CoCalc -- F1.qti

masterit / banks / clontz-diff-eq / build / qti-bank / F1.qti
²⁰⁸⁶⁵ views
<?xml version='1.0' encoding='UTF-8'?>
<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="F1">
    <qtimetadata>
      <qtimetadatafield><fieldlabel>bank_title</fieldlabel><fieldentry>Differential Equations -- F1</fieldentry></qtimetadatafield>
    </qtimetadata>
  <item ident="F1-4371" title="F1 | Direction fields for first-order ODEs | ver. 4371"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(sin(y/2), (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \sin\left(\frac{1}{2} \, {y}\right)" alt="{y'} = \sin\left(\frac{1}{2} \, {y}\right)" title="{y'} = \sin\left(\frac{1}{2} \, {y}\right)" data-latex="{y'} = \sin\left(\frac{1}{2} \, {y}\right)"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= -3" alt="t= -3" title="t= -3" data-latex="t= -3"/>. </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'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -1 )= -2" alt="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -1 )= -2" title="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -1 )= -2" data-latex="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -1 )= -2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(sin(y/2), (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Csin%5Cleft(%5Cfrac%7B1%7D%7B2%7D%20%5C,%20%7By%7D%5Cright)" alt="{y'} = \sin\left(\frac{1}{2} \, {y}\right)" title="{y'} = \sin\left(\frac{1}{2} \, {y}\right)" data-latex="{y'} = \sin\left(\frac{1}{2} \, {y}\right)"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%20-3" alt="t= -3" title="t= -3" data-latex="t= -3"&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?%7By'%7D%20=%20%5Csin%5Cleft(%5Cfrac%7B1%7D%7B2%7D%20%5C,%20%7By%7D%5Cright)%20%5Chspace%7B2em%7D%20y(%20-1%20)=%20-2" alt="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -1 )= -2" title="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -1 )= -2" data-latex="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( -3 )\approx -2.0" alt="y_p( -3 )\approx -2.0" title="y_p( -3 )\approx -2.0" data-latex="y_p( -3 )\approx -2.0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%20-3%20)%5Capprox%20-2.0" alt="y_p( -3 )\approx -2.0" title="y_p( -3 )\approx -2.0" data-latex="y_p( -3 )\approx -2.0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-1233" title="F1 | Direction fields for first-order ODEs | ver. 1233"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(sin(t+y), (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \sin\left({y} + t\right)" alt="{y'} = \sin\left({y} + t\right)" title="{y'} = \sin\left({y} + t\right)" data-latex="{y'} = \sin\left({y} + t\right)"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= -2" alt="t= -2" title="t= -2" data-latex="t= -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'} = \sin\left({y} + t\right) \hspace{2em} y( 0 )= 1" alt="{y'} = \sin\left({y} + t\right) \hspace{2em} y( 0 )= 1" title="{y'} = \sin\left({y} + t\right) \hspace{2em} y( 0 )= 1" data-latex="{y'} = \sin\left({y} + t\right) \hspace{2em} y( 0 )= 1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(sin(t+y), (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Csin%5Cleft(%7By%7D%20+%20t%5Cright)" alt="{y'} = \sin\left({y} + t\right)" title="{y'} = \sin\left({y} + t\right)" data-latex="{y'} = \sin\left({y} + t\right)"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%20-2" alt="t= -2" title="t= -2" data-latex="t= -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?%7By'%7D%20=%20%5Csin%5Cleft(%7By%7D%20+%20t%5Cright)%20%5Chspace%7B2em%7D%20y(%200%20)=%201" alt="{y'} = \sin\left({y} + t\right) \hspace{2em} y( 0 )= 1" title="{y'} = \sin\left({y} + t\right) \hspace{2em} y( 0 )= 1" data-latex="{y'} = \sin\left({y} + t\right) \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( -2 )\approx 1.0" alt="y_p( -2 )\approx 1.0" title="y_p( -2 )\approx 1.0" data-latex="y_p( -2 )\approx 1.0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%20-2%20)%5Capprox%201.0" alt="y_p( -2 )\approx 1.0" title="y_p( -2 )\approx 1.0" data-latex="y_p( -2 )\approx 1.0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-0018" title="F1 | Direction fields for first-order ODEs | ver. 0018"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(t*y/9+t/3, (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \frac{1}{9} \, t {y} + \frac{1}{3} \, t" alt="{y'} = \frac{1}{9} \, t {y} + \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, t {y} + \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, t {y} + \frac{1}{3} \, t"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 2" alt="t= 2" title="t= 2" data-latex="t= 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'} = \frac{1}{9} \, t {y} + \frac{1}{3} \, t \hspace{2em} y( 0 )= -2" alt="{y'} = \frac{1}{9} \, t {y} + \frac{1}{3} \, t \hspace{2em} y( 0 )= -2" title="{y'} = \frac{1}{9} \, t {y} + \frac{1}{3} \, t \hspace{2em} y( 0 )= -2" data-latex="{y'} = \frac{1}{9} \, t {y} + \frac{1}{3} \, t \hspace{2em} y( 0 )= -2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(t*y/9+t/3, (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20t%20%7By%7D%20+%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t" alt="{y'} = \frac{1}{9} \, t {y} + \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, t {y} + \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, t {y} + \frac{1}{3} \, t"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%202" alt="t= 2" title="t= 2" data-latex="t= 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?