t^{2} y + 2 \, x y^{2} - 2 \hspace{2em} x( 1 )= 1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-2%20%5C,%20t%5E%7B2%7D%20x%20+%203%20%5C,%20x%20y%20-%203%20%5Chspace%7B2em%7D%20y(%201%20)=%20-1" alt="y'= -2 \, t^{2} x + 3 \, x y - 3 \hspace{2em} y( 1 )= -1" title="y'= -2 \, t^{2} x + 3 \, x y - 3 \hspace{2em} y( 1 )= -1" data-latex="y'= -2 \, t^{2} x + 3 \, x y - 3 \hspace{2em} y( 1 )= -1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.1 )\approx 1.40" alt="x( 1.1 )\approx 1.40" title="x( 1.1 )\approx 1.40" data-latex="x( 1.1 )\approx 1.40"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.1 )\approx -1.80" alt="y( 1.1 )\approx -1.80" title="y( 1.1 )\approx -1.80" data-latex="y( 1.1 )\approx -1.80"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.2 )\approx 2.98" alt="x( 1.2 )\approx 2.98" title="x( 1.2 )\approx 2.98" data-latex="x( 1.2 )\approx 2.98"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.2 )\approx -3.20" alt="y( 1.2 )\approx -3.20" title="y( 1.2 )\approx -3.20" data-latex="y( 1.2 )\approx -3.20"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.1%20)%5Capprox%201.40" alt="x( 1.1 )\approx 1.40" title="x( 1.1 )\approx 1.40" data-latex="x( 1.1 )\approx 1.40"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.1%20)%5Capprox%20-1.80" alt="y( 1.1 )\approx -1.80" title="y( 1.1 )\approx -1.80" data-latex="y( 1.1 )\approx -1.80"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.2%20)%5Capprox%202.98" alt="x( 1.2 )\approx 2.98" title="x( 1.2 )\approx 2.98" data-latex="x( 1.2 )\approx 2.98"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.2%20)%5Capprox%20-3.20" alt="y( 1.2 )\approx -3.20" title="y( 1.2 )\approx -3.20" data-latex="y( 1.2 )\approx -3.20"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-7690" title="N2 | Euler's method for approximating IVP solutions | ver. 7690"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.80 )" alt="x( -0.80 )" title="x( -0.80 )" data-latex="x( -0.80 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.80 )" alt="y( -0.80 )" title="y( -0.80 )" data-latex="y( -0.80 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= t^{2} y + 4 \, x^{2} y - 2 \hspace{2em} x( -1 )= 0" alt="x'= t^{2} y + 4 \, x^{2} y - 2 \hspace{2em} x( -1 )= 0" title="x'= t^{2} y + 4 \, x^{2} y - 2 \hspace{2em} x( -1 )= 0" data-latex="x'= t^{2} y + 4 \, x^{2} y - 2 \hspace{2em} x( -1 )= 0"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -3 \, x^{2} y + 2 \, t x \hspace{2em} y( -1 )= 2" alt="y'= -3 \, x^{2} y + 2 \, t x \hspace{2em} y( -1 )= 2" title="y'= -3 \, x^{2} y + 2 \, t x \hspace{2em} y( -1 )= 2" data-latex="y'= -3 \, x^{2} y + 2 \, t x \hspace{2em} y( -1 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.80%20)" alt="x( -0.80 )" title="x( -0.80 )" data-latex="x( -0.80 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.80%20)" alt="y( -0.80 )" title="y( -0.80 )" data-latex="y( -0.80 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20t%5E%7B2%7D%20y%20+%204%20%5C,%20x%5E%7B2%7D%20y%20-%202%20%5Chspace%7B2em%7D%20x(%20-1%20)=%200" alt="x'= t^{2} y + 4 \, x^{2} y - 2 \hspace{2em} x( -1 )= 0" title="x'= t^{2} y + 4 \, x^{2} y - 2 \hspace{2em} x( -1 )= 0" data-latex="x'= t^{2} y + 4 \, x^{2} y - 2 \hspace{2em} x( -1 )= 0"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-3%20%5C,%20x%5E%7B2%7D%20y%20+%202%20%5C,%20t%20x%20%5Chspace%7B2em%7D%20y(%20-1%20)=%202" alt="y'= -3 \, x^{2} y + 2 \, t x \hspace{2em} y( -1 )= 2" title="y'= -3 \, x^{2} y + 2 \, t x \hspace{2em} y( -1 )= 2" data-latex="y'= -3 \, x^{2} y + 2 \, t x \hspace{2em} y( -1 )= 2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.90 )\approx 0.000" alt="x( -0.90 )\approx 0.000" title="x( -0.90 )\approx 0.000" data-latex="x( -0.90 )\approx 0.000"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.90 )\approx 2.00" alt="y( -0.90 )\approx 2.00" title="y( -0.90 )\approx 2.00" data-latex="y( -0.90 )\approx 2.00"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.80 )\approx -0.0379" alt="x( -0.80 )\approx -0.0379" title="x( -0.80 )\approx -0.0379" data-latex="x( -0.80 )\approx -0.0379"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.80 )\approx 2.00" alt="y( -0.80 )\approx 2.00" title="y( -0.80 )\approx 2.00" data-latex="y( -0.80 )\approx 2.00"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.90%20)%5Capprox%200.000" alt="x( -0.90 )\approx 0.000" title="x( -0.90 )\approx 0.000" data-latex="x( -0.90 )\approx 0.000"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.90%20)%5Capprox%202.00" alt="y( -0.90 )\approx 2.00" title="y( -0.90 )\approx 2.00" data-latex="y( -0.90 )\approx 2.00"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.80%20)%5Capprox%20-0.0379" alt="x( -0.80 )\approx -0.0379" title="x( -0.80 )\approx -0.0379" data-latex="x( -0.80 )\approx -0.0379"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.80%20)%5Capprox%202.00" alt="y( -0.80 )\approx 2.00" title="y( -0.80 )\approx 2.00" data-latex="y( -0.80 )\approx 2.00"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-1166" title="N2 | Euler's method for approximating IVP solutions | ver. 1166"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.8 )" alt="x( -1.8 )" title="x( -1.8 )" data-latex="x( -1.8 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.8 )" alt="y( -1.8 )" title="y( -1.8 )" data-latex="y( -1.8 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 2 \, t^{2} y + x y^{2} + 3 \hspace{2em} x( -2 )= -1" alt="x'= 2 \, t^{2} y + x y^{2} + 3 \hspace{2em} x( -2 )= -1" title="x'= 2 \, t^{2} y + x y^{2} + 3 \hspace{2em} x( -2 )= -1" data-latex="x'= 2 \, t^{2} y + x y^{2} + 3 \hspace{2em} x( -2 )= -1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 4 \, t^{2} y^{2} - x^{2} y^{2} - 2 \hspace{2em} y( -2 )= 2" alt="y'= 4 \, t^{2} y^{2} - x^{2} y^{2} - 2 \hspace{2em} y( -2 )= 2" title="y'= 4 \, t^{2} y^{2} - x^{2} y^{2} - 2 \hspace{2em} y( -2 )= 2" data-latex="y'= 4 \, t^{2} y^{2} - x^{2} y^{2} - 2 \hspace{2em} y( -2 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.8%20)" alt="x( -1.8 )" title="x( -1.8 )" data-latex="x( -1.8 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.8%20)" alt="y( -1.8 )" title="y( -1.8 )" data-latex="y( -1.8 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%202%20%5C,%20t%5E%7B2%7D%20y%20+%20x%20y%5E%7B2%7D%20+%203%20%5Chspace%7B2em%7D%20x(%20-2%20)=%20-1" alt="x'= 2 \, t^{2} y + x y^{2} + 3 \hspace{2em} x( -2 )= -1" title="x'= 2 \, t^{2} y + x y^{2} + 3 \hspace{2em} x( -2 )= -1" data-latex="x'= 2 \, t^{2} y + x y^{2} + 3 \hspace{2em} x( -2 )= -1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%204%20%5C,%20t%5E%7B2%7D%20y%5E%7B2%7D%20-%20x%5E%7B2%7D%20y%5E%7B2%7D%20-%202%20%5Chspace%7B2em%7D%20y(%20-2%20)=%202" alt="y'= 4 \, t^{2} y^{2} - x^{2} y^{2} - 2 \hspace{2em} y( -2 )= 2" title="y'= 4 \, t^{2} y^{2} - x^{2} y^{2} - 2 \hspace{2em} y( -2 )= 2" data-latex="y'= 4 \, t^{2} y^{2} - x^{2} y^{2} - 2 \hspace{2em} y( -2 )= 2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.9 )\approx 0.500" alt="x( -1.9 )\approx 0.500" title="x( -1.9 )\approx 0.500" data-latex="x( -1.9 )\approx 0.500"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.9 )\approx 7.80" alt="y( -1.9 )\approx 7.80" title="y( -1.9 )\approx 7.80" data-latex="y( -1.9 )\approx 7.80"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.8 )\approx 9.47" alt="x( -1.8 )\approx 9.47" title="x( -1.8 )\approx 9.47" data-latex="x( -1.8 )\approx 9.47"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.8 )\approx 93.9" alt="y( -1.8 )\approx 93.9" title="y( -1.8 )\approx 93.9" data-latex="y( -1.8 )\approx 93.9"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.9%20)%5Capprox%200.500" alt="x( -1.9 )\approx 0.500" title="x( -1.9 )\approx 0.500" data-latex="x( -1.9 )\approx 0.500"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.9%20)%5Capprox%207.80" alt="y( -1.9 )\approx 7.80" title="y( -1.9 )\approx 7.80" data-latex="y( -1.9 )\approx 7.80"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.8%20)%5Capprox%209.47" alt="x( -1.8 )\approx 9.47" title="x( -1.8 )\approx 9.47" data-latex="x( -1.8 )\approx 9.47"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.8%20)%5Capprox%2093.9" alt="y( -1.8 )\approx 93.9" title="y( -1.8 )\approx 93.9" data-latex="y( -1.8 )\approx 93.9"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-7031" title="N2 | Euler's method for approximating IVP solutions | ver. 7031"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.80 )" alt="x( -0.80 )" title="x( -0.80 )" data-latex="x( -0.80 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.80 )" alt="y( -0.80 )" title="y( -0.80 )" data-latex="y( -0.80 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -2 \, x^{2} y + 2 \, t y^{2} + 1 \hspace{2em} x( -1 )= 1" alt="x'= -2 \, x^{2} y + 2 \, t y^{2} + 1 \hspace{2em} x( -1 )= 1" title="x'= -2 \, x^{2} y + 2 \, t y^{2} + 1 \hspace{2em} x( -1 )= 1" data-latex="x'= -2 \, x^{2} y + 2 \, t y^{2} + 1 \hspace{2em} x( -1 )= 1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 2 \, t^{2} x^{2} + x y^{2} - 1 \hspace{2em} y( -1 )= -2" alt="y'= 2 \, t^{2} x^{2} + x y^{2} - 1 \hspace{2em} y( -1 )= -2" title="y'= 2 \, t^{2} x^{2} + x y^{2} - 1 \hspace{2em} y( -1 )= -2" data-latex="y'= 2 \, t^{2} x^{2} + x y^{2} - 1 \hspace{2em} y( -1 )= -2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.80%20)" alt="x( -0.80 )" title="x( -0.80 )" data-latex="x( -0.80 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.80%20)" alt="y( -0.80 )" title="y( -0.80 )" data-latex="y( -0.80 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-2%20%5C,%20x%5E%7B2%7D%20y%20+%202%20%5C,%20t%20y%5E%7B2%7D%20+%201%20%5Chspace%7B2em%7D%20x(%20-1%20)=%201" alt="x'= -2 \, x^{2} y + 2 \, t y^{2} + 1 \hspace{2em} x( -1 )= 1" title="x'= -2 \, x^{2} y + 2 \, t y^{2} + 1 \hspace{2em} x( -1 )= 1" data-latex="x'= -2 \, x^{2} y + 2 \, t y^{2} + 1 \hspace{2em} x( -1 )= 1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%202%20%5C,%20t%5E%7B2%7D%20x%5E%7B2%7D%20+%20x%20y%5E%7B2%7D%20-%201%20%5Chspace%7B2em%7D%20y(%20-1%20)=%20-2" alt="y'= 2 \, t^{2} x^{2} + x y^{2} - 1 \hspace{2em} y( -1 )= -2" title="y'= 2 \, t^{2} x^{2} + x y^{2} - 1 \hspace{2em} y( -1 )= -2" data-latex="y'= 2 \, t^{2} x^{2} + x y^{2} - 1 \hspace{2em} y( -1 )= -2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.90 )\approx 0.700" alt="x( -0.90 )\approx 0.700" title="x( -0.90 )\approx 0.700" data-latex="x( -0.90 )\approx 0.700"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.90 )\approx -1.50" alt="y( -0.90 )\approx -1.50" title="y( -0.90 )\approx -1.50" data-latex="y( -0.90 )\approx -1.50"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.80 )\approx 0.542" alt="x( -0.80 )\approx 0.542" title="x( -0.80 )\approx 0.542" data-latex="x( -0.80 )\approx 0.542"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.80 )\approx -1.36" alt="y( -0.80 )\approx -1.36" title="y( -0.80 )\approx -1.36" data-latex="y( -0.80 )\approx -1.36"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.90%20)%5Capprox%200.700" alt="x( -0.90 )\approx 0.700" title="x( -0.90 )\approx 0.700" data-latex="x( -0.90 )\approx 0.700"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.90%20)%5Capprox%20-1.50" alt="y( -0.90 )\approx -1.50" title="y( -0.90 )\approx -1.50" data-latex="y( -0.90 )\approx -1.50"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.80%20)%5Capprox%200.542" alt="x( -0.80 )\approx 0.542" title="x( -0.80 )\approx 0.542" data-latex="x( -0.80 )\approx 0.542"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.80%20)%5Capprox%20-1.36" alt="y( -0.80 )\approx -1.36" title="y( -0.80 )\approx -1.36" data-latex="y( -0.80 )\approx -1.36"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-3305" title="N2 | Euler's method for approximating IVP solutions | ver. 3305"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.8 )" alt="x( -1.8 )" title="x( -1.8 )" data-latex="x( -1.8 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.8 )" alt="y( -1.8 )" title="y( -1.8 )" data-latex="y( -1.8 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -3 \, t^{2} x^{2} - 4 \, t y + 2 \hspace{2em} x( -2 )= 0" alt="x'= -3 \, t^{2} x^{2} - 4 \, t y + 2 \hspace{2em} x( -2 )= 0" title="x'= -3 \, t^{2} x^{2} - 4 \, t y + 2 \hspace{2em} x( -2 )= 0" data-latex="x'= -3 \, t^{2} x^{2} - 4 \, t y + 2 \hspace{2em} x( -2 )= 0"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -3 \, x^{2} y^{2} - 3 \, t^{2} y + 2 \hspace{2em} y( -2 )= 1" alt="y'= -3 \, x^{2} y^{2} - 3 \, t^{2} y + 2 \hspace{2em} y( -2 )= 1" title="y'= -3 \, x^{2} y^{2} - 3 \, t^{2} y + 2 \hspace{2em} y( -2 )= 1" data-latex="y'= -3 \, x^{2} y^{2} - 3 \, t^{2} y + 2 \hspace{2em} y( -2 )= 1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.8%20)" alt="x( -1.8 )" title="x( -1.8 )" data-latex="x( -1.8 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.8%20)" alt="y( -1.8 )" title="y( -1.8 )" data-latex="y( -1.8 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-3%20%5C,%20t%5E%7B2%7D%20x%5E%7B2%7D%20-%204%20%5C,%20t%20y%20+%202%20%5Chspace%7B2em%7D%20x(%20-2%20)=%200" alt="x'= -3 \, t^{2} x^{2} - 4 \, t y + 2 \hspace{2em} x( -2 )= 0" title="x'= -3 \, t^{2} x^{2} - 4 \, t y + 2 \hspace{2em} x( -2 )= 0" data-latex="x'= -3 \, t^{2} x^{2} - 4 \, t y + 2 \hspace{2em} x( -2 )= 0"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-3%20%5C,%20x%5E%7B2%7D%20y%5E%7B2%7D%20-%203%20%5C,%20t%5E%7B2%7D%20y%20+%202%20%5Chspace%7B2em%7D%20y(%20-2%20)=%201" alt="y'= -3 \, x^{2} y^{2} - 3 \, t^{2} y + 2 \hspace{2em} y( -2 )= 1" title="y'= -3 \, x^{2} y^{2} - 3 \, t^{2} y + 2 \hspace{2em} y( -2 )= 1" data-latex="y'= -3 \, x^{2} y^{2} - 3 \, t^{2} y + 2 \hspace{2em} y( -2 )= 1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.9 )\approx 1.00" alt="x( -1.9 )\approx 1.00" title="x( -1.9 )\approx 1.00" data-latex="x( -1.9 )\approx 1.00"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.9 )\approx 0.000" alt="y( -1.9 )\approx 0.000" title="y( -1.9 )\approx 0.000" data-latex="y( -1.9 )\approx 0.000"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.8 )\approx 0.116" alt="x( -1.8 )\approx 0.116" title="x( -1.8 )\approx 0.116" data-latex="x( -1.8 )\approx 0.116"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.8 )\approx 0.200" alt="y( -1.8 )\approx 0.200" title="y( -1.8 )\approx 0.200" data-latex="y( -1.8 )\approx 0.200"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.9%20)%5Capprox%201.00" alt="x( -1.9 )\approx 1.00" title="x( -1.9 )\approx 1.00" data-latex="x( -1.9 )\approx 1.00"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.9%20)%5Capprox%200.000" alt="y( -1.9 )\approx 0.000" title="y( -1.9 )\approx 0.000" data-latex="y( -1.9 )\approx 0.000"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.8%20)%5Capprox%200.116" alt="x( -1.8 )\approx 0.116" title="x( -1.8 )\approx 0.116" data-latex="x( -1.8 )\approx 0.116"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.8%20)%5Capprox%200.200" alt="y( -1.8 )\approx 0.200" title="y( -1.8 )\approx 0.200" data-latex="y( -1.8 )\approx 0.200"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-9976" title="N2 | Euler's method for approximating IVP solutions | ver. 9976"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.80 )" alt="x( -0.80 )" title="x( -0.80 )" data-latex="x( -0.80 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.80 )" alt="y( -0.80 )" title="y( -0.80 )" data-latex="y( -0.80 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 2 \, x^{2} y - 3 \, t y + 1 \hspace{2em} x( -1 )= -1" alt="x'= 2 \, x^{2} y - 3 \, t y + 1 \hspace{2em} x( -1 )= -1" title="x'= 2 \, x^{2} y - 3 \, t y + 1 \hspace{2em} x( -1 )= -1" data-latex="x'= 2 \, x^{2} y - 3 \, t y + 1 \hspace{2em} x( -1 )= -1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -3 \, x^{2} y^{2} + 3 \, t y - 2 \hspace{2em} y( -1 )= 1" alt="y'= -3 \, x^{2} y^{2} + 3 \, t y - 2 \hspace{2em} y( -1 )= 1" title="y'= -3 \, x^{2} y^{2} + 3 \, t y - 2 \hspace{2em} y( -1 )= 1" data-latex="y'= -3 \, x^{2} y^{2} + 3 \, t y - 2 \hspace{2em} y( -1 )= 1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.80%20)" alt="x( -0.80 )" title="x( -0.80 )" data-latex="x( -0.80 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.80%20)" alt="y( -0.80 )" title="y( -0.80 )" data-latex="y( -0.80 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%202%20%5C,%20x%5E%7B2%7D%20y%20-%203%20%5C,%20t%20y%20+%201%20%5Chspace%7B2em%7D%20x(%20-1%20)=%20-1" alt="x'= 2 \, x^{2} y - 3 \, t y + 1 \hspace{2em} x( -1 )= -1" title="x'= 2 \, x^{2} y - 3 \, t y + 1 \hspace{2em} x( -1 )= -1" data-latex="x'= 2 \, x^{2} y - 3 \, t y + 1 \hspace{2em} x( -1 )= -1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-3%20%5C,%20x%5E%7B2%7D%20y%5E%7B2%7D%20+%203%20%5C,%20t%20y%20-%202%20%5Chspace%7B2em%7D%20y(%20-1%20)=%201" alt="y'= -3 \, x^{2} y^{2} + 3 \, t y - 2 \hspace{2em} y( -1 )= 1" title="y'= -3 \, x^{2} y^{2} + 3 \, t y - 2 \hspace{2em} y( -1 )= 1" data-latex="y'= -3 \, x^{2} y^{2} + 3 \, t y - 2 \hspace{2em} y( -1 )= 1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.90 )\approx -0.400" alt="x( -0.90 )\approx -0.400" title="x( -0.90 )\approx -0.400" data-latex="x( -0.90 )\approx -0.400"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.90 )\approx 0.200" alt="y( -0.90 )\approx 0.200" title="y( -0.90 )\approx 0.200" data-latex="y( -0.90 )\approx 0.200"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.80 )\approx -0.240" alt="x( -0.80 )\approx -0.240" title="x( -0.80 )\approx -0.240" data-latex="x( -0.80 )\approx -0.240"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.80 )\approx -0.0562" alt="y( -0.80 )\approx -0.0562" title="y( -0.80 )\approx -0.0562" data-latex="y( -0.80 )\approx -0.0562"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.90%20)%5Capprox%20-0.400" alt="x( -0.90 )\approx -0.400" title="x( -0.90 )\approx -0.400" data-latex="x( -0.90 )\approx -0.400"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.90%20)%5Capprox%200.200" alt="y( -0.90 )\approx 0.200" title="y( -0.90 )\approx 0.200" data-latex="y( -0.90 )\approx 0.200"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.80%20)%5Capprox%20-0.240" alt="x( -0.80 )\approx -0.240" title="x( -0.80 )\approx -0.240" data-latex="x( -0.80 )\approx -0.240"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.80%20)%5Capprox%20-0.0562" alt="y( -0.80 )\approx -0.0562" title="y( -0.80 )\approx -0.0562" data-latex="y( -0.80 )\approx -0.0562"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-1855" title="N2 | Euler's method for approximating IVP solutions | ver. 1855"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.20 )" alt="x( 0.20 )" title="x( 0.20 )" data-latex="x( 0.20 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.20 )" alt="y( 0.20 )" title="y( 0.20 )" data-latex="y( 0.20 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -3 \, t^{2} x + 3 \, t y - 1 \hspace{2em} x( 0 )= -1" alt="x'= -3 \, t^{2} x + 3 \, t y - 1 \hspace{2em} x( 0 )= -1" title="x'= -3 \, t^{2} x + 3 \, t y - 1 \hspace{2em} x( 0 )= -1" data-latex="x'= -3 \, t^{2} x + 3 \, t y - 1 \hspace{2em} x( 0 )= -1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -2 \, t^{2} x + 2 \, x y - 2 \hspace{2em} y( 0 )= 0" alt="y'= -2 \, t^{2} x + 2 \, x y - 2 \hspace{2em} y( 0 )= 0" title="y'= -2 \, t^{2} x + 2 \, x y - 2 \hspace{2em} y( 0 )= 0" data-latex="y'= -2 \, t^{2} x + 2 \, x y - 2 \hspace{2em} y( 0 )= 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.20%20)" alt="x( 0.20 )" title="x( 0.20 )" data-latex="x( 0.20 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.20%20)" alt="y( 0.20 )" title="y( 0.20 )" data-latex="y( 0.20 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-3%20%5C,%20t%5E%7B2%7D%20x%20+%203%20%5C,%20t%20y%20-%201%20%5Chspace%7B2em%7D%20x(%200%20)=%20-1" alt="x'= -3 \, t^{2} x + 3 \, t y - 1 \hspace{2em} x( 0 )= -1" title="x'= -3 \, t^{2} x + 3 \, t y - 1 \hspace{2em} x( 0 )= -1" data-latex="x'= -3 \, t^{2} x + 3 \, t y - 1 \hspace{2em} x( 0 )= -1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-2%20%5C,%20t%5E%7B2%7D%20x%20+%202%20%5C,%20x%20y%20-%202%20%5Chspace%7B2em%7D%20y(%200%20)=%200" alt="y'= -2 \, t^{2} x + 2 \, x y - 2 \hspace{2em} y( 0 )= 0" title="y'= -2 \, t^{2} x + 2 \, x y - 2 \hspace{2em} y( 0 )= 0" data-latex="y'= -2 \, t^{2} x + 2 \, x y - 2 \hspace{2em} y( 0 )= 0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.10 )\approx -1.10" alt="x( 0.10 )\approx -1.10" title="x( 0.10 )\approx -1.10" data-latex="x( 0.10 )\approx -1.10"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.10 )\approx -0.200" alt="y( 0.10 )\approx -0.200" title="y( 0.10 )\approx -0.200" data-latex="y( 0.10 )\approx -0.200"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.20 )\approx -1.20" alt="x( 0.20 )\approx -1.20" title="x( 0.20 )\approx -1.20" data-latex="x( 0.20 )\approx -1.20"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.20 )\approx -0.354" alt="y( 0.20 )\approx -0.354" title="y( 0.20 )\approx -0.354" data-latex="y( 0.20 )\approx -0.354"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.10%20)%5Capprox%20-1.10" alt="x( 0.10 )\approx -1.10" title="x( 0.10 )\approx -1.10" data-latex="x( 0.10 )\approx -1.10"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.10%20)%5Capprox%20-0.200" alt="y( 0.10 )\approx -0.200" title="y( 0.10 )\approx -0.200" data-latex="y( 0.10 )\approx -0.200"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.20%20)%5Capprox%20-1.20" alt="x( 0.20 )\approx -1.20" title="x( 0.20 )\approx -1.20" data-latex="x( 0.20 )\approx -1.20"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.20%20)%5Capprox%20-0.354" alt="y( 0.20 )\approx -0.354" title="y( 0.20 )\approx -0.354" data-latex="y( 0.20 )\approx -0.354"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-1578" title="N2 | Euler's method for approximating IVP solutions | ver. 1578"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.2 )" alt="x( 1.2 )" title="x( 1.2 )" data-latex="x( 1.2 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.2 )" alt="y( 1.2 )" title="y( 1.2 )" data-latex="y( 1.2 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 3 \, t^{2} y^{2} - t^{2} x + 1 \hspace{2em} x( 1 )= 1" alt="x'= 3 \, t^{2} y^{2} - t^{2} x + 1 \hspace{2em} x( 1 )= 1" title="x'= 3 \, t^{2} y^{2} - t^{2} x + 1 \hspace{2em} x( 1 )= 1" data-latex="x'= 3 \, t^{2} y^{2} - t^{2} x + 1 \hspace{2em} x( 1 )= 1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -t^{2} y^{2} + t x + 3 \hspace{2em} y( 1 )= -2" alt="y'= -t^{2} y^{2} + t x + 3 \hspace{2em} y( 1 )= -2" title="y'= -t^{2} y^{2} + t x + 3 \hspace{2em} y( 1 )= -2" data-latex="y'= -t^{2} y^{2} + t x + 3 \hspace{2em} y( 1 )= -2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.2%20)" alt="x( 1.2 )" title="x( 1.2 )" data-latex="x( 1.2 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.2%20)" alt="y( 1.2 )" title="y( 1.2 )" data-latex="y( 1.2 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%203%20%5C,%20t%5E%7B2%7D%20y%5E%7B2%7D%20-%20t%5E%7B2%7D%20x%20+%201%20%5Chspace%7B2em%7D%20x(%201%20)=%201" alt="x'= 3 \, t^{2} y^{2} - t^{2} x + 1 \hspace{2em} x( 1 )= 1" title="x'= 3 \, t^{2} y^{2} - t^{2} x + 1 \hspace{2em} x( 1 )= 1" data-latex="x'= 3 \, t^{2} y^{2} - t^{2} x + 1 \hspace{2em} x( 1 )= 1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-t%5E%7B2%7D%20y%5E%7B2%7D%20+%20t%20x%20+%203%20%5Chspace%7B2em%7D%20y(%201%20)=%20-2" alt="y'= -t^{2} y^{2} + t x + 3 \hspace{2em} y( 1 )= -2" title="y'= -t^{2} y^{2} + t x + 3 \hspace{2em} y( 1 )= -2" data-latex="y'= -t^{2} y^{2} + t x + 3 \hspace{2em} y( 1 )= -2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.1 )\approx 2.20" alt="x( 1.1 )\approx 2.20" title="x( 1.1 )\approx 2.20" data-latex="x( 1.1 )\approx 2.20"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.1 )\approx -2.00" alt="y( 1.1 )\approx -2.00" title="y( 1.1 )\approx -2.00" data-latex="y( 1.1 )\approx -2.00"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.2 )\approx 3.48" alt="x( 1.2 )\approx 3.48" title="x( 1.2 )\approx 3.48" data-latex="x( 1.2 )\approx 3.48"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.2 )\approx -1.94" alt="y( 1.2 )\approx -1.94" title="y( 1.2 )\approx -1.94" data-latex="y( 1.2 )\approx -1.94"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.1%20)%5Capprox%202.20" alt="x( 1.1 )\approx 2.20" title="x( 1.1 )\approx 2.20" data-latex="x( 1.1 )\approx 2.20"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.1%20)%5Capprox%20-2.00" alt="y( 1.1 )\approx -2.00" title="y( 1.1 )\approx -2.00" data-latex="y( 1.1 )\approx -2.00"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.2%20)%5Capprox%203.48" alt="x( 1.2 )\approx 3.48" title="x( 1.2 )\approx 3.48" data-latex="x( 1.2 )\approx 3.48"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.2%20)%5Capprox%20-1.94" alt="y( 1.2 )\approx -1.94" title="y( 1.2 )\approx -1.94" data-latex="y( 1.2 )\approx -1.94"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-4170" title="N2 | Euler's method for approximating IVP solutions | ver. 4170"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.8 )" alt="x( -1.8 )" title="x( -1.8 )" data-latex="x( -1.8 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.8 )" alt="y( -1.8 )" title="y( -1.8 )" data-latex="y( -1.8 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 2 \, t x - 2 \, t y - 3 \hspace{2em} x( -2 )= -2" alt="x'= 2 \, t x - 2 \, t y - 3 \hspace{2em} x( -2 )= -2" title="x'= 2 \, t x - 2 \, t y - 3 \hspace{2em} x( -2 )= -2" data-latex="x'= 2 \, t x - 2 \, t y - 3 \hspace{2em} x( -2 )= -2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -t^{2} y + 2 \, x y^{2} - 2 \hspace{2em} y( -2 )= -2" alt="y'= -t^{2} y + 2 \, x y^{2} - 2 \hspace{2em} y( -2 )= -2" title="y'= -t^{2} y + 2 \, x y^{2} - 2 \hspace{2em} y( -2 )= -2" data-latex="y'= -t^{2} y + 2 \, x y^{2} - 2 \hspace{2em} y( -2 )= -2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.8%20)" alt="x( -1.8 )" title="x( -1.8 )" data-latex="x( -1.8 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.8%20)" alt="y( -1.8 )" title="y( -1.8 )" data-latex="y( -1.8 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%202%20%5C,%20t%20x%20-%202%20%5C,%20t%20y%20-%203%20%5Chspace%7B2em%7D%20x(%20-2%20)=%20-2" alt="x'= 2 \, t x - 2 \, t y - 3 \hspace{2em} x( -2 )= -2" title="x'= 2 \, t x - 2 \, t y - 3 \hspace{2em} x( -2 )= -2" data-latex="x'= 2 \, t x - 2 \, t y - 3 \hspace{2em} x( -2 )= -2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-t%5E%7B2%7D%20y%20+%202%20%5C,%20x%20y%5E%7B2%7D%20-%202%20%5Chspace%7B2em%7D%20y(%20-2%20)=%20-2" alt="y'= -t^{2} y + 2 \, x y^{2} - 2 \hspace{2em} y( -2 )= -2" title="y'= -t^{2} y + 2 \, x y^{2} - 2 \hspace{2em} y( -2 )= -2" data-latex="y'= -t^{2} y + 2 \, x y^{2} - 2 \hspace{2em} y( -2 )= -2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.9 )\approx -2.30" alt="x( -1.9 )\approx -2.30" title="x( -1.9 )\approx -2.30" data-latex="x( -1.9 )\approx -2.30"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.9 )\approx -3.00" alt="y( -1.9 )\approx -3.00" title="y( -1.9 )\approx -3.00" data-latex="y( -1.9 )\approx -3.00"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.8 )\approx -2.87" alt="x( -1.8 )\approx -2.87" title="x( -1.8 )\approx -2.87" data-latex="x( -1.8 )\approx -2.87"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.8 )\approx -6.27" alt="y( -1.8 )\approx -6.27" title="y( -1.8 )\approx -6.27" data-latex="y( -1.8 )\approx -6.27"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.9%20)%5Capprox%20-2.30" alt="x( -1.9 )\approx -2.30" title="x( -1.9 )\approx -2.30" data-latex="x( -1.9 )\approx -2.30"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.9%20)%5Capprox%20-3.00" alt="y( -1.9 )\approx -3.00" title="y( -1.9 )\approx -3.00" data-latex="y( -1.9 )\approx -3.00"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.8%20)%5Capprox%20-2.87" alt="x( -1.8 )\approx -2.87" title="x( -1.8 )\approx -2.87" data-latex="x( -1.8 )\approx -2.87"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.8%20)%5Capprox%20-6.27" alt="y( -1.8 )\approx -6.27" title="y( -1.8 )\approx -6.27" data-latex="y( -1.8 )\approx -6.27"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-6848" title="N2 | Euler's method for approximating IVP solutions | ver. 6848"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.2 )" alt="x( 1.2 )" title="x( 1.2 )" data-latex="x( 1.2 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.2 )" alt="y( 1.2 )" title="y( 1.2 )" data-latex="y( 1.2 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 4 \, x^{2} y^{2} - t x + 3 \hspace{2em} x( 1 )= 0" alt="x'= 4 \, x^{2} y^{2} - t x + 3 \hspace{2em} x( 1 )= 0" title="x'= 4 \, x^{2} y^{2} - t x + 3 \hspace{2em} x( 1 )= 0" data-latex="x'= 4 \, x^{2} y^{2} - t x + 3 \hspace{2em} x( 1 )= 0"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= t^{2} y^{2} - 3 \, t^{2} x - 2 \hspace{2em} y( 1 )= 2" alt="y'= t^{2} y^{2} - 3 \, t^{2} x - 2 \hspace{2em} y( 1 )= 2" title="y'= t^{2} y^{2} - 3 \, t^{2} x - 2 \hspace{2em} y( 1 )= 2" data-latex="y'= t^{2} y^{2} - 3 \, t^{2} x - 2 \hspace{2em} y( 1 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.2%20)" alt="x( 1.2 )" title="x( 1.2 )" data-latex="x( 1.2 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.2%20)" alt="y( 1.2 )" title="y( 1.2 )" data-latex="y( 1.2 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%204%20%5C,%20x%5E%7B2%7D%20y%5E%7B2%7D%20-%20t%20x%20+%203%20%5Chspace%7B2em%7D%20x(%201%20)=%200" alt="x'= 4 \, x^{2} y^{2} - t x + 3 \hspace{2em} x( 1 )= 0" title="x'= 4 \, x^{2} y^{2} - t x + 3 \hspace{2em} x( 1 )= 0" data-latex="x'= 4 \, x^{2} y^{2} - t x + 3 \hspace{2em} x( 1 )= 0"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20t%5E%7B2%7D%20y%5E%7B2%7D%20-%203%20%5C,%20t%5E%7B2%7D%20x%20-%202%20%5Chspace%7B2em%7D%20y(%201%20)=%202" alt="y'= t^{2} y^{2} - 3 \, t^{2} x - 2 \hspace{2em} y( 1 )= 2" title="y'= t^{2} y^{2} - 3 \, t^{2} x - 2 \hspace{2em} y( 1 )= 2" data-latex="y'= t^{2} y^{2} - 3 \, t^{2} x - 2 \hspace{2em} y( 1 )= 2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.1 )\approx 0.300" alt="x( 1.1 )\approx 0.300" title="x( 1.1 )\approx 0.300" data-latex="x( 1.1 )\approx 0.300"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.1 )\approx 2.20" alt="y( 1.1 )\approx 2.20" title="y( 1.1 )\approx 2.20" data-latex="y( 1.1 )\approx 2.20"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.2 )\approx 0.741" alt="x( 1.2 )\approx 0.741" title="x( 1.2 )\approx 0.741" data-latex="x( 1.2 )\approx 0.741"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.2 )\approx 2.48" alt="y( 1.2 )\approx 2.48" title="y( 1.2 )\approx 2.48" data-latex="y( 1.2 )\approx 2.48"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.1%20)%5Capprox%200.300" alt="x( 1.1 )\approx 0.300" title="x( 1.1 )\approx 0.300" data-latex="x( 1.1 )\approx 0.300"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.1%20)%5Capprox%202.20" alt="y( 1.1 )\approx 2.20" title="y( 1.1 )\approx 2.20" data-latex="y( 1.1 )\approx 2.20"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.2%20)%5Capprox%200.741" alt="x( 1.2 )\approx 0.741" title="x( 1.2 )\approx 0.741" data-latex="x( 1.2 )\approx 0.741"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.2%20)%5Capprox%202.48" alt="y( 1.2 )\approx 2.48" title="y( 1.2 )\approx 2.48" data-latex="y( 1.2 )\approx 2.48"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-6124" title="N2 | Euler's method for approximating IVP solutions | ver. 6124"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.2 )" alt="x( 1.2 )" title="x( 1.2 )" data-latex="x( 1.2 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.2 )" alt="y( 1.2 )" title="y( 1.2 )" data-latex="y( 1.2 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 4 \, t^{2} y^{2} - 2 \, t x^{2} - 2 \hspace{2em} x( 1 )= -1" alt="x'= 4 \, t^{2} y^{2} - 2 \, t x^{2} - 2 \hspace{2em} x( 1 )= -1" title="x'= 4 \, t^{2} y^{2} - 2 \, t x^{2} - 2 \hspace{2em} x( 1 )= -1" data-latex="x'= 4 \, t^{2} y^{2} - 2 \, t x^{2} - 2 \hspace{2em} x( 1 )= -1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 4 \, x y^{2} + 2 \, t y \hspace{2em} y( 1 )= -1" alt="y'= 4 \, x y^{2} + 2 \, t y \hspace{2em} y( 1 )= -1" title="y'= 4 \, x y^{2} + 2 \, t y \hspace{2em} y( 1 )= -1" data-latex="y'= 4 \, x y^{2} + 2 \, t y \hspace{2em} y( 1 )= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.2%20)" alt="x( 1.2 )" title="x( 1.2 )" data-latex="x( 1.2 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.2%20)" alt="y( 1.2 )" title="y( 1.2 )" data-latex="y( 1.2 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%204%20%5C,%20t%5E%7B2%7D%20y%5E%7B2%7D%20-%202%20%5C,%20t%20x%5E%7B2%7D%20-%202%20%5Chspace%7B2em%7D%20x(%201%20)=%20-1" alt="x'= 4 \, t^{2} y^{2} - 2 \, t x^{2} - 2 \hspace{2em} x( 1 )= -1" title="x'= 4 \, t^{2} y^{2} - 2 \, t x^{2} - 2 \hspace{2em} x( 1 )= -1" data-latex="x'= 4 \, t^{2} y^{2} - 2 \, t x^{2} - 2 \hspace{2em} x( 1 )= -1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%204%20%5C,%20x%20y%5E%7B2%7D%20+%202%20%5C,%20t%20y%20%5Chspace%7B2em%7D%20y(%201%20)=%20-1" alt="y'= 4 \, x y^{2} + 2 \, t y \hspace{2em} y( 1 )= -1" title="y'= 4 \, x y^{2} + 2 \, t y \hspace{2em} y( 1 )= -1" data-latex="y'= 4 \, x y^{2} + 2 \, t y \hspace{2em} y( 1 )= -1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.1 )\approx -1.00" alt="x( 1.1 )\approx -1.00" title="x( 1.1 )\approx -1.00" data-latex="x( 1.1 )\approx -1.00"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.1 )\approx -1.60" alt="y( 1.1 )\approx -1.60" title="y( 1.1 )\approx -1.60" data-latex="y( 1.1 )\approx -1.60"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.2 )\approx -0.183" alt="x( 1.2 )\approx -0.183" title="x( 1.2 )\approx -0.183" data-latex="x( 1.2 )\approx -0.183"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.2 )\approx -2.98" alt="y( 1.2 )\approx -2.98" title="y( 1.2 )\approx -2.98" data-latex="y( 1.2 )\approx -2.98"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.1%20)%5Capprox%20-1.00" alt="x( 1.1 )\approx -1.00" title="x( 1.1 )\approx -1.00" data-latex="x( 1.1 )\approx -1.00"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.1%20)%5Capprox%20-1.60" alt="y( 1.1 )\approx -1.60" title="y( 1.1 )\approx -1.60" data-latex="y( 1.1 )\approx -1.60"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.2%20)%5Capprox%20-0.183" alt="x( 1.2 )\approx -0.183" title="x( 1.2 )\approx -0.183" data-latex="x( 1.2 )\approx -0.183"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.2%20)%5Capprox%20-2.98" alt="y( 1.2 )\approx -2.98" title="y( 1.2 )\approx -2.98" data-latex="y( 1.2 )\approx -2.98"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-6077" title="N2 | Euler's method for approximating IVP solutions | ver. 6077"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.2 )" alt="x( 1.2 )" title="x( 1.2 )" data-latex="x( 1.2 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.2 )" alt="y( 1.2 )" title="y( 1.2 )" data-latex="y( 1.2 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 4 \, t x - 2 \, x y + 1 \hspace{2em} x( 1 )= 2" alt="x'= 4 \, t x - 2 \, x y + 1 \hspace{2em} x( 1 )= 2" title="x'= 4 \, t x - 2 \, x y + 1 \hspace{2em} x( 1 )= 2" data-latex="x'= 4 \, t x - 2 \, x y + 1 \hspace{2em} x( 1 )= 2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -4 \, x^{2} y^{2} - 3 \, t^{2} y + 1 \hspace{2em} y( 1 )= -1" alt="y'= -4 \, x^{2} y^{2} - 3 \, t^{2} y + 1 \hspace{2em} y( 1 )= -1" title="y'= -4 \, x^{2} y^{2} - 3 \, t^{2} y + 1 \hspace{2em} y( 1 )= -1" data-latex="y'= -4 \, x^{2} y^{2} - 3 \, t^{2} y + 1 \hspace{2em} y( 1 )= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.2%20)" alt="x( 1.2 )" title="x( 1.2 )" data-latex="x( 1.2 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.2%20)" alt="y( 1.2 )" title="y( 1.2 )" data-latex="y( 1.2 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%204%20%5C,%20t%20x%20-%202%20%5C,%20x%20y%20+%201%20%5Chspace%7B2em%7D%20x(%201%20)=%202" alt="x'= 4 \, t x - 2 \, x y + 1 \hspace{2em} x( 1 )= 2" title="x'= 4 \, t x - 2 \, x y + 1 \hspace{2em} x( 1 )= 2" data-latex="x'= 4 \, t x - 2 \, x y + 1 \hspace{2em} x( 1 )= 2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-4%20%5C,%20x%5E%7B2%7D%20y%5E%7B2%7D%20-%203%20%5C,%20t%5E%7B2%7D%20y%20+%201%20%5Chspace%7B2em%7D%20y(%201%20)=%20-1" alt="y'= -4 \, x^{2} y^{2} - 3 \, t^{2} y + 1 \hspace{2em} y( 1 )= -1" title="y'= -4 \, x^{2} y^{2} - 3 \, t^{2} y + 1 \hspace{2em} y( 1 )= -1" data-latex="y'= -4 \, x^{2} y^{2} - 3 \, t^{2} y + 1 \hspace{2em} y( 1 )= -1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.1 )\approx 3.30" alt="x( 1.1 )\approx 3.30" title="x( 1.1 )\approx 3.30" data-latex="x( 1.1 )\approx 3.30"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.1 )\approx -2.20" alt="y( 1.1 )\approx -2.20" title="y( 1.1 )\approx -2.20" data-latex="y( 1.1 )\approx -2.20"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.2 )\approx 6.30" alt="x( 1.2 )\approx 6.30" title="x( 1.2 )\approx 6.30" data-latex="x( 1.2 )\approx 6.30"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.2 )\approx -22.3" alt="y( 1.2 )\approx -22.3" title="y( 1.2 )\approx -22.3" data-latex="y( 1.2 )\approx -22.3"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.1%20)%5Capprox%203.30" alt="x( 1.1 )\approx 3.30" title="x( 1.1 )\approx 3.30" data-latex="x( 1.1 )\approx 3.30"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.1%20)%5Capprox%20-2.20" alt="y( 1.1 )\approx -2.20" title="y( 1.1 )\approx -2.20" data-latex="y( 1.1 )\approx -2.20"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.2%20)%5Capprox%206.30" alt="x( 1.2 )\approx 6.30" title="x( 1.2 )\approx 6.30" data-latex="x( 1.2 )\approx 6.30"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.2%20)%5Capprox%20-22.3" alt="y( 1.2 )\approx -22.3" title="y( 1.2 )\approx -22.3" data-latex="y( 1.2 )\approx -22.3"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-5247" title="N2 | Euler's method for approximating IVP solutions | ver. 5247"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 2.2 )" alt="x( 2.2 )" title="x( 2.2 )" data-latex="x( 2.2 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 2.2 )" alt="y( 2.2 )" title="y( 2.2 )" data-latex="y( 2.2 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -2 \, x^{2} y^{2} - 3 \, t^{2} x - 2 \hspace{2em} x( 2 )= 0" alt="x'= -2 \, x^{2} y^{2} - 3 \, t^{2} x - 2 \hspace{2em} x( 2 )= 0" title="x'= -2 \, x^{2} y^{2} - 3 \, t^{2} x - 2 \hspace{2em} x( 2 )= 0" data-latex="x'= -2 \, x^{2} y^{2} - 3 \, t^{2} x - 2 \hspace{2em} x( 2 )= 0"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -2 \, t^{2} y^{2} - x^{2} y + 3 \hspace{2em} y( 2 )= 2" alt="y'= -2 \, t^{2} y^{2} - x^{2} y + 3 \hspace{2em} y( 2 )= 2" title="y'= -2 \, t^{2} y^{2} - x^{2} y + 3 \hspace{2em} y( 2 )= 2" data-latex="y'= -2 \, t^{2} y^{2} - x^{2} y + 3 \hspace{2em} y( 2 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%202.2%20)" alt="x( 2.2 )" title="x( 2.2 )" data-latex="x( 2.2 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%202.2%20)" alt="y( 2.2 )" title="y( 2.2 )" data-latex="y( 2.2 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-2%20%5C,%20x%5E%7B2%7D%20y%5E%7B2%7D%20-%203%20%5C,%20t%5E%7B2%7D%20x%20-%202%20%5Chspace%7B2em%7D%20x(%202%20)=%200" alt="x'= -2 \, x^{2} y^{2} - 3 \, t^{2} x - 2 \hspace{2em} x( 2 )= 0" title="x'= -2 \, x^{2} y^{2} - 3 \, t^{2} x - 2 \hspace{2em} x( 2 )= 0" data-latex="x'= -2 \, x^{2} y^{2} - 3 \, t^{2} x - 2 \hspace{2em} x( 2 )= 0"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-2%20%5C,%20t%5E%7B2%7D%20y%5E%7B2%7D%20-%20x%5E%7B2%7D%20y%20+%203%20%5Chspace%7B2em%7D%20y(%202%20)=%202" alt="y'= -2 \, t^{2} y^{2} - x^{2} y + 3 \hspace{2em} y( 2 )= 2" title="y'= -2 \, t^{2} y^{2} - x^{2} y + 3 \hspace{2em} y( 2 )= 2" data-latex="y'= -2 \, t^{2} y^{2} - x^{2} y + 3 \hspace{2em} y( 2 )= 2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 2.1 )\approx -0.200" alt="x( 2.1 )\approx -0.200" title="x( 2.1 )\approx -0.200" data-latex="x( 2.1 )\approx -0.200"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 2.1 )\approx -0.898" alt="y( 2.1 )\approx -0.898" title="y( 2.1 )\approx -0.898" data-latex="y( 2.1 )\approx -0.898"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 2.2 )\approx -0.141" alt="x( 2.2 )\approx -0.141" title="x( 2.2 )\approx -0.141" data-latex="x( 2.2 )\approx -0.141"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 2.2 )\approx -1.31" alt="y( 2.2 )\approx -1.31" title="y( 2.2 )\approx -1.31" data-latex="y( 2.2 )\approx -1.31"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%202.1%20)%5Capprox%20-0.200" alt="x( 2.1 )\approx -0.200" title="x( 2.1 )\approx -0.200" data-latex="x( 2.1 )\approx -0.200"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%202.1%20)%5Capprox%20-0.898" alt="y( 2.1 )\approx -0.898" title="y( 2.1 )\approx -0.898" data-latex="y( 2.1 )\approx -0.898"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%202.2%20)%5Capprox%20-0.141" alt="x( 2.2 )\approx -0.141" title="x( 2.2 )\approx -0.141" data-latex="x( 2.2 )\approx -0.141"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%202.2%20)%5Capprox%20-1.31" alt="y( 2.2 )\approx -1.31" title="y( 2.2 )\approx -1.31" data-latex="y( 2.2 )\approx -1.31"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-7402" title="N2 | Euler's method for approximating IVP solutions | ver. 7402"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.2 )" alt="x( 1.2 )" title="x( 1.2 )" data-latex="x( 1.2 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.2 )" alt="y( 1.2 )" title="y( 1.2 )" data-latex="y( 1.2 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= t^{2} x^{2} + t y^{2} - 3 \hspace{2em} x( 1 )= 0" alt="x'= t^{2} x^{2} + t y^{2} - 3 \hspace{2em} x( 1 )= 0" title="x'= t^{2} x^{2} + t y^{2} - 3 \hspace{2em} x( 1 )= 0" data-latex="x'= t^{2} x^{2} + t y^{2} - 3 \hspace{2em} x( 1 )= 0"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -4 \, t^{2} x - 2 \, x^{2} y \hspace{2em} y( 1 )= 1" alt="y'= -4 \, t^{2} x - 2 \, x^{2} y \hspace{2em} y( 1 )= 1" title="y'= -4 \, t^{2} x - 2 \, x^{2} y \hspace{2em} y( 1 )= 1" data-latex="y'= -4 \, t^{2} x - 2 \, x^{2} y \hspace{2em} y( 1 )= 1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.2%20)" alt="x( 1.2 )" title="x( 1.2 )" data-latex="x( 1.2 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.2%20)" alt="y( 1.2 )" title="y( 1.2 )" data-latex="y( 1.2 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20t%5E%7B2%7D%20x%5E%7B2%7D%20+%20t%20y%5E%7B2%7D%20-%203%20%5Chspace%7B2em%7D%20x(%201%20)=%200" alt="x'= t^{2} x^{2} + t y^{2} - 3 \hspace{2em} x( 1 )= 0" title="x'= t^{2} x^{2} + t y^{2} - 3 \hspace{2em} x( 1 )= 0" data-latex="x'= t^{2} x^{2} + t y^{2} - 3 \hspace{2em} x( 1 )= 0"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-4%20%5C,%20t%5E%7B2%7D%20x%20-%202%20%5C,%20x%5E%7B2%7D%20y%20%5Chspace%7B2em%7D%20y(%201%20)=%201" alt="y'= -4 \, t^{2} x - 2 \, x^{2} y \hspace{2em} y( 1 )= 1" title="y'= -4 \, t^{2} x - 2 \, x^{2} y \hspace{2em} y( 1 )= 1" data-latex="y'= -4 \, t^{2} x - 2 \, x^{2} y \hspace{2em} y( 1 )= 1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.1 )\approx -0.200" alt="x( 1.1 )\approx -0.200" title="x( 1.1 )\approx -0.200" data-latex="x( 1.1 )\approx -0.200"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.1 )\approx 1.00" alt="y( 1.1 )\approx 1.00" title="y( 1.1 )\approx 1.00" data-latex="y( 1.1 )\approx 1.00"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.2 )\approx -0.385" alt="x( 1.2 )\approx -0.385" title="x( 1.2 )\approx -0.385" data-latex="x( 1.2 )\approx -0.385"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.2 )\approx 1.09" alt="y( 1.2 )\approx 1.09" title="y( 1.2 )\approx 1.09" data-latex="y( 1.2 )\approx 1.09"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.1%20)%5Capprox%20-0.200" alt="x( 1.1 )\approx -0.200" title="x( 1.1 )\approx -0.200" data-latex="x( 1.1 )\approx -0.200"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.1%20)%5Capprox%201.00" alt="y( 1.1 )\approx 1.00" title="y( 1.1 )\approx 1.00" data-latex="y( 1.1 )\approx 1.00"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.2%20)%5Capprox%20-0.385" alt="x( 1.2 )\approx -0.385" title="x( 1.2 )\approx -0.385" data-latex="x( 1.2 )\approx -0.385"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.2%20)%5Capprox%201.09" alt="y( 1.2 )\approx 1.09" title="y( 1.2 )\approx 1.09" data-latex="y( 1.2 )\approx 1.09"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-1592" title="N2 | Euler's method for approximating IVP solutions | ver. 1592"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.8 )" alt="x( -1.8 )" title="x( -1.8 )" data-latex="x( -1.8 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.8 )" alt="y( -1.8 )" title="y( -1.8 )" data-latex="y( -1.8 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 2 \, t^{2} x^{2} - 2 \, x^{2} y^{2} - 1 \hspace{2em} x( -2 )= -2" alt="x'= 2 \, t^{2} x^{2} - 2 \, x^{2} y^{2} - 1 \hspace{2em} x( -2 )= -2" title="x'= 2 \, t^{2} x^{2} - 2 \, x^{2} y^{2} - 1 \hspace{2em} x( -2 )= -2" data-latex="x'= 2 \, t^{2} x^{2} - 2 \, x^{2} y^{2} - 1 \hspace{2em} x( -2 )= -2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -2 \, x^{2} y^{2} + 2 \, t y^{2} - 1 \hspace{2em} y( -2 )= 2" alt="y'= -2 \, x^{2} y^{2} + 2 \, t y^{2} - 1 \hspace{2em} y( -2 )= 2" title="y'= -2 \, x^{2} y^{2} + 2 \, t y^{2} - 1 \hspace{2em} y( -2 )= 2" data-latex="y'= -2 \, x^{2} y^{2} + 2 \, t y^{2} - 1 \hspace{2em} y( -2 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.8%20)" alt="x( -1.8 )" title="x( -1.8 )" data-latex="x( -1.8 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.8%20)" alt="y( -1.8 )" title="y( -1.8 )" data-latex="y( -1.8 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%202%20%5C,%20t%5E%7B2%7D%20x%5E%7B2%7D%20-%202%20%5C,%20x%5E%7B2%7D%20y%5E%7B2%7D%20-%201%20%5Chspace%7B2em%7D%20x(%20-2%20)=%20-2" alt="x'= 2 \, t^{2} x^{2} - 2 \, x^{2} y^{2} - 1 \hspace{2em} x( -2 )= -2" title="x'= 2 \, t^{2} x^{2} - 2 \, x^{2} y^{2} - 1 \hspace{2em} x( -2 )= -2" data-latex="x'= 2 \, t^{2} x^{2} - 2 \, x^{2} y^{2} - 1 \hspace{2em} x( -2 )= -2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-2%20%5C,%20x%5E%7B2%7D%20y%5E%7B2%7D%20+%202%20%5C,%20t%20y%5E%7B2%7D%20-%201%20%5Chspace%7B2em%7D%20y(%20-2%20)=%202" alt="y'= -2 \, x^{2} y^{2} + 2 \, t y^{2} - 1 \hspace{2em} y( -2 )= 2" title="y'= -2 \, x^{2} y^{2} + 2 \, t y^{2} - 1 \hspace{2em} y( -2 )= 2" data-latex="y'= -2 \, x^{2} y^{2} + 2 \, t y^{2} - 1 \hspace{2em} y( -2 )= 2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.9 )\approx -2.10" alt="x( -1.9 )\approx -2.10" title="x( -1.9 )\approx -2.10" data-latex="x( -1.9 )\approx -2.10"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.9 )\approx -2.90" alt="y( -1.9 )\approx -2.90" title="y( -1.9 )\approx -2.90" data-latex="y( -1.9 )\approx -2.90"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.8 )\approx -6.44" alt="x( -1.8 )\approx -6.44" title="x( -1.8 )\approx -6.44" data-latex="x( -1.8 )\approx -6.44"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.8 )\approx -13.6" alt="y( -1.8 )\approx -13.6" title="y( -1.8 )\approx -13.6" data-latex="y( -1.8 )\approx -13.6"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.9%20)%5Capprox%20-2.10" alt="x( -1.9 )\approx -2.10" title="x( -1.9 )\approx -2.10" data-latex="x( -1.9 )\approx -2.10"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.9%20)%5Capprox%20-2.90" alt="y( -1.9 )\approx -2.90" title="y( -1.9 )\approx -2.90" data-latex="y( -1.9 )\approx -2.90"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.8%20)%5Capprox%20-6.44" alt="x( -1.8 )\approx -6.44" title="x( -1.8 )\approx -6.44" data-latex="x( -1.8 )\approx -6.44"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.8%20)%5Capprox%20-13.6" alt="y( -1.8 )\approx -13.6" title="y( -1.8 )\approx -13.6" data-latex="y( -1.8 )\approx -13.6"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-0651" title="N2 | Euler's method for approximating IVP solutions | ver. 0651"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.20 )" alt="x( 0.20 )" title="x( 0.20 )" data-latex="x( 0.20 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.20 )" alt="y( 0.20 )" title="y( 0.20 )" data-latex="y( 0.20 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= t^{2} y - 2 \, x y^{2} - 3 \hspace{2em} x( 0 )= 2" alt="x'= t^{2} y - 2 \, x y^{2} - 3 \hspace{2em} x( 0 )= 2" title="x'= t^{2} y - 2 \, x y^{2} - 3 \hspace{2em} x( 0 )= 2" data-latex="x'= t^{2} y - 2 \, x y^{2} - 3 \hspace{2em} x( 0 )= 2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -3 \, t x - 4 \, t y + 2 \hspace{2em} y( 0 )= -1" alt="y'= -3 \, t x - 4 \, t y + 2 \hspace{2em} y( 0 )= -1" title="y'= -3 \, t x - 4 \, t y + 2 \hspace{2em} y( 0 )= -1" data-latex="y'= -3 \, t x - 4 \, t y + 2 \hspace{2em} y( 0 )= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.20%20)" alt="x( 0.20 )" title="x( 0.20 )" data-latex="x( 0.20 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.20%20)" alt="y( 0.20 )" title="y( 0.20 )" data-latex="y( 0.20 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20t%5E%7B2%7D%20y%20-%202%20%5C,%20x%20y%5E%7B2%7D%20-%203%20%5Chspace%7B2em%7D%20x(%200%20)=%202" alt="x'= t^{2} y - 2 \, x y^{2} - 3 \hspace{2em} x( 0 )= 2" title="x'= t^{2} y - 2 \, x y^{2} - 3 \hspace{2em} x( 0 )= 2" data-latex="x'= t^{2} y - 2 \, x y^{2} - 3 \hspace{2em} x( 0 )= 2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-3%20%5C,%20t%20x%20-%204%20%5C,%20t%20y%20+%202%20%5Chspace%7B2em%7D%20y(%200%20)=%20-1" alt="y'= -3 \, t x - 4 \, t y + 2 \hspace{2em} y( 0 )= -1" title="y'= -3 \, t x - 4 \, t y + 2 \hspace{2em} y( 0 )= -1" data-latex="y'= -3 \, t x - 4 \, t y + 2 \hspace{2em} y( 0 )= -1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.10 )\approx 1.30" alt="x( 0.10 )\approx 1.30" title="x( 0.10 )\approx 1.30" data-latex="x( 0.10 )\approx 1.30"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.10 )\approx -0.800" alt="y( 0.10 )\approx -0.800" title="y( 0.10 )\approx -0.800" data-latex="y( 0.10 )\approx -0.800"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.20 )\approx 0.834" alt="x( 0.20 )\approx 0.834" title="x( 0.20 )\approx 0.834" data-latex="x( 0.20 )\approx 0.834"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.20 )\approx -0.607" alt="y( 0.20 )\approx -0.607" title="y( 0.20 )\approx -0.607" data-latex="y( 0.20 )\approx -0.607"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.10%20)%5Capprox%201.30" alt="x( 0.10 )\approx 1.30" title="x( 0.10 )\approx 1.30" data-latex="x( 0.10 )\approx 1.30"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.10%20)%5Capprox%20-0.800" alt="y( 0.10 )\approx -0.800" title="y( 0.10 )\approx -0.800" data-latex="y( 0.10 )\approx -0.800"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.20%20)%5Capprox%200.834" alt="x( 0.20 )\approx 0.834" title="x( 0.20 )\approx 0.834" data-latex="x( 0.20 )\approx 0.834"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.20%20)%5Capprox%20-0.607" alt="y( 0.20 )\approx -0.607" title="y( 0.20 )\approx -0.607" data-latex="y( 0.20 )\approx -0.607"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-1529" title="N2 | Euler's method for approximating IVP solutions | ver. 1529"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.20 )" alt="x( 0.20 )" title="x( 0.20 )" data-latex="x( 0.20 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.20 )" alt="y( 0.20 )" title="y( 0.20 )" data-latex="y( 0.20 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 3 \, t^{2} x^{2} + 4 \, t^{2} y^{2} + 1 \hspace{2em} x( 0 )= 1" alt="x'= 3 \, t^{2} x^{2} + 4 \, t^{2} y^{2} + 1 \hspace{2em} x( 0 )= 1" title="x'= 3 \, t^{2} x^{2} + 4 \, t^{2} y^{2} + 1 \hspace{2em} x( 0 )= 1" data-latex="x'= 3 \, t^{2} x^{2} + 4 \, t^{2} y^{2} + 1 \hspace{2em} x( 0 )= 1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -2 \, t^{2} x - 3 \, t y^{2} + 3 \hspace{2em} y( 0 )= 2" alt="y'= -2 \, t^{2} x - 3 \, t y^{2} + 3 \hspace{2em} y( 0 )= 2" title="y'= -2 \, t^{2} x - 3 \, t y^{2} + 3 \hspace{2em} y( 0 )= 2" data-latex="y'= -2 \, t^{2} x - 3 \, t y^{2} + 3 \hspace{2em} y( 0 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.20%20)" alt="x( 0.20 )" title="x( 0.20 )" data-latex="x( 0.20 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.20%20)" alt="y( 0.20 )" title="y( 0.20 )" data-latex="y( 0.20 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%203%20%5C,%20t%5E%7B2%7D%20x%5E%7B2%7D%20+%204%20%5C,%20t%5E%7B2%7D%20y%5E%7B2%7D%20+%201%20%5Chspace%7B2em%7D%20x(%200%20)=%201" alt="x'= 3 \, t^{2} x^{2} + 4 \, t^{2} y^{2} + 1 \hspace{2em} x( 0 )= 1" title="x'= 3 \, t^{2} x^{2} + 4 \, t^{2} y^{2} + 1 \hspace{2em} x( 0 )= 1" data-latex="x'= 3 \, t^{2} x^{2} + 4 \, t^{2} y^{2} + 1 \hspace{2em} x( 0 )= 1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-2%20%5C,%20t%5E%7B2%7D%20x%20-%203%20%5C,%20t%20y%5E%7B2%7D%20+%203%20%5Chspace%7B2em%7D%20y(%200%20)=%202" alt="y'= -2 \, t^{2} x - 3 \, t y^{2} + 3 \hspace{2em} y( 0 )= 2" title="y'= -2 \, t^{2} x - 3 \, t y^{2} + 3 \hspace{2em} y( 0 )= 2" data-latex="y'= -2 \, t^{2} x - 3 \, t y^{2} + 3 \hspace{2em} y( 0 )= 2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.10 )\approx 1.10" alt="x( 0.10 )\approx 1.10" title="x( 0.10 )\approx 1.10" data-latex="x( 0.10 )\approx 1.10"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.10 )\approx 2.30" alt="y( 0.10 )\approx 2.30" title="y( 0.10 )\approx 2.30" data-latex="y( 0.10 )\approx 2.30"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.20 )\approx 1.22" alt="x( 0.20 )\approx 1.22" title="x( 0.20 )\approx 1.22" data-latex="x( 0.20 )\approx 1.22"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.20 )\approx 2.44" alt="y( 0.20 )\approx 2.44" title="y( 0.20 )\approx 2.44" data-latex="y( 0.20 )\approx 2.44"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.10%20)%5Capprox%201.10" alt="x( 0.10 )\approx 1.10" title="x( 0.10 )\approx 1.10" data-latex="x( 0.10 )\approx 1.10"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.10%20)%5Capprox%202.30" alt="y( 0.10 )\approx 2.30" title="y( 0.10 )\approx 2.30" data-latex="y( 0.10 )\approx 2.30"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.20%20)%5Capprox%201.22" alt="x( 0.20 )\approx 1.22" title="x( 0.20 )\approx 1.22" data-latex="x( 0.20 )\approx 1.22"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.20%20)%5Capprox%202.44" alt="y( 0.20 )\approx 2.44" title="y( 0.20 )\approx 2.44" data-latex="y( 0.20 )\approx 2.44"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-1201" title="N2 | Euler's method for approximating IVP solutions | ver. 1201"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.8 )" alt="x( -1.8 )" title="x( -1.8 )" data-latex="x( -1.8 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.8 )" alt="y( -1.8 )" title="y( -1.8 )" data-latex="y( -1.8 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -4 \, t^{2} y^{2} + 2 \, x^{2} y + 3 \hspace{2em} x( -2 )= 1" alt="x'= -4 \, t^{2} y^{2} + 2 \, x^{2} y + 3 \hspace{2em} x( -2 )= 1" title="x'= -4 \, t^{2} y^{2} + 2 \, x^{2} y + 3 \hspace{2em} x( -2 )= 1" data-latex="x'= -4 \, t^{2} y^{2} + 2 \, x^{2} y + 3 \hspace{2em} x( -2 )= 1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -3 \, t x^{2} + 2 \, x^{2} y - 2 \hspace{2em} y( -2 )= 2" alt="y'= -3 \, t x^{2} + 2 \, x^{2} y - 2 \hspace{2em} y( -2 )= 2" title="y'= -3 \, t x^{2} + 2 \, x^{2} y - 2 \hspace{2em} y( -2 )= 2" data-latex="y'= -3 \, t x^{2} + 2 \, x^{2} y - 2 \hspace{2em} y( -2 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.8%20)" alt="x( -1.8 )" title="x( -1.8 )" data-latex="x( -1.8 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.8%20)" alt="y( -1.8 )" title="y( -1.8 )" data-latex="y( -1.8 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-4%20%5C,%20t%5E%7B2%7D%20y%5E%7B2%7D%20+%202%20%5C,%20x%5E%7B2%7D%20y%20+%203%20%5Chspace%7B2em%7D%20x(%20-2%20)=%201" alt="x'= -4 \, t^{2} y^{2} + 2 \, x^{2} y + 3 \hspace{2em} x( -2 )= 1" title="x'= -4 \, t^{2} y^{2} + 2 \, x^{2} y + 3 \hspace{2em} x( -2 )= 1" data-latex="x'= -4 \, t^{2} y^{2} + 2 \, x^{2} y + 3 \hspace{2em} x( -2 )= 1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-3%20%5C,%20t%20x%5E%7B2%7D%20+%202%20%5C,%20x%5E%7B2%7D%20y%20-%202%20%5Chspace%7B2em%7D%20y(%20-2%20)=%202" alt="y'= -3 \, t x^{2} + 2 \, x^{2} y - 2 \hspace{2em} y( -2 )= 2" title="y'= -3 \, t x^{2} + 2 \, x^{2} y - 2 \hspace{2em} y( -2 )= 2" data-latex="y'= -3 \, t x^{2} + 2 \, x^{2} y - 2 \hspace{2em} y( -2 )= 2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.9 )\approx -4.70" alt="x( -1.9 )\approx -4.70" title="x( -1.9 )\approx -4.70" data-latex="x( -1.9 )\approx -4.70"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.9 )\approx 2.80" alt="y( -1.9 )\approx 2.80" title="y( -1.9 )\approx 2.80" data-latex="y( -1.9 )\approx 2.80"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.8 )\approx -3.38" alt="x( -1.8 )\approx -3.38" title="x( -1.8 )\approx -3.38" data-latex="x( -1.8 )\approx -3.38"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.8 )\approx 27.5" alt="y( -1.8 )\approx 27.5" title="y( -1.8 )\approx 27.5" data-latex="y( -1.8 )\approx 27.5"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.9%20)%5Capprox%20-4.70" alt="x( -1.9 )\approx -4.70" title="x( -1.9 )\approx -4.70" data-latex="x( -1.9 )\approx -4.70"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.9%20)%5Capprox%202.80" alt="y( -1.9 )\approx 2.80" title="y( -1.9 )\approx 2.80" data-latex="y( -1.9 )\approx 2.80"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.8%20)%5Capprox%20-3.38" alt="x( -1.8 )\approx -3.38" title="x( -1.8 )\approx -3.38" data-latex="x( -1.8 )\approx -3.38"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.8%20)%5Capprox%2027.5" alt="y( -1.8 )\approx 27.5" title="y( -1.8 )\approx 27.5" data-latex="y( -1.8 )\approx 27.5"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-0297" title="N2 | Euler's method for approximating IVP solutions | ver. 0297"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.2 )" alt="x( 1.2 )" title="x( 1.2 )" data-latex="x( 1.2 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.2 )" alt="y( 1.2 )" title="y( 1.2 )" data-latex="y( 1.2 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= t x + 2 \, x y - 3 \hspace{2em} x( 1 )= -1" alt="x'= t x + 2 \, x y - 3 \hspace{2em} x( 1 )= -1" title="x'= t x + 2 \, x y - 3 \hspace{2em} x( 1 )= -1" data-latex="x'= t x + 2 \, x y - 3 \hspace{2em} x( 1 )= -1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -3 \, t^{2} x^{2} + 4 \, x y - 1 \hspace{2em} y( 1 )= -2" alt="y'= -3 \, t^{2} x^{2} + 4 \, x y - 1 \hspace{2em} y( 1 )= -2" title="y'= -3 \, t^{2} x^{2} + 4 \, x y - 1 \hspace{2em} y( 1 )= -2" data-latex="y'= -3 \, t^{2} x^{2} + 4 \, x y - 1 \hspace{2em} y( 1 )= -2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.2%20)" alt="x( 1.2 )" title="x( 1.2 )" data-latex="x( 1.2 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.2%20)" alt="y( 1.2 )" title="y( 1.2 )" data-latex="y( 1.2 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20t%20x%20+%202%20%5C,%20x%20y%20-%203%20%5Chspace%7B2em%7D%20x(%201%20)=%20-1" alt="x'= t x + 2 \, x y - 3 \hspace{2em} x( 1 )= -1" title="x'= t x + 2 \, x y - 3 \hspace{2em} x( 1 )= -1" data-latex="x'= t x + 2 \, x y - 3 \hspace{2em} x( 1 )= -1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-3%20%5C,%20t%5E%7B2%7D%20x%5E%7B2%7D%20+%204%20%5C,%20x%20y%20-%201%20%5Chspace%7B2em%7D%20y(%201%20)=%20-2" alt="y'= -3 \, t^{2} x^{2} + 4 \, x y - 1 \hspace{2em} y( 1 )= -2" title="y'= -3 \, t^{2} x^{2} + 4 \, x y - 1 \hspace{2em} y( 1 )= -2" data-latex="y'= -3 \, t^{2} x^{2} + 4 \, x y - 1 \hspace{2em} y( 1 )= -2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.1 )\approx -1.00" alt="x( 1.1 )\approx -1.00" title="x( 1.1 )\approx -1.00" data-latex="x( 1.1 )\approx -1.00"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.1 )\approx -1.60" alt="y( 1.1 )\approx -1.60" title="y( 1.1 )\approx -1.60" data-latex="y( 1.1 )\approx -1.60"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.2 )\approx -1.09" alt="x( 1.2 )\approx -1.09" title="x( 1.2 )\approx -1.09" data-latex="x( 1.2 )\approx -1.09"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.2 )\approx -1.42" alt="y( 1.2 )\approx -1.42" title="y( 1.2 )\approx -1.42" data-latex="y( 1.2 )\approx -1.42"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.1%20)%5Capprox%20-1.00" alt="x( 1.1 )\approx -1.00" title="x( 1.1 )\approx -1.00" data-latex="x( 1.1 )\approx -1.00"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.1%20)%5Capprox%20-1.60" alt="y( 1.1 )\approx -1.60" title="y( 1.1 )\approx -1.60" data-latex="y( 1.1 )\approx -1.60"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.2%20)%5Capprox%20-1.09" alt="x( 1.2 )\approx -1.09" title="x( 1.2 )\approx -1.09" data-latex="x( 1.2 )\approx -1.09"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.2%20)%5Capprox%20-1.42" alt="y( 1.2 )\approx -1.42" title="y( 1.2 )\approx -1.42" data-latex="y( 1.2 )\approx -1.42"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-5416" title="N2 | Euler's method for approximating IVP solutions | ver. 5416"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.2 )" alt="x( 1.2 )" title="x( 1.2 )" data-latex="x( 1.2 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.2 )" alt="y( 1.2 )" title="y( 1.2 )" data-latex="y( 1.2 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -3 \, t^{2} x - 3 \, t y + 2 \hspace{2em} x( 1 )= 1" alt="x'= -3 \, t^{2} x - 3 \, t y + 2 \hspace{2em} x( 1 )= 1" title="x'= -3 \, t^{2} x - 3 \, t y + 2 \hspace{2em} x( 1 )= 1" data-latex="x'= -3 \, t^{2} x - 3 \, t y + 2 \hspace{2em} x( 1 )= 1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -3 \, t x^{2} - 2 \, t^{2} y - 2 \hspace{2em} y( 1 )= 0" alt="y'= -3 \, t x^{2} - 2 \, t^{2} y - 2 \hspace{2em} y( 1 )= 0" title="y'= -3 \, t x^{2} - 2 \, t^{2} y - 2 \hspace{2em} y( 1 )= 0" data-latex="y'= -3 \, t x^{2} - 2 \, t^{2} y - 2 \hspace{2em} y( 1 )= 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.2%20)" alt="x( 1.2 )" title="x( 1.2 )" data-latex="x( 1.2 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.2%20)" alt="y( 1.2 )" title="y( 1.2 )" data-latex="y( 1.2 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-3%20%5C,%20t%5E%7B2%7D%20x%20-%203%20%5C,%20t%20y%20+%202%20%5Chspace%7B2em%7D%20x(%201%20)=%201" alt="x'= -3 \, t^{2} x - 3 \, t y + 2 \hspace{2em} x( 1 )= 1" title="x'= -3 \, t^{2} x - 3 \, t y + 2 \hspace{2em} x( 1 )= 1" data-latex="x'= -3 \, t^{2} x - 3 \, t y + 2 \hspace{2em} x( 1 )= 1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-3%20%5C,%20t%20x%5E%7B2%7D%20-%202%20%5C,%20t%5E%7B2%7D%20y%20-%202%20%5Chspace%7B2em%7D%20y(%201%20)=%200" alt="y'= -3 \, t x^{2} - 2 \, t^{2} y - 2 \hspace{2em} y( 1 )= 0" title="y'= -3 \, t x^{2} - 2 \, t^{2} y - 2 \hspace{2em} y( 1 )= 0" data-latex="y'= -3 \, t x^{2} - 2 \, t^{2} y - 2 \hspace{2em} y( 1 )= 0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.1 )\approx 0.900" alt="x( 1.1 )\approx 0.900" title="x( 1.1 )\approx 0.900" data-latex="x( 1.1 )\approx 0.900"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.1 )\approx -0.500" alt="y( 1.1 )\approx -0.500" title="y( 1.1 )\approx -0.500" data-latex="y( 1.1 )\approx -0.500"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.2 )\approx 0.938" alt="x( 1.2 )\approx 0.938" title="x( 1.2 )\approx 0.938" data-latex="x( 1.2 )\approx 0.938"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.2 )\approx -0.847" alt="y( 1.2 )\approx -0.847" title="y( 1.2 )\approx -0.847" data-latex="y( 1.2 )\approx -0.847"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.1%20)%5Capprox%200.900" alt="x( 1.1 )\approx 0.900" title="x( 1.1 )\approx 0.900" data-latex="x( 1.1 )\approx 0.900"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.1%20)%5Capprox%20-0.500" alt="y( 1.1 )\approx -0.500" title="y( 1.1 )\approx -0.500" data-latex="y( 1.1 )\approx -0.500"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.2%20)%5Capprox%200.938" alt="x( 1.2 )\approx 0.938" title="x( 1.2 )\approx 0.938" data-latex="x( 1.2 )\approx 0.938"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.2%20)%5Capprox%20-0.847" alt="y( 1.2 )\approx -0.847" title="y( 1.2 )\approx -0.847" data-latex="y( 1.2 )\approx -0.847"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-5318" title="N2 | Euler's method for approximating IVP solutions | ver. 5318"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.20 )" alt="x( 0.20 )" title="x( 0.20 )" data-latex="x( 0.20 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.20 )" alt="y( 0.20 )" title="y( 0.20 )" data-latex="y( 0.20 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -3 \, t^{2} x + 2 \, x^{2} y + 3 \hspace{2em} x( 0 )= 0" alt="x'= -3 \, t^{2} x + 2 \, x^{2} y + 3 \hspace{2em} x( 0 )= 0" title="x'= -3 \, t^{2} x + 2 \, x^{2} y + 3 \hspace{2em} x( 0 )= 0" data-latex="x'= -3 \, t^{2} x + 2 \, x^{2} y + 3 \hspace{2em} x( 0 )= 0"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 2 \, t^{2} x^{2} + 4 \, t^{2} y^{2} - 2 \hspace{2em} y( 0 )= -2" alt="y'= 2 \, t^{2} x^{2} + 4 \, t^{2} y^{2} - 2 \hspace{2em} y( 0 )= -2" title="y'= 2 \, t^{2} x^{2} + 4 \, t^{2} y^{2} - 2 \hspace{2em} y( 0 )= -2" data-latex="y'= 2 \, t^{2} x^{2} + 4 \, t^{2} y^{2} - 2 \hspace{2em} y( 0 )= -2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.20%20)" alt="x( 0.20 )" title="x( 0.20 )" data-latex="x( 0.20 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.20%20)" alt="y( 0.20 )" title="y( 0.20 )" data-latex="y( 0.20 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-3%20%5C,%20t%5E%7B2%7D%20x%20+%202%20%5C,%20x%5E%7B2%7D%20y%20+%203%20%5Chspace%7B2em%7D%20x(%200%20)=%200" alt="x'= -3 \, t^{2} x + 2 \, x^{2} y + 3 \hspace{2em} x( 0 )= 0" title="x'= -3 \, t^{2} x + 2 \, x^{2} y + 3 \hspace{2em} x( 0 )= 0" data-latex="x'= -3 \, t^{2} x + 2 \, x^{2} y + 3 \hspace{2em} x( 0 )= 0"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%202%20%5C,%20t%5E%7B2%7D%20x%5E%7B2%7D%20+%204%20%5C,%20t%5E%7B2%7D%20y%5E%7B2%7D%20-%202%20%5Chspace%7B2em%7D%20y(%200%20)=%20-2" alt="y'= 2 \, t^{2} x^{2} + 4 \, t^{2} y^{2} - 2 \hspace{2em} y( 0 )= -2" title="y'= 2 \, t^{2} x^{2} + 4 \, t^{2} y^{2} - 2 \hspace{2em} y( 0 )= -2" data-latex="y'= 2 \, t^{2} x^{2} + 4 \, t^{2} y^{2} - 2 \hspace{2em} y( 0 )= -2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.10 )\approx 0.300" alt="x( 0.10 )\approx 0.300" title="x( 0.10 )\approx 0.300" data-latex="x( 0.10 )\approx 0.300"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.10 )\approx -2.20" alt="y( 0.10 )\approx -2.20" title="y( 0.10 )\approx -2.20" data-latex="y( 0.10 )\approx -2.20"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.20 )\approx 0.560" alt="x( 0.20 )\approx 0.560" title="x( 0.20 )\approx 0.560" data-latex="x( 0.20 )\approx 0.560"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.20 )\approx -2.38" alt="y( 0.20 )\approx -2.38" title="y( 0.20 )\approx -2.38" data-latex="y( 0.20 )\approx -2.38"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.10%20)%5Capprox%200.300" alt="x( 0.10 )\approx 0.300" title="x( 0.10 )\approx 0.300" data-latex="x( 0.10 )\approx 0.300"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.10%20)%5Capprox%20-2.20" alt="y( 0.10 )\approx -2.20" title="y( 0.10 )\approx -2.20" data-latex="y( 0.10 )\approx -2.20"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.20%20)%5Capprox%200.560" alt="x( 0.20 )\approx 0.560" title="x( 0.20 )\approx 0.560" data-latex="x( 0.20 )\approx 0.560"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.20%20)%5Capprox%20-2.38" alt="y( 0.20 )\approx -2.38" title="y( 0.20 )\approx -2.38" data-latex="y( 0.20 )\approx -2.38"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-3629" title="N2 | Euler's method for approximating IVP solutions | ver. 3629"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.80 )" alt="x( -0.80 )" title="x( -0.80 )" data-latex="x( -0.80 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.80 )" alt="y( -0.80 )" title="y( -0.80 )" data-latex="y( -0.80 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= t x^{2} - 4 \, t^{2} y + 1 \hspace{2em} x( -1 )= 1" alt="x'= t x^{2} - 4 \, t^{2} y + 1 \hspace{2em} x( -1 )= 1" title="x'= t x^{2} - 4 \, t^{2} y + 1 \hspace{2em} x( -1 )= 1" data-latex="x'= t x^{2} - 4 \, t^{2} y + 1 \hspace{2em} x( -1 )= 1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -2 \, t y^{2} - 2 \, t x - 3 \hspace{2em} y( -1 )= -1" alt="y'= -2 \, t y^{2} - 2 \, t x - 3 \hspace{2em} y( -1 )= -1" title="y'= -2 \, t y^{2} - 2 \, t x - 3 \hspace{2em} y( -1 )= -1" data-latex="y'= -2 \, t y^{2} - 2 \, t x - 3 \hspace{2em} y( -1 )= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.80%20)" alt="x( -0.80 )" title="x( -0.80 )" data-latex="x( -0.80 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.80%20)" alt="y( -0.80 )" title="y( -0.80 )" data-latex="y( -0.80 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20t%20x%5E%7B2%7D%20-%204%20%5C,%20t%5E%7B2%7D%20y%20+%201%20%5Chspace%7B2em%7D%20x(%20-1%20)=%201" alt="x'= t x^{2} - 4 \, t^{2} y + 1 \hspace{2em} x( -1 )= 1" title="x'= t x^{2} - 4 \, t^{2} y + 1 \hspace{2em} x( -1 )= 1" data-latex="x'= t x^{2} - 4 \, t^{2} y + 1 \hspace{2em} x( -1 )= 1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-2%20%5C,%20t%20y%5E%7B2%7D%20-%202%20%5C,%20t%20x%20-%203%20%5Chspace%7B2em%7D%20y(%20-1%20)=%20-1" alt="y'= -2 \, t y^{2} - 2 \, t x - 3 \hspace{2em} y( -1 )= -1" title="y'= -2 \, t y^{2} - 2 \, t x - 3 \hspace{2em} y( -1 )= -1" data-latex="y'= -2 \, t y^{2} - 2 \, t x - 3 \hspace{2em} y( -1 )= -1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.90 )\approx 1.40" alt="x( -0.90 )\approx 1.40" title="x( -0.90 )\approx 1.40" data-latex="x( -0.90 )\approx 1.40"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.90 )\approx -0.900" alt="y( -0.90 )\approx -0.900" title="y( -0.90 )\approx -0.900" data-latex="y( -0.90 )\approx -0.900"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.80 )\approx 1.62" alt="x( -0.80 )\approx 1.62" title="x( -0.80 )\approx 1.62" data-latex="x( -0.80 )\approx 1.62"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.80 )\approx -0.802" alt="y( -0.80 )\approx -0.802" title="y( -0.80 )\approx -0.802" data-latex="y( -0.80 )\approx -0.802"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.90%20)%5Capprox%201.40" alt="x( -0.90 )\approx 1.40" title="x( -0.90 )\approx 1.40" data-latex="x( -0.90 )\approx 1.40"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.90%20)%5Capprox%20-0.900" alt="y( -0.90 )\approx -0.900" title="y( -0.90 )\approx -0.900" data-latex="y( -0.90 )\approx -0.900"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.80%20)%5Capprox%201.62" alt="x( -0.80 )\approx 1.62" title="x( -0.80 )\approx 1.62" data-latex="x( -0.80 )\approx 1.62"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.80%20)%5Capprox%20-0.802" alt="y( -0.80 )\approx -0.802" title="y( -0.80 )\approx -0.802" data-latex="y( -0.80 )\approx -0.802"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-6104" title="N2 | Euler's method for approximating IVP solutions | ver. 6104"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 2.2 )" alt="x( 2.2 )" title="x( 2.2 )" data-latex="x( 2.2 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 2.2 )" alt="y( 2.2 )" title="y( 2.2 )" data-latex="y( 2.2 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 2 \, x^{2} y + t x - 1 \hspace{2em} x( 2 )= -1" alt="x'= 2 \, x^{2} y + t x - 1 \hspace{2em} x( 2 )= -1" title="x'= 2 \, x^{2} y + t x - 1 \hspace{2em} x( 2 )= -1" data-latex="x'= 2 \, x^{2} y + t x - 1 \hspace{2em} x( 2 )= -1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 2 \, t^{2} x - 4 \, t y \hspace{2em} y( 2 )= 1" alt="y'= 2 \, t^{2} x - 4 \, t y \hspace{2em} y( 2 )= 1" title="y'= 2 \, t^{2} x - 4 \, t y \hspace{2em} y( 2 )= 1" data-latex="y'= 2 \, t^{2} x - 4 \, t y \hspace{2em} y( 2 )= 1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%202.2%20)" alt="x( 2.2 )" title="x( 2.2 )" data-latex="x( 2.2 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%202.2%20)" alt="y( 2.2 )" title="y( 2.2 )" data-latex="y( 2.2 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%202%20%5C,%20x%5E%7B2%7D%20y%20+%20t%20x%20-%201%20%5Chspace%7B2em%7D%20x(%202%20)=%20-1" alt="x'= 2 \, x^{2} y + t x - 1 \hspace{2em} x( 2 )= -1" title="x'= 2 \, x^{2} y + t x - 1 \hspace{2em} x( 2 )= -1" data-latex="x'= 2 \, x^{2} y + t x - 1 \hspace{2em} x( 2 )= -1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%202%20%5C,%20t%5E%7B2%7D%20x%20-%204%20%5C,%20t%20y%20%5Chspace%7B2em%7D%20y(%202%20)=%201" alt="y'= 2 \, t^{2} x - 4 \, t y \hspace{2em} y( 2 )= 1" title="y'= 2 \, t^{2} x - 4 \, t y \hspace{2em} y( 2 )= 1" data-latex="y'= 2 \, t^{2} x - 4 \, t y \hspace{2em} y( 2 )= 1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 2.1 )\approx -1.10" alt="x( 2.1 )\approx -1.10" title="x( 2.1 )\approx -1.10" data-latex="x( 2.1 )\approx -1.10"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 2.1 )\approx -0.600" alt="y( 2.1 )\approx -0.600" title="y( 2.1 )\approx -0.600" data-latex="y( 2.1 )\approx -0.600"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 2.2 )\approx -1.58" alt="x( 2.2 )\approx -1.58" title="x( 2.2 )\approx -1.58" data-latex="x( 2.2 )\approx -1.58"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 2.2 )\approx -1.07" alt="y( 2.2 )\approx -1.07" title="y( 2.2 )\approx -1.07" data-latex="y( 2.2 )\approx -1.07"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%202.1%20)%5Capprox%20-1.10" alt="x( 2.1 )\approx -1.10" title="x( 2.1 )\approx -1.10" data-latex="x( 2.1 )\approx -1.10"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%202.1%20)%5Capprox%20-0.600" alt="y( 2.1 )\approx -0.600" title="y( 2.1 )\approx -0.600" data-latex="y( 2.1 )\approx -0.600"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%202.2%20)%5Capprox%20-1.58" alt="x( 2.2 )\approx -1.58" title="x( 2.2 )\approx -1.58" data-latex="x( 2.2 )\approx -1.58"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%202.2%20)%5Capprox%20-1.07" alt="y( 2.2 )\approx -1.07" title="y( 2.2 )\approx -1.07" data-latex="y( 2.2 )\approx -1.07"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-4007" title="N2 | Euler's method for approximating IVP solutions | ver. 4007"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 2.2 )" alt="x( 2.2 )" title="x( 2.2 )" data-latex="x( 2.2 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 2.2 )" alt="y( 2.2 )" title="y( 2.2 )" data-latex="y( 2.2 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 4 \, t^{2} x + 2 \, t y^{2} - 3 \hspace{2em} x( 2 )= 0" alt="x'= 4 \, t^{2} x + 2 \, t y^{2} - 3 \hspace{2em} x( 2 )= 0" title="x'= 4 \, t^{2} x + 2 \, t y^{2} - 3 \hspace{2em} x( 2 )= 0" data-latex="x'= 4 \, t^{2} x + 2 \, t y^{2} - 3 \hspace{2em} x( 2 )= 0"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= t x^{2} - 4 \, t^{2} y - 1 \hspace{2em} y( 2 )= 1" alt="y'= t x^{2} - 4 \, t^{2} y - 1 \hspace{2em} y( 2 )= 1" title="y'= t x^{2} - 4 \, t^{2} y - 1 \hspace{2em} y( 2 )= 1" data-latex="y'= t x^{2} - 4 \, t^{2} y - 1 \hspace{2em} y( 2 )= 1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%202.2%20)" alt="x( 2.2 )" title="x( 2.2 )" data-latex="x( 2.2 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%202.2%20)" alt="y( 2.2 )" title="y( 2.2 )" data-latex="y( 2.2 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%204%20%5C,%20t%5E%7B2%7D%20x%20+%202%20%5C,%20t%20y%5E%7B2%7D%20-%203%20%5Chspace%7B2em%7D%20x(%202%20)=%200" alt="x'= 4 \, t^{2} x + 2 \, t y^{2} - 3 \hspace{2em} x( 2 )= 0" title="x'= 4 \, t^{2} x + 2 \, t y^{2} - 3 \hspace{2em} x( 2 )= 0" data-latex="x'= 4 \, t^{2} x + 2 \, t y^{2} - 3 \hspace{2em} x( 2 )= 0"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20t%20x%5E%7B2%7D%20-%204%20%5C,%20t%5E%7B2%7D%20y%20-%201%20%5Chspace%7B2em%7D%20y(%202%20)=%201" alt="y'= t x^{2} - 4 \, t^{2} y - 1 \hspace{2em} y( 2 )= 1" title="y'= t x^{2} - 4 \, t^{2} y - 1 \hspace{2em} y( 2 )= 1" data-latex="y'= t x^{2} - 4 \, t^{2} y - 1 \hspace{2em} y( 2 )= 1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 2.1 )\approx 0.100" alt="x( 2.1 )\approx 0.100" title="x( 2.1 )\approx 0.100" data-latex="x( 2.1 )\approx 0.100"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 2.1 )\approx -0.699" alt="y( 2.1 )\approx -0.699" title="y( 2.1 )\approx -0.699" data-latex="y( 2.1 )\approx -0.699"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 2.2 )\approx 0.182" alt="x( 2.2 )\approx 0.182" title="x( 2.2 )\approx 0.182" data-latex="x( 2.2 )\approx 0.182"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 2.2 )\approx 0.436" alt="y( 2.2 )\approx 0.436" title="y( 2.2 )\approx 0.436" data-latex="y( 2.2 )\approx 0.436"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%202.1%20)%5Capprox%200.100" alt="x( 2.1 )\approx 0.100" title="x( 2.1 )\approx 0.100" data-latex="x( 2.1 )\approx 0.100"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%202.1%20)%5Capprox%20-0.699" alt="y( 2.1 )\approx -0.699" title="y( 2.1 )\approx -0.699" data-latex="y( 2.1 )\approx -0.699"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%202.2%20)%5Capprox%200.182" alt="x( 2.2 )\approx 0.182" title="x( 2.2 )\approx 0.182" data-latex="x( 2.2 )\approx 0.182"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%202.2%20)%5Capprox%200.436" alt="y( 2.2 )\approx 0.436" title="y( 2.2 )\approx 0.436" data-latex="y( 2.2 )\approx 0.436"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-6361" title="N2 | Euler's method for approximating IVP solutions | ver. 6361"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 2.2 )" alt="x( 2.2 )" title="x( 2.2 )" data-latex="x( 2.2 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 2.2 )" alt="y( 2.2 )" title="y( 2.2 )" data-latex="y( 2.2 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -3 \, t^{2} x - 4 \, x y + 3 \hspace{2em} x( 2 )= 0" alt="x'= -3 \, t^{2} x - 4 \, x y + 3 \hspace{2em} x( 2 )= 0" title="x'= -3 \, t^{2} x - 4 \, x y + 3 \hspace{2em} x( 2 )= 0" data-latex="x'= -3 \, t^{2} x - 4 \, x y + 3 \hspace{2em} x( 2 )= 0"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -t^{2} x^{2} - t^{2} y^{2} + 1 \hspace{2em} y( 2 )= -1" alt="y'= -t^{2} x^{2} - t^{2} y^{2} + 1 \hspace{2em} y( 2 )= -1" title="y'= -t^{2} x^{2} - t^{2} y^{2} + 1 \hspace{2em} y( 2 )= -1" data-latex="y'= -t^{2} x^{2} - t^{2} y^{2} + 1 \hspace{2em} y( 2 )= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%202.2%20)" alt="x( 2.2 )" title="x( 2.2 )" data-latex="x( 2.2 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%202.2%20)" alt="y( 2.2 )" title="y( 2.2 )" data-latex="y( 2.2 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-3%20%5C,%20t%5E%7B2%7D%20x%20-%204%20%5C,%20x%20y%20+%203%20%5Chspace%7B2em%7D%20x(%202%20)=%200" alt="x'= -3 \, t^{2} x - 4 \, x y + 3 \hspace{2em} x( 2 )= 0" title="x'= -3 \, t^{2} x - 4 \, x y + 3 \hspace{2em} x( 2 )= 0" data-latex="x'= -3 \, t^{2} x - 4 \, x y + 3 \hspace{2em} x( 2 )= 0"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-t%5E%7B2%7D%20x%5E%7B2%7D%20-%20t%5E%7B2%7D%20y%5E%7B2%7D%20+%201%20%5Chspace%7B2em%7D%20y(%202%20)=%20-1" alt="y'= -t^{2} x^{2} - t^{2} y^{2} + 1 \hspace{2em} y( 2 )= -1" title="y'= -t^{2} x^{2} - t^{2} y^{2} + 1 \hspace{2em} y( 2 )= -1" data-latex="y'= -t^{2} x^{2} - t^{2} y^{2} + 1 \hspace{2em} y( 2 )= -1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 2.1 )\approx 0.300" alt="x( 2.1 )\approx 0.300" title="x( 2.1 )\approx 0.300" data-latex="x( 2.1 )\approx 0.300"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 2.1 )\approx -1.30" alt="y( 2.1 )\approx -1.30" title="y( 2.1 )\approx -1.30" data-latex="y( 2.1 )\approx -1.30"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 2.2 )\approx 0.359" alt="x( 2.2 )\approx 0.359" title="x( 2.2 )\approx 0.359" data-latex="x( 2.2 )\approx 0.359"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 2.2 )\approx -1.99" alt="y( 2.2 )\approx -1.99" title="y( 2.2 )\approx -1.99" data-latex="y( 2.2 )\approx -1.99"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%202.1%20)%5Capprox%200.300" alt="x( 2.1 )\approx 0.300" title="x( 2.1 )\approx 0.300" data-latex="x( 2.1 )\approx 0.300"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%202.1%20)%5Capprox%20-1.30" alt="y( 2.1 )\approx -1.30" title="y( 2.1 )\approx -1.30" data-latex="y( 2.1 )\approx -1.30"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%202.2%20)%5Capprox%200.359" alt="x( 2.2 )\approx 0.359" title="x( 2.2 )\approx 0.359" data-latex="x( 2.2 )\approx 0.359"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%202.2%20)%5Capprox%20-1.99" alt="y( 2.2 )\approx -1.99" title="y( 2.2 )\approx -1.99" data-latex="y( 2.2 )\approx -1.99"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-4277" title="N2 | Euler's method for approximating IVP solutions | ver. 4277"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.8 )" alt="x( -1.8 )" title="x( -1.8 )" data-latex="x( -1.8 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.8 )" alt="y( -1.8 )" title="y( -1.8 )" data-latex="y( -1.8 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -3 \, t^{2} y^{2} + 4 \, x^{2} y^{2} - 2 \hspace{2em} x( -2 )= 0" alt="x'= -3 \, t^{2} y^{2} + 4 \, x^{2} y^{2} - 2 \hspace{2em} x( -2 )= 0" title="x'= -3 \, t^{2} y^{2} + 4 \, x^{2} y^{2} - 2 \hspace{2em} x( -2 )= 0" data-latex="x'= -3 \, t^{2} y^{2} + 4 \, x^{2} y^{2} - 2 \hspace{2em} x( -2 )= 0"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 3 \, x^{2} y - t y + 2 \hspace{2em} y( -2 )= 2" alt="y'= 3 \, x^{2} y - t y + 2 \hspace{2em} y( -2 )= 2" title="y'= 3 \, x^{2} y - t y + 2 \hspace{2em} y( -2 )= 2" data-latex="y'= 3 \, x^{2} y - t y + 2 \hspace{2em} y( -2 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.8%20)" alt="x( -1.8 )" title="x( -1.8 )" data-latex="x( -1.8 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.8%20)" alt="y( -1.8 )" title="y( -1.8 )" data-latex="y( -1.8 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-3%20%5C,%20t%5E%7B2%7D%20y%5E%7B2%7D%20+%204%20%5C,%20x%5E%7B2%7D%20y%5E%7B2%7D%20-%202%20%5Chspace%7B2em%7D%20x(%20-2%20)=%200" alt="x'= -3 \, t^{2} y^{2} + 4 \, x^{2} y^{2} - 2 \hspace{2em} x( -2 )= 0" title="x'= -3 \, t^{2} y^{2} + 4 \, x^{2} y^{2} - 2 \hspace{2em} x( -2 )= 0" data-latex="x'= -3 \, t^{2} y^{2} + 4 \, x^{2} y^{2} - 2 \hspace{2em} x( -2 )= 0"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%203%20%5C,%20x%5E%7B2%7D%20y%20-%20t%20y%20+%202%20%5Chspace%7B2em%7D%20y(%20-2%20)=%202" alt="y'= 3 \, x^{2} y - t y + 2 \hspace{2em} y( -2 )= 2" title="y'= 3 \, x^{2} y - t y + 2 \hspace{2em} y( -2 )= 2" data-latex="y'= 3 \, x^{2} y - t y + 2 \hspace{2em} y( -2 )= 2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.9 )\approx -5.00" alt="x( -1.9 )\approx -5.00" title="x( -1.9 )\approx -5.00" data-latex="x( -1.9 )\approx -5.00"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.9 )\approx 2.60" alt="y( -1.9 )\approx 2.60" title="y( -1.9 )\approx 2.60" data-latex="y( -1.9 )\approx 2.60"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.8 )\approx 55.2" alt="x( -1.8 )\approx 55.2" title="x( -1.8 )\approx 55.2" data-latex="x( -1.8 )\approx 55.2"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.8 )\approx 22.8" alt="y( -1.8 )\approx 22.8" title="y( -1.8 )\approx 22.8" data-latex="y( -1.8 )\approx 22.8"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.9%20)%5Capprox%20-5.00" alt="x( -1.9 )\approx -5.00" title="x( -1.9 )\approx -5.00" data-latex="x( -1.9 )\approx -5.00"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.9%20)%5Capprox%202.60" alt="y( -1.9 )\approx 2.60" title="y( -1.9 )\approx 2.60" data-latex="y( -1.9 )\approx 2.60"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.8%20)%5Capprox%2055.2" alt="x( -1.8 )\approx 55.2" title="x( -1.8 )\approx 55.2" data-latex="x( -1.8 )\approx 55.2"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.8%20)%5Capprox%2022.8" alt="y( -1.8 )\approx 22.8" title="y( -1.8 )\approx 22.8" data-latex="y( -1.8 )\approx 22.8"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-3073" title="N2 | Euler's method for approximating IVP solutions | ver. 3073"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.8 )" alt="x( -1.8 )" title="x( -1.8 )" data-latex="x( -1.8 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.8 )" alt="y( -1.8 )" title="y( -1.8 )" data-latex="y( -1.8 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 2 \, x^{2} y^{2} - 4 \, t^{2} y + 2 \hspace{2em} x( -2 )= 0" alt="x'= 2 \, x^{2} y^{2} - 4 \, t^{2} y + 2 \hspace{2em} x( -2 )= 0" title="x'= 2 \, x^{2} y^{2} - 4 \, t^{2} y + 2 \hspace{2em} x( -2 )= 0" data-latex="x'= 2 \, x^{2} y^{2} - 4 \, t^{2} y + 2 \hspace{2em} x( -2 )= 0"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -t x + 3 \, t y - 2 \hspace{2em} y( -2 )= 2" alt="y'= -t x + 3 \, t y - 2 \hspace{2em} y( -2 )= 2" title="y'= -t x + 3 \, t y - 2 \hspace{2em} y( -2 )= 2" data-latex="y'= -t x + 3 \, t y - 2 \hspace{2em} y( -2 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.8%20)" alt="x( -1.8 )" title="x( -1.8 )" data-latex="x( -1.8 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.8%20)" alt="y( -1.8 )" title="y( -1.8 )" data-latex="y( -1.8 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%202%20%5C,%20x%5E%7B2%7D%20y%5E%7B2%7D%20-%204%20%5C,%20t%5E%7B2%7D%20y%20+%202%20%5Chspace%7B2em%7D%20x(%20-2%20)=%200" alt="x'= 2 \, x^{2} y^{2} - 4 \, t^{2} y + 2 \hspace{2em} x( -2 )= 0" title="x'= 2 \, x^{2} y^{2} - 4 \, t^{2} y + 2 \hspace{2em} x( -2 )= 0" data-latex="x'= 2 \, x^{2} y^{2} - 4 \, t^{2} y + 2 \hspace{2em} x( -2 )= 0"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-t%20x%20+%203%20%5C,%20t%20y%20-%202%20%5Chspace%7B2em%7D%20y(%20-2%20)=%202" alt="y'= -t x + 3 \, t y - 2 \hspace{2em} y( -2 )= 2" title="y'= -t x + 3 \, t y - 2 \hspace{2em} y( -2 )= 2" data-latex="y'= -t x + 3 \, t y - 2 \hspace{2em} y( -2 )= 2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.9 )\approx -3.00" alt="x( -1.9 )\approx -3.00" title="x( -1.9 )\approx -3.00" data-latex="x( -1.9 )\approx -3.00"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.9 )\approx 0.600" alt="y( -1.9 )\approx 0.600" title="y( -1.9 )\approx 0.600" data-latex="y( -1.9 )\approx 0.600"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.8 )\approx -3.02" alt="x( -1.8 )\approx -3.02" title="x( -1.8 )\approx -3.02" data-latex="x( -1.8 )\approx -3.02"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.8 )\approx -0.512" alt="y( -1.8 )\approx -0.512" title="y( -1.8 )\approx -0.512" data-latex="y( -1.8 )\approx -0.512"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.9%20)%5Capprox%20-3.00" alt="x( -1.9 )\approx -3.00" title="x( -1.9 )\approx -3.00" data-latex="x( -1.9 )\approx -3.00"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.9%20)%5Capprox%200.600" alt="y( -1.9 )\approx 0.600" title="y( -1.9 )\approx 0.600" data-latex="y( -1.9 )\approx 0.600"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.8%20)%5Capprox%20-3.02" alt="x( -1.8 )\approx -3.02" title="x( -1.8 )\approx -3.02" data-latex="x( -1.8 )\approx -3.02"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.8%20)%5Capprox%20-0.512" alt="y( -1.8 )\approx -0.512" title="y( -1.8 )\approx -0.512" data-latex="y( -1.8 )\approx -0.512"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-7510" title="N2 | Euler's method for approximating IVP solutions | ver. 7510"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.20 )" alt="x( 0.20 )" title="x( 0.20 )" data-latex="x( 0.20 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.20 )" alt="y( 0.20 )" title="y( 0.20 )" data-latex="y( 0.20 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 4 \, t^{2} y + 2 \, x y - 3 \hspace{2em} x( 0 )= -1" alt="x'= 4 \, t^{2} y + 2 \, x y - 3 \hspace{2em} x( 0 )= -1" title="x'= 4 \, t^{2} y + 2 \, x y - 3 \hspace{2em} x( 0 )= -1" data-latex="x'= 4 \, t^{2} y + 2 \, x y - 3 \hspace{2em} x( 0 )= -1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= t^{2} y^{2} - 3 \, x^{2} y^{2} + 2 \hspace{2em} y( 0 )= -1" alt="y'= t^{2} y^{2} - 3 \, x^{2} y^{2} + 2 \hspace{2em} y( 0 )= -1" title="y'= t^{2} y^{2} - 3 \, x^{2} y^{2} + 2 \hspace{2em} y( 0 )= -1" data-latex="y'= t^{2} y^{2} - 3 \, x^{2} y^{2} + 2 \hspace{2em} y( 0 )= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.20%20)" alt="x( 0.20 )" title="x( 0.20 )" data-latex="x( 0.20 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.20%20)" alt="y( 0.20 )" title="y( 0.20 )" data-latex="y( 0.20 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%204%20%5C,%20t%5E%7B2%7D%20y%20+%202%20%5C,%20x%20y%20-%203%20%5Chspace%7B2em%7D%20x(%200%20)=%20-1" alt="x'= 4 \, t^{2} y + 2 \, x y - 3 \hspace{2em} x( 0 )= -1" title="x'= 4 \, t^{2} y + 2 \, x y - 3 \hspace{2em} x( 0 )= -1" data-latex="x'= 4 \, t^{2} y + 2 \, x y - 3 \hspace{2em} x( 0 )= -1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20t%5E%7B2%7D%20y%5E%7B2%7D%20-%203%20%5C,%20x%5E%7B2%7D%20y%5E%7B2%7D%20+%202%20%5Chspace%7B2em%7D%20y(%200%20)=%20-1" alt="y'= t^{2} y^{2} - 3 \, x^{2} y^{2} + 2 \hspace{2em} y( 0 )= -1" title="y'= t^{2} y^{2} - 3 \, x^{2} y^{2} + 2 \hspace{2em} y( 0 )= -1" data-latex="y'= t^{2} y^{2} - 3 \, x^{2} y^{2} + 2 \hspace{2em} y( 0 )= -1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.10 )\approx -1.10" alt="x( 0.10 )\approx -1.10" title="x( 0.10 )\approx -1.10" data-latex="x( 0.10 )\approx -1.10"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.10 )\approx -1.10" alt="y( 0.10 )\approx -1.10" title="y( 0.10 )\approx -1.10" data-latex="y( 0.10 )\approx -1.10"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.20 )\approx -1.16" alt="x( 0.20 )\approx -1.16" title="x( 0.20 )\approx -1.16" data-latex="x( 0.20 )\approx -1.16"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.20 )\approx -1.34" alt="y( 0.20 )\approx -1.34" title="y( 0.20 )\approx -1.34" data-latex="y( 0.20 )\approx -1.34"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.10%20)%5Capprox%20-1.10" alt="x( 0.10 )\approx -1.10" title="x( 0.10 )\approx -1.10" data-latex="x( 0.10 )\approx -1.10"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.10%20)%5Capprox%20-1.10" alt="y( 0.10 )\approx -1.10" title="y( 0.10 )\approx -1.10" data-latex="y( 0.10 )\approx -1.10"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.20%20)%5Capprox%20-1.16" alt="x( 0.20 )\approx -1.16" title="x( 0.20 )\approx -1.16" data-latex="x( 0.20 )\approx -1.16"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.20%20)%5Capprox%20-1.34" alt="y( 0.20 )\approx -1.34" title="y( 0.20 )\approx -1.34" data-latex="y( 0.20 )\approx -1.34"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-4494" title="N2 | Euler's method for approximating IVP solutions | ver. 4494"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.20 )" alt="x( 0.20 )" title="x( 0.20 )" data-latex="x( 0.20 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.20 )" alt="y( 0.20 )" title="y( 0.20 )" data-latex="y( 0.20 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 4 \, x y^{2} - 4 \, t x + 1 \hspace{2em} x( 0 )= 2" alt="x'= 4 \, x y^{2} - 4 \, t x + 1 \hspace{2em} x( 0 )= 2" title="x'= 4 \, x y^{2} - 4 \, t x + 1 \hspace{2em} x( 0 )= 2" data-latex="x'= 4 \, x y^{2} - 4 \, t x + 1 \hspace{2em} x( 0 )= 2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= x^{2} y^{2} - 4 \, t x + 1 \hspace{2em} y( 0 )= -2" alt="y'= x^{2} y^{2} - 4 \, t x + 1 \hspace{2em} y( 0 )= -2" title="y'= x^{2} y^{2} - 4 \, t x + 1 \hspace{2em} y( 0 )= -2" data-latex="y'= x^{2} y^{2} - 4 \, t x + 1 \hspace{2em} y( 0 )= -2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.20%20)" alt="x( 0.20 )" title="x( 0.20 )" data-latex="x( 0.20 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.20%20)" alt="y( 0.20 )" title="y( 0.20 )" data-latex="y( 0.20 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%204%20%5C,%20x%20y%5E%7B2%7D%20-%204%20%5C,%20t%20x%20+%201%20%5Chspace%7B2em%7D%20x(%200%20)=%202" alt="x'= 4 \, x y^{2} - 4 \, t x + 1 \hspace{2em} x( 0 )= 2" title="x'= 4 \, x y^{2} - 4 \, t x + 1 \hspace{2em} x( 0 )= 2" data-latex="x'= 4 \, x y^{2} - 4 \, t x + 1 \hspace{2em} x( 0 )= 2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20x%5E%7B2%7D%20y%5E%7B2%7D%20-%204%20%5C,%20t%20x%20+%201%20%5Chspace%7B2em%7D%20y(%200%20)=%20-2" alt="y'= x^{2} y^{2} - 4 \, t x + 1 \hspace{2em} y( 0 )= -2" title="y'= x^{2} y^{2} - 4 \, t x + 1 \hspace{2em} y( 0 )= -2" data-latex="y'= x^{2} y^{2} - 4 \, t x + 1 \hspace{2em} y( 0 )= -2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.10 )\approx 5.30" alt="x( 0.10 )\approx 5.30" title="x( 0.10 )\approx 5.30" data-latex="x( 0.10 )\approx 5.30"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.10 )\approx -0.301" alt="y( 0.10 )\approx -0.301" title="y( 0.10 )\approx -0.301" data-latex="y( 0.10 )\approx -0.301"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.20 )\approx 5.38" alt="x( 0.20 )\approx 5.38" title="x( 0.20 )\approx 5.38" data-latex="x( 0.20 )\approx 5.38"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.20 )\approx -0.159" alt="y( 0.20 )\approx -0.159" title="y( 0.20 )\approx -0.159" data-latex="y( 0.20 )\approx -0.159"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.10%20)%5Capprox%205.30" alt="x( 0.10 )\approx 5.30" title="x( 0.10 )\approx 5.30" data-latex="x( 0.10 )\approx 5.30"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.10%20)%5Capprox%20-0.301" alt="y( 0.10 )\approx -0.301" title="y( 0.10 )\approx -0.301" data-latex="y( 0.10 )\approx -0.301"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.20%20)%5Capprox%205.38" alt="x( 0.20 )\approx 5.38" title="x( 0.20 )\approx 5.38" data-latex="x( 0.20 )\approx 5.38"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.20%20)%5Capprox%20-0.159" alt="y( 0.20 )\approx -0.159" title="y( 0.20 )\approx -0.159" data-latex="y( 0.20 )\approx -0.159"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-5754" title="N2 | Euler's method for approximating IVP solutions | ver. 5754"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 2.2 )" alt="x( 2.2 )" title="x( 2.2 )" data-latex="x( 2.2 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 2.2 )" alt="y( 2.2 )" title="y( 2.2 )" data-latex="y( 2.2 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -4 \, t x^{2} + 3 \, t y^{2} - 1 \hspace{2em} x( 2 )= 0" alt="x'= -4 \, t x^{2} + 3 \, t y^{2} - 1 \hspace{2em} x( 2 )= 0" title="x'= -4 \, t x^{2} + 3 \, t y^{2} - 1 \hspace{2em} x( 2 )= 0" data-latex="x'= -4 \, t x^{2} + 3 \, t y^{2} - 1 \hspace{2em} x( 2 )= 0"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 3 \, t^{2} x^{2} + x^{2} y^{2} - 3 \hspace{2em} y( 2 )= -1" alt="y'= 3 \, t^{2} x^{2} + x^{2} y^{2} - 3 \hspace{2em} y( 2 )= -1" title="y'= 3 \, t^{2} x^{2} + x^{2} y^{2} - 3 \hspace{2em} y( 2 )= -1" data-latex="y'= 3 \, t^{2} x^{2} + x^{2} y^{2} - 3 \hspace{2em} y( 2 )= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%202.2%20)" alt="x( 2.2 )" title="x( 2.2 )" data-latex="x( 2.2 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%202.2%20)" alt="y( 2.2 )" title="y( 2.2 )" data-latex="y( 2.2 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-4%20%5C,%20t%20x%5E%7B2%7D%20+%203%20%5C,%20t%20y%5E%7B2%7D%20-%201%20%5Chspace%7B2em%7D%20x(%202%20)=%200" alt="x'= -4 \, t x^{2} + 3 \, t y^{2} - 1 \hspace{2em} x( 2 )= 0" title="x'= -4 \, t x^{2} + 3 \, t y^{2} - 1 \hspace{2em} x( 2 )= 0" data-latex="x'= -4 \, t x^{2} + 3 \, t y^{2} - 1 \hspace{2em} x( 2 )= 0"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%203%20%5C,%20t%5E%7B2%7D%20x%5E%7B2%7D%20+%20x%5E%7B2%7D%20y%5E%7B2%7D%20-%203%20%5Chspace%7B2em%7D%20y(%202%20)=%20-1" alt="y'= 3 \, t^{2} x^{2} + x^{2} y^{2} - 3 \hspace{2em} y( 2 )= -1" title="y'= 3 \, t^{2} x^{2} + x^{2} y^{2} - 3 \hspace{2em} y( 2 )= -1" data-latex="y'= 3 \, t^{2} x^{2} + x^{2} y^{2} - 3 \hspace{2em} y( 2 )= -1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 2.1 )\approx 0.500" alt="x( 2.1 )\approx 0.500" title="x( 2.1 )\approx 0.500" data-latex="x( 2.1 )\approx 0.500"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 2.1 )\approx -1.30" alt="y( 2.1 )\approx -1.30" title="y( 2.1 )\approx -1.30" data-latex="y( 2.1 )\approx -1.30"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 2.2 )\approx 1.25" alt="x( 2.2 )\approx 1.25" title="x( 2.2 )\approx 1.25" data-latex="x( 2.2 )\approx 1.25"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 2.2 )\approx -1.23" alt="y( 2.2 )\approx -1.23" title="y( 2.2 )\approx -1.23" data-latex="y( 2.2 )\approx -1.23"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%202.1%20)%5Capprox%200.500" alt="x( 2.1 )\approx 0.500" title="x( 2.1 )\approx 0.500" data-latex="x( 2.1 )\approx 0.500"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%202.1%20)%5Capprox%20-1.30" alt="y( 2.1 )\approx -1.30" title="y( 2.1 )\approx -1.30" data-latex="y( 2.1 )\approx -1.30"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%202.2%20)%5Capprox%201.25" alt="x( 2.2 )\approx 1.25" title="x( 2.2 )\approx 1.25" data-latex="x( 2.2 )\approx 1.25"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%202.2%20)%5Capprox%20-1.23" alt="y( 2.2 )\approx -1.23" title="y( 2.2 )\approx -1.23" data-latex="y( 2.2 )\approx -1.23"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-9664" title="N2 | Euler's method for approximating IVP solutions | ver. 9664"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.80 )" alt="x( -0.80 )" title="x( -0.80 )" data-latex="x( -0.80 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.80 )" alt="y( -0.80 )" title="y( -0.80 )" data-latex="y( -0.80 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 3 \, t x^{2} + 3 \, x y^{2} \hspace{2em} x( -1 )= 1" alt="x'= 3 \, t x^{2} + 3 \, x y^{2} \hspace{2em} x( -1 )= 1" title="x'= 3 \, t x^{2} + 3 \, x y^{2} \hspace{2em} x( -1 )= 1" data-latex="x'= 3 \, t x^{2} + 3 \, x y^{2} \hspace{2em} x( -1 )= 1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -3 \, t^{2} y^{2} + 2 \, x^{2} y - 3 \hspace{2em} y( -1 )= 2" alt="y'= -3 \, t^{2} y^{2} + 2 \, x^{2} y - 3 \hspace{2em} y( -1 )= 2" title="y'= -3 \, t^{2} y^{2} + 2 \, x^{2} y - 3 \hspace{2em} y( -1 )= 2" data-latex="y'= -3 \, t^{2} y^{2} + 2 \, x^{2} y - 3 \hspace{2em} y( -1 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.80%20)" alt="x( -0.80 )" title="x( -0.80 )" data-latex="x( -0.80 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.80%20)" alt="y( -0.80 )" title="y( -0.80 )" data-latex="y( -0.80 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%203%20%5C,%20t%20x%5E%7B2%7D%20+%203%20%5C,%20x%20y%5E%7B2%7D%20%5Chspace%7B2em%7D%20x(%20-1%20)=%201" alt="x'= 3 \, t x^{2} + 3 \, x y^{2} \hspace{2em} x( -1 )= 1" title="x'= 3 \, t x^{2} + 3 \, x y^{2} \hspace{2em} x( -1 )= 1" data-latex="x'= 3 \, t x^{2} + 3 \, x y^{2} \hspace{2em} x( -1 )= 1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-3%20%5C,%20t%5E%7B2%7D%20y%5E%7B2%7D%20+%202%20%5C,%20x%5E%7B2%7D%20y%20-%203%20%5Chspace%7B2em%7D%20y(%20-1%20)=%202" alt="y'= -3 \, t^{2} y^{2} + 2 \, x^{2} y - 3 \hspace{2em} y( -1 )= 2" title="y'= -3 \, t^{2} y^{2} + 2 \, x^{2} y - 3 \hspace{2em} y( -1 )= 2" data-latex="y'= -3 \, t^{2} y^{2} + 2 \, x^{2} y - 3 \hspace{2em} y( -1 )= 2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.90 )\approx 1.90" alt="x( -0.90 )\approx 1.90" title="x( -0.90 )\approx 1.90" data-latex="x( -0.90 )\approx 1.90"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.90 )\approx 0.900" alt="y( -0.90 )\approx 0.900" title="y( -0.90 )\approx 0.900" data-latex="y( -0.90 )\approx 0.900"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.80 )\approx 1.39" alt="x( -0.80 )\approx 1.39" title="x( -0.80 )\approx 1.39" data-latex="x( -0.80 )\approx 1.39"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.80 )\approx 1.05" alt="y( -0.80 )\approx 1.05" title="y( -0.80 )\approx 1.05" data-latex="y( -0.80 )\approx 1.05"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.90%20)%5Capprox%201.90" alt="x( -0.90 )\approx 1.90" title="x( -0.90 )\approx 1.90" data-latex="x( -0.90 )\approx 1.90"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.90%20)%5Capprox%200.900" alt="y( -0.90 )\approx 0.900" title="y( -0.90 )\approx 0.900" data-latex="y( -0.90 )\approx 0.900"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.80%20)%5Capprox%201.39" alt="x( -0.80 )\approx 1.39" title="x( -0.80 )\approx 1.39" data-latex="x( -0.80 )\approx 1.39"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.80%20)%5Capprox%201.05" alt="y( -0.80 )\approx 1.05" title="y( -0.80 )\approx 1.05" data-latex="y( -0.80 )\approx 1.05"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-7757" title="N2 | Euler's method for approximating IVP solutions | ver. 7757"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.20 )" alt="x( 0.20 )" title="x( 0.20 )" data-latex="x( 0.20 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.20 )" alt="y( 0.20 )" title="y( 0.20 )" data-latex="y( 0.20 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -4 \, x^{2} y - 2 \, t y + 3 \hspace{2em} x( 0 )= 1" alt="x'= -4 \, x^{2} y - 2 \, t y + 3 \hspace{2em} x( 0 )= 1" title="x'= -4 \, x^{2} y - 2 \, t y + 3 \hspace{2em} x( 0 )= 1" data-latex="x'= -4 \, x^{2} y - 2 \, t y + 3 \hspace{2em} x( 0 )= 1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 2 \, t x^{2} + 2 \, x y^{2} \hspace{2em} y( 0 )= 1" alt="y'= 2 \, t x^{2} + 2 \, x y^{2} \hspace{2em} y( 0 )= 1" title="y'= 2 \, t x^{2} + 2 \, x y^{2} \hspace{2em} y( 0 )= 1" data-latex="y'= 2 \, t x^{2} + 2 \, x y^{2} \hspace{2em} y( 0 )= 1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.20%20)" alt="x( 0.20 )" title="x( 0.20 )" data-latex="x( 0.20 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.20%20)" alt="y( 0.20 )" title="y( 0.20 )" data-latex="y( 0.20 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-4%20%5C,%20x%5E%7B2%7D%20y%20-%202%20%5C,%20t%20y%20+%203%20%5Chspace%7B2em%7D%20x(%200%20)=%201" alt="x'= -4 \, x^{2} y - 2 \, t y + 3 \hspace{2em} x( 0 )= 1" title="x'= -4 \, x^{2} y - 2 \, t y + 3 \hspace{2em} x( 0 )= 1" data-latex="x'= -4 \, x^{2} y - 2 \, t y + 3 \hspace{2em} x( 0 )= 1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%202%20%5C,%20t%20x%5E%7B2%7D%20+%202%20%5C,%20x%20y%5E%7B2%7D%20%5Chspace%7B2em%7D%20y(%200%20)=%201" alt="y'= 2 \, t x^{2} + 2 \, x y^{2} \hspace{2em} y( 0 )= 1" title="y'= 2 \, t x^{2} + 2 \, x y^{2} \hspace{2em} y( 0 )= 1" data-latex="y'= 2 \, t x^{2} + 2 \, x y^{2} \hspace{2em} y( 0 )= 1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.10 )\approx 0.900" alt="x( 0.10 )\approx 0.900" title="x( 0.10 )\approx 0.900" data-latex="x( 0.10 )\approx 0.900"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.10 )\approx 1.20" alt="y( 0.10 )\approx 1.20" title="y( 0.10 )\approx 1.20" data-latex="y( 0.10 )\approx 1.20"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.20 )\approx 0.787" alt="x( 0.20 )\approx 0.787" title="x( 0.20 )\approx 0.787" data-latex="x( 0.20 )\approx 0.787"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.20 )\approx 1.47" alt="y( 0.20 )\approx 1.47" title="y( 0.20 )\approx 1.47" data-latex="y( 0.20 )\approx 1.47"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.10%20)%5Capprox%200.900" alt="x( 0.10 )\approx 0.900" title="x( 0.10 )\approx 0.900" data-latex="x( 0.10 )\approx 0.900"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.10%20)%5Capprox%201.20" alt="y( 0.10 )\approx 1.20" title="y( 0.10 )\approx 1.20" data-latex="y( 0.10 )\approx 1.20"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.20%20)%5Capprox%200.787" alt="x( 0.20 )\approx 0.787" title="x( 0.20 )\approx 0.787" data-latex="x( 0.20 )\approx 0.787"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.20%20)%5Capprox%201.47" alt="y( 0.20 )\approx 1.47" title="y( 0.20 )\approx 1.47" data-latex="y( 0.20 )\approx 1.47"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-3812" title="N2 | Euler's method for approximating IVP solutions | ver. 3812"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.8 )" alt="x( -1.8 )" title="x( -1.8 )" data-latex="x( -1.8 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.8 )" alt="y( -1.8 )" title="y( -1.8 )" data-latex="y( -1.8 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -t^{2} y - 3 \, x y^{2} - 3 \hspace{2em} x( -2 )= -1" alt="x'= -t^{2} y - 3 \, x y^{2} - 3 \hspace{2em} x( -2 )= -1" title="x'= -t^{2} y - 3 \, x y^{2} - 3 \hspace{2em} x( -2 )= -1" data-latex="x'= -t^{2} y - 3 \, x y^{2} - 3 \hspace{2em} x( -2 )= -1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -3 \, t^{2} x + t y + 3 \hspace{2em} y( -2 )= 1" alt="y'= -3 \, t^{2} x + t y + 3 \hspace{2em} y( -2 )= 1" title="y'= -3 \, t^{2} x + t y + 3 \hspace{2em} y( -2 )= 1" data-latex="y'= -3 \, t^{2} x + t y + 3 \hspace{2em} y( -2 )= 1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.8%20)" alt="x( -1.8 )" title="x( -1.8 )" data-latex="x( -1.8 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.8%20)" alt="y( -1.8 )" title="y( -1.8 )" data-latex="y( -1.8 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-t%5E%7B2%7D%20y%20-%203%20%5C,%20x%20y%5E%7B2%7D%20-%203%20%5Chspace%7B2em%7D%20x(%20-2%20)=%20-1" alt="x'= -t^{2} y - 3 \, x y^{2} - 3 \hspace{2em} x( -2 )= -1" title="x'= -t^{2} y - 3 \, x y^{2} - 3 \hspace{2em} x( -2 )= -1" data-latex="x'= -t^{2} y - 3 \, x y^{2} - 3 \hspace{2em} x( -2 )= -1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-3%20%5C,%20t%5E%7B2%7D%20x%20+%20t%20y%20+%203%20%5Chspace%7B2em%7D%20y(%20-2%20)=%201" alt="y'= -3 \, t^{2} x + t y + 3 \hspace{2em} y( -2 )= 1" title="y'= -3 \, t^{2} x + t y + 3 \hspace{2em} y( -2 )= 1" data-latex="y'= -3 \, t^{2} x + t y + 3 \hspace{2em} y( -2 )= 1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.9 )\approx -1.40" alt="x( -1.9 )\approx -1.40" title="x( -1.9 )\approx -1.40" data-latex="x( -1.9 )\approx -1.40"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.9 )\approx 2.30" alt="y( -1.9 )\approx 2.30" title="y( -1.9 )\approx 2.30" data-latex="y( -1.9 )\approx 2.30"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.8 )\approx -0.314" alt="x( -1.8 )\approx -0.314" title="x( -1.8 )\approx -0.314" data-latex="x( -1.8 )\approx -0.314"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.8 )\approx 3.68" alt="y( -1.8 )\approx 3.68" title="y( -1.8 )\approx 3.68" data-latex="y( -1.8 )\approx 3.68"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.9%20)%5Capprox%20-1.40" alt="x( -1.9 )\approx -1.40" title="x( -1.9 )\approx -1.40" data-latex="x( -1.9 )\approx -1.40"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.9%20)%5Capprox%202.30" alt="y( -1.9 )\approx 2.30" title="y( -1.9 )\approx 2.30" data-latex="y( -1.9 )\approx 2.30"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.8%20)%5Capprox%20-0.314" alt="x( -1.8 )\approx -0.314" title="x( -1.8 )\approx -0.314" data-latex="x( -1.8 )\approx -0.314"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.8%20)%5Capprox%203.68" alt="y( -1.8 )\approx 3.68" title="y( -1.8 )\approx 3.68" data-latex="y( -1.8 )\approx 3.68"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-3062" title="N2 | Euler's method for approximating IVP solutions | ver. 3062"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.8 )" alt="x( -1.8 )" title="x( -1.8 )" data-latex="x( -1.8 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.8 )" alt="y( -1.8 )" title="y( -1.8 )" data-latex="y( -1.8 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 2 \, t^{2} y + x^{2} y - 2 \hspace{2em} x( -2 )= -2" alt="x'= 2 \, t^{2} y + x^{2} y - 2 \hspace{2em} x( -2 )= -2" title="x'= 2 \, t^{2} y + x^{2} y - 2 \hspace{2em} x( -2 )= -2" data-latex="x'= 2 \, t^{2} y + x^{2} y - 2 \hspace{2em} x( -2 )= -2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -3 \, t^{2} x - x^{2} y - 3 \hspace{2em} y( -2 )= 0" alt="y'= -3 \, t^{2} x - x^{2} y - 3 \hspace{2em} y( -2 )= 0" title="y'= -3 \, t^{2} x - x^{2} y - 3 \hspace{2em} y( -2 )= 0" data-latex="y'= -3 \, t^{2} x - x^{2} y - 3 \hspace{2em} y( -2 )= 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.8%20)" alt="x( -1.8 )" title="x( -1.8 )" data-latex="x( -1.8 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.8%20)" alt="y( -1.8 )" title="y( -1.8 )" data-latex="y( -1.8 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%202%20%5C,%20t%5E%7B2%7D%20y%20+%20x%5E%7B2%7D%20y%20-%202%20%5Chspace%7B2em%7D%20x(%20-2%20)=%20-2" alt="x'= 2 \, t^{2} y + x^{2} y - 2 \hspace{2em} x( -2 )= -2" title="x'= 2 \, t^{2} y + x^{2} y - 2 \hspace{2em} x( -2 )= -2" data-latex="x'= 2 \, t^{2} y + x^{2} y - 2 \hspace{2em} x( -2 )= -2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-3%20%5C,%20t%5E%7B2%7D%20x%20-%20x%5E%7B2%7D%20y%20-%203%20%5Chspace%7B2em%7D%20y(%20-2%20)=%200" alt="y'= -3 \, t^{2} x - x^{2} y - 3 \hspace{2em} y( -2 )= 0" title="y'= -3 \, t^{2} x - x^{2} y - 3 \hspace{2em} y( -2 )= 0" data-latex="y'= -3 \, t^{2} x - x^{2} y - 3 \hspace{2em} y( -2 )= 0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.9 )\approx -2.20" alt="x( -1.9 )\approx -2.20" title="x( -1.9 )\approx -2.20" data-latex="x( -1.9 )\approx -2.20"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.9 )\approx 2.10" alt="y( -1.9 )\approx 2.10" title="y( -1.9 )\approx 2.10" data-latex="y( -1.9 )\approx 2.10"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.8 )\approx 0.133" alt="x( -1.8 )\approx 0.133" title="x( -1.8 )\approx 0.133" data-latex="x( -1.8 )\approx 0.133"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.8 )\approx 3.17" alt="y( -1.8 )\approx 3.17" title="y( -1.8 )\approx 3.17" data-latex="y( -1.8 )\approx 3.17"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.9%20)%5Capprox%20-2.20" alt="x( -1.9 )\approx -2.20" title="x( -1.9 )\approx -2.20" data-latex="x( -1.9 )\approx -2.20"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.9%20)%5Capprox%202.10" alt="y( -1.9 )\approx 2.10" title="y( -1.9 )\approx 2.10" data-latex="y( -1.9 )\approx 2.10"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.8%20)%5Capprox%200.133" alt="x( -1.8 )\approx 0.133" title="x( -1.8 )\approx 0.133" data-latex="x( -1.8 )\approx 0.133"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.8%20)%5Capprox%203.17" alt="y( -1.8 )\approx 3.17" title="y( -1.8 )\approx 3.17" data-latex="y( -1.8 )\approx 3.17"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-2716" title="N2 | Euler's method for approximating IVP solutions | ver. 2716"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.80 )" alt="x( -0.80 )" title="x( -0.80 )" data-latex="x( -0.80 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.80 )" alt="y( -0.80 )" title="y( -0.80 )" data-latex="y( -0.80 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -x y^{2} + 2 \, t x + 3 \hspace{2em} x( -1 )= 1" alt="x'= -x y^{2} + 2 \, t x + 3 \hspace{2em} x( -1 )= 1" title="x'= -x y^{2} + 2 \, t x + 3 \hspace{2em} x( -1 )= 1" data-latex="x'= -x y^{2} + 2 \, t x + 3 \hspace{2em} x( -1 )= 1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -4 \, x^{2} y^{2} - 2 \, t^{2} x + 1 \hspace{2em} y( -1 )= 2" alt="y'= -4 \, x^{2} y^{2} - 2 \, t^{2} x + 1 \hspace{2em} y( -1 )= 2" title="y'= -4 \, x^{2} y^{2} - 2 \, t^{2} x + 1 \hspace{2em} y( -1 )= 2" data-latex="y'= -4 \, x^{2} y^{2} - 2 \, t^{2} x + 1 \hspace{2em} y( -1 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.80%20)" alt="x( -0.80 )" title="x( -0.80 )" data-latex="x( -0.80 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.80%20)" alt="y( -0.80 )" title="y( -0.80 )" data-latex="y( -0.80 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-x%20y%5E%7B2%7D%20+%202%20%5C,%20t%20x%20+%203%20%5Chspace%7B2em%7D%20x(%20-1%20)=%201" alt="x'= -x y^{2} + 2 \, t x + 3 \hspace{2em} x( -1 )= 1" title="x'= -x y^{2} + 2 \, t x + 3 \hspace{2em} x( -1 )= 1" data-latex="x'= -x y^{2} + 2 \, t x + 3 \hspace{2em} x( -1 )= 1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-4%20%5C,%20x%5E%7B2%7D%20y%5E%7B2%7D%20-%202%20%5C,%20t%5E%7B2%7D%20x%20+%201%20%5Chspace%7B2em%7D%20y(%20-1%20)=%202" alt="y'= -4 \, x^{2} y^{2} - 2 \, t^{2} x + 1 \hspace{2em} y( -1 )= 2" title="y'= -4 \, x^{2} y^{2} - 2 \, t^{2} x + 1 \hspace{2em} y( -1 )= 2" data-latex="y'= -4 \, x^{2} y^{2} - 2 \, t^{2} x + 1 \hspace{2em} y( -1 )= 2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.90 )\approx 0.700" alt="x( -0.90 )\approx 0.700" title="x( -0.90 )\approx 0.700" data-latex="x( -0.90 )\approx 0.700"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.90 )\approx 0.301" alt="y( -0.90 )\approx 0.301" title="y( -0.90 )\approx 0.301" data-latex="y( -0.90 )\approx 0.301"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.80 )\approx 0.867" alt="x( -0.80 )\approx 0.867" title="x( -0.80 )\approx 0.867" data-latex="x( -0.80 )\approx 0.867"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.80 )\approx 0.270" alt="y( -0.80 )\approx 0.270" title="y( -0.80 )\approx 0.270" data-latex="y( -0.80 )\approx 0.270"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.90%20)%5Capprox%200.700" alt="x( -0.90 )\approx 0.700" title="x( -0.90 )\approx 0.700" data-latex="x( -0.90 )\approx 0.700"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.90%20)%5Capprox%200.301" alt="y( -0.90 )\approx 0.301" title="y( -0.90 )\approx 0.301" data-latex="y( -0.90 )\approx 0.301"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.80%20)%5Capprox%200.867" alt="x( -0.80 )\approx 0.867" title="x( -0.80 )\approx 0.867" data-latex="x( -0.80 )\approx 0.867"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.80%20)%5Capprox%200.270" alt="y( -0.80 )\approx 0.270" title="y( -0.80 )\approx 0.270" data-latex="y( -0.80 )\approx 0.270"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-5176" title="N2 | Euler's method for approximating IVP solutions | ver. 5176"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 2.2 )" alt="x( 2.2 )" title="x( 2.2 )" data-latex="x( 2.2 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 2.2 )" alt="y( 2.2 )" title="y( 2.2 )" data-latex="y( 2.2 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -4 \, t x^{2} - 2 \, t y^{2} + 2 \hspace{2em} x( 2 )= 1" alt="x'= -4 \, t x^{2} - 2 \, t y^{2} + 2 \hspace{2em} x( 2 )= 1" title="x'= -4 \, t x^{2} - 2 \, t y^{2} + 2 \hspace{2em} x( 2 )= 1" data-latex="x'= -4 \, t x^{2} - 2 \, t y^{2} + 2 \hspace{2em} x( 2 )= 1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= t^{2} y + 3 \, x y + 1 \hspace{2em} y( 2 )= 1" alt="y'= t^{2} y + 3 \, x y + 1 \hspace{2em} y( 2 )= 1" title="y'= t^{2} y + 3 \, x y + 1 \hspace{2em} y( 2 )= 1" data-latex="y'= t^{2} y + 3 \, x y + 1 \hspace{2em} y( 2 )= 1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%202.2%20)" alt="x( 2.2 )" title="x( 2.2 )" data-latex="x( 2.2 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%202.2%20)" alt="y( 2.2 )" title="y( 2.2 )" data-latex="y( 2.2 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-4%20%5C,%20t%20x%5E%7B2%7D%20-%202%20%5C,%20t%20y%5E%7B2%7D%20+%202%20%5Chspace%7B2em%7D%20x(%202%20)=%201" alt="x'= -4 \, t x^{2} - 2 \, t y^{2} + 2 \hspace{2em} x( 2 )= 1" title="x'= -4 \, t x^{2} - 2 \, t y^{2} + 2 \hspace{2em} x( 2 )= 1" data-latex="x'= -4 \, t x^{2} - 2 \, t y^{2} + 2 \hspace{2em} x( 2 )= 1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20t%5E%7B2%7D%20y%20+%203%20%5C,%20x%20y%20+%201%20%5Chspace%7B2em%7D%20y(%202%20)=%201" alt="y'= t^{2} y + 3 \, x y + 1 \hspace{2em} y( 2 )= 1" title="y'= t^{2} y + 3 \, x y + 1 \hspace{2em} y( 2 )= 1" data-latex="y'= t^{2} y + 3 \, x y + 1 \hspace{2em} y( 2 )= 1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 2.1 )\approx 0.000" alt="x( 2.1 )\approx 0.000" title="x( 2.1 )\approx 0.000" data-latex="x( 2.1 )\approx 0.000"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 2.1 )\approx 1.80" alt="y( 2.1 )\approx 1.80" title="y( 2.1 )\approx 1.80" data-latex="y( 2.1 )\approx 1.80"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 2.2 )\approx -1.16" alt="x( 2.2 )\approx -1.16" title="x( 2.2 )\approx -1.16" data-latex="x( 2.2 )\approx -1.16"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 2.2 )\approx 2.70" alt="y( 2.2 )\approx 2.70" title="y( 2.2 )\approx 2.70" data-latex="y( 2.2 )\approx 2.70"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%202.1%20)%5Capprox%200.000" alt="x( 2.1 )\approx 0.000" title="x( 2.1 )\approx 0.000" data-latex="x( 2.1 )\approx 0.000"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%202.1%20)%5Capprox%201.80" alt="y( 2.1 )\approx 1.80" title="y( 2.1 )\approx 1.80" data-latex="y( 2.1 )\approx 1.80"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%202.2%20)%5Capprox%20-1.16" alt="x( 2.2 )\approx -1.16" title="x( 2.2 )\approx -1.16" data-latex="x( 2.2 )\approx -1.16"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%202.2%20)%5Capprox%202.70" alt="y( 2.2 )\approx 2.70" title="y( 2.2 )\approx 2.70" data-latex="y( 2.2 )\approx 2.70"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-6971" title="N2 | Euler's method for approximating IVP solutions | ver. 6971"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.2 )" alt="x( 1.2 )" title="x( 1.2 )" data-latex="x( 1.2 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.2 )" alt="y( 1.2 )" title="y( 1.2 )" data-latex="y( 1.2 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= t^{2} x^{2} + 4 \, x^{2} y^{2} + 2 \hspace{2em} x( 1 )= 1" alt="x'= t^{2} x^{2} + 4 \, x^{2} y^{2} + 2 \hspace{2em} x( 1 )= 1" title="x'= t^{2} x^{2} + 4 \, x^{2} y^{2} + 2 \hspace{2em} x( 1 )= 1" data-latex="x'= t^{2} x^{2} + 4 \, x^{2} y^{2} + 2 \hspace{2em} x( 1 )= 1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -4 \, t^{2} x - 3 \, t^{2} y - 1 \hspace{2em} y( 1 )= 0" alt="y'= -4 \, t^{2} x - 3 \, t^{2} y - 1 \hspace{2em} y( 1 )= 0" title="y'= -4 \, t^{2} x - 3 \, t^{2} y - 1 \hspace{2em} y( 1 )= 0" data-latex="y'= -4 \, t^{2} x - 3 \, t^{2} y - 1 \hspace{2em} y( 1 )= 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.2%20)" alt="x( 1.2 )" title="x( 1.2 )" data-latex="x( 1.2 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.2%20)" alt="y( 1.2 )" title="y( 1.2 )" data-latex="y( 1.2 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20t%5E%7B2%7D%20x%5E%7B2%7D%20+%204%20%5C,%20x%5E%7B2%7D%20y%5E%7B2%7D%20+%202%20%5Chspace%7B2em%7D%20x(%201%20)=%201" alt="x'= t^{2} x^{2} + 4 \, x^{2} y^{2} + 2 \hspace{2em} x( 1 )= 1" title="x'= t^{2} x^{2} + 4 \, x^{2} y^{2} + 2 \hspace{2em} x( 1 )= 1" data-latex="x'= t^{2} x^{2} + 4 \, x^{2} y^{2} + 2 \hspace{2em} x( 1 )= 1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-4%20%5C,%20t%5E%7B2%7D%20x%20-%203%20%5C,%20t%5E%7B2%7D%20y%20-%201%20%5Chspace%7B2em%7D%20y(%201%20)=%200" alt="y'= -4 \, t^{2} x - 3 \, t^{2} y - 1 \hspace{2em} y( 1 )= 0" title="y'= -4 \, t^{2} x - 3 \, t^{2} y - 1 \hspace{2em} y( 1 )= 0" data-latex="y'= -4 \, t^{2} x - 3 \, t^{2} y - 1 \hspace{2em} y( 1 )= 0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.1 )\approx 1.30" alt="x( 1.1 )\approx 1.30" title="x( 1.1 )\approx 1.30" data-latex="x( 1.1 )\approx 1.30"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.1 )\approx -0.500" alt="y( 1.1 )\approx -0.500" title="y( 1.1 )\approx -0.500" data-latex="y( 1.1 )\approx -0.500"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.2 )\approx 1.88" alt="x( 1.2 )\approx 1.88" title="x( 1.2 )\approx 1.88" data-latex="x( 1.2 )\approx 1.88"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.2 )\approx -1.05" alt="y( 1.2 )\approx -1.05" title="y( 1.2 )\approx -1.05" data-latex="y( 1.2 )\approx -1.05"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.1%20)%5Capprox%201.30" alt="x( 1.1 )\approx 1.30" title="x( 1.1 )\approx 1.30" data-latex="x( 1.1 )\approx 1.30"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.1%20)%5Capprox%20-0.500" alt="y( 1.1 )\approx -0.500" title="y( 1.1 )\approx -0.500" data-latex="y( 1.1 )\approx -0.500"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.2%20)%5Capprox%201.88" alt="x( 1.2 )\approx 1.88" title="x( 1.2 )\approx 1.88" data-latex="x( 1.2 )\approx 1.88"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.2%20)%5Capprox%20-1.05" alt="y( 1.2 )\approx -1.05" title="y( 1.2 )\approx -1.05" data-latex="y( 1.2 )\approx -1.05"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-4458" title="N2 | Euler's method for approximating IVP solutions | ver. 4458"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.80 )" alt="x( -0.80 )" title="x( -0.80 )" data-latex="x( -0.80 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.80 )" alt="y( -0.80 )" title="y( -0.80 )" data-latex="y( -0.80 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 4 \, x^{2} y^{2} - 3 \, t y - 1 \hspace{2em} x( -1 )= -2" alt="x'= 4 \, x^{2} y^{2} - 3 \, t y - 1 \hspace{2em} x( -1 )= -2" title="x'= 4 \, x^{2} y^{2} - 3 \, t y - 1 \hspace{2em} x( -1 )= -2" data-latex="x'= 4 \, x^{2} y^{2} - 3 \, t y - 1 \hspace{2em} x( -1 )= -2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -t x^{2} + 2 \, t y^{2} - 2 \hspace{2em} y( -1 )= 0" alt="y'= -t x^{2} + 2 \, t y^{2} - 2 \hspace{2em} y( -1 )= 0" title="y'= -t x^{2} + 2 \, t y^{2} - 2 \hspace{2em} y( -1 )= 0" data-latex="y'= -t x^{2} + 2 \, t y^{2} - 2 \hspace{2em} y( -1 )= 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.80%20)" alt="x( -0.80 )" title="x( -0.80 )" data-latex="x( -0.80 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.80%20)" alt="y( -0.80 )" title="y( -0.80 )" data-latex="y( -0.80 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%204%20%5C,%20x%5E%7B2%7D%20y%5E%7B2%7D%20-%203%20%5C,%20t%20y%20-%201%20%5Chspace%7B2em%7D%20x(%20-1%20)=%20-2" alt="x'= 4 \, x^{2} y^{2} - 3 \, t y - 1 \hspace{2em} x( -1 )= -2" title="x'= 4 \, x^{2} y^{2} - 3 \, t y - 1 \hspace{2em} x( -1 )= -2" data-latex="x'= 4 \, x^{2} y^{2} - 3 \, t y - 1 \hspace{2em} x( -1 )= -2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-t%20x%5E%7B2%7D%20+%202%20%5C,%20t%20y%5E%7B2%7D%20-%202%20%5Chspace%7B2em%7D%20y(%20-1%20)=%200" alt="y'= -t x^{2} + 2 \, t y^{2} - 2 \hspace{2em} y( -1 )= 0" title="y'= -t x^{2} + 2 \, t y^{2} - 2 \hspace{2em} y( -1 )= 0" data-latex="y'= -t x^{2} + 2 \, t y^{2} - 2 \hspace{2em} y( -1 )= 0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.90 )\approx -2.10" alt="x( -0.90 )\approx -2.10" title="x( -0.90 )\approx -2.10" data-latex="x( -0.90 )\approx -2.10"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.90 )\approx 0.200" alt="y( -0.90 )\approx 0.200" title="y( -0.90 )\approx 0.200" data-latex="y( -0.90 )\approx 0.200"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.80 )\approx -2.08" alt="x( -0.80 )\approx -2.08" title="x( -0.80 )\approx -2.08" data-latex="x( -0.80 )\approx -2.08"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.80 )\approx 0.390" alt="y( -0.80 )\approx 0.390" title="y( -0.80 )\approx 0.390" data-latex="y( -0.80 )\approx 0.390"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.90%20)%5Capprox%20-2.10" alt="x( -0.90 )\approx -2.10" title="x( -0.90 )\approx -2.10" data-latex="x( -0.90 )\approx -2.10"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.90%20)%5Capprox%200.200" alt="y( -0.90 )\approx 0.200" title="y( -0.90 )\approx 0.200" data-latex="y( -0.90 )\approx 0.200"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.80%20)%5Capprox%20-2.08" alt="x( -0.80 )\approx -2.08" title="x( -0.80 )\approx -2.08" data-latex="x( -0.80 )\approx -2.08"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.80%20)%5Capprox%200.390" alt="y( -0.80 )\approx 0.390" title="y( -0.80 )\approx 0.390" data-latex="y( -0.80 )\approx 0.390"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-6606" title="N2 | Euler's method for approximating IVP solutions | ver. 6606"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 2.2 )" alt="x( 2.2 )" title="x( 2.2 )" data-latex="x( 2.2 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 2.2 )" alt="y( 2.2 )" title="y( 2.2 )" data-latex="y( 2.2 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -4 \, t x^{2} + 4 \, t y^{2} - 2 \hspace{2em} x( 2 )= -1" alt="x'= -4 \, t x^{2} + 4 \, t y^{2} - 2 \hspace{2em} x( 2 )= -1" title="x'= -4 \, t x^{2} + 4 \, t y^{2} - 2 \hspace{2em} x( 2 )= -1" data-latex="x'= -4 \, t x^{2} + 4 \, t y^{2} - 2 \hspace{2em} x( 2 )= -1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 4 \, t^{2} y^{2} + 4 \, x y - 2 \hspace{2em} y( 2 )= -1" alt="y'= 4 \, t^{2} y^{2} + 4 \, x y - 2 \hspace{2em} y( 2 )= -1" title="y'= 4 \, t^{2} y^{2} + 4 \, x y - 2 \hspace{2em} y( 2 )= -1" data-latex="y'= 4 \, t^{2} y^{2} + 4 \, x y - 2 \hspace{2em} y( 2 )= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%202.2%20)" alt="x( 2.2 )" title="x( 2.2 )" data-latex="x( 2.2 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%202.2%20)" alt="y( 2.2 )" title="y( 2.2 )" data-latex="y( 2.2 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-4%20%5C,%20t%20x%5E%7B2%7D%20+%204%20%5C,%20t%20y%5E%7B2%7D%20-%202%20%5Chspace%7B2em%7D%20x(%202%20)=%20-1" alt="x'= -4 \, t x^{2} + 4 \, t y^{2} - 2 \hspace{2em} x( 2 )= -1" title="x'= -4 \, t x^{2} + 4 \, t y^{2} - 2 \hspace{2em} x( 2 )= -1" data-latex="x'= -4 \, t x^{2} + 4 \, t y^{2} - 2 \hspace{2em} x( 2 )= -1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%204%20%5C,%20t%5E%7B2%7D%20y%5E%7B2%7D%20+%204%20%5C,%20x%20y%20-%202%20%5Chspace%7B2em%7D%20y(%202%20)=%20-1" alt="y'= 4 \, t^{2} y^{2} + 4 \, x y - 2 \hspace{2em} y( 2 )= -1" title="y'= 4 \, t^{2} y^{2} + 4 \, x y - 2 \hspace{2em} y( 2 )= -1" data-latex="y'= 4 \, t^{2} y^{2} + 4 \, x y - 2 \hspace{2em} y( 2 )= -1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 2.1 )\approx -1.20" alt="x( 2.1 )\approx -1.20" title="x( 2.1 )\approx -1.20" data-latex="x( 2.1 )\approx -1.20"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 2.1 )\approx 0.799" alt="y( 2.1 )\approx 0.799" title="y( 2.1 )\approx 0.799" data-latex="y( 2.1 )\approx 0.799"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 2.2 )\approx -2.07" alt="x( 2.2 )\approx -2.07" title="x( 2.2 )\approx -2.07" data-latex="x( 2.2 )\approx -2.07"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 2.2 )\approx 1.34" alt="y( 2.2 )\approx 1.34" title="y( 2.2 )\approx 1.34" data-latex="y( 2.2 )\approx 1.34"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%202.1%20)%5Capprox%20-1.20" alt="x( 2.1 )\approx -1.20" title="x( 2.1 )\approx -1.20" data-latex="x( 2.1 )\approx -1.20"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%202.1%20)%5Capprox%200.799" alt="y( 2.1 )\approx 0.799" title="y( 2.1 )\approx 0.799" data-latex="y( 2.1 )\approx 0.799"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%202.2%20)%5Capprox%20-2.07" alt="x( 2.2 )\approx -2.07" title="x( 2.2 )\approx -2.07" data-latex="x( 2.2 )\approx -2.07"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%202.2%20)%5Capprox%201.34" alt="y( 2.2 )\approx 1.34" title="y( 2.2 )\approx 1.34" data-latex="y( 2.2 )\approx 1.34"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-2769" title="N2 | Euler's method for approximating IVP solutions | ver. 2769"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.80 )" alt="x( -0.80 )" title="x( -0.80 )" data-latex="x( -0.80 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.80 )" alt="y( -0.80 )" title="y( -0.80 )" data-latex="y( -0.80 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 4 \, t^{2} y - 2 \, x y - 1 \hspace{2em} x( -1 )= 0" alt="x'= 4 \, t^{2} y - 2 \, x y - 1 \hspace{2em} x( -1 )= 0" title="x'= 4 \, t^{2} y - 2 \, x y - 1 \hspace{2em} x( -1 )= 0" data-latex="x'= 4 \, t^{2} y - 2 \, x y - 1 \hspace{2em} x( -1 )= 0"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -t^{2} x^{2} + t y + 3 \hspace{2em} y( -1 )= -1" alt="y'= -t^{2} x^{2} + t y + 3 \hspace{2em} y( -1 )= -1" title="y'= -t^{2} x^{2} + t y + 3 \hspace{2em} y( -1 )= -1" data-latex="y'= -t^{2} x^{2} + t y + 3 \hspace{2em} y( -1 )= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.80%20)" alt="x( -0.80 )" title="x( -0.80 )" data-latex="x( -0.80 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.80%20)" alt="y( -0.80 )" title="y( -0.80 )" data-latex="y( -0.80 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%204%20%5C,%20t%5E%7B2%7D%20y%20-%202%20%5C,%20x%20y%20-%201%20%5Chspace%7B2em%7D%20x(%20-1%20)=%200" alt="x'= 4 \, t^{2} y - 2 \, x y - 1 \hspace{2em} x( -1 )= 0" title="x'= 4 \, t^{2} y - 2 \, x y - 1 \hspace{2em} x( -1 )= 0" data-latex="x'= 4 \, t^{2} y - 2 \, x y - 1 \hspace{2em} x( -1 )= 0"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-t%5E%7B2%7D%20x%5E%7B2%7D%20+%20t%20y%20+%203%20%5Chspace%7B2em%7D%20y(%20-1%20)=%20-1" alt="y'= -t^{2} x^{2} + t y + 3 \hspace{2em} y( -1 )= -1" title="y'= -t^{2} x^{2} + t y + 3 \hspace{2em} y( -1 )= -1" data-latex="y'= -t^{2} x^{2} + t y + 3 \hspace{2em} y( -1 )= -1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.90 )\approx -0.500" alt="x( -0.90 )\approx -0.500" title="x( -0.90 )\approx -0.500" data-latex="x( -0.90 )\approx -0.500"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.90 )\approx -0.600" alt="y( -0.90 )\approx -0.600" title="y( -0.90 )\approx -0.600" data-latex="y( -0.90 )\approx -0.600"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.80 )\approx -0.854" alt="x( -0.80 )\approx -0.854" title="x( -0.80 )\approx -0.854" data-latex="x( -0.80 )\approx -0.854"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.80 )\approx -0.266" alt="y( -0.80 )\approx -0.266" title="y( -0.80 )\approx -0.266" data-latex="y( -0.80 )\approx -0.266"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.90%20)%5Capprox%20-0.500" alt="x( -0.90 )\approx -0.500" title="x( -0.90 )\approx -0.500" data-latex="x( -0.90 )\approx -0.500"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.90%20)%5Capprox%20-0.600" alt="y( -0.90 )\approx -0.600" title="y( -0.90 )\approx -0.600" data-latex="y( -0.90 )\approx -0.600"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.80%20)%5Capprox%20-0.854" alt="x( -0.80 )\approx -0.854" title="x( -0.80 )\approx -0.854" data-latex="x( -0.80 )\approx -0.854"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.80%20)%5Capprox%20-0.266" alt="y( -0.80 )\approx -0.266" title="y( -0.80 )\approx -0.266" data-latex="y( -0.80 )\approx -0.266"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-6509" title="N2 | Euler's method for approximating IVP solutions | ver. 6509"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.20 )" alt="x( 0.20 )" title="x( 0.20 )" data-latex="x( 0.20 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.20 )" alt="y( 0.20 )" title="y( 0.20 )" data-latex="y( 0.20 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -2 \, t y^{2} + 4 \, t x \hspace{2em} x( 0 )= 1" alt="x'= -2 \, t y^{2} + 4 \, t x \hspace{2em} x( 0 )= 1" title="x'= -2 \, t y^{2} + 4 \, t x \hspace{2em} x( 0 )= 1" data-latex="x'= -2 \, t y^{2} + 4 \, t x \hspace{2em} x( 0 )= 1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -2 \, t^{2} y^{2} + 4 \, x^{2} y - 1 \hspace{2em} y( 0 )= 2" alt="y'= -2 \, t^{2} y^{2} + 4 \, x^{2} y - 1 \hspace{2em} y( 0 )= 2" title="y'= -2 \, t^{2} y^{2} + 4 \, x^{2} y - 1 \hspace{2em} y( 0 )= 2" data-latex="y'= -2 \, t^{2} y^{2} + 4 \, x^{2} y - 1 \hspace{2em} y( 0 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.20%20)" alt="x( 0.20 )" title="x( 0.20 )" data-latex="x( 0.20 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.20%20)" alt="y( 0.20 )" title="y( 0.20 )" data-latex="y( 0.20 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-2%20%5C,%20t%20y%5E%7B2%7D%20+%204%20%5C,%20t%20x%20%5Chspace%7B2em%7D%20x(%200%20)=%201" alt="x'= -2 \, t y^{2} + 4 \, t x \hspace{2em} x( 0 )= 1" title="x'= -2 \, t y^{2} + 4 \, t x \hspace{2em} x( 0 )= 1" data-latex="x'= -2 \, t y^{2} + 4 \, t x \hspace{2em} x( 0 )= 1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-2%20%5C,%20t%5E%7B2%7D%20y%5E%7B2%7D%20+%204%20%5C,%20x%5E%7B2%7D%20y%20-%201%20%5Chspace%7B2em%7D%20y(%200%20)=%202" alt="y'= -2 \, t^{2} y^{2} + 4 \, x^{2} y - 1 \hspace{2em} y( 0 )= 2" title="y'= -2 \, t^{2} y^{2} + 4 \, x^{2} y - 1 \hspace{2em} y( 0 )= 2" data-latex="y'= -2 \, t^{2} y^{2} + 4 \, x^{2} y - 1 \hspace{2em} y( 0 )= 2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.10 )\approx 1.00" alt="x( 0.10 )\approx 1.00" title="x( 0.10 )\approx 1.00" data-latex="x( 0.10 )\approx 1.00"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.10 )\approx 2.70" alt="y( 0.10 )\approx 2.70" title="y( 0.10 )\approx 2.70" data-latex="y( 0.10 )\approx 2.70"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.20 )\approx 0.895" alt="x( 0.20 )\approx 0.895" title="x( 0.20 )\approx 0.895" data-latex="x( 0.20 )\approx 0.895"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.20 )\approx 3.66" alt="y( 0.20 )\approx 3.66" title="y( 0.20 )\approx 3.66" data-latex="y( 0.20 )\approx 3.66"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.10%20)%5Capprox%201.00" alt="x( 0.10 )\approx 1.00" title="x( 0.10 )\approx 1.00" data-latex="x( 0.10 )\approx 1.00"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.10%20)%5Capprox%202.70" alt="y( 0.10 )\approx 2.70" title="y( 0.10 )\approx 2.70" data-latex="y( 0.10 )\approx 2.70"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.20%20)%5Capprox%200.895" alt="x( 0.20 )\approx 0.895" title="x( 0.20 )\approx 0.895" data-latex="x( 0.20 )\approx 0.895"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.20%20)%5Capprox%203.66" alt="y( 0.20 )\approx 3.66" title="y( 0.20 )\approx 3.66" data-latex="y( 0.20 )\approx 3.66"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-4141" title="N2 | Euler's method for approximating IVP solutions | ver. 4141"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.20 )" alt="x( 0.20 )" title="x( 0.20 )" data-latex="x( 0.20 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.20 )" alt="y( 0.20 )" title="y( 0.20 )" data-latex="y( 0.20 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -2 \, t^{2} x^{2} - 4 \, x^{2} y + 2 \hspace{2em} x( 0 )= -1" alt="x'= -2 \, t^{2} x^{2} - 4 \, x^{2} y + 2 \hspace{2em} x( 0 )= -1" title="x'= -2 \, t^{2} x^{2} - 4 \, x^{2} y + 2 \hspace{2em} x( 0 )= -1" data-latex="x'= -2 \, t^{2} x^{2} - 4 \, x^{2} y + 2 \hspace{2em} x( 0 )= -1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -4 \, x y^{2} + t x + 2 \hspace{2em} y( 0 )= 1" alt="y'= -4 \, x y^{2} + t x + 2 \hspace{2em} y( 0 )= 1" title="y'= -4 \, x y^{2} + t x + 2 \hspace{2em} y( 0 )= 1" data-latex="y'= -4 \, x y^{2} + t x + 2 \hspace{2em} y( 0 )= 1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.20%20)" alt="x( 0.20 )" title="x( 0.20 )" data-latex="x( 0.20 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.20%20)" alt="y( 0.20 )" title="y( 0.20 )" data-latex="y( 0.20 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-2%20%5C,%20t%5E%7B2%7D%20x%5E%7B2%7D%20-%204%20%5C,%20x%5E%7B2%7D%20y%20+%202%20%5Chspace%7B2em%7D%20x(%200%20)=%20-1" alt="x'= -2 \, t^{2} x^{2} - 4 \, x^{2} y + 2 \hspace{2em} x( 0 )= -1" title="x'= -2 \, t^{2} x^{2} - 4 \, x^{2} y + 2 \hspace{2em} x( 0 )= -1" data-latex="x'= -2 \, t^{2} x^{2} - 4 \, x^{2} y + 2 \hspace{2em} x( 0 )= -1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-4%20%5C,%20x%20y%5E%7B2%7D%20+%20t%20x%20+%202%20%5Chspace%7B2em%7D%20y(%200%20)=%201" alt="y'= -4 \, x y^{2} + t x + 2 \hspace{2em} y( 0 )= 1" title="y'= -4 \, x y^{2} + t x + 2 \hspace{2em} y( 0 )= 1" data-latex="y'= -4 \, x y^{2} + t x + 2 \hspace{2em} y( 0 )= 1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.10 )\approx -1.20" alt="x( 0.10 )\approx -1.20" title="x( 0.10 )\approx -1.20" data-latex="x( 0.10 )\approx -1.20"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.10 )\approx 1.60" alt="y( 0.10 )\approx 1.60" title="y( 0.10 )\approx 1.60" data-latex="y( 0.10 )\approx 1.60"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.20 )\approx -1.92" alt="x( 0.20 )\approx -1.92" title="x( 0.20 )\approx -1.92" data-latex="x( 0.20 )\approx -1.92"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.20 )\approx 3.02" alt="y( 0.20 )\approx 3.02" title="y( 0.20 )\approx 3.02" data-latex="y( 0.20 )\approx 3.02"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.10%20)%5Capprox%20-1.20" alt="x( 0.10 )\approx -1.20" title="x( 0.10 )\approx -1.20" data-latex="x( 0.10 )\approx -1.20"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.10%20)%5Capprox%201.60" alt="y( 0.10 )\approx 1.60" title="y( 0.10 )\approx 1.60" data-latex="y( 0.10 )\approx 1.60"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.20%20)%5Capprox%20-1.92" alt="x( 0.20 )\approx -1.92" title="x( 0.20 )\approx -1.92" data-latex="x( 0.20 )\approx -1.92"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.20%20)%5Capprox%203.02" alt="y( 0.20 )\approx 3.02" title="y( 0.20 )\approx 3.02" data-latex="y( 0.20 )\approx 3.02"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-7557" title="N2 | Euler's method for approximating IVP solutions | ver. 7557"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 2.2 )" alt="x( 2.2 )" title="x( 2.2 )" data-latex="x( 2.2 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 2.2 )" alt="y( 2.2 )" title="y( 2.2 )" data-latex="y( 2.2 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 3 \, t y^{2} - t x + 1 \hspace{2em} x( 2 )= -1" alt="x'= 3 \, t y^{2} - t x + 1 \hspace{2em} x( 2 )= -1" title="x'= 3 \, t y^{2} - t x + 1 \hspace{2em} x( 2 )= -1" data-latex="x'= 3 \, t y^{2} - t x + 1 \hspace{2em} x( 2 )= -1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 4 \, t x - t y + 1 \hspace{2em} y( 2 )= 0" alt="y'= 4 \, t x - t y + 1 \hspace{2em} y( 2 )= 0" title="y'= 4 \, t x - t y + 1 \hspace{2em} y( 2 )= 0" data-latex="y'= 4 \, t x - t y + 1 \hspace{2em} y( 2 )= 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%202.2%20)" alt="x( 2.2 )" title="x( 2.2 )" data-latex="x( 2.2 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%202.2%20)" alt="y( 2.2 )" title="y( 2.2 )" data-latex="y( 2.2 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%203%20%5C,%20t%20y%5E%7B2%7D%20-%20t%20x%20+%201%20%5Chspace%7B2em%7D%20x(%202%20)=%20-1" alt="x'= 3 \, t y^{2} - t x + 1 \hspace{2em} x( 2 )= -1" title="x'= 3 \, t y^{2} - t x + 1 \hspace{2em} x( 2 )= -1" data-latex="x'= 3 \, t y^{2} - t x + 1 \hspace{2em} x( 2 )= -1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%204%20%5C,%20t%20x%20-%20t%20y%20+%201%20%5Chspace%7B2em%7D%20y(%202%20)=%200" alt="y'= 4 \, t x - t y + 1 \hspace{2em} y( 2 )= 0" title="y'= 4 \, t x - t y + 1 \hspace{2em} y( 2 )= 0" data-latex="y'= 4 \, t x - t y + 1 \hspace{2em} y( 2 )= 0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 2.1 )\approx -0.700" alt="x( 2.1 )\approx -0.700" title="x( 2.1 )\approx -0.700" data-latex="x( 2.1 )\approx -0.700"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 2.1 )\approx -0.700" alt="y( 2.1 )\approx -0.700" title="y( 2.1 )\approx -0.700" data-latex="y( 2.1 )\approx -0.700"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 2.2 )\approx -0.145" alt="x( 2.2 )\approx -0.145" title="x( 2.2 )\approx -0.145" data-latex="x( 2.2 )\approx -0.145"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 2.2 )\approx -1.04" alt="y( 2.2 )\approx -1.04" title="y( 2.2 )\approx -1.04" data-latex="y( 2.2 )\approx -1.04"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%202.1%20)%5Capprox%20-0.700" alt="x( 2.1 )\approx -0.700" title="x( 2.1 )\approx -0.700" data-latex="x( 2.1 )\approx -0.700"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%202.1%20)%5Capprox%20-0.700" alt="y( 2.1 )\approx -0.700" title="y( 2.1 )\approx -0.700" data-latex="y( 2.1 )\approx -0.700"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%202.2%20)%5Capprox%20-0.145" alt="x( 2.2 )\approx -0.145" title="x( 2.2 )\approx -0.145" data-latex="x( 2.2 )\approx -0.145"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%202.2%20)%5Capprox%20-1.04" alt="y( 2.2 )\approx -1.04" title="y( 2.2 )\approx -1.04" data-latex="y( 2.2 )\approx -1.04"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-3264" title="N2 | Euler's method for approximating IVP solutions | ver. 3264"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.2 )" alt="x( 1.2 )" title="x( 1.2 )" data-latex="x( 1.2 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.2 )" alt="y( 1.2 )" title="y( 1.2 )" data-latex="y( 1.2 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 4 \, t^{2} y^{2} + 4 \, t x - 2 \hspace{2em} x( 1 )= -2" alt="x'= 4 \, t^{2} y^{2} + 4 \, t x - 2 \hspace{2em} x( 1 )= -2" title="x'= 4 \, t^{2} y^{2} + 4 \, t x - 2 \hspace{2em} x( 1 )= -2" data-latex="x'= 4 \, t^{2} y^{2} + 4 \, t x - 2 \hspace{2em} x( 1 )= -2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 4 \, t^{2} x^{2} - 4 \, x^{2} y^{2} - 3 \hspace{2em} y( 1 )= -2" alt="y'= 4 \, t^{2} x^{2} - 4 \, x^{2} y^{2} - 3 \hspace{2em} y( 1 )= -2" title="y'= 4 \, t^{2} x^{2} - 4 \, x^{2} y^{2} - 3 \hspace{2em} y( 1 )= -2" data-latex="y'= 4 \, t^{2} x^{2} - 4 \, x^{2} y^{2} - 3 \hspace{2em} y( 1 )= -2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.2%20)" alt="x( 1.2 )" title="x( 1.2 )" data-latex="x( 1.2 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.2%20)" alt="y( 1.2 )" title="y( 1.2 )" data-latex="y( 1.2 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%204%20%5C,%20t%5E%7B2%7D%20y%5E%7B2%7D%20+%204%20%5C,%20t%20x%20-%202%20%5Chspace%7B2em%7D%20x(%201%20)=%20-2" alt="x'= 4 \, t^{2} y^{2} + 4 \, t x - 2 \hspace{2em} x( 1 )= -2" title="x'= 4 \, t^{2} y^{2} + 4 \, t x - 2 \hspace{2em} x( 1 )= -2" data-latex="x'= 4 \, t^{2} y^{2} + 4 \, t x - 2 \hspace{2em} x( 1 )= -2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%204%20%5C,%20t%5E%7B2%7D%20x%5E%7B2%7D%20-%204%20%5C,%20x%5E%7B2%7D%20y%5E%7B2%7D%20-%203%20%5Chspace%7B2em%7D%20y(%201%20)=%20-2" alt="y'= 4 \, t^{2} x^{2} - 4 \, x^{2} y^{2} - 3 \hspace{2em} y( 1 )= -2" title="y'= 4 \, t^{2} x^{2} - 4 \, x^{2} y^{2} - 3 \hspace{2em} y( 1 )= -2" data-latex="y'= 4 \, t^{2} x^{2} - 4 \, x^{2} y^{2} - 3 \hspace{2em} y( 1 )= -2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.1 )\approx -1.40" alt="x( 1.1 )\approx -1.40" title="x( 1.1 )\approx -1.40" data-latex="x( 1.1 )\approx -1.40"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.1 )\approx -7.10" alt="y( 1.1 )\approx -7.10" title="y( 1.1 )\approx -7.10" data-latex="y( 1.1 )\approx -7.10"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.2 )\approx 22.2" alt="x( 1.2 )\approx 22.2" title="x( 1.2 )\approx 22.2" data-latex="x( 1.2 )\approx 22.2"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.2 )\approx -46.0" alt="y( 1.2 )\approx -46.0" title="y( 1.2 )\approx -46.0" data-latex="y( 1.2 )\approx -46.0"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.1%20)%5Capprox%20-1.40" alt="x( 1.1 )\approx -1.40" title="x( 1.1 )\approx -1.40" data-latex="x( 1.1 )\approx -1.40"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.1%20)%5Capprox%20-7.10" alt="y( 1.1 )\approx -7.10" title="y( 1.1 )\approx -7.10" data-latex="y( 1.1 )\approx -7.10"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.2%20)%5Capprox%2022.2" alt="x( 1.2 )\approx 22.2" title="x( 1.2 )\approx 22.2" data-latex="x( 1.2 )\approx 22.2"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.2%20)%5Capprox%20-46.0" alt="y( 1.2 )\approx -46.0" title="y( 1.2 )\approx -46.0" data-latex="y( 1.2 )\approx -46.0"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-9628" title="N2 | Euler's method for approximating IVP solutions | ver. 9628"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.8 )" alt="x( -1.8 )" title="x( -1.8 )" data-latex="x( -1.8 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.8 )" alt="y( -1.8 )" title="y( -1.8 )" data-latex="y( -1.8 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -2 \, t x^{2} + t^{2} y + 1 \hspace{2em} x( -2 )= -1" alt="x'= -2 \, t x^{2} + t^{2} y + 1 \hspace{2em} x( -2 )= -1" title="x'= -2 \, t x^{2} + t^{2} y + 1 \hspace{2em} x( -2 )= -1" data-latex="x'= -2 \, t x^{2} + t^{2} y + 1 \hspace{2em} x( -2 )= -1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -4 \, t^{2} x^{2} + 4 \, x^{2} y \hspace{2em} y( -2 )= 2" alt="y'= -4 \, t^{2} x^{2} + 4 \, x^{2} y \hspace{2em} y( -2 )= 2" title="y'= -4 \, t^{2} x^{2} + 4 \, x^{2} y \hspace{2em} y( -2 )= 2" data-latex="y'= -4 \, t^{2} x^{2} + 4 \, x^{2} y \hspace{2em} y( -2 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.8%20)" alt="x( -1.8 )" title="x( -1.8 )" data-latex="x( -1.8 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.8%20)" alt="y( -1.8 )" title="y( -1.8 )" data-latex="y( -1.8 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-2%20%5C,%20t%20x%5E%7B2%7D%20+%20t%5E%7B2%7D%20y%20+%201%20%5Chspace%7B2em%7D%20x(%20-2%20)=%20-1" alt="x'= -2 \, t x^{2} + t^{2} y + 1 \hspace{2em} x( -2 )= -1" title="x'= -2 \, t x^{2} + t^{2} y + 1 \hspace{2em} x( -2 )= -1" data-latex="x'= -2 \, t x^{2} + t^{2} y + 1 \hspace{2em} x( -2 )= -1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-4%20%5C,%20t%5E%7B2%7D%20x%5E%7B2%7D%20+%204%20%5C,%20x%5E%7B2%7D%20y%20%5Chspace%7B2em%7D%20y(%20-2%20)=%202" alt="y'= -4 \, t^{2} x^{2} + 4 \, x^{2} y \hspace{2em} y( -2 )= 2" title="y'= -4 \, t^{2} x^{2} + 4 \, x^{2} y \hspace{2em} y( -2 )= 2" data-latex="y'= -4 \, t^{2} x^{2} + 4 \, x^{2} y \hspace{2em} y( -2 )= 2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.9 )\approx 0.299" alt="x( -1.9 )\approx 0.299" title="x( -1.9 )\approx 0.299" data-latex="x( -1.9 )\approx 0.299"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.9 )\approx 1.20" alt="y( -1.9 )\approx 1.20" title="y( -1.9 )\approx 1.20" data-latex="y( -1.9 )\approx 1.20"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.8 )\approx 0.866" alt="x( -1.8 )\approx 0.866" title="x( -1.8 )\approx 0.866" data-latex="x( -1.8 )\approx 0.866"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.8 )\approx 1.11" alt="y( -1.8 )\approx 1.11" title="y( -1.8 )\approx 1.11" data-latex="y( -1.8 )\approx 1.11"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.9%20)%5Capprox%200.299" alt="x( -1.9 )\approx 0.299" title="x( -1.9 )\approx 0.299" data-latex="x( -1.9 )\approx 0.299"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.9%20)%5Capprox%201.20" alt="y( -1.9 )\approx 1.20" title="y( -1.9 )\approx 1.20" data-latex="y( -1.9 )\approx 1.20"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.8%20)%5Capprox%200.866" alt="x( -1.8 )\approx 0.866" title="x( -1.8 )\approx 0.866" data-latex="x( -1.8 )\approx 0.866"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.8%20)%5Capprox%201.11" alt="y( -1.8 )\approx 1.11" title="y( -1.8 )\approx 1.11" data-latex="y( -1.8 )\approx 1.11"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-9040" title="N2 | Euler's method for approximating IVP solutions | ver. 9040"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.2 )" alt="x( 1.2 )" title="x( 1.2 )" data-latex="x( 1.2 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.2 )" alt="y( 1.2 )" title="y( 1.2 )" data-latex="y( 1.2 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 4 \, x^{2} y^{2} - t x^{2} - 1 \hspace{2em} x( 1 )= 0" alt="x'= 4 \, x^{2} y^{2} - t x^{2} - 1 \hspace{2em} x( 1 )= 0" title="x'= 4 \, x^{2} y^{2} - t x^{2} - 1 \hspace{2em} x( 1 )= 0" data-latex="x'= 4 \, x^{2} y^{2} - t x^{2} - 1 \hspace{2em} x( 1 )= 0"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= t^{2} x^{2} - 3 \, t^{2} y + 2 \hspace{2em} y( 1 )= -2" alt="y'= t^{2} x^{2} - 3 \, t^{2} y + 2 \hspace{2em} y( 1 )= -2" title="y'= t^{2} x^{2} - 3 \, t^{2} y + 2 \hspace{2em} y( 1 )= -2" data-latex="y'= t^{2} x^{2} - 3 \, t^{2} y + 2 \hspace{2em} y( 1 )= -2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.2%20)" alt="x( 1.2 )" title="x( 1.2 )" data-latex="x( 1.2 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.2%20)" alt="y( 1.2 )" title="y( 1.2 )" data-latex="y( 1.2 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%204%20%5C,%20x%5E%7B2%7D%20y%5E%7B2%7D%20-%20t%20x%5E%7B2%7D%20-%201%20%5Chspace%7B2em%7D%20x(%201%20)=%200" alt="x'= 4 \, x^{2} y^{2} - t x^{2} - 1 \hspace{2em} x( 1 )= 0" title="x'= 4 \, x^{2} y^{2} - t x^{2} - 1 \hspace{2em} x( 1 )= 0" data-latex="x'= 4 \, x^{2} y^{2} - t x^{2} - 1 \hspace{2em} x( 1 )= 0"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20t%5E%7B2%7D%20x%5E%7B2%7D%20-%203%20%5C,%20t%5E%7B2%7D%20y%20+%202%20%5Chspace%7B2em%7D%20y(%201%20)=%20-2" alt="y'= t^{2} x^{2} - 3 \, t^{2} y + 2 \hspace{2em} y( 1 )= -2" title="y'= t^{2} x^{2} - 3 \, t^{2} y + 2 \hspace{2em} y( 1 )= -2" data-latex="y'= t^{2} x^{2} - 3 \, t^{2} y + 2 \hspace{2em} y( 1 )= -2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.1 )\approx -0.100" alt="x( 1.1 )\approx -0.100" title="x( 1.1 )\approx -0.100" data-latex="x( 1.1 )\approx -0.100"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.1 )\approx -1.20" alt="y( 1.1 )\approx -1.20" title="y( 1.1 )\approx -1.20" data-latex="y( 1.1 )\approx -1.20"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.2 )\approx -0.195" alt="x( 1.2 )\approx -0.195" title="x( 1.2 )\approx -0.195" data-latex="x( 1.2 )\approx -0.195"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.2 )\approx -0.563" alt="y( 1.2 )\approx -0.563" title="y( 1.2 )\approx -0.563" data-latex="y( 1.2 )\approx -0.563"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.1%20)%5Capprox%20-0.100" alt="x( 1.1 )\approx -0.100" title="x( 1.1 )\approx -0.100" data-latex="x( 1.1 )\approx -0.100"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.1%20)%5Capprox%20-1.20" alt="y( 1.1 )\approx -1.20" title="y( 1.1 )\approx -1.20" data-latex="y( 1.1 )\approx -1.20"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.2%20)%5Capprox%20-0.195" alt="x( 1.2 )\approx -0.195" title="x( 1.2 )\approx -0.195" data-latex="x( 1.2 )\approx -0.195"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.2%20)%5Capprox%20-0.563" alt="y( 1.2 )\approx -0.563" title="y( 1.2 )\approx -0.563" data-latex="y( 1.2 )\approx -0.563"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-7499" title="N2 | Euler's method for approximating IVP solutions | ver. 7499"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.8 )" alt="x( -1.8 )" title="x( -1.8 )" data-latex="x( -1.8 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.8 )" alt="y( -1.8 )" title="y( -1.8 )" data-latex="y( -1.8 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 2 \, t^{2} y^{2} - 3 \, x y - 1 \hspace{2em} x( -2 )= 2" alt="x'= 2 \, t^{2} y^{2} - 3 \, x y - 1 \hspace{2em} x( -2 )= 2" title="x'= 2 \, t^{2} y^{2} - 3 \, x y - 1 \hspace{2em} x( -2 )= 2" data-latex="x'= 2 \, t^{2} y^{2} - 3 \, x y - 1 \hspace{2em} x( -2 )= 2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 4 \, t y^{2} + x y^{2} - 3 \hspace{2em} y( -2 )= 1" alt="y'= 4 \, t y^{2} + x y^{2} - 3 \hspace{2em} y( -2 )= 1" title="y'= 4 \, t y^{2} + x y^{2} - 3 \hspace{2em} y( -2 )= 1" data-latex="y'= 4 \, t y^{2} + x y^{2} - 3 \hspace{2em} y( -2 )= 1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.8%20)" alt="x( -1.8 )" title="x( -1.8 )" data-latex="x( -1.8 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.8%20)" alt="y( -1.8 )" title="y( -1.8 )" data-latex="y( -1.8 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%202%20%5C,%20t%5E%7B2%7D%20y%5E%7B2%7D%20-%203%20%5C,%20x%20y%20-%201%20%5Chspace%7B2em%7D%20x(%20-2%20)=%202" alt="x'= 2 \, t^{2} y^{2} - 3 \, x y - 1 \hspace{2em} x( -2 )= 2" title="x'= 2 \, t^{2} y^{2} - 3 \, x y - 1 \hspace{2em} x( -2 )= 2" data-latex="x'= 2 \, t^{2} y^{2} - 3 \, x y - 1 \hspace{2em} x( -2 )= 2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%204%20%5C,%20t%20y%5E%7B2%7D%20+%20x%20y%5E%7B2%7D%20-%203%20%5Chspace%7B2em%7D%20y(%20-2%20)=%201" alt="y'= 4 \, t y^{2} + x y^{2} - 3 \hspace{2em} y( -2 )= 1" title="y'= 4 \, t y^{2} + x y^{2} - 3 \hspace{2em} y( -2 )= 1" data-latex="y'= 4 \, t y^{2} + x y^{2} - 3 \hspace{2em} y( -2 )= 1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.9 )\approx 2.10" alt="x( -1.9 )\approx 2.10" title="x( -1.9 )\approx 2.10" data-latex="x( -1.9 )\approx 2.10"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.9 )\approx 0.101" alt="y( -1.9 )\approx 0.101" title="y( -1.9 )\approx 0.101" data-latex="y( -1.9 )\approx 0.101"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.8 )\approx 1.95" alt="x( -1.8 )\approx 1.95" title="x( -1.8 )\approx 1.95" data-latex="x( -1.8 )\approx 1.95"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.8 )\approx -0.205" alt="y( -1.8 )\approx -0.205" title="y( -1.8 )\approx -0.205" data-latex="y( -1.8 )\approx -0.205"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.9%20)%5Capprox%202.10" alt="x( -1.9 )\approx 2.10" title="x( -1.9 )\approx 2.10" data-latex="x( -1.9 )\approx 2.10"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.9%20)%5Capprox%200.101" alt="y( -1.9 )\approx 0.101" title="y( -1.9 )\approx 0.101" data-latex="y( -1.9 )\approx 0.101"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.8%20)%5Capprox%201.95" alt="x( -1.8 )\approx 1.95" title="x( -1.8 )\approx 1.95" data-latex="x( -1.8 )\approx 1.95"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.8%20)%5Capprox%20-0.205" alt="y( -1.8 )\approx -0.205" title="y( -1.8 )\approx -0.205" data-latex="y( -1.8 )\approx -0.205"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-4012" title="N2 | Euler's method for approximating IVP solutions | ver. 4012"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.2 )" alt="x( 1.2 )" title="x( 1.2 )" data-latex="x( 1.2 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.2 )" alt="y( 1.2 )" title="y( 1.2 )" data-latex="y( 1.2 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -2 \, x y^{2} - 3 \, t x - 2 \hspace{2em} x( 1 )= -2" alt="x'= -2 \, x y^{2} - 3 \, t x - 2 \hspace{2em} x( 1 )= -2" title="x'= -2 \, x y^{2} - 3 \, t x - 2 \hspace{2em} x( 1 )= -2" data-latex="x'= -2 \, x y^{2} - 3 \, t x - 2 \hspace{2em} x( 1 )= -2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 3 \, t^{2} x^{2} - 4 \, t^{2} y^{2} + 2 \hspace{2em} y( 1 )= 0" alt="y'= 3 \, t^{2} x^{2} - 4 \, t^{2} y^{2} + 2 \hspace{2em} y( 1 )= 0" title="y'= 3 \, t^{2} x^{2} - 4 \, t^{2} y^{2} + 2 \hspace{2em} y( 1 )= 0" data-latex="y'= 3 \, t^{2} x^{2} - 4 \, t^{2} y^{2} + 2 \hspace{2em} y( 1 )= 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.2%20)" alt="x( 1.2 )" title="x( 1.2 )" data-latex="x( 1.2 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.2%20)" alt="y( 1.2 )" title="y( 1.2 )" data-latex="y( 1.2 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-2%20%5C,%20x%20y%5E%7B2%7D%20-%203%20%5C,%20t%20x%20-%202%20%5Chspace%7B2em%7D%20x(%201%20)=%20-2" alt="x'= -2 \, x y^{2} - 3 \, t x - 2 \hspace{2em} x( 1 )= -2" title="x'= -2 \, x y^{2} - 3 \, t x - 2 \hspace{2em} x( 1 )= -2" data-latex="x'= -2 \, x y^{2} - 3 \, t x - 2 \hspace{2em} x( 1 )= -2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%203%20%5C,%20t%5E%7B2%7D%20x%5E%7B2%7D%20-%204%20%5C,%20t%5E%7B2%7D%20y%5E%7B2%7D%20+%202%20%5Chspace%7B2em%7D%20y(%201%20)=%200" alt="y'= 3 \, t^{2} x^{2} - 4 \, t^{2} y^{2} + 2 \hspace{2em} y( 1 )= 0" title="y'= 3 \, t^{2} x^{2} - 4 \, t^{2} y^{2} + 2 \hspace{2em} y( 1 )= 0" data-latex="y'= 3 \, t^{2} x^{2} - 4 \, t^{2} y^{2} + 2 \hspace{2em} y( 1 )= 0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.1 )\approx -1.60" alt="x( 1.1 )\approx -1.60" title="x( 1.1 )\approx -1.60" data-latex="x( 1.1 )\approx -1.60"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.1 )\approx 1.40" alt="y( 1.1 )\approx 1.40" title="y( 1.1 )\approx 1.40" data-latex="y( 1.1 )\approx 1.40"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.2 )\approx -0.644" alt="x( 1.2 )\approx -0.644" title="x( 1.2 )\approx -0.644" data-latex="x( 1.2 )\approx -0.644"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.2 )\approx 1.58" alt="y( 1.2 )\approx 1.58" title="y( 1.2 )\approx 1.58" data-latex="y( 1.2 )\approx 1.58"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.1%20)%5Capprox%20-1.60" alt="x( 1.1 )\approx -1.60" title="x( 1.1 )\approx -1.60" data-latex="x( 1.1 )\approx -1.60"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.1%20)%5Capprox%201.40" alt="y( 1.1 )\approx 1.40" title="y( 1.1 )\approx 1.40" data-latex="y( 1.1 )\approx 1.40"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.2%20)%5Capprox%20-0.644" alt="x( 1.2 )\approx -0.644" title="x( 1.2 )\approx -0.644" data-latex="x( 1.2 )\approx -0.644"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.2%20)%5Capprox%201.58" alt="y( 1.2 )\approx 1.58" title="y( 1.2 )\approx 1.58" data-latex="y( 1.2 )\approx 1.58"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-6371" title="N2 | Euler's method for approximating IVP solutions | ver. 6371"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.2 )" alt="x( 1.2 )" title="x( 1.2 )" data-latex="x( 1.2 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.2 )" alt="y( 1.2 )" title="y( 1.2 )" data-latex="y( 1.2 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 4 \, t^{2} y^{2} + 4 \, x y + 2 \hspace{2em} x( 1 )= -2" alt="x'= 4 \, t^{2} y^{2} + 4 \, x y + 2 \hspace{2em} x( 1 )= -2" title="x'= 4 \, t^{2} y^{2} + 4 \, x y + 2 \hspace{2em} x( 1 )= -2" data-latex="x'= 4 \, t^{2} y^{2} + 4 \, x y + 2 \hspace{2em} x( 1 )= -2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -3 \, t^{2} y^{2} - 4 \, t x + 3 \hspace{2em} y( 1 )= 2" alt="y'= -3 \, t^{2} y^{2} - 4 \, t x + 3 \hspace{2em} y( 1 )= 2" title="y'= -3 \, t^{2} y^{2} - 4 \, t x + 3 \hspace{2em} y( 1 )= 2" data-latex="y'= -3 \, t^{2} y^{2} - 4 \, t x + 3 \hspace{2em} y( 1 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.2%20)" alt="x( 1.2 )" title="x( 1.2 )" data-latex="x( 1.2 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.2%20)" alt="y( 1.2 )" title="y( 1.2 )" data-latex="y( 1.2 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%204%20%5C,%20t%5E%7B2%7D%20y%5E%7B2%7D%20+%204%20%5C,%20x%20y%20+%202%20%5Chspace%7B2em%7D%20x(%201%20)=%20-2" alt="x'= 4 \, t^{2} y^{2} + 4 \, x y + 2 \hspace{2em} x( 1 )= -2" title="x'= 4 \, t^{2} y^{2} + 4 \, x y + 2 \hspace{2em} x( 1 )= -2" data-latex="x'= 4 \, t^{2} y^{2} + 4 \, x y + 2 \hspace{2em} x( 1 )= -2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-3%20%5C,%20t%5E%7B2%7D%20y%5E%7B2%7D%20-%204%20%5C,%20t%20x%20+%203%20%5Chspace%7B2em%7D%20y(%201%20)=%202" alt="y'= -3 \, t^{2} y^{2} - 4 \, t x + 3 \hspace{2em} y( 1 )= 2" title="y'= -3 \, t^{2} y^{2} - 4 \, t x + 3 \hspace{2em} y( 1 )= 2" data-latex="y'= -3 \, t^{2} y^{2} - 4 \, t x + 3 \hspace{2em} y( 1 )= 2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.1 )\approx -1.80" alt="x( 1.1 )\approx -1.80" title="x( 1.1 )\approx -1.80" data-latex="x( 1.1 )\approx -1.80"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.1 )\approx 1.90" alt="y( 1.1 )\approx 1.90" title="y( 1.1 )\approx 1.90" data-latex="y( 1.1 )\approx 1.90"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.2 )\approx -1.22" alt="x( 1.2 )\approx -1.22" title="x( 1.2 )\approx -1.22" data-latex="x( 1.2 )\approx -1.22"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.2 )\approx 1.68" alt="y( 1.2 )\approx 1.68" title="y( 1.2 )\approx 1.68" data-latex="y( 1.2 )\approx 1.68"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.1%20)%5Capprox%20-1.80" alt="x( 1.1 )\approx -1.80" title="x( 1.1 )\approx -1.80" data-latex="x( 1.1 )\approx -1.80"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.1%20)%5Capprox%201.90" alt="y( 1.1 )\approx 1.90" title="y( 1.1 )\approx 1.90" data-latex="y( 1.1 )\approx 1.90"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.2%20)%5Capprox%20-1.22" alt="x( 1.2 )\approx -1.22" title="x( 1.2 )\approx -1.22" data-latex="x( 1.2 )\approx -1.22"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.2%20)%5Capprox%201.68" alt="y( 1.2 )\approx 1.68" title="y( 1.2 )\approx 1.68" data-latex="y( 1.2 )\approx 1.68"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-9402" title="N2 | Euler's method for approximating IVP solutions | ver. 9402"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.20 )" alt="x( 0.20 )" title="x( 0.20 )" data-latex="x( 0.20 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.20 )" alt="y( 0.20 )" title="y( 0.20 )" data-latex="y( 0.20 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -3 \, t y^{2} + 3 \, x y^{2} + 1 \hspace{2em} x( 0 )= 0" alt="x'= -3 \, t y^{2} + 3 \, x y^{2} + 1 \hspace{2em} x( 0 )= 0" title="x'= -3 \, t y^{2} + 3 \, x y^{2} + 1 \hspace{2em} x( 0 )= 0" data-latex="x'= -3 \, t y^{2} + 3 \, x y^{2} + 1 \hspace{2em} x( 0 )= 0"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -t^{2} x + 3 \, t y^{2} + 2 \hspace{2em} y( 0 )= 0" alt="y'= -t^{2} x + 3 \, t y^{2} + 2 \hspace{2em} y( 0 )= 0" title="y'= -t^{2} x + 3 \, t y^{2} + 2 \hspace{2em} y( 0 )= 0" data-latex="y'= -t^{2} x + 3 \, t y^{2} + 2 \hspace{2em} y( 0 )= 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.20%20)" alt="x( 0.20 )" title="x( 0.20 )" data-latex="x( 0.20 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.20%20)" alt="y( 0.20 )" title="y( 0.20 )" data-latex="y( 0.20 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-3%20%5C,%20t%20y%5E%7B2%7D%20+%203%20%5C,%20x%20y%5E%7B2%7D%20+%201%20%5Chspace%7B2em%7D%20x(%200%20)=%200" alt="x'= -3 \, t y^{2} + 3 \, x y^{2} + 1 \hspace{2em} x( 0 )= 0" title="x'= -3 \, t y^{2} + 3 \, x y^{2} + 1 \hspace{2em} x( 0 )= 0" data-latex="x'= -3 \, t y^{2} + 3 \, x y^{2} + 1 \hspace{2em} x( 0 )= 0"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-t%5E%7B2%7D%20x%20+%203%20%5C,%20t%20y%5E%7B2%7D%20+%202%20%5Chspace%7B2em%7D%20y(%200%20)=%200" alt="y'= -t^{2} x + 3 \, t y^{2} + 2 \hspace{2em} y( 0 )= 0" title="y'= -t^{2} x + 3 \, t y^{2} + 2 \hspace{2em} y( 0 )= 0" data-latex="y'= -t^{2} x + 3 \, t y^{2} + 2 \hspace{2em} y( 0 )= 0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.10 )\approx 0.100" alt="x( 0.10 )\approx 0.100" title="x( 0.10 )\approx 0.100" data-latex="x( 0.10 )\approx 0.100"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.10 )\approx 0.200" alt="y( 0.10 )\approx 0.200" title="y( 0.10 )\approx 0.200" data-latex="y( 0.10 )\approx 0.200"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.20 )\approx 0.200" alt="x( 0.20 )\approx 0.200" title="x( 0.20 )\approx 0.200" data-latex="x( 0.20 )\approx 0.200"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.20 )\approx 0.401" alt="y( 0.20 )\approx 0.401" title="y( 0.20 )\approx 0.401" data-latex="y( 0.20 )\approx 0.401"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.10%20)%5Capprox%200.100" alt="x( 0.10 )\approx 0.100" title="x( 0.10 )\approx 0.100" data-latex="x( 0.10 )\approx 0.100"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.10%20)%5Capprox%200.200" alt="y( 0.10 )\approx 0.200" title="y( 0.10 )\approx 0.200" data-latex="y( 0.10 )\approx 0.200"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.20%20)%5Capprox%200.200" alt="x( 0.20 )\approx 0.200" title="x( 0.20 )\approx 0.200" data-latex="x( 0.20 )\approx 0.200"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.20%20)%5Capprox%200.401" alt="y( 0.20 )\approx 0.401" title="y( 0.20 )\approx 0.401" data-latex="y( 0.20 )\approx 0.401"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-0226" title="N2 | Euler's method for approximating IVP solutions | ver. 0226"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.20 )" alt="x( 0.20 )" title="x( 0.20 )" data-latex="x( 0.20 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.20 )" alt="y( 0.20 )" title="y( 0.20 )" data-latex="y( 0.20 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -2 \, t^{2} x^{2} - x y - 1 \hspace{2em} x( 0 )= 1" alt="x'= -2 \, t^{2} x^{2} - x y - 1 \hspace{2em} x( 0 )= 1" title="x'= -2 \, t^{2} x^{2} - x y - 1 \hspace{2em} x( 0 )= 1" data-latex="x'= -2 \, t^{2} x^{2} - x y - 1 \hspace{2em} x( 0 )= 1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -2 \, t^{2} y - 2 \, x y + 3 \hspace{2em} y( 0 )= 2" alt="y'= -2 \, t^{2} y - 2 \, x y + 3 \hspace{2em} y( 0 )= 2" title="y'= -2 \, t^{2} y - 2 \, x y + 3 \hspace{2em} y( 0 )= 2" data-latex="y'= -2 \, t^{2} y - 2 \, x y + 3 \hspace{2em} y( 0 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.20%20)" alt="x( 0.20 )" title="x( 0.20 )" data-latex="x( 0.20 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.20%20)" alt="y( 0.20 )" title="y( 0.20 )" data-latex="y( 0.20 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-2%20%5C,%20t%5E%7B2%7D%20x%5E%7B2%7D%20-%20x%20y%20-%201%20%5Chspace%7B2em%7D%20x(%200%20)=%201" alt="x'= -2 \, t^{2} x^{2} - x y - 1 \hspace{2em} x( 0 )= 1" title="x'= -2 \, t^{2} x^{2} - x y - 1 \hspace{2em} x( 0 )= 1" data-latex="x'= -2 \, t^{2} x^{2} - x y - 1 \hspace{2em} x( 0 )= 1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-2%20%5C,%20t%5E%7B2%7D%20y%20-%202%20%5C,%20x%20y%20+%203%20%5Chspace%7B2em%7D%20y(%200%20)=%202" alt="y'= -2 \, t^{2} y - 2 \, x y + 3 \hspace{2em} y( 0 )= 2" title="y'= -2 \, t^{2} y - 2 \, x y + 3 \hspace{2em} y( 0 )= 2" data-latex="y'= -2 \, t^{2} y - 2 \, x y + 3 \hspace{2em} y( 0 )= 2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.10 )\approx 0.700" alt="x( 0.10 )\approx 0.700" title="x( 0.10 )\approx 0.700" data-latex="x( 0.10 )\approx 0.700"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.10 )\approx 1.90" alt="y( 0.10 )\approx 1.90" title="y( 0.10 )\approx 1.90" data-latex="y( 0.10 )\approx 1.90"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.20 )\approx 0.466" alt="x( 0.20 )\approx 0.466" title="x( 0.20 )\approx 0.466" data-latex="x( 0.20 )\approx 0.466"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.20 )\approx 1.93" alt="y( 0.20 )\approx 1.93" title="y( 0.20 )\approx 1.93" data-latex="y( 0.20 )\approx 1.93"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.10%20)%5Capprox%200.700" alt="x( 0.10 )\approx 0.700" title="x( 0.10 )\approx 0.700" data-latex="x( 0.10 )\approx 0.700"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.10%20)%5Capprox%201.90" alt="y( 0.10 )\approx 1.90" title="y( 0.10 )\approx 1.90" data-latex="y( 0.10 )\approx 1.90"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.20%20)%5Capprox%200.466" alt="x( 0.20 )\approx 0.466" title="x( 0.20 )\approx 0.466" data-latex="x( 0.20 )\approx 0.466"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.20%20)%5Capprox%201.93" alt="y( 0.20 )\approx 1.93" title="y( 0.20 )\approx 1.93" data-latex="y( 0.20 )\approx 1.93"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-2093" title="N2 | Euler's method for approximating IVP solutions | ver. 2093"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.2 )" alt="x( 1.2 )" title="x( 1.2 )" data-latex="x( 1.2 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.2 )" alt="y( 1.2 )" title="y( 1.2 )" data-latex="y( 1.2 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 2 \, t y^{2} - 2 \, t x - 3 \hspace{2em} x( 1 )= -2" alt="x'= 2 \, t y^{2} - 2 \, t x - 3 \hspace{2em} x( 1 )= -2" title="x'= 2 \, t y^{2} - 2 \, t x - 3 \hspace{2em} x( 1 )= -2" data-latex="x'= 2 \, t y^{2} - 2 \, t x - 3 \hspace{2em} x( 1 )= -2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -2 \, t^{2} x^{2} - 3 \, t^{2} y \hspace{2em} y( 1 )= -2" alt="y'= -2 \, t^{2} x^{2} - 3 \, t^{2} y \hspace{2em} y( 1 )= -2" title="y'= -2 \, t^{2} x^{2} - 3 \, t^{2} y \hspace{2em} y( 1 )= -2" data-latex="y'= -2 \, t^{2} x^{2} - 3 \, t^{2} y \hspace{2em} y( 1 )= -2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.2%20)" alt="x( 1.2 )" title="x( 1.2 )" data-latex="x( 1.2 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.2%20)" alt="y( 1.2 )" title="y( 1.2 )" data-latex="y( 1.2 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%202%20%5C,%20t%20y%5E%7B2%7D%20-%202%20%5C,%20t%20x%20-%203%20%5Chspace%7B2em%7D%20x(%201%20)=%20-2" alt="x'= 2 \, t y^{2} - 2 \, t x - 3 \hspace{2em} x( 1 )= -2" title="x'= 2 \, t y^{2} - 2 \, t x - 3 \hspace{2em} x( 1 )= -2" data-latex="x'= 2 \, t y^{2} - 2 \, t x - 3 \hspace{2em} x( 1 )= -2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-2%20%5C,%20t%5E%7B2%7D%20x%5E%7B2%7D%20-%203%20%5C,%20t%5E%7B2%7D%20y%20%5Chspace%7B2em%7D%20y(%201%20)=%20-2" alt="y'= -2 \, t^{2} x^{2} - 3 \, t^{2} y \hspace{2em} y( 1 )= -2" title="y'= -2 \, t^{2} x^{2} - 3 \, t^{2} y \hspace{2em} y( 1 )= -2" data-latex="y'= -2 \, t^{2} x^{2} - 3 \, t^{2} y \hspace{2em} y( 1 )= -2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.1 )\approx -1.10" alt="x( 1.1 )\approx -1.10" title="x( 1.1 )\approx -1.10" data-latex="x( 1.1 )\approx -1.10"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.1 )\approx -2.20" alt="y( 1.1 )\approx -2.20" title="y( 1.1 )\approx -2.20" data-latex="y( 1.1 )\approx -2.20"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.2 )\approx -0.0957" alt="x( 1.2 )\approx -0.0957" title="x( 1.2 )\approx -0.0957" data-latex="x( 1.2 )\approx -0.0957"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.2 )\approx -1.70" alt="y( 1.2 )\approx -1.70" title="y( 1.2 )\approx -1.70" data-latex="y( 1.2 )\approx -1.70"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.1%20)%5Capprox%20-1.10" alt="x( 1.1 )\approx -1.10" title="x( 1.1 )\approx -1.10" data-latex="x( 1.1 )\approx -1.10"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.1%20)%5Capprox%20-2.20" alt="y( 1.1 )\approx -2.20" title="y( 1.1 )\approx -2.20" data-latex="y( 1.1 )\approx -2.20"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.2%20)%5Capprox%20-0.0957" alt="x( 1.2 )\approx -0.0957" title="x( 1.2 )\approx -0.0957" data-latex="x( 1.2 )\approx -0.0957"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.2%20)%5Capprox%20-1.70" alt="y( 1.2 )\approx -1.70" title="y( 1.2 )\approx -1.70" data-latex="y( 1.2 )\approx -1.70"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-5358" title="N2 | Euler's method for approximating IVP solutions | ver. 5358"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.80 )" alt="x( -0.80 )" title="x( -0.80 )" data-latex="x( -0.80 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.80 )" alt="y( -0.80 )" title="y( -0.80 )" data-latex="y( -0.80 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 4 \, x^{2} y^{2} + t y \hspace{2em} x( -1 )= 2" alt="x'= 4 \, x^{2} y^{2} + t y \hspace{2em} x( -1 )= 2" title="x'= 4 \, x^{2} y^{2} + t y \hspace{2em} x( -1 )= 2" data-latex="x'= 4 \, x^{2} y^{2} + t y \hspace{2em} x( -1 )= 2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -t x^{2} - 2 \, t^{2} y - 1 \hspace{2em} y( -1 )= 0" alt="y'= -t x^{2} - 2 \, t^{2} y - 1 \hspace{2em} y( -1 )= 0" title="y'= -t x^{2} - 2 \, t^{2} y - 1 \hspace{2em} y( -1 )= 0" data-latex="y'= -t x^{2} - 2 \, t^{2} y - 1 \hspace{2em} y( -1 )= 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.80%20)" alt="x( -0.80 )" title="x( -0.80 )" data-latex="x( -0.80 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.80%20)" alt="y( -0.80 )" title="y( -0.80 )" data-latex="y( -0.80 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%204%20%5C,%20x%5E%7B2%7D%20y%5E%7B2%7D%20+%20t%20y%20%5Chspace%7B2em%7D%20x(%20-1%20)=%202" alt="x'= 4 \, x^{2} y^{2} + t y \hspace{2em} x( -1 )= 2" title="x'= 4 \, x^{2} y^{2} + t y \hspace{2em} x( -1 )= 2" data-latex="x'= 4 \, x^{2} y^{2} + t y \hspace{2em} x( -1 )= 2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-t%20x%5E%7B2%7D%20-%202%20%5C,%20t%5E%7B2%7D%20y%20-%201%20%5Chspace%7B2em%7D%20y(%20-1%20)=%200" alt="y'= -t x^{2} - 2 \, t^{2} y - 1 \hspace{2em} y( -1 )= 0" title="y'= -t x^{2} - 2 \, t^{2} y - 1 \hspace{2em} y( -1 )= 0" data-latex="y'= -t x^{2} - 2 \, t^{2} y - 1 \hspace{2em} y( -1 )= 0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.90 )\approx 2.00" alt="x( -0.90 )\approx 2.00" title="x( -0.90 )\approx 2.00" data-latex="x( -0.90 )\approx 2.00"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.90 )\approx 0.300" alt="y( -0.90 )\approx 0.300" title="y( -0.90 )\approx 0.300" data-latex="y( -0.90 )\approx 0.300"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.80 )\approx 2.12" alt="x( -0.80 )\approx 2.12" title="x( -0.80 )\approx 2.12" data-latex="x( -0.80 )\approx 2.12"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.80 )\approx 0.512" alt="y( -0.80 )\approx 0.512" title="y( -0.80 )\approx 0.512" data-latex="y( -0.80 )\approx 0.512"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.90%20)%5Capprox%202.00" alt="x( -0.90 )\approx 2.00" title="x( -0.90 )\approx 2.00" data-latex="x( -0.90 )\approx 2.00"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.90%20)%5Capprox%200.300" alt="y( -0.90 )\approx 0.300" title="y( -0.90 )\approx 0.300" data-latex="y( -0.90 )\approx 0.300"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.80%20)%5Capprox%202.12" alt="x( -0.80 )\approx 2.12" title="x( -0.80 )\approx 2.12" data-latex="x( -0.80 )\approx 2.12"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.80%20)%5Capprox%200.512" alt="y( -0.80 )\approx 0.512" title="y( -0.80 )\approx 0.512" data-latex="y( -0.80 )\approx 0.512"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-1953" title="N2 | Euler's method for approximating IVP solutions | ver. 1953"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.2 )" alt="x( 1.2 )" title="x( 1.2 )" data-latex="x( 1.2 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.2 )" alt="y( 1.2 )" title="y( 1.2 )" data-latex="y( 1.2 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 4 \, t y - 3 \, x y \hspace{2em} x( 1 )= -2" alt="x'= 4 \, t y - 3 \, x y \hspace{2em} x( 1 )= -2" title="x'= 4 \, t y - 3 \, x y \hspace{2em} x( 1 )= -2" data-latex="x'= 4 \, t y - 3 \, x y \hspace{2em} x( 1 )= -2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 3 \, t y^{2} - t x + 2 \hspace{2em} y( 1 )= -1" alt="y'= 3 \, t y^{2} - t x + 2 \hspace{2em} y( 1 )= -1" title="y'= 3 \, t y^{2} - t x + 2 \hspace{2em} y( 1 )= -1" data-latex="y'= 3 \, t y^{2} - t x + 2 \hspace{2em} y( 1 )= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.2%20)" alt="x( 1.2 )" title="x( 1.2 )" data-latex="x( 1.2 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.2%20)" alt="y( 1.2 )" title="y( 1.2 )" data-latex="y( 1.2 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%204%20%5C,%20t%20y%20-%203%20%5C,%20x%20y%20%5Chspace%7B2em%7D%20x(%201%20)=%20-2" alt="x'= 4 \, t y - 3 \, x y \hspace{2em} x( 1 )= -2" title="x'= 4 \, t y - 3 \, x y \hspace{2em} x( 1 )= -2" data-latex="x'= 4 \, t y - 3 \, x y \hspace{2em} x( 1 )= -2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%203%20%5C,%20t%20y%5E%7B2%7D%20-%20t%20x%20+%202%20%5Chspace%7B2em%7D%20y(%201%20)=%20-1" alt="y'= 3 \, t y^{2} - t x + 2 \hspace{2em} y( 1 )= -1" title="y'= 3 \, t y^{2} - t x + 2 \hspace{2em} y( 1 )= -1" data-latex="y'= 3 \, t y^{2} - t x + 2 \hspace{2em} y( 1 )= -1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.1 )\approx -3.00" alt="x( 1.1 )\approx -3.00" title="x( 1.1 )\approx -3.00" data-latex="x( 1.1 )\approx -3.00"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.1 )\approx -0.300" alt="y( 1.1 )\approx -0.300" title="y( 1.1 )\approx -0.300" data-latex="y( 1.1 )\approx -0.300"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.2 )\approx -3.40" alt="x( 1.2 )\approx -3.40" title="x( 1.2 )\approx -3.40" data-latex="x( 1.2 )\approx -3.40"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.2 )\approx 0.260" alt="y( 1.2 )\approx 0.260" title="y( 1.2 )\approx 0.260" data-latex="y( 1.2 )\approx 0.260"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.1%20)%5Capprox%20-3.00" alt="x( 1.1 )\approx -3.00" title="x( 1.1 )\approx -3.00" data-latex="x( 1.1 )\approx -3.00"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.1%20)%5Capprox%20-0.300" alt="y( 1.1 )\approx -0.300" title="y( 1.1 )\approx -0.300" data-latex="y( 1.1 )\approx -0.300"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.2%20)%5Capprox%20-3.40" alt="x( 1.2 )\approx -3.40" title="x( 1.2 )\approx -3.40" data-latex="x( 1.2 )\approx -3.40"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.2%20)%5Capprox%200.260" alt="y( 1.2 )\approx 0.260" title="y( 1.2 )\approx 0.260" data-latex="y( 1.2 )\approx 0.260"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-3023" title="N2 | Euler's method for approximating IVP solutions | ver. 3023"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 2.2 )" alt="x( 2.2 )" title="x( 2.2 )" data-latex="x( 2.2 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 2.2 )" alt="y( 2.2 )" title="y( 2.2 )" data-latex="y( 2.2 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -3 \, t x^{2} - 4 \, x y + 2 \hspace{2em} x( 2 )= -1" alt="x'= -3 \, t x^{2} - 4 \, x y + 2 \hspace{2em} x( 2 )= -1" title="x'= -3 \, t x^{2} - 4 \, x y + 2 \hspace{2em} x( 2 )= -1" data-latex="x'= -3 \, t x^{2} - 4 \, x y + 2 \hspace{2em} x( 2 )= -1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= t^{2} x^{2} - 4 \, x y - 2 \hspace{2em} y( 2 )= 2" alt="y'= t^{2} x^{2} - 4 \, x y - 2 \hspace{2em} y( 2 )= 2" title="y'= t^{2} x^{2} - 4 \, x y - 2 \hspace{2em} y( 2 )= 2" data-latex="y'= t^{2} x^{2} - 4 \, x y - 2 \hspace{2em} y( 2 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%202.2%20)" alt="x( 2.2 )" title="x( 2.2 )" data-latex="x( 2.2 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%202.2%20)" alt="y( 2.2 )" title="y( 2.2 )" data-latex="y( 2.2 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-3%20%5C,%20t%20x%5E%7B2%7D%20-%204%20%5C,%20x%20y%20+%202%20%5Chspace%7B2em%7D%20x(%202%20)=%20-1" alt="x'= -3 \, t x^{2} - 4 \, x y + 2 \hspace{2em} x( 2 )= -1" title="x'= -3 \, t x^{2} - 4 \, x y + 2 \hspace{2em} x( 2 )= -1" data-latex="x'= -3 \, t x^{2} - 4 \, x y + 2 \hspace{2em} x( 2 )= -1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20t%5E%7B2%7D%20x%5E%7B2%7D%20-%204%20%5C,%20x%20y%20-%202%20%5Chspace%7B2em%7D%20y(%202%20)=%202" alt="y'= t^{2} x^{2} - 4 \, x y - 2 \hspace{2em} y( 2 )= 2" title="y'= t^{2} x^{2} - 4 \, x y - 2 \hspace{2em} y( 2 )= 2" data-latex="y'= t^{2} x^{2} - 4 \, x y - 2 \hspace{2em} y( 2 )= 2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 2.1 )\approx -0.600" alt="x( 2.1 )\approx -0.600" title="x( 2.1 )\approx -0.600" data-latex="x( 2.1 )\approx -0.600"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 2.1 )\approx 3.00" alt="y( 2.1 )\approx 3.00" title="y( 2.1 )\approx 3.00" data-latex="y( 2.1 )\approx 3.00"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 2.2 )\approx 0.0928" alt="x( 2.2 )\approx 0.0928" title="x( 2.2 )\approx 0.0928" data-latex="x( 2.2 )\approx 0.0928"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 2.2 )\approx 3.68" alt="y( 2.2 )\approx 3.68" title="y( 2.2 )\approx 3.68" data-latex="y( 2.2 )\approx 3.68"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%202.1%20)%5Capprox%20-0.600" alt="x( 2.1 )\approx -0.600" title="x( 2.1 )\approx -0.600" data-latex="x( 2.1 )\approx -0.600"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%202.1%20)%5Capprox%203.00" alt="y( 2.1 )\approx 3.00" title="y( 2.1 )\approx 3.00" data-latex="y( 2.1 )\approx 3.00"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%202.2%20)%5Capprox%200.0928" alt="x( 2.2 )\approx 0.0928" title="x( 2.2 )\approx 0.0928" data-latex="x( 2.2 )\approx 0.0928"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%202.2%20)%5Capprox%203.68" alt="y( 2.2 )\approx 3.68" title="y( 2.2 )\approx 3.68" data-latex="y( 2.2 )\approx 3.68"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-4817" title="N2 | Euler's method for approximating IVP solutions | ver. 4817"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.2 )" alt="x( 1.2 )" title="x( 1.2 )" data-latex="x( 1.2 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.2 )" alt="y( 1.2 )" title="y( 1.2 )" data-latex="y( 1.2 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -t^{2} x + 4 \, t y + 2 \hspace{2em} x( 1 )= 1" alt="x'= -t^{2} x + 4 \, t y + 2 \hspace{2em} x( 1 )= 1" title="x'= -t^{2} x + 4 \, t y + 2 \hspace{2em} x( 1 )= 1" data-latex="x'= -t^{2} x + 4 \, t y + 2 \hspace{2em} x( 1 )= 1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -2 \, t^{2} x + 2 \, x y^{2} - 2 \hspace{2em} y( 1 )= 2" alt="y'= -2 \, t^{2} x + 2 \, x y^{2} - 2 \hspace{2em} y( 1 )= 2" title="y'= -2 \, t^{2} x + 2 \, x y^{2} - 2 \hspace{2em} y( 1 )= 2" data-latex="y'= -2 \, t^{2} x + 2 \, x y^{2} - 2 \hspace{2em} y( 1 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.2%20)" alt="x( 1.2 )" title="x( 1.2 )" data-latex="x( 1.2 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.2%20)" alt="y( 1.2 )" title="y( 1.2 )" data-latex="y( 1.2 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-t%5E%7B2%7D%20x%20+%204%20%5C,%20t%20y%20+%202%20%5Chspace%7B2em%7D%20x(%201%20)=%201" alt="x'= -t^{2} x + 4 \, t y + 2 \hspace{2em} x( 1 )= 1" title="x'= -t^{2} x + 4 \, t y + 2 \hspace{2em} x( 1 )= 1" data-latex="x'= -t^{2} x + 4 \, t y + 2 \hspace{2em} x( 1 )= 1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-2%20%5C,%20t%5E%7B2%7D%20x%20+%202%20%5C,%20x%20y%5E%7B2%7D%20-%202%20%5Chspace%7B2em%7D%20y(%201%20)=%202" alt="y'= -2 \, t^{2} x + 2 \, x y^{2} - 2 \hspace{2em} y( 1 )= 2" title="y'= -2 \, t^{2} x + 2 \, x y^{2} - 2 \hspace{2em} y( 1 )= 2" data-latex="y'= -2 \, t^{2} x + 2 \, x y^{2} - 2 \hspace{2em} y( 1 )= 2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.1 )\approx 1.90" alt="x( 1.1 )\approx 1.90" title="x( 1.1 )\approx 1.90" data-latex="x( 1.1 )\approx 1.90"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.1 )\approx 2.40" alt="y( 1.1 )\approx 2.40" title="y( 1.1 )\approx 2.40" data-latex="y( 1.1 )\approx 2.40"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.2 )\approx 2.92" alt="x( 1.2 )\approx 2.92" title="x( 1.2 )\approx 2.92" data-latex="x( 1.2 )\approx 2.92"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.2 )\approx 3.92" alt="y( 1.2 )\approx 3.92" title="y( 1.2 )\approx 3.92" data-latex="y( 1.2 )\approx 3.92"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.1%20)%5Capprox%201.90" alt="x( 1.1 )\approx 1.90" title="x( 1.1 )\approx 1.90" data-latex="x( 1.1 )\approx 1.90"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.1%20)%5Capprox%202.40" alt="y( 1.1 )\approx 2.40" title="y( 1.1 )\approx 2.40" data-latex="y( 1.1 )\approx 2.40"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.2%20)%5Capprox%202.92" alt="x( 1.2 )\approx 2.92" title="x( 1.2 )\approx 2.92" data-latex="x( 1.2 )\approx 2.92"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.2%20)%5Capprox%203.92" alt="y( 1.2 )\approx 3.92" title="y( 1.2 )\approx 3.92" data-latex="y( 1.2 )\approx 3.92"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-9729" title="N2 | Euler's method for approximating IVP solutions | ver. 9729"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 2.2 )" alt="x( 2.2 )" title="x( 2.2 )" data-latex="x( 2.2 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 2.2 )" alt="y( 2.2 )" title="y( 2.2 )" data-latex="y( 2.2 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -t^{2} x^{2} - 4 \, x^{2} y + 2 \hspace{2em} x( 2 )= 1" alt="x'= -t^{2} x^{2} - 4 \, x^{2} y + 2 \hspace{2em} x( 2 )= 1" title="x'= -t^{2} x^{2} - 4 \, x^{2} y + 2 \hspace{2em} x( 2 )= 1" data-latex="x'= -t^{2} x^{2} - 4 \, x^{2} y + 2 \hspace{2em} x( 2 )= 1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -2 \, x y^{2} + 3 \, t y - 3 \hspace{2em} y( 2 )= 2" alt="y'= -2 \, x y^{2} + 3 \, t y - 3 \hspace{2em} y( 2 )= 2" title="y'= -2 \, x y^{2} + 3 \, t y - 3 \hspace{2em} y( 2 )= 2" data-latex="y'= -2 \, x y^{2} + 3 \, t y - 3 \hspace{2em} y( 2 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%202.2%20)" alt="x( 2.2 )" title="x( 2.2 )" data-latex="x( 2.2 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%202.2%20)" alt="y( 2.2 )" title="y( 2.2 )" data-latex="y( 2.2 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-t%5E%7B2%7D%20x%5E%7B2%7D%20-%204%20%5C,%20x%5E%7B2%7D%20y%20+%202%20%5Chspace%7B2em%7D%20x(%202%20)=%201" alt="x'= -t^{2} x^{2} - 4 \, x^{2} y + 2 \hspace{2em} x( 2 )= 1" title="x'= -t^{2} x^{2} - 4 \, x^{2} y + 2 \hspace{2em} x( 2 )= 1" data-latex="x'= -t^{2} x^{2} - 4 \, x^{2} y + 2 \hspace{2em} x( 2 )= 1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-2%20%5C,%20x%20y%5E%7B2%7D%20+%203%20%5C,%20t%20y%20-%203%20%5Chspace%7B2em%7D%20y(%202%20)=%202" alt="y'= -2 \, x y^{2} + 3 \, t y - 3 \hspace{2em} y( 2 )= 2" title="y'= -2 \, x y^{2} + 3 \, t y - 3 \hspace{2em} y( 2 )= 2" data-latex="y'= -2 \, x y^{2} + 3 \, t y - 3 \hspace{2em} y( 2 )= 2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 2.1 )\approx 0.000" alt="x( 2.1 )\approx 0.000" title="x( 2.1 )\approx 0.000" data-latex="x( 2.1 )\approx 0.000"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 2.1 )\approx 2.10" alt="y( 2.1 )\approx 2.10" title="y( 2.1 )\approx 2.10" data-latex="y( 2.1 )\approx 2.10"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 2.2 )\approx 0.200" alt="x( 2.2 )\approx 0.200" title="x( 2.2 )\approx 0.200" data-latex="x( 2.2 )\approx 0.200"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 2.2 )\approx 3.12" alt="y( 2.2 )\approx 3.12" title="y( 2.2 )\approx 3.12" data-latex="y( 2.2 )\approx 3.12"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%202.1%20)%5Capprox%200.000" alt="x( 2.1 )\approx 0.000" title="x( 2.1 )\approx 0.000" data-latex="x( 2.1 )\approx 0.000"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%202.1%20)%5Capprox%202.10" alt="y( 2.1 )\approx 2.10" title="y( 2.1 )\approx 2.10" data-latex="y( 2.1 )\approx 2.10"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%202.2%20)%5Capprox%200.200" alt="x( 2.2 )\approx 0.200" title="x( 2.2 )\approx 0.200" data-latex="x( 2.2 )\approx 0.200"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%202.2%20)%5Capprox%203.12" alt="y( 2.2 )\approx 3.12" title="y( 2.2 )\approx 3.12" data-latex="y( 2.2 )\approx 3.12"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-3309" title="N2 | Euler's method for approximating IVP solutions | ver. 3309"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.8 )" alt="x( -1.8 )" title="x( -1.8 )" data-latex="x( -1.8 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.8 )" alt="y( -1.8 )" title="y( -1.8 )" data-latex="y( -1.8 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -3 \, t x^{2} - 2 \, t y - 1 \hspace{2em} x( -2 )= 2" alt="x'= -3 \, t x^{2} - 2 \, t y - 1 \hspace{2em} x( -2 )= 2" title="x'= -3 \, t x^{2} - 2 \, t y - 1 \hspace{2em} x( -2 )= 2" data-latex="x'= -3 \, t x^{2} - 2 \, t y - 1 \hspace{2em} x( -2 )= 2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -t^{2} y - x y^{2} - 2 \hspace{2em} y( -2 )= -1" alt="y'= -t^{2} y - x y^{2} - 2 \hspace{2em} y( -2 )= -1" title="y'= -t^{2} y - x y^{2} - 2 \hspace{2em} y( -2 )= -1" data-latex="y'= -t^{2} y - x y^{2} - 2 \hspace{2em} y( -2 )= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.8%20)" alt="x( -1.8 )" title="x( -1.8 )" data-latex="x( -1.8 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.8%20)" alt="y( -1.8 )" title="y( -1.8 )" data-latex="y( -1.8 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-3%20%5C,%20t%20x%5E%7B2%7D%20-%202%20%5C,%20t%20y%20-%201%20%5Chspace%7B2em%7D%20x(%20-2%20)=%202" alt="x'= -3 \, t x^{2} - 2 \, t y - 1 \hspace{2em} x( -2 )= 2" title="x'= -3 \, t x^{2} - 2 \, t y - 1 \hspace{2em} x( -2 )= 2" data-latex="x'= -3 \, t x^{2} - 2 \, t y - 1 \hspace{2em} x( -2 )= 2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-t%5E%7B2%7D%20y%20-%20x%20y%5E%7B2%7D%20-%202%20%5Chspace%7B2em%7D%20y(%20-2%20)=%20-1" alt="y'= -t^{2} y - x y^{2} - 2 \hspace{2em} y( -2 )= -1" title="y'= -t^{2} y - x y^{2} - 2 \hspace{2em} y( -2 )= -1" data-latex="y'= -t^{2} y - x y^{2} - 2 \hspace{2em} y( -2 )= -1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.9 )\approx 3.90" alt="x( -1.9 )\approx 3.90" title="x( -1.9 )\approx 3.90" data-latex="x( -1.9 )\approx 3.90"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.9 )\approx -1.00" alt="y( -1.9 )\approx -1.00" title="y( -1.9 )\approx -1.00" data-latex="y( -1.9 )\approx -1.00"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.8 )\approx 12.1" alt="x( -1.8 )\approx 12.1" title="x( -1.8 )\approx 12.1" data-latex="x( -1.8 )\approx 12.1"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.8 )\approx -1.23" alt="y( -1.8 )\approx -1.23" title="y( -1.8 )\approx -1.23" data-latex="y( -1.8 )\approx -1.23"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.9%20)%5Capprox%203.90" alt="x( -1.9 )\approx 3.90" title="x( -1.9 )\approx 3.90" data-latex="x( -1.9 )\approx 3.90"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.9%20)%5Capprox%20-1.00" alt="y( -1.9 )\approx -1.00" title="y( -1.9 )\approx -1.00" data-latex="y( -1.9 )\approx -1.00"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.8%20)%5Capprox%2012.1" alt="x( -1.8 )\approx 12.1" title="x( -1.8 )\approx 12.1" data-latex="x( -1.8 )\approx 12.1"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.8%20)%5Capprox%20-1.23" alt="y( -1.8 )\approx -1.23" title="y( -1.8 )\approx -1.23" data-latex="y( -1.8 )\approx -1.23"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-1113" title="N2 | Euler's method for approximating IVP solutions | ver. 1113"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.20 )" alt="x( 0.20 )" title="x( 0.20 )" data-latex="x( 0.20 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.20 )" alt="y( 0.20 )" title="y( 0.20 )" data-latex="y( 0.20 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 4 \, x^{2} y + t y \hspace{2em} x( 0 )= 1" alt="x'= 4 \, x^{2} y + t y \hspace{2em} x( 0 )= 1" title="x'= 4 \, x^{2} y + t y \hspace{2em} x( 0 )= 1" data-latex="x'= 4 \, x^{2} y + t y \hspace{2em} x( 0 )= 1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 4 \, t^{2} x - 4 \, t y^{2} + 1 \hspace{2em} y( 0 )= -1" alt="y'= 4 \, t^{2} x - 4 \, t y^{2} + 1 \hspace{2em} y( 0 )= -1" title="y'= 4 \, t^{2} x - 4 \, t y^{2} + 1 \hspace{2em} y( 0 )= -1" data-latex="y'= 4 \, t^{2} x - 4 \, t y^{2} + 1 \hspace{2em} y( 0 )= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.20%20)" alt="x( 0.20 )" title="x( 0.20 )" data-latex="x( 0.20 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.20%20)" alt="y( 0.20 )" title="y( 0.20 )" data-latex="y( 0.20 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%204%20%5C,%20x%5E%7B2%7D%20y%20+%20t%20y%20%5Chspace%7B2em%7D%20x(%200%20)=%201" alt="x'= 4 \, x^{2} y + t y \hspace{2em} x( 0 )= 1" title="x'= 4 \, x^{2} y + t y \hspace{2em} x( 0 )= 1" data-latex="x'= 4 \, x^{2} y + t y \hspace{2em} x( 0 )= 1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%204%20%5C,%20t%5E%7B2%7D%20x%20-%204%20%5C,%20t%20y%5E%7B2%7D%20+%201%20%5Chspace%7B2em%7D%20y(%200%20)=%20-1" alt="y'= 4 \, t^{2} x - 4 \, t y^{2} + 1 \hspace{2em} y( 0 )= -1" title="y'= 4 \, t^{2} x - 4 \, t y^{2} + 1 \hspace{2em} y( 0 )= -1" data-latex="y'= 4 \, t^{2} x - 4 \, t y^{2} + 1 \hspace{2em} y( 0 )= -1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.10 )\approx 0.600" alt="x( 0.10 )\approx 0.600" title="x( 0.10 )\approx 0.600" data-latex="x( 0.10 )\approx 0.600"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.10 )\approx -0.900" alt="y( 0.10 )\approx -0.900" title="y( 0.10 )\approx -0.900" data-latex="y( 0.10 )\approx -0.900"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.20 )\approx 0.461" alt="x( 0.20 )\approx 0.461" title="x( 0.20 )\approx 0.461" data-latex="x( 0.20 )\approx 0.461"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.20 )\approx -0.830" alt="y( 0.20 )\approx -0.830" title="y( 0.20 )\approx -0.830" data-latex="y( 0.20 )\approx -0.830"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.10%20)%5Capprox%200.600" alt="x( 0.10 )\approx 0.600" title="x( 0.10 )\approx 0.600" data-latex="x( 0.10 )\approx 0.600"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.10%20)%5Capprox%20-0.900" alt="y( 0.10 )\approx -0.900" title="y( 0.10 )\approx -0.900" data-latex="y( 0.10 )\approx -0.900"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.20%20)%5Capprox%200.461" alt="x( 0.20 )\approx 0.461" title="x( 0.20 )\approx 0.461" data-latex="x( 0.20 )\approx 0.461"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.20%20)%5Capprox%20-0.830" alt="y( 0.20 )\approx -0.830" title="y( 0.20 )\approx -0.830" data-latex="y( 0.20 )\approx -0.830"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-1036" title="N2 | Euler's method for approximating IVP solutions | ver. 1036"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.80 )" alt="x( -0.80 )" title="x( -0.80 )" data-latex="x( -0.80 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.80 )" alt="y( -0.80 )" title="y( -0.80 )" data-latex="y( -0.80 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 3 \, t^{2} x - x y + 2 \hspace{2em} x( -1 )= -2" alt="x'= 3 \, t^{2} x - x y + 2 \hspace{2em} x( -1 )= -2" title="x'= 3 \, t^{2} x - x y + 2 \hspace{2em} x( -1 )= -2" data-latex="x'= 3 \, t^{2} x - x y + 2 \hspace{2em} x( -1 )= -2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 3 \, t^{2} x^{2} + 2 \, x y - 2 \hspace{2em} y( -1 )= 0" alt="y'= 3 \, t^{2} x^{2} + 2 \, x y - 2 \hspace{2em} y( -1 )= 0" title="y'= 3 \, t^{2} x^{2} + 2 \, x y - 2 \hspace{2em} y( -1 )= 0" data-latex="y'= 3 \, t^{2} x^{2} + 2 \, x y - 2 \hspace{2em} y( -1 )= 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.80%20)" alt="x( -0.80 )" title="x( -0.80 )" data-latex="x( -0.80 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.80%20)" alt="y( -0.80 )" title="y( -0.80 )" data-latex="y( -0.80 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%203%20%5C,%20t%5E%7B2%7D%20x%20-%20x%20y%20+%202%20%5Chspace%7B2em%7D%20x(%20-1%20)=%20-2" alt="x'= 3 \, t^{2} x - x y + 2 \hspace{2em} x( -1 )= -2" title="x'= 3 \, t^{2} x - x y + 2 \hspace{2em} x( -1 )= -2" data-latex="x'= 3 \, t^{2} x - x y + 2 \hspace{2em} x( -1 )= -2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%203%20%5C,%20t%5E%7B2%7D%20x%5E%7B2%7D%20+%202%20%5C,%20x%20y%20-%202%20%5Chspace%7B2em%7D%20y(%20-1%20)=%200" alt="y'= 3 \, t^{2} x^{2} + 2 \, x y - 2 \hspace{2em} y( -1 )= 0" title="y'= 3 \, t^{2} x^{2} + 2 \, x y - 2 \hspace{2em} y( -1 )= 0" data-latex="y'= 3 \, t^{2} x^{2} + 2 \, x y - 2 \hspace{2em} y( -1 )= 0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.90 )\approx -2.40" alt="x( -0.90 )\approx -2.40" title="x( -0.90 )\approx -2.40" data-latex="x( -0.90 )\approx -2.40"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.90 )\approx 1.00" alt="y( -0.90 )\approx 1.00" title="y( -0.90 )\approx 1.00" data-latex="y( -0.90 )\approx 1.00"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.80 )\approx -2.54" alt="x( -0.80 )\approx -2.54" title="x( -0.80 )\approx -2.54" data-latex="x( -0.80 )\approx -2.54"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.80 )\approx 1.72" alt="y( -0.80 )\approx 1.72" title="y( -0.80 )\approx 1.72" data-latex="y( -0.80 )\approx 1.72"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.90%20)%5Capprox%20-2.40" alt="x( -0.90 )\approx -2.40" title="x( -0.90 )\approx -2.40" data-latex="x( -0.90 )\approx -2.40"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.90%20)%5Capprox%201.00" alt="y( -0.90 )\approx 1.00" title="y( -0.90 )\approx 1.00" data-latex="y( -0.90 )\approx 1.00"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.80%20)%5Capprox%20-2.54" alt="x( -0.80 )\approx -2.54" title="x( -0.80 )\approx -2.54" data-latex="x( -0.80 )\approx -2.54"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.80%20)%5Capprox%201.72" alt="y( -0.80 )\approx 1.72" title="y( -0.80 )\approx 1.72" data-latex="y( -0.80 )\approx 1.72"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-3696" title="N2 | Euler's method for approximating IVP solutions | ver. 3696"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.20 )" alt="x( 0.20 )" title="x( 0.20 )" data-latex="x( 0.20 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.20 )" alt="y( 0.20 )" title="y( 0.20 )" data-latex="y( 0.20 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -t^{2} x^{2} - 3 \, t y - 3 \hspace{2em} x( 0 )= 0" alt="x'= -t^{2} x^{2} - 3 \, t y - 3 \hspace{2em} x( 0 )= 0" title="x'= -t^{2} x^{2} - 3 \, t y - 3 \hspace{2em} x( 0 )= 0" data-latex="x'= -t^{2} x^{2} - 3 \, t y - 3 \hspace{2em} x( 0 )= 0"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -2 \, t^{2} y^{2} + 4 \, x^{2} y^{2} - 3 \hspace{2em} y( 0 )= -2" alt="y'= -2 \, t^{2} y^{2} + 4 \, x^{2} y^{2} - 3 \hspace{2em} y( 0 )= -2" title="y'= -2 \, t^{2} y^{2} + 4 \, x^{2} y^{2} - 3 \hspace{2em} y( 0 )= -2" data-latex="y'= -2 \, t^{2} y^{2} + 4 \, x^{2} y^{2} - 3 \hspace{2em} y( 0 )= -2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.20%20)" alt="x( 0.20 )" title="x( 0.20 )" data-latex="x( 0.20 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.20%20)" alt="y( 0.20 )" title="y( 0.20 )" data-latex="y( 0.20 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-t%5E%7B2%7D%20x%5E%7B2%7D%20-%203%20%5C,%20t%20y%20-%203%20%5Chspace%7B2em%7D%20x(%200%20)=%200" alt="x'= -t^{2} x^{2} - 3 \, t y - 3 \hspace{2em} x( 0 )= 0" title="x'= -t^{2} x^{2} - 3 \, t y - 3 \hspace{2em} x( 0 )= 0" data-latex="x'= -t^{2} x^{2} - 3 \, t y - 3 \hspace{2em} x( 0 )= 0"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-2%20%5C,%20t%5E%7B2%7D%20y%5E%7B2%7D%20+%204%20%5C,%20x%5E%7B2%7D%20y%5E%7B2%7D%20-%203%20%5Chspace%7B2em%7D%20y(%200%20)=%20-2" alt="y'= -2 \, t^{2} y^{2} + 4 \, x^{2} y^{2} - 3 \hspace{2em} y( 0 )= -2" title="y'= -2 \, t^{2} y^{2} + 4 \, x^{2} y^{2} - 3 \hspace{2em} y( 0 )= -2" data-latex="y'= -2 \, t^{2} y^{2} + 4 \, x^{2} y^{2} - 3 \hspace{2em} y( 0 )= -2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.10 )\approx -0.300" alt="x( 0.10 )\approx -0.300" title="x( 0.10 )\approx -0.300" data-latex="x( 0.10 )\approx -0.300"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.10 )\approx -2.30" alt="y( 0.10 )\approx -2.30" title="y( 0.10 )\approx -2.30" data-latex="y( 0.10 )\approx -2.30"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.20 )\approx -0.531" alt="x( 0.20 )\approx -0.531" title="x( 0.20 )\approx -0.531" data-latex="x( 0.20 )\approx -0.531"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.20 )\approx -2.42" alt="y( 0.20 )\approx -2.42" title="y( 0.20 )\approx -2.42" data-latex="y( 0.20 )\approx -2.42"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.10%20)%5Capprox%20-0.300" alt="x( 0.10 )\approx -0.300" title="x( 0.10 )\approx -0.300" data-latex="x( 0.10 )\approx -0.300"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.10%20)%5Capprox%20-2.30" alt="y( 0.10 )\approx -2.30" title="y( 0.10 )\approx -2.30" data-latex="y( 0.10 )\approx -2.30"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.20%20)%5Capprox%20-0.531" alt="x( 0.20 )\approx -0.531" title="x( 0.20 )\approx -0.531" data-latex="x( 0.20 )\approx -0.531"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.20%20)%5Capprox%20-2.42" alt="y( 0.20 )\approx -2.42" title="y( 0.20 )\approx -2.42" data-latex="y( 0.20 )\approx -2.42"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-3841" title="N2 | Euler's method for approximating IVP solutions | ver. 3841"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.80 )" alt="x( -0.80 )" title="x( -0.80 )" data-latex="x( -0.80 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.80 )" alt="y( -0.80 )" title="y( -0.80 )" data-latex="y( -0.80 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 2 \, t^{2} y + 3 \, x^{2} y - 3 \hspace{2em} x( -1 )= 2" alt="x'= 2 \, t^{2} y + 3 \, x^{2} y - 3 \hspace{2em} x( -1 )= 2" title="x'= 2 \, t^{2} y + 3 \, x^{2} y - 3 \hspace{2em} x( -1 )= 2" data-latex="x'= 2 \, t^{2} y + 3 \, x^{2} y - 3 \hspace{2em} x( -1 )= 2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 3 \, t^{2} y + x^{2} y + 3 \hspace{2em} y( -1 )= -2" alt="y'= 3 \, t^{2} y + x^{2} y + 3 \hspace{2em} y( -1 )= -2" title="y'= 3 \, t^{2} y + x^{2} y + 3 \hspace{2em} y( -1 )= -2" data-latex="y'= 3 \, t^{2} y + x^{2} y + 3 \hspace{2em} y( -1 )= -2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.80%20)" alt="x( -0.80 )" title="x( -0.80 )" data-latex="x( -0.80 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.80%20)" alt="y( -0.80 )" title="y( -0.80 )" data-latex="y( -0.80 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%202%20%5C,%20t%5E%7B2%7D%20y%20+%203%20%5C,%20x%5E%7B2%7D%20y%20-%203%20%5Chspace%7B2em%7D%20x(%20-1%20)=%202" alt="x'= 2 \, t^{2} y + 3 \, x^{2} y - 3 \hspace{2em} x( -1 )= 2" title="x'= 2 \, t^{2} y + 3 \, x^{2} y - 3 \hspace{2em} x( -1 )= 2" data-latex="x'= 2 \, t^{2} y + 3 \, x^{2} y - 3 \hspace{2em} x( -1 )= 2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%203%20%5C,%20t%5E%7B2%7D%20y%20+%20x%5E%7B2%7D%20y%20+%203%20%5Chspace%7B2em%7D%20y(%20-1%20)=%20-2" alt="y'= 3 \, t^{2} y + x^{2} y + 3 \hspace{2em} y( -1 )= -2" title="y'= 3 \, t^{2} y + x^{2} y + 3 \hspace{2em} y( -1 )= -2" data-latex="y'= 3 \, t^{2} y + x^{2} y + 3 \hspace{2em} y( -1 )= -2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.90 )\approx -1.10" alt="x( -0.90 )\approx -1.10" title="x( -0.90 )\approx -1.10" data-latex="x( -0.90 )\approx -1.10"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.90 )\approx -3.10" alt="y( -0.90 )\approx -3.10" title="y( -0.90 )\approx -3.10" data-latex="y( -0.90 )\approx -3.10"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.80 )\approx -3.02" alt="x( -0.80 )\approx -3.02" title="x( -0.80 )\approx -3.02" data-latex="x( -0.80 )\approx -3.02"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.80 )\approx -3.93" alt="y( -0.80 )\approx -3.93" title="y( -0.80 )\approx -3.93" data-latex="y( -0.80 )\approx -3.93"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.90%20)%5Capprox%20-1.10" alt="x( -0.90 )\approx -1.10" title="x( -0.90 )\approx -1.10" data-latex="x( -0.90 )\approx -1.10"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.90%20)%5Capprox%20-3.10" alt="y( -0.90 )\approx -3.10" title="y( -0.90 )\approx -3.10" data-latex="y( -0.90 )\approx -3.10"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.80%20)%5Capprox%20-3.02" alt="x( -0.80 )\approx -3.02" title="x( -0.80 )\approx -3.02" data-latex="x( -0.80 )\approx -3.02"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.80%20)%5Capprox%20-3.93" alt="y( -0.80 )\approx -3.93" title="y( -0.80 )\approx -3.93" data-latex="y( -0.80 )\approx -3.93"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-7727" title="N2 | Euler's method for approximating IVP solutions | ver. 7727"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.20 )" alt="x( 0.20 )" title="x( 0.20 )" data-latex="x( 0.20 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.20 )" alt="y( 0.20 )" title="y( 0.20 )" data-latex="y( 0.20 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 4 \, t^{2} y - 4 \, t x + 3 \hspace{2em} x( 0 )= 0" alt="x'= 4 \, t^{2} y - 4 \, t x + 3 \hspace{2em} x( 0 )= 0" title="x'= 4 \, t^{2} y - 4 \, t x + 3 \hspace{2em} x( 0 )= 0" data-latex="x'= 4 \, t^{2} y - 4 \, t x + 3 \hspace{2em} x( 0 )= 0"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -t^{2} y^{2} - 2 \, x^{2} y + 1 \hspace{2em} y( 0 )= 2" alt="y'= -t^{2} y^{2} - 2 \, x^{2} y + 1 \hspace{2em} y( 0 )= 2" title="y'= -t^{2} y^{2} - 2 \, x^{2} y + 1 \hspace{2em} y( 0 )= 2" data-latex="y'= -t^{2} y^{2} - 2 \, x^{2} y + 1 \hspace{2em} y( 0 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.20%20)" alt="x( 0.20 )" title="x( 0.20 )" data-latex="x( 0.20 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.20%20)" alt="y( 0.20 )" title="y( 0.20 )" data-latex="y( 0.20 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%204%20%5C,%20t%5E%7B2%7D%20y%20-%204%20%5C,%20t%20x%20+%203%20%5Chspace%7B2em%7D%20x(%200%20)=%200" alt="x'= 4 \, t^{2} y - 4 \, t x + 3 \hspace{2em} x( 0 )= 0" title="x'= 4 \, t^{2} y - 4 \, t x + 3 \hspace{2em} x( 0 )= 0" data-latex="x'= 4 \, t^{2} y - 4 \, t x + 3 \hspace{2em} x( 0 )= 0"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-t%5E%7B2%7D%20y%5E%7B2%7D%20-%202%20%5C,%20x%5E%7B2%7D%20y%20+%201%20%5Chspace%7B2em%7D%20y(%200%20)=%202" alt="y'= -t^{2} y^{2} - 2 \, x^{2} y + 1 \hspace{2em} y( 0 )= 2" title="y'= -t^{2} y^{2} - 2 \, x^{2} y + 1 \hspace{2em} y( 0 )= 2" data-latex="y'= -t^{2} y^{2} - 2 \, x^{2} y + 1 \hspace{2em} y( 0 )= 2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.10 )\approx 0.300" alt="x( 0.10 )\approx 0.300" title="x( 0.10 )\approx 0.300" data-latex="x( 0.10 )\approx 0.300"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.10 )\approx 2.10" alt="y( 0.10 )\approx 2.10" title="y( 0.10 )\approx 2.10" data-latex="y( 0.10 )\approx 2.10"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.20 )\approx 0.596" alt="x( 0.20 )\approx 0.596" title="x( 0.20 )\approx 0.596" data-latex="x( 0.20 )\approx 0.596"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.20 )\approx 2.16" alt="y( 0.20 )\approx 2.16" title="y( 0.20 )\approx 2.16" data-latex="y( 0.20 )\approx 2.16"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.10%20)%5Capprox%200.300" alt="x( 0.10 )\approx 0.300" title="x( 0.10 )\approx 0.300" data-latex="x( 0.10 )\approx 0.300"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.10%20)%5Capprox%202.10" alt="y( 0.10 )\approx 2.10" title="y( 0.10 )\approx 2.10" data-latex="y( 0.10 )\approx 2.10"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.20%20)%5Capprox%200.596" alt="x( 0.20 )\approx 0.596" title="x( 0.20 )\approx 0.596" data-latex="x( 0.20 )\approx 0.596"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.20%20)%5Capprox%202.16" alt="y( 0.20 )\approx 2.16" title="y( 0.20 )\approx 2.16" data-latex="y( 0.20 )\approx 2.16"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-5430" title="N2 | Euler's method for approximating IVP solutions | ver. 5430"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.20 )" alt="x( 0.20 )" title="x( 0.20 )" data-latex="x( 0.20 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.20 )" alt="y( 0.20 )" title="y( 0.20 )" data-latex="y( 0.20 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -2 \, t^{2} y^{2} + x^{2} y^{2} \hspace{2em} x( 0 )= -2" alt="x'= -2 \, t^{2} y^{2} + x^{2} y^{2} \hspace{2em} x( 0 )= -2" title="x'= -2 \, t^{2} y^{2} + x^{2} y^{2} \hspace{2em} x( 0 )= -2" data-latex="x'= -2 \, t^{2} y^{2} + x^{2} y^{2} \hspace{2em} x( 0 )= -2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -t^{2} x + x y - 2 \hspace{2em} y( 0 )= 1" alt="y'= -t^{2} x + x y - 2 \hspace{2em} y( 0 )= 1" title="y'= -t^{2} x + x y - 2 \hspace{2em} y( 0 )= 1" data-latex="y'= -t^{2} x + x y - 2 \hspace{2em} y( 0 )= 1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.20%20)" alt="x( 0.20 )" title="x( 0.20 )" data-latex="x( 0.20 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.20%20)" alt="y( 0.20 )" title="y( 0.20 )" data-latex="y( 0.20 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-2%20%5C,%20t%5E%7B2%7D%20y%5E%7B2%7D%20+%20x%5E%7B2%7D%20y%5E%7B2%7D%20%5Chspace%7B2em%7D%20x(%200%20)=%20-2" alt="x'= -2 \, t^{2} y^{2} + x^{2} y^{2} \hspace{2em} x( 0 )= -2" title="x'= -2 \, t^{2} y^{2} + x^{2} y^{2} \hspace{2em} x( 0 )= -2" data-latex="x'= -2 \, t^{2} y^{2} + x^{2} y^{2} \hspace{2em} x( 0 )= -2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-t%5E%7B2%7D%20x%20+%20x%20y%20-%202%20%5Chspace%7B2em%7D%20y(%200%20)=%201" alt="y'= -t^{2} x + x y - 2 \hspace{2em} y( 0 )= 1" title="y'= -t^{2} x + x y - 2 \hspace{2em} y( 0 )= 1" data-latex="y'= -t^{2} x + x y - 2 \hspace{2em} y( 0 )= 1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.10 )\approx -1.60" alt="x( 0.10 )\approx -1.60" title="x( 0.10 )\approx -1.60" data-latex="x( 0.10 )\approx -1.60"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.10 )\approx 0.600" alt="y( 0.10 )\approx 0.600" title="y( 0.10 )\approx 0.600" data-latex="y( 0.10 )\approx 0.600"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.20 )\approx -1.51" alt="x( 0.20 )\approx -1.51" title="x( 0.20 )\approx -1.51" data-latex="x( 0.20 )\approx -1.51"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.20 )\approx 0.305" alt="y( 0.20 )\approx 0.305" title="y( 0.20 )\approx 0.305" data-latex="y( 0.20 )\approx 0.305"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.10%20)%5Capprox%20-1.60" alt="x( 0.10 )\approx -1.60" title="x( 0.10 )\approx -1.60" data-latex="x( 0.10 )\approx -1.60"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.10%20)%5Capprox%200.600" alt="y( 0.10 )\approx 0.600" title="y( 0.10 )\approx 0.600" data-latex="y( 0.10 )\approx 0.600"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.20%20)%5Capprox%20-1.51" alt="x( 0.20 )\approx -1.51" title="x( 0.20 )\approx -1.51" data-latex="x( 0.20 )\approx -1.51"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.20%20)%5Capprox%200.305" alt="y( 0.20 )\approx 0.305" title="y( 0.20 )\approx 0.305" data-latex="y( 0.20 )\approx 0.305"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-8933" title="N2 | Euler's method for approximating IVP solutions | ver. 8933"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 2.2 )" alt="x( 2.2 )" title="x( 2.2 )" data-latex="x( 2.2 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 2.2 )" alt="y( 2.2 )" title="y( 2.2 )" data-latex="y( 2.2 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -3 \, t^{2} x^{2} - x y^{2} - 2 \hspace{2em} x( 2 )= 0" alt="x'= -3 \, t^{2} x^{2} - x y^{2} - 2 \hspace{2em} x( 2 )= 0" title="x'= -3 \, t^{2} x^{2} - x y^{2} - 2 \hspace{2em} x( 2 )= 0" data-latex="x'= -3 \, t^{2} x^{2} - x y^{2} - 2 \hspace{2em} x( 2 )= 0"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 4 \, x^{2} y^{2} - 3 \, t y - 1 \hspace{2em} y( 2 )= -2" alt="y'= 4 \, x^{2} y^{2} - 3 \, t y - 1 \hspace{2em} y( 2 )= -2" title="y'= 4 \, x^{2} y^{2} - 3 \, t y - 1 \hspace{2em} y( 2 )= -2" data-latex="y'= 4 \, x^{2} y^{2} - 3 \, t y - 1 \hspace{2em} y( 2 )= -2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%202.2%20)" alt="x( 2.2 )" title="x( 2.2 )" data-latex="x( 2.2 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%202.2%20)" alt="y( 2.2 )" title="y( 2.2 )" data-latex="y( 2.2 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-3%20%5C,%20t%5E%7B2%7D%20x%5E%7B2%7D%20-%20x%20y%5E%7B2%7D%20-%202%20%5Chspace%7B2em%7D%20x(%202%20)=%200" alt="x'= -3 \, t^{2} x^{2} - x y^{2} - 2 \hspace{2em} x( 2 )= 0" title="x'= -3 \, t^{2} x^{2} - x y^{2} - 2 \hspace{2em} x( 2 )= 0" data-latex="x'= -3 \, t^{2} x^{2} - x y^{2} - 2 \hspace{2em} x( 2 )= 0"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%204%20%5C,%20x%5E%7B2%7D%20y%5E%7B2%7D%20-%203%20%5C,%20t%20y%20-%201%20%5Chspace%7B2em%7D%20y(%202%20)=%20-2" alt="y'= 4 \, x^{2} y^{2} - 3 \, t y - 1 \hspace{2em} y( 2 )= -2" title="y'= 4 \, x^{2} y^{2} - 3 \, t y - 1 \hspace{2em} y( 2 )= -2" data-latex="y'= 4 \, x^{2} y^{2} - 3 \, t y - 1 \hspace{2em} y( 2 )= -2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 2.1 )\approx -0.200" alt="x( 2.1 )\approx -0.200" title="x( 2.1 )\approx -0.200" data-latex="x( 2.1 )\approx -0.200"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 2.1 )\approx -0.900" alt="y( 2.1 )\approx -0.900" title="y( 2.1 )\approx -0.900" data-latex="y( 2.1 )\approx -0.900"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 2.2 )\approx -0.437" alt="x( 2.2 )\approx -0.437" title="x( 2.2 )\approx -0.437" data-latex="x( 2.2 )\approx -0.437"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 2.2 )\approx -0.419" alt="y( 2.2 )\approx -0.419" title="y( 2.2 )\approx -0.419" data-latex="y( 2.2 )\approx -0.419"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%202.1%20)%5Capprox%20-0.200" alt="x( 2.1 )\approx -0.200" title="x( 2.1 )\approx -0.200" data-latex="x( 2.1 )\approx -0.200"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%202.1%20)%5Capprox%20-0.900" alt="y( 2.1 )\approx -0.900" title="y( 2.1 )\approx -0.900" data-latex="y( 2.1 )\approx -0.900"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%202.2%20)%5Capprox%20-0.437" alt="x( 