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20t%20%7By%7D%20+%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20%5Chspace%7B2em%7D%20y(%200%20)=%20-2" alt="{y'} = \frac{1}{9} \, t {y} + \frac{1}{3} \, t \hspace{2em} y( 0 )= -2" title="{y'} = \frac{1}{9} \, t {y} + \frac{1}{3} \, t \hspace{2em} y( 0 )= -2" data-latex="{y'} = \frac{1}{9} \, t {y} + \frac{1}{3} \, t \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 2 )\approx -1.8" alt="y_p( 2 )\approx -1.8" title="y_p( 2 )\approx -1.8" data-latex="y_p( 2 )\approx -1.8"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%202%20)%5Capprox%20-1.8" alt="y_p( 2 )\approx -1.8" title="y_p( 2 )\approx -1.8" data-latex="y_p( 2 )\approx -1.8"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-1974" title="F1 | Direction fields for first-order ODEs | ver. 1974"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(t/3-y/15, (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t" alt="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t" title="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t" data-latex="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= -1" alt="t= -1" title="t= -1" data-latex="t= -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'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( 1 )= 2" alt="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( 1 )= 2" title="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( 1 )= 2" data-latex="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( 1 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(t/3-y/15, (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20-%5Cfrac%7B1%7D%7B15%7D%20%5C,%20%7By%7D%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t" alt="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t" title="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t" data-latex="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%20-1" alt="t= -1" title="t= -1" data-latex="t= -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?%7By'%7D%20=%20-%5Cfrac%7B1%7D%7B15%7D%20%5C,%20%7By%7D%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20%5Chspace%7B2em%7D%20y(%201%20)=%202" alt="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( 1 )= 2" title="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( 1 )= 2" data-latex="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( -1 )\approx 2.0" alt="y_p( -1 )\approx 2.0" title="y_p( -1 )\approx 2.0" data-latex="y_p( -1 )\approx 2.0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%20-1%20)%5Capprox%202.0" alt="y_p( -1 )\approx 2.0" title="y_p( -1 )\approx 2.0" data-latex="y_p( -1 )\approx 2.0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-2482" title="F1 | Direction fields for first-order ODEs | ver. 2482"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(t*y/9+t/3, (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" alt="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 0" alt="t= 0" title="t= 0" data-latex="t= 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'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( -2 )= 2" alt="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( -2 )= 2" title="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( -2 )= 2" data-latex="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( -2 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(t*y/9+t/3, (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20%7By%7D%20t%20+%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t" alt="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%200" alt="t= 0" title="t= 0" data-latex="t= 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?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20%7By%7D%20t%20+%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20%5Chspace%7B2em%7D%20y(%20-2%20)=%202" alt="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( -2 )= 2" title="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( -2 )= 2" data-latex="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 0 )\approx 1.0" alt="y_p( 0 )\approx 1.0" title="y_p( 0 )\approx 1.0" data-latex="y_p( 0 )\approx 1.0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%200%20)%5Capprox%201.0" alt="y_p( 0 )\approx 1.0" title="y_p( 0 )\approx 1.0" data-latex="y_p( 0 )\approx 1.0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-0688" title="F1 | Direction fields for first-order ODEs | ver. 