2.2 )\approx -0.437" title="x( 2.2 )\approx -0.437" data-latex="x( 2.2 )\approx -0.437"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%202.2%20)%5Capprox%20-0.419" alt="y( 2.2 )\approx -0.419" title="y( 2.2 )\approx -0.419" data-latex="y( 2.2 )\approx -0.419"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-4848" title="N2 | Euler's method for approximating IVP solutions | ver. 4848"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.8 )" alt="x( -1.8 )" title="x( -1.8 )" data-latex="x( -1.8 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.8 )" alt="y( -1.8 )" title="y( -1.8 )" data-latex="y( -1.8 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= t^{2} y - t x - 1 \hspace{2em} x( -2 )= 1" alt="x'= t^{2} y - t x - 1 \hspace{2em} x( -2 )= 1" title="x'= t^{2} y - t x - 1 \hspace{2em} x( -2 )= 1" data-latex="x'= t^{2} y - t x - 1 \hspace{2em} x( -2 )= 1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 4 \, x y^{2} + 4 \, t x - 3 \hspace{2em} y( -2 )= -1" alt="y'= 4 \, x y^{2} + 4 \, t x - 3 \hspace{2em} y( -2 )= -1" title="y'= 4 \, x y^{2} + 4 \, t x - 3 \hspace{2em} y( -2 )= -1" data-latex="y'= 4 \, x y^{2} + 4 \, t x - 3 \hspace{2em} y( -2 )= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.8%20)" alt="x( -1.8 )" title="x( -1.8 )" data-latex="x( -1.8 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.8%20)" alt="y( -1.8 )" title="y( -1.8 )" data-latex="y( -1.8 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20t%5E%7B2%7D%20y%20-%20t%20x%20-%201%20%5Chspace%7B2em%7D%20x(%20-2%20)=%201" alt="x'= t^{2} y - t x - 1 \hspace{2em} x( -2 )= 1" title="x'= t^{2} y - t x - 1 \hspace{2em} x( -2 )= 1" data-latex="x'= t^{2} y - t x - 1 \hspace{2em} x( -2 )= 1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%204%20%5C,%20x%20y%5E%7B2%7D%20+%204%20%5C,%20t%20x%20-%203%20%5Chspace%7B2em%7D%20y(%20-2%20)=%20-1" alt="y'= 4 \, x y^{2} + 4 \, t x - 3 \hspace{2em} y( -2 )= -1" title="y'= 4 \, x y^{2} + 4 \, t x - 3 \hspace{2em} y( -2 )= -1" data-latex="y'= 4 \, x y^{2} + 4 \, t x - 3 \hspace{2em} y( -2 )= -1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.9 )\approx 0.700" alt="x( -1.9 )\approx 0.700" title="x( -1.9 )\approx 0.700" data-latex="x( -1.9 )\approx 0.700"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.9 )\approx -1.70" alt="y( -1.9 )\approx -1.70" title="y( -1.9 )\approx -1.70" data-latex="y( -1.9 )\approx -1.70"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.8 )\approx 0.119" alt="x( -1.8 )\approx 0.119" title="x( -1.8 )\approx 0.119" data-latex="x( -1.8 )\approx 0.119"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.8 )\approx -1.72" alt="y( -1.8 )\approx -1.72" title="y( -1.8 )\approx -1.72" data-latex="y( -1.8 )\approx -1.72"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.9%20)%5Capprox%200.700" alt="x( -1.9 )\approx 0.700" title="x( -1.9 )\approx 0.700" data-latex="x( -1.9 )\approx 0.700"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.9%20)%5Capprox%20-1.70" alt="y( -1.9 )\approx -1.70" title="y( -1.9 )\approx -1.70" data-latex="y( -1.9 )\approx -1.70"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.8%20)%5Capprox%200.119" alt="x( -1.8 )\approx 0.119" title="x( -1.8 )\approx 0.119" data-latex="x( -1.8 )\approx 0.119"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.8%20)%5Capprox%20-1.72" alt="y( -1.8 )\approx -1.72" title="y( -1.8 )\approx -1.72" data-latex="y( -1.8 )\approx -1.72"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-9723" title="N2 | Euler's method for approximating IVP solutions | ver. 9723"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 2.2 )" alt="x( 2.2 )" title="x( 2.2 )" data-latex="x( 2.2 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 2.2 )" alt="y( 2.2 )" title="y( 2.2 )" data-latex="y( 2.2 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -4 \, x^{2} y + 2 \, t x - 3 \hspace{2em} x( 2 )= -2" alt="x'= -4 \, x^{2} y + 2 \, t x - 3 \hspace{2em} x( 2 )= -2" title="x'= -4 \, x^{2} y + 2 \, t x - 3 \hspace{2em} x( 2 )= -2" data-latex="x'= -4 \, x^{2} y + 2 \, t x - 3 \hspace{2em} x( 2 )= -2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -2 \, t x^{2} - 4 \, x y^{2} - 1 \hspace{2em} y( 2 )= 0" alt="y'= -2 \, t x^{2} - 4 \, x y^{2} - 1 \hspace{2em} y( 2 )= 0" title="y'= -2 \, t x^{2} - 4 \, x y^{2} - 1 \hspace{2em} y( 2 )= 0" data-latex="y'= -2 \, t x^{2} - 4 \, x y^{2} - 1 \hspace{2em} y( 2 )= 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%202.2%20)" alt="x( 2.2 )" title="x( 2.2 )" data-latex="x( 2.2 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%202.2%20)" alt="y( 2.2 )" title="y( 2.2 )" data-latex="y( 2.2 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-4%20%5C,%20x%5E%7B2%7D%20y%20+%202%20%5C,%20t%20x%20-%203%20%5Chspace%7B2em%7D%20x(%202%20)=%20-2" alt="x'= -4 \, x^{2} y + 2 \, t x - 3 \hspace{2em} x( 2 )= -2" title="x'= -4 \, x^{2} y + 2 \, t x - 3 \hspace{2em} x( 2 )= -2" data-latex="x'= -4 \, x^{2} y + 2 \, t x - 3 \hspace{2em} x( 2 )= -2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-2%20%5C,%20t%20x%5E%7B2%7D%20-%204%20%5C,%20x%20y%5E%7B2%7D%20-%201%20%5Chspace%7B2em%7D%20y(%202%20)=%200" alt="y'= -2 \, t x^{2} - 4 \, x y^{2} - 1 \hspace{2em} y( 2 )= 0" title="y'= -2 \, t x^{2} - 4 \, x y^{2} - 1 \hspace{2em} y( 2 )= 0" data-latex="y'= -2 \, t x^{2} - 4 \, x y^{2} - 1 \hspace{2em} y( 2 )= 0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 2.1 )\approx -3.10" alt="x( 2.1 )\approx -3.10" title="x( 2.1 )\approx -3.10" data-latex="x( 2.1 )\approx -3.10"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 2.1 )\approx -1.70" alt="y( 2.1 )\approx -1.70" title="y( 2.1 )\approx -1.70" data-latex="y( 2.1 )\approx -1.70"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 2.2 )\approx 1.84" alt="x( 2.2 )\approx 1.84" title="x( 2.2 )\approx 1.84" data-latex="x( 2.2 )\approx 1.84"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 2.2 )\approx -2.26" alt="y( 2.2 )\approx -2.26" title="y( 2.2 )\approx -2.26" data-latex="y( 2.2 )\approx -2.26"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%202.1%20)%5Capprox%20-3.10" alt="x( 2.1 )\approx -3.10" title="x( 2.1 )\approx -3.10" data-latex="x( 2.1 )\approx -3.10"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%202.1%20)%5Capprox%20-1.70" alt="y( 2.1 )\approx -1.70" title="y( 2.1 )\approx -1.70" data-latex="y( 2.1 )\approx -1.70"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%202.2%20)%5Capprox%201.84" alt="x( 2.2 )\approx 1.84" title="x( 2.2 )\approx 1.84" data-latex="x( 2.2 )\approx 1.84"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%202.2%20)%5Capprox%20-2.26" alt="y( 2.2 )\approx -2.26" title="y( 2.2 )\approx -2.26" data-latex="y( 2.2 )\approx -2.26"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-3390" title="N2 | Euler's method for approximating IVP solutions | ver. 3390"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.20 )" alt="x( 0.20 )" title="x( 0.20 )" data-latex="x( 0.20 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.20 )" alt="y( 0.20 )" title="y( 0.20 )" data-latex="y( 0.20 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -3 \, t x - 4 \, t y - 3 \hspace{2em} x( 0 )= 2" alt="x'= -3 \, t x - 4 \, t y - 3 \hspace{2em} x( 0 )= 2" title="x'= -3 \, t x - 4 \, t y - 3 \hspace{2em} x( 0 )= 2" data-latex="x'= -3 \, t x - 4 \, t y - 3 \hspace{2em} x( 0 )= 2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 4 \, x y^{2} + 4 \, t y - 2 \hspace{2em} y( 0 )= 0" alt="y'= 4 \, x y^{2} + 4 \, t y - 2 \hspace{2em} y( 0 )= 0" title="y'= 4 \, x y^{2} + 4 \, t y - 2 \hspace{2em} y( 0 )= 0" data-latex="y'= 4 \, x y^{2} + 4 \, t y - 2 \hspace{2em} y( 0 )= 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.20%20)" alt="x( 0.20 )" title="x( 0.20 )" data-latex="x( 0.20 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.20%20)" alt="y( 0.20 )" title="y( 0.20 )" data-latex="y( 0.20 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-3%20%5C,%20t%20x%20-%204%20%5C,%20t%20y%20-%203%20%5Chspace%7B2em%7D%20x(%200%20)=%202" alt="x'= -3 \, t x - 4 \, t y - 3 \hspace{2em} x( 0 )= 2" title="x'= -3 \, t x - 4 \, t y - 3 \hspace{2em} x( 0 )= 2" data-latex="x'= -3 \, t x - 4 \, t y - 3 \hspace{2em} x( 0 )= 2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%204%20%5C,%20x%20y%5E%7B2%7D%20+%204%20%5C,%20t%20y%20-%202%20%5Chspace%7B2em%7D%20y(%200%20)=%200" alt="y'= 4 \, x y^{2} + 4 \, t y - 2 \hspace{2em} y( 0 )= 0" title="y'= 4 \, x y^{2} + 4 \, t y - 2 \hspace{2em} y( 0 )= 0" data-latex="y'= 4 \, x y^{2} + 4 \, t y - 2 \hspace{2em} y( 0 )= 0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.10 )\approx 1.70" alt="x( 0.10 )\approx 1.70" title="x( 0.10 )\approx 1.70" data-latex="x( 0.10 )\approx 1.70"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.10 )\approx -0.200" alt="y( 0.10 )\approx -0.200" title="y( 0.10 )\approx -0.200" data-latex="y( 0.10 )\approx -0.200"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.20 )\approx 1.36" alt="x( 0.20 )\approx 1.36" title="x( 0.20 )\approx 1.36" data-latex="x( 0.20 )\approx 1.36"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.20 )\approx -0.381" alt="y( 0.20 )\approx -0.381" title="y( 0.20 )\approx -0.381" data-latex="y( 0.20 )\approx -0.381"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.10%20)%5Capprox%201.70" alt="x( 0.10 )\approx 1.70" title="x( 0.10 )\approx 1.70" data-latex="x( 0.10 )\approx 1.70"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.10%20)%5Capprox%20-0.200" alt="y( 0.10 )\approx -0.200" title="y( 0.10 )\approx -0.200" data-latex="y( 0.10 )\approx -0.200"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.20%20)%5Capprox%201.36" alt="x( 0.20 )\approx 1.36" title="x( 0.20 )\approx 1.36" data-latex="x( 0.20 )\approx 1.36"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.20%20)%5Capprox%20-0.381" alt="y( 0.20 )\approx -0.381" title="y( 0.20 )\approx -0.381" data-latex="y( 0.20 )\approx -0.381"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-7979" title="N2 | Euler's method for approximating IVP solutions | ver. 7979"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.8 )" alt="x( -1.8 )" title="x( -1.8 )" data-latex="x( -1.8 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.8 )" alt="y( -1.8 )" title="y( -1.8 )" data-latex="y( -1.8 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 3 \, t^{2} y^{2} - 2 \, x^{2} y + 2 \hspace{2em} x( -2 )= 1" alt="x'= 3 \, t^{2} y^{2} - 2 \, x^{2} y + 2 \hspace{2em} x( -2 )= 1" title="x'= 3 \, t^{2} y^{2} - 2 \, x^{2} y + 2 \hspace{2em} x( -2 )= 1" data-latex="x'= 3 \, t^{2} y^{2} - 2 \, x^{2} y + 2 \hspace{2em} x( -2 )= 1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -2 \, t^{2} x - 3 \, x^{2} y - 1 \hspace{2em} y( -2 )= -2" alt="y'= -2 \, t^{2} x - 3 \, x^{2} y - 1 \hspace{2em} y( -2 )= -2" title="y'= -2 \, t^{2} x - 3 \, x^{2} y - 1 \hspace{2em} y( -2 )= -2" data-latex="y'= -2 \, t^{2} x - 3 \, x^{2} y - 1 \hspace{2em} y( -2 )= -2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.8%20)" alt="x( -1.8 )" title="x( -1.8 )" data-latex="x( -1.8 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.8%20)" alt="y( -1.8 )" title="y( -1.8 )" data-latex="y( -1.8 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%203%20%5C,%20t%5E%7B2%7D%20y%5E%7B2%7D%20-%202%20%5C,%20x%5E%7B2%7D%20y%20+%202%20%5Chspace%7B2em%7D%20x(%20-2%20)=%201" alt="x'= 3 \, t^{2} y^{2} - 2 \, x^{2} y + 2 \hspace{2em} x( -2 )= 1" title="x'= 3 \, t^{2} y^{2} - 2 \, x^{2} y + 2 \hspace{2em} x( -2 )= 1" data-latex="x'= 3 \, t^{2} y^{2} - 2 \, x^{2} y + 2 \hspace{2em} x( -2 )= 1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-2%20%5C,%20t%5E%7B2%7D%20x%20-%203%20%5C,%20x%5E%7B2%7D%20y%20-%201%20%5Chspace%7B2em%7D%20y(%20-2%20)=%20-2" alt="y'= -2 \, t^{2} x - 3 \, x^{2} y - 1 \hspace{2em} y( -2 )= -2" title="y'= -2 \, t^{2} x - 3 \, x^{2} y - 1 \hspace{2em} y( -2 )= -2" data-latex="y'= -2 \, t^{2} x - 3 \, x^{2} y - 1 \hspace{2em} y( -2 )= -2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.9 )\approx 6.40" alt="x( -1.9 )\approx 6.40" title="x( -1.9 )\approx 6.40" data-latex="x( -1.9 )\approx 6.40"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.9 )\approx -2.30" alt="y( -1.9 )\approx -2.30" title="y( -1.9 )\approx -2.30" data-latex="y( -1.9 )\approx -2.30"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.8 )\approx 31.2" alt="x( -1.8 )\approx 31.2" title="x( -1.8 )\approx 31.2" data-latex="x( -1.8 )\approx 31.2"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.8 )\approx 21.2" alt="y( -1.8 )\approx 21.2" title="y( -1.8 )\approx 21.2" data-latex="y( -1.8 )\approx 21.2"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.9%20)%5Capprox%206.40" alt="x( -1.9 )\approx 6.40" title="x( -1.9 )\approx 6.40" data-latex="x( -1.9 )\approx 6.40"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.9%20)%5Capprox%20-2.30" alt="y( -1.9 )\approx -2.30" title="y( -1.9 )\approx -2.30" data-latex="y( -1.9 )\approx -2.30"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.8%20)%5Capprox%2031.2" alt="x( -1.8 )\approx 31.2" title="x( -1.8 )\approx 31.2" data-latex="x( -1.8 )\approx 31.2"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.8%20)%5Capprox%2021.2" alt="y( -1.8 )\approx 21.2" title="y( -1.8 )\approx 21.2" data-latex="y( -1.8 )\approx 21.2"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-9715" title="N2 | Euler's method for approximating IVP solutions | ver. 9715"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.20 )" alt="x( 0.20 )" title="x( 0.20 )" data-latex="x( 0.20 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.20 )" alt="y( 0.20 )" title="y( 0.20 )" data-latex="y( 0.20 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -x^{2} y^{2} + 3 \, t x^{2} - 3 \hspace{2em} x( 0 )= -1" alt="x'= -x^{2} y^{2} + 3 \, t x^{2} - 3 \hspace{2em} x( 0 )= -1" title="x'= -x^{2} y^{2} + 3 \, t x^{2} - 3 \hspace{2em} x( 0 )= -1" data-latex="x'= -x^{2} y^{2} + 3 \, t x^{2} - 3 \hspace{2em} x( 0 )= -1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -4 \, t^{2} x^{2} - 3 \, t^{2} y + 1 \hspace{2em} y( 0 )= 1" alt="y'= -4 \, t^{2} x^{2} - 3 \, t^{2} y + 1 \hspace{2em} y( 0 )= 1" title="y'= -4 \, t^{2} x^{2} - 3 \, t^{2} y + 1 \hspace{2em} y( 0 )= 1" data-latex="y'= -4 \, t^{2} x^{2} - 3 \, t^{2} y + 1 \hspace{2em} y( 0 )= 1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.20%20)" alt="x( 0.20 )" title="x( 0.20 )" data-latex="x( 0.20 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.20%20)" alt="y( 0.20 )" title="y( 0.20 )" data-latex="y( 0.20 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-x%5E%7B2%7D%20y%5E%7B2%7D%20+%203%20%5C,%20t%20x%5E%7B2%7D%20-%203%20%5Chspace%7B2em%7D%20x(%200%20)=%20-1" alt="x'= -x^{2} y^{2} + 3 \, t x^{2} - 3 \hspace{2em} x( 0 )= -1" title="x'= -x^{2} y^{2} + 3 \, t x^{2} - 3 \hspace{2em} x( 0 )= -1" data-latex="x'= -x^{2} y^{2} + 3 \, t x^{2} - 3 \hspace{2em} x( 0 )= -1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-4%20%5C,%20t%5E%7B2%7D%20x%5E%7B2%7D%20-%203%20%5C,%20t%5E%7B2%7D%20y%20+%201%20%5Chspace%7B2em%7D%20y(%200%20)=%201" alt="y'= -4 \, t^{2} x^{2} - 3 \, t^{2} y + 1 \hspace{2em} y( 0 )= 1" title="y'= -4 \, t^{2} x^{2} - 3 \, t^{2} y + 1 \hspace{2em} y( 0 )= 1" data-latex="y'= -4 \, t^{2} x^{2} - 3 \, t^{2} y + 1 \hspace{2em} y( 0 )= 1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.10 )\approx -1.40" alt="x( 0.10 )\approx -1.40" title="x( 0.10 )\approx -1.40" data-latex="x( 0.10 )\approx -1.40"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.10 )\approx 1.10" alt="y( 0.10 )\approx 1.10" title="y( 0.10 )\approx 1.10" data-latex="y( 0.10 )\approx 1.10"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.20 )\approx -1.88" alt="x( 0.20 )\approx -1.88" title="x( 0.20 )\approx -1.88" data-latex="x( 0.20 )\approx -1.88"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.20 )\approx 1.19" alt="y( 0.20 )\approx 1.19" title="y( 0.20 )\approx 1.19" data-latex="y( 0.20 )\approx 1.19"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.10%20)%5Capprox%20-1.40" alt="x( 0.10 )\approx -1.40" title="x( 0.10 )\approx -1.40" data-latex="x( 0.10 )\approx -1.40"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.10%20)%5Capprox%201.10" alt="y( 0.10 )\approx 1.10" title="y( 0.10 )\approx 1.10" data-latex="y( 0.10 )\approx 1.10"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%200.20%20)%5Capprox%20-1.88" alt="x( 0.20 )\approx -1.88" title="x( 0.20 )\approx -1.88" data-latex="x( 0.20 )\approx -1.88"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.20%20)%5Capprox%201.19" alt="y( 0.20 )\approx 1.19" title="y( 0.20 )\approx 1.19" data-latex="y( 0.20 )\approx 1.19"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-7674" title="N2 | Euler's method for approximating IVP solutions | ver. 7674"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.80 )" alt="x( -0.80 )" title="x( -0.80 )" data-latex="x( -0.80 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.80 )" alt="y( -0.80 )" title="y( -0.80 )" data-latex="y( -0.80 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -4 \, t^{2} y^{2} + 4 \, x^{2} y^{2} - 2 \hspace{2em} x( -1 )= -2" alt="x'= -4 \, t^{2} y^{2} + 4 \, x^{2} y^{2} - 2 \hspace{2em} x( -1 )= -2" title="x'= -4 \, t^{2} y^{2} + 4 \, x^{2} y^{2} - 2 \hspace{2em} x( -1 )= -2" data-latex="x'= -4 \, t^{2} y^{2} + 4 \, x^{2} y^{2} - 2 \hspace{2em} x( -1 )= -2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 2 \, x^{2} y^{2} - 4 \, t x^{2} - 3 \hspace{2em} y( -1 )= -1" alt="y'= 2 \, x^{2} y^{2} - 4 \, t x^{2} - 3 \hspace{2em} y( -1 )= -1" title="y'= 2 \, x^{2} y^{2} - 4 \, t x^{2} - 3 \hspace{2em} y( -1 )= -1" data-latex="y'= 2 \, x^{2} y^{2} - 4 \, t x^{2} - 3 \hspace{2em} y( -1 )= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.80%20)" alt="x( -0.80 )" title="x( -0.80 )" data-latex="x( -0.80 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.80%20)" alt="y( -0.80 )" title="y( -0.80 )" data-latex="y( -0.80 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-4%20%5C,%20t%5E%7B2%7D%20y%5E%7B2%7D%20+%204%20%5C,%20x%5E%7B2%7D%20y%5E%7B2%7D%20-%202%20%5Chspace%7B2em%7D%20x(%20-1%20)=%20-2" alt="x'= -4 \, t^{2} y^{2} + 4 \, x^{2} y^{2} - 2 \hspace{2em} x( -1 )= -2" title="x'= -4 \, t^{2} y^{2} + 4 \, x^{2} y^{2} - 2 \hspace{2em} x( -1 )= -2" data-latex="x'= -4 \, t^{2} y^{2} + 4 \, x^{2} y^{2} - 2 \hspace{2em} x( -1 )= -2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%202%20%5C,%20x%5E%7B2%7D%20y%5E%7B2%7D%20-%204%20%5C,%20t%20x%5E%7B2%7D%20-%203%20%5Chspace%7B2em%7D%20y(%20-1%20)=%20-1" alt="y'= 2 \, x^{2} y^{2} - 4 \, t x^{2} - 3 \hspace{2em} y( -1 )= -1" title="y'= 2 \, x^{2} y^{2} - 4 \, t x^{2} - 3 \hspace{2em} y( -1 )= -1" data-latex="y'= 2 \, x^{2} y^{2} - 4 \, t x^{2} - 3 \hspace{2em} y( -1 )= -1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.90 )\approx -1.00" alt="x( -0.90 )\approx -1.00" title="x( -0.90 )\approx -1.00" data-latex="x( -0.90 )\approx -1.00"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.90 )\approx 1.10" alt="y( -0.90 )\approx 1.10" title="y( -0.90 )\approx 1.10" data-latex="y( -0.90 )\approx 1.10"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.80 )\approx -1.11" alt="x( -0.80 )\approx -1.11" title="x( -0.80 )\approx -1.11" data-latex="x( -0.80 )\approx -1.11"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.80 )\approx 1.40" alt="y( -0.80 )\approx 1.40" title="y( -0.80 )\approx 1.40" data-latex="y( -0.80 )\approx 1.40"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.90%20)%5Capprox%20-1.00" alt="x( -0.90 )\approx -1.00" title="x( -0.90 )\approx -1.00" data-latex="x( -0.90 )\approx -1.00"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.90%20)%5Capprox%201.10" alt="y( -0.90 )\approx 1.10" title="y( -0.90 )\approx 1.10" data-latex="y( -0.90 )\approx 1.10"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.80%20)%5Capprox%20-1.11" alt="x( -0.80 )\approx -1.11" title="x( -0.80 )\approx -1.11" data-latex="x( -0.80 )\approx -1.11"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.80%20)%5Capprox%201.40" alt="y( -0.80 )\approx 1.40" title="y( -0.80 )\approx 1.40" data-latex="y( -0.80 )\approx 1.40"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-0324" title="N2 | Euler's method for approximating IVP solutions | ver. 0324"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.8 )" alt="x( -1.8 )" title="x( -1.8 )" data-latex="x( -1.8 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.8 )" alt="y( -1.8 )" title="y( -1.8 )" data-latex="y( -1.8 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -x y^{2} + t y \hspace{2em} x( -2 )= 1" alt="x'= -x y^{2} + t y \hspace{2em} x( -2 )= 1" title="x'= -x y^{2} + t y \hspace{2em} x( -2 )= 1" data-latex="x'= -x y^{2} + t y \hspace{2em} x( -2 )= 1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -2 \, t x^{2} + 4 \, x y^{2} \hspace{2em} y( -2 )= -1" alt="y'= -2 \, t x^{2} + 4 \, x y^{2} \hspace{2em} y( -2 )= -1" title="y'= -2 \, t x^{2} + 4 \, x y^{2} \hspace{2em} y( -2 )= -1" data-latex="y'= -2 \, t x^{2} + 4 \, x y^{2} \hspace{2em} y( -2 )= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.8%20)" alt="x( -1.8 )" title="x( -1.8 )" data-latex="x( -1.8 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.8%20)" alt="y( -1.8 )" title="y( -1.8 )" data-latex="y( -1.8 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-x%20y%5E%7B2%7D%20+%20t%20y%20%5Chspace%7B2em%7D%20x(%20-2%20)=%201" alt="x'= -x y^{2} + t y \hspace{2em} x( -2 )= 1" title="x'= -x y^{2} + t y \hspace{2em} x( -2 )= 1" data-latex="x'= -x y^{2} + t y \hspace{2em} x( -2 )= 1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-2%20%5C,%20t%20x%5E%7B2%7D%20+%204%20%5C,%20x%20y%5E%7B2%7D%20%5Chspace%7B2em%7D%20y(%20-2%20)=%20-1" alt="y'= -2 \, t x^{2} + 4 \, x y^{2} \hspace{2em} y( -2 )= -1" title="y'= -2 \, t x^{2} + 4 \, x y^{2} \hspace{2em} y( -2 )= -1" data-latex="y'= -2 \, t x^{2} + 4 \, x y^{2} \hspace{2em} y( -2 )= -1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.9 )\approx 1.10" alt="x( -1.9 )\approx 1.10" title="x( -1.9 )\approx 1.10" data-latex="x( -1.9 )\approx 1.10"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.9 )\approx -0.200" alt="y( -1.9 )\approx -0.200" title="y( -1.9 )\approx -0.200" data-latex="y( -1.9 )\approx -0.200"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -1.8 )\approx 1.13" alt="x( -1.8 )\approx 1.13" title="x( -1.8 )\approx 1.13" data-latex="x( -1.8 )\approx 1.13"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1.8 )\approx 0.277" alt="y( -1.8 )\approx 0.277" title="y( -1.8 )\approx 0.277" data-latex="y( -1.8 )\approx 0.277"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.9%20)%5Capprox%201.10" alt="x( -1.9 )\approx 1.10" title="x( -1.9 )\approx 1.10" data-latex="x( -1.9 )\approx 1.10"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.9%20)%5Capprox%20-0.200" alt="y( -1.9 )\approx -0.200" title="y( -1.9 )\approx -0.200" data-latex="y( -1.9 )\approx -0.200"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-1.8%20)%5Capprox%201.13" alt="x( -1.8 )\approx 1.13" title="x( -1.8 )\approx 1.13" data-latex="x( -1.8 )\approx 1.13"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1.8%20)%5Capprox%200.277" alt="y( -1.8 )\approx 0.277" title="y( -1.8 )\approx 0.277" data-latex="y( -1.8 )\approx 0.277"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-2566" title="N2 | Euler's method for approximating IVP solutions | ver. 2566"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.2 )" alt="x( 1.2 )" title="x( 1.2 )" data-latex="x( 1.2 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.2 )" alt="y( 1.2 )" title="y( 1.2 )" data-latex="y( 1.2 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -2 \, t^{2} x + 4 \, x y^{2} + 2 \hspace{2em} x( 1 )= 0" alt="x'= -2 \, t^{2} x + 4 \, x y^{2} + 2 \hspace{2em} x( 1 )= 0" title="x'= -2 \, t^{2} x + 4 \, x y^{2} + 2 \hspace{2em} x( 1 )= 0" data-latex="x'= -2 \, t^{2} x + 4 \, x y^{2} + 2 \hspace{2em} x( 1 )= 0"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 3 \, t^{2} y^{2} - 4 \, x y^{2} + 1 \hspace{2em} y( 1 )= 0" alt="y'= 3 \, t^{2} y^{2} - 4 \, x y^{2} + 1 \hspace{2em} y( 1 )= 0" title="y'= 3 \, t^{2} y^{2} - 4 \, x y^{2} + 1 \hspace{2em} y( 1 )= 0" data-latex="y'= 3 \, t^{2} y^{2} - 4 \, x y^{2} + 1 \hspace{2em} y( 1 )= 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.2%20)" alt="x( 1.2 )" title="x( 1.2 )" data-latex="x( 1.2 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.2%20)" alt="y( 1.2 )" title="y( 1.2 )" data-latex="y( 1.2 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-2%20%5C,%20t%5E%7B2%7D%20x%20+%204%20%5C,%20x%20y%5E%7B2%7D%20+%202%20%5Chspace%7B2em%7D%20x(%201%20)=%200" alt="x'= -2 \, t^{2} x + 4 \, x y^{2} + 2 \hspace{2em} x( 1 )= 0" title="x'= -2 \, t^{2} x + 4 \, x y^{2} + 2 \hspace{2em} x( 1 )= 0" data-latex="x'= -2 \, t^{2} x + 4 \, x y^{2} + 2 \hspace{2em} x( 1 )= 0"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%203%20%5C,%20t%5E%7B2%7D%20y%5E%7B2%7D%20-%204%20%5C,%20x%20y%5E%7B2%7D%20+%201%20%5Chspace%7B2em%7D%20y(%201%20)=%200" alt="y'= 3 \, t^{2} y^{2} - 4 \, x y^{2} + 1 \hspace{2em} y( 1 )= 0" title="y'= 3 \, t^{2} y^{2} - 4 \, x y^{2} + 1 \hspace{2em} y( 1 )= 0" data-latex="y'= 3 \, t^{2} y^{2} - 4 \, x y^{2} + 1 \hspace{2em} y( 1 )= 0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.1 )\approx 0.200" alt="x( 1.1 )\approx 0.200" title="x( 1.1 )\approx 0.200" data-latex="x( 1.1 )\approx 0.200"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.1 )\approx 0.100" alt="y( 1.1 )\approx 0.100" title="y( 1.1 )\approx 0.100" data-latex="y( 1.1 )\approx 0.100"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 1.2 )\approx 0.353" alt="x( 1.2 )\approx 0.353" title="x( 1.2 )\approx 0.353" data-latex="x( 1.2 )\approx 0.353"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1.2 )\approx 0.203" alt="y( 1.2 )\approx 0.203" title="y( 1.2 )\approx 0.203" data-latex="y( 1.2 )\approx 0.203"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.1%20)%5Capprox%200.200" alt="x( 1.1 )\approx 0.200" title="x( 1.1 )\approx 0.200" data-latex="x( 1.1 )\approx 0.200"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.1%20)%5Capprox%200.100" alt="y( 1.1 )\approx 0.100" title="y( 1.1 )\approx 0.100" data-latex="y( 1.1 )\approx 0.100"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%201.2%20)%5Capprox%200.353" alt="x( 1.2 )\approx 0.353" title="x( 1.2 )\approx 0.353" data-latex="x( 1.2 )\approx 0.353"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201.2%20)%5Capprox%200.203" alt="y( 1.2 )\approx 0.203" title="y( 1.2 )\approx 0.203" data-latex="y( 1.2 )\approx 0.203"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="N2-6413" title="N2 | Euler's method for approximating IVP solutions | ver. 6413"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>N2.</strong></p><p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.80 )" alt="x( -0.80 )" title="x( -0.80 )" data-latex="x( -0.80 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.80 )" alt="y( -0.80 )" title="y( -0.80 )" data-latex="y( -0.80 )"/> given the following system of IVPs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 3 \, t y^{2} - 3 \, t x - 3 \hspace{2em} x( -1 )= -2" alt="x'= 3 \, t y^{2} - 3 \, t x - 3 \hspace{2em} x( -1 )= -2" title="x'= 3 \, t y^{2} - 3 \, t x - 3 \hspace{2em} x( -1 )= -2" data-latex="x'= 3 \, t y^{2} - 3 \, t x - 3 \hspace{2em} x( -1 )= -2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -3 \, t y^{2} - 2 \, t x - 3 \hspace{2em} y( -1 )= -1" alt="y'= -3 \, t y^{2} - 2 \, t x - 3 \hspace{2em} y( -1 )= -1" title="y'= -3 \, t y^{2} - 2 \, t x - 3 \hspace{2em} y( -1 )= -1" data-latex="y'= -3 \, t y^{2} - 2 \, t x - 3 \hspace{2em} y( -1 )= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.80%20)" alt="x( -0.80 )" title="x( -0.80 )" data-latex="x( -0.80 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.80%20)" alt="y( -0.80 )" title="y( -0.80 )" data-latex="y( -0.80 )"&gt; given the following system of IVPs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%203%20%5C,%20t%20y%5E%7B2%7D%20-%203%20%5C,%20t%20x%20-%203%20%5Chspace%7B2em%7D%20x(%20-1%20)=%20-2" alt="x'= 3 \, t y^{2} - 3 \, t x - 3 \hspace{2em} x( -1 )= -2" title="x'= 3 \, t y^{2} - 3 \, t x - 3 \hspace{2em} x( -1 )= -2" data-latex="x'= 3 \, t y^{2} - 3 \, t x - 3 \hspace{2em} x( -1 )= -2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-3%20%5C,%20t%20y%5E%7B2%7D%20-%202%20%5C,%20t%20x%20-%203%20%5Chspace%7B2em%7D%20y(%20-1%20)=%20-1" alt="y'= -3 \, t y^{2} - 2 \, t x - 3 \hspace{2em} y( -1 )= -1" title="y'= -3 \, t y^{2} - 2 \, t x - 3 \hspace{2em} y( -1 )= -1" data-latex="y'= -3 \, t y^{2} - 2 \, t x - 3 \hspace{2em} y( -1 )= -1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.90 )\approx -3.20" alt="x( -0.90 )\approx -3.20" title="x( -0.90 )\approx -3.20" data-latex="x( -0.90 )\approx -3.20"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.90 )\approx -1.40" alt="y( -0.90 )\approx -1.40" title="y( -0.90 )\approx -1.40" data-latex="y( -0.90 )\approx -1.40"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.80 )\approx -4.89" alt="x( -0.80 )\approx -4.89" title="x( -0.80 )\approx -4.89" data-latex="x( -0.80 )\approx -4.89"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.80 )\approx -1.75" alt="y( -0.80 )\approx -1.75" title="y( -0.80 )\approx -1.75" data-latex="y( -0.80 )\approx -1.75"/></li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.90%20)%5Capprox%20-3.20" alt="x( -0.90 )\approx -3.20" title="x( -0.90 )\approx -3.20" data-latex="x( -0.90 )\approx -3.20"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.90%20)%5Capprox%20-1.40" alt="y( -0.90 )\approx -1.40" title="y( -0.90 )\approx -1.40" data-latex="y( -0.90 )\approx -1.40"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.80%20)%5Capprox%20-4.89" alt="x( -0.80 )\approx -4.89" title="x( -0.80 )\approx -4.89" data-latex="x( -0.80 )\approx -4.89"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.80%20)%5Capprox%20-1.75" alt="y( -0.80 )\approx -1.75" title="y( -0.80 )\approx -1.75" data-latex="y( -0.80 )\approx -1.75"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item></objectbank>
</questestinterop>