0688"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(cos(y/2), (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \cos\left(\frac{1}{2} \, {y}\right)" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right)"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 1" alt="t= 1" title="t= 1" data-latex="t= 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'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -1 )= 0" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -1 )= 0" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -1 )= 0" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -1 )= 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(cos(y/2), (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Ccos%5Cleft(%5Cfrac%7B1%7D%7B2%7D%20%5C,%20%7By%7D%5Cright)" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right)"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%201" alt="t= 1" title="t= 1" data-latex="t= 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?%7By'%7D%20=%20%5Ccos%5Cleft(%5Cfrac%7B1%7D%7B2%7D%20%5C,%20%7By%7D%5Cright)%20%5Chspace%7B2em%7D%20y(%20-1%20)=%200" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -1 )= 0" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -1 )= 0" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 1 )\approx 1.7" alt="y_p( 1 )\approx 1.7" title="y_p( 1 )\approx 1.7" data-latex="y_p( 1 )\approx 1.7"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%201%20)%5Capprox%201.7" alt="y_p( 1 )\approx 1.7" title="y_p( 1 )\approx 1.7" data-latex="y_p( 1 )\approx 1.7"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-9488" title="F1 | Direction fields for first-order ODEs | ver. 9488"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(t*y/9+t/3, (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" alt="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 0" alt="t= 0" title="t= 0" data-latex="t= 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'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( 2 )= -2" alt="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( 2 )= -2" title="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( 2 )= -2" data-latex="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( 2 )= -2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(t*y/9+t/3, (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20%7By%7D%20t%20+%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t" alt="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%200" alt="t= 0" title="t= 0" data-latex="t= 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?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20%7By%7D%20t%20+%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20%5Chspace%7B2em%7D%20y(%202%20)=%20-2" alt="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( 2 )= -2" title="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( 2 )= -2" data-latex="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 0 )\approx -2.0" alt="y_p( 0 )\approx -2.0" title="y_p( 0 )\approx -2.0" data-latex="y_p( 0 )\approx -2.0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%200%20)%5Capprox%20-2.0" alt="y_p( 0 )\approx -2.0" title="y_p( 0 )\approx -2.0" data-latex="y_p( 0 )\approx -2.0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-6113" title="F1 | Direction fields for first-order ODEs | ver. 6113"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(cos(t+y), (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \cos\left({y} + t\right)" alt="{y'} = \cos\left({y} + t\right)" title="{y'} = \cos\left({y} + t\right)" data-latex="{y'} = \cos\left({y} + t\right)"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 0" alt="t= 0" title="t= 0" data-latex="t= 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'} = \cos\left({y} + t\right) \hspace{2em} y( 2 )= -1" alt="{y'} = \cos\left({y} + t\right) \hspace{2em} y( 2 )= -1" title="{y'} = \cos\left({y} + t\right) \hspace{2em} y( 2 )= -1" data-latex="{y'} = \cos\left({y} + t\right) \hspace{2em} y( 2 )= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(cos(t+y), (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Ccos%5Cleft(%7By%7D%20+%20t%5Cright)" alt="{y'} = \cos\left({y} + t\right)" title="{y'} = \cos\left({y} + t\right)" data-latex="{y'} = \cos\left({y} + t\right)"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%200" alt="t= 0" title="t= 0" data-latex="t= 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?%7By'%7D%20=%20%5Ccos%5Cleft(%7By%7D%20+%20t%5Cright)%20%5Chspace%7B2em%7D%20y(%202%20)=%20-1" alt="{y'} = \cos\left({y} + t\right) \hspace{2em} y( 2 )= -1" title="{y'} = \cos\left({y} + t\right) \hspace{2em} y( 2 )= -1" data-latex="{y'} = \cos\left({y} + t\right) \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 0 )\approx -1.0" alt="y_p( 0 )\approx -1.0" title="y_p( 0 )\approx -1.0" data-latex="y_p( 0 )\approx -1.0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%200%20)%5Capprox%20-1.0" alt="y_p( 0 )\approx -1.0" title="y_p( 0 )\approx -1.0" data-latex="y_p( 0 )\approx -1.0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-8703" title="F1 | Direction fields for first-order ODEs | ver. 8703"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(cos(t+y), (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \cos\left({y} + t\right)" alt="{y'} = \cos\left({y} + t\right)" title="{y'} = \cos\left({y} + t\right)" data-latex="{y'} = \cos\left({y} + t\right)"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= -4" alt="t= -4" title="t= -4" data-latex="t= -4"/>. </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'} = \cos\left({y} + t\right) \hspace{2em} y( -2 )= 2" alt="{y'} = \cos\left({y} + t\right) \hspace{2em} y( -2 )= 2" title="{y'} = \cos\left({y} + t\right) \hspace{2em} y( -2 )= 2" data-latex="{y'} = \cos\left({y} + t\right) \hspace{2em} y( -2 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(cos(t+y), (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Ccos%5Cleft(%7By%7D%20+%20t%5Cright)" alt="{y'} = \cos\left({y} + t\right)" title="{y'} = \cos\left({y} + t\right)" data-latex="{y'} = \cos\left({y} + t\right)"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%20-4" alt="t= -4" title="t= -4" data-latex="t= -4"&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?%7By'%7D%20=%20%5Ccos%5Cleft(%7By%7D%20+%20t%5Cright)%20%5Chspace%7B2em%7D%20y(%20-2%20)=%202" alt="{y'} = \cos\left({y} + t\right) \hspace{2em} y( -2 )= 2" title="{y'} = \cos\left({y} + t\right) \hspace{2em} y( -2 )= 2" data-latex="{y'} = \cos\left({y} + t\right) \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( -4 )\approx 2.0" alt="y_p( -4 )\approx 2.0" title="y_p( -4 )\approx 2.0" data-latex="y_p( -4 )\approx 2.0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%20-4%20)%5Capprox%202.0" alt="y_p( -4 )\approx 2.0" title="y_p( -4 )\approx 2.0" data-latex="y_p( -4 )\approx 2.0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-9039" title="F1 | Direction fields for first-order ODEs | ver. 9039"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(t*y/9-t/3, (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" alt="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= -2" alt="t= -2" title="t= -2" data-latex="t= -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'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( 0 )= 2" alt="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( 0 )= 2" title="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( 0 )= 2" data-latex="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( 0 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(t*y/9-t/3, (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20%7By%7D%20t%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t" alt="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%20-2" alt="t= -2" title="t= -2" data-latex="t= -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?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20%7By%7D%20t%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20%5Chspace%7B2em%7D%20y(%200%20)=%202" alt="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( 0 )= 2" title="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( 0 )= 2" data-latex="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( -2 )\approx 2.0" alt="y_p( -2 )\approx 2.0" title="y_p( -2 )\approx 2.0" data-latex="y_p( -2 )\approx 2.0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%20-2%20)%5Capprox%202.0" alt="y_p( -2 )\approx 2.0" title="y_p( -2 )\approx 2.0" data-latex="y_p( -2 )\approx 2.0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-9430" title="F1 | Direction fields for first-order ODEs | ver. 9430"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(cos(t+y), (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \cos\left({y} + t\right)" alt="{y'} = \cos\left({y} + t\right)" title="{y'} = \cos\left({y} + t\right)" data-latex="{y'} = \cos\left({y} + t\right)"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= -1" alt="t= -1" title="t= -1" data-latex="t= -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'} = \cos\left({y} + t\right) \hspace{2em} y( 1 )= -2" alt="{y'} = \cos\left({y} + t\right) \hspace{2em} y( 1 )= -2" title="{y'} = \cos\left({y} + t\right) \hspace{2em} y( 1 )= -2" data-latex="{y'} = \cos\left({y} + t\right) \hspace{2em} y( 1 )= -2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(cos(t+y), (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Ccos%5Cleft(%7By%7D%20+%20t%5Cright)" alt="{y'} = \cos\left({y} + t\right)" title="{y'} = \cos\left({y} + t\right)" data-latex="{y'} = \cos\left({y} + t\right)"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%20-1" alt="t= -1" title="t= -1" data-latex="t= -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?%7By'%7D%20=%20%5Ccos%5Cleft(%7By%7D%20+%20t%5Cright)%20%5Chspace%7B2em%7D%20y(%201%20)=%20-2" alt="{y'} = \cos\left({y} + t\right) \hspace{2em} y( 1 )= -2" title="{y'} = \cos\left({y} + t\right) \hspace{2em} y( 1 )= -2" data-latex="{y'} = \cos\left({y} + t\right) \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( -1 )\approx -2.0" alt="y_p( -1 )\approx -2.0" title="y_p( -1 )\approx -2.0" data-latex="y_p( -1 )\approx -2.0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%20-1%20)%5Capprox%20-2.0" alt="y_p( -1 )\approx -2.0" title="y_p( -1 )\approx -2.0" data-latex="y_p( -1 )\approx -2.0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-6854" title="F1 | Direction fields for first-order ODEs | ver. 6854"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(t*y/9-t/3, (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" alt="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 4" alt="t= 4" title="t= 4" data-latex="t= 4"/>. </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'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( 2 )= 1" alt="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( 2 )= 1" title="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( 2 )= 1" data-latex="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( 2 )= 1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(t*y/9-t/3, (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20%7By%7D%20t%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t" alt="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%204" alt="t= 4" title="t= 4" data-latex="t= 4"&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?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20%7By%7D%20t%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20%5Chspace%7B2em%7D%20y(%202%20)=%201" alt="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( 2 )= 1" title="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( 2 )= 1" data-latex="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 4 )\approx -0.90" alt="y_p( 4 )\approx -0.90" title="y_p( 4 )\approx -0.90" data-latex="y_p( 4 )\approx -0.90"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%204%20)%5Capprox%20-0.90" alt="y_p( 4 )\approx -0.90" title="y_p( 4 )\approx -0.90" data-latex="y_p( 4 )\approx -0.90"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-2450" title="F1 | Direction fields for first-order ODEs | ver. 2450"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(y/15-t/3, (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t" alt="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t" title="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 3" alt="t= 3" title="t= 3" data-latex="t= 3"/>. </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'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( 1 )= 2" alt="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( 1 )= 2" title="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( 1 )= 2" data-latex="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( 1 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(y/15-t/3, (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B15%7D%20%5C,%20%7By%7D%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t" alt="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t" title="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%203" alt="t= 3" title="t= 3" data-latex="t= 3"&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?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B15%7D%20%5C,%20%7By%7D%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20%5Chspace%7B2em%7D%20y(%201%20)=%202" alt="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( 1 )= 2" title="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( 1 )= 2" data-latex="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 3 )\approx 0.88" alt="y_p( 3 )\approx 0.88" title="y_p( 3 )\approx 0.88" data-latex="y_p( 3 )\approx 0.88"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%203%20)%5Capprox%200.88" alt="y_p( 3 )\approx 0.88" title="y_p( 3 )\approx 0.88" data-latex="y_p( 3 )\approx 0.88"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-5124" title="F1 | Direction fields for first-order ODEs | ver. 5124"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(y/15-t/3, (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t" alt="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t" title="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 0" alt="t= 0" title="t= 0" data-latex="t= 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'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( -2 )= -1" alt="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( -2 )= -1" title="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( -2 )= -1" data-latex="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( -2 )= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(y/15-t/3, (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B15%7D%20%5C,%20%7By%7D%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t" alt="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t" title="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%200" alt="t= 0" title="t= 0" data-latex="t= 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?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B15%7D%20%5C,%20%7By%7D%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20%5Chspace%7B2em%7D%20y(%20-2%20)=%20-1" alt="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( -2 )= -1" title="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( -2 )= -1" data-latex="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 0 )\approx -0.41" alt="y_p( 0 )\approx -0.41" title="y_p( 0 )\approx -0.41" data-latex="y_p( 0 )\approx -0.41"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%200%20)%5Capprox%20-0.41" alt="y_p( 0 )\approx -0.41" title="y_p( 0 )\approx -0.41" data-latex="y_p( 0 )\approx -0.41"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-0367" title="F1 | Direction fields for first-order ODEs | ver. 0367"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(t*y/9-t/3, (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \frac{1}{9} \, t {y} - \frac{1}{3} \, t" alt="{y'} = \frac{1}{9} \, t {y} - \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, t {y} - \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, t {y} - \frac{1}{3} \, t"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 1" alt="t= 1" title="t= 1" data-latex="t= 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'} = \frac{1}{9} \, t {y} - \frac{1}{3} \, t \hspace{2em} y( -1 )= 1" alt="{y'} = \frac{1}{9} \, t {y} - \frac{1}{3} \, t \hspace{2em} y( -1 )= 1" title="{y'} = \frac{1}{9} \, t {y} - \frac{1}{3} \, t \hspace{2em} y( -1 )= 1" data-latex="{y'} = \frac{1}{9} \, t {y} - \frac{1}{3} \, t \hspace{2em} y( -1 )= 1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(t*y/9-t/3, (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20t%20%7By%7D%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t" alt="{y'} = \frac{1}{9} \, t {y} - \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, t {y} - \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, t {y} - \frac{1}{3} \, t"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%201" alt="t= 1" title="t= 1" data-latex="t= 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?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20t%20%7By%7D%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20%5Chspace%7B2em%7D%20y(%20-1%20)=%201" alt="{y'} = \frac{1}{9} \, t {y} - \frac{1}{3} \, t \hspace{2em} y( -1 )= 1" title="{y'} = \frac{1}{9} \, t {y} - \frac{1}{3} \, t \hspace{2em} y( -1 )= 1" data-latex="{y'} = \frac{1}{9} \, t {y} - \frac{1}{3} \, t \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 1 )\approx 1.0" alt="y_p( 1 )\approx 1.0" title="y_p( 1 )\approx 1.0" data-latex="y_p( 1 )\approx 1.0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%201%20)%5Capprox%201.0" alt="y_p( 1 )\approx 1.0" title="y_p( 1 )\approx 1.0" data-latex="y_p( 1 )\approx 1.0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-1316" title="F1 | Direction fields for first-order ODEs | ver. 1316"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(cos(t+y), (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \cos\left({y} + t\right)" alt="{y'} = \cos\left({y} + t\right)" title="{y'} = \cos\left({y} + t\right)" data-latex="{y'} = \cos\left({y} + t\right)"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 3" alt="t= 3" title="t= 3" data-latex="t= 3"/>. </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'} = \cos\left({y} + t\right) \hspace{2em} y( 1 )= -1" alt="{y'} = \cos\left({y} + t\right) \hspace{2em} y( 1 )= -1" title="{y'} = \cos\left({y} + t\right) \hspace{2em} y( 1 )= -1" data-latex="{y'} = \cos\left({y} + t\right) \hspace{2em} y( 1 )= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(cos(t+y), (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Ccos%5Cleft(%7By%7D%20+%20t%5Cright)" alt="{y'} = \cos\left({y} + t\right)" title="{y'} = \cos\left({y} + t\right)" data-latex="{y'} = \cos\left({y} + t\right)"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%203" alt="t= 3" title="t= 3" data-latex="t= 3"&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?%7By'%7D%20=%20%5Ccos%5Cleft(%7By%7D%20+%20t%5Cright)%20%5Chspace%7B2em%7D%20y(%201%20)=%20-1" alt="{y'} = \cos\left({y} + t\right) \hspace{2em} y( 1 )= -1" title="{y'} = \cos\left({y} + t\right) \hspace{2em} y( 1 )= -1" data-latex="{y'} = \cos\left({y} + t\right) \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 3 )\approx -0.79" alt="y_p( 3 )\approx -0.79" title="y_p( 3 )\approx -0.79" data-latex="y_p( 3 )\approx -0.79"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%203%20)%5Capprox%20-0.79" alt="y_p( 3 )\approx -0.79" title="y_p( 3 )\approx -0.79" data-latex="y_p( 3 )\approx -0.79"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-4675" title="F1 | Direction fields for first-order ODEs | ver. 4675"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(y/15-t/3, (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y}" alt="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y}" title="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y}" data-latex="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y}"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= -4" alt="t= -4" title="t= -4" data-latex="t= -4"/>. </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'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y} \hspace{2em} y( -2 )= 2" alt="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y} \hspace{2em} y( -2 )= 2" title="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y} \hspace{2em} y( -2 )= 2" data-latex="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {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;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(y/15-t/3, (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20-%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20+%20%5Cfrac%7B1%7D%7B15%7D%20%5C,%20%7By%7D" alt="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y}" title="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y}" data-latex="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y}"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%20-4" alt="t= -4" title="t= -4" data-latex="t= -4"&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?%7By'%7D%20=%20-%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20+%20%5Cfrac%7B1%7D%7B15%7D%20%5C,%20%7By%7D%20%5Chspace%7B2em%7D%20y(%20-2%20)=%202" alt="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y} \hspace{2em} y( -2 )= 2" title="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y} \hspace{2em} y( -2 )= 2" data-latex="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( -4 )\approx 2.0" alt="y_p( -4 )\approx 2.0" title="y_p( -4 )\approx 2.0" data-latex="y_p( -4 )\approx 2.0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%20-4%20)%5Capprox%202.0" alt="y_p( -4 )\approx 2.0" title="y_p( -4 )\approx 2.0" data-latex="y_p( -4 )\approx 2.0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-9697" title="F1 | Direction fields for first-order ODEs | ver. 9697"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(t*y/9-t/3, (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" alt="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= -3" alt="t= -3" title="t= -3" data-latex="t= -3"/>. </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'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( -1 )= 1" alt="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( -1 )= 1" title="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( -1 )= 1" data-latex="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( -1 )= 1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(t*y/9-t/3, (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20%7By%7D%20t%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t" alt="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%20-3" alt="t= -3" title="t= -3" data-latex="t= -3"&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?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20%7By%7D%20t%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20%5Chspace%7B2em%7D%20y(%20-1%20)=%201" alt="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( -1 )= 1" title="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( -1 )= 1" data-latex="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( -3 )\approx 1.0" alt="y_p( -3 )\approx 1.0" title="y_p( -3 )\approx 1.0" data-latex="y_p( -3 )\approx 1.0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%20-3%20)%5Capprox%201.0" alt="y_p( -3 )\approx 1.0" title="y_p( -3 )\approx 1.0" data-latex="y_p( -3 )\approx 1.0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-5269" title="F1 | Direction fields for first-order ODEs | ver. 5269"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(sin(t+y), (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \sin\left(t + {y}\right)" alt="{y'} = \sin\left(t + {y}\right)" title="{y'} = \sin\left(t + {y}\right)" data-latex="{y'} = \sin\left(t + {y}\right)"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 4" alt="t= 4" title="t= 4" data-latex="t= 4"/>. </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'} = \sin\left(t + {y}\right) \hspace{2em} y( 2 )= -1" alt="{y'} = \sin\left(t + {y}\right) \hspace{2em} y( 2 )= -1" title="{y'} = \sin\left(t + {y}\right) \hspace{2em} y( 2 )= -1" data-latex="{y'} = \sin\left(t + {y}\right) \hspace{2em} y( 2 )= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(sin(t+y), (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Csin%5Cleft(t%20+%20%7By%7D%5Cright)" alt="{y'} = \sin\left(t + {y}\right)" title="{y'} = \sin\left(t + {y}\right)" data-latex="{y'} = \sin\left(t + {y}\right)"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%204" alt="t= 4" title="t= 4" data-latex="t= 4"&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?%7By'%7D%20=%20%5Csin%5Cleft(t%20+%20%7By%7D%5Cright)%20%5Chspace%7B2em%7D%20y(%202%20)=%20-1" alt="{y'} = \sin\left(t + {y}\right) \hspace{2em} y( 2 )= -1" title="{y'} = \sin\left(t + {y}\right) \hspace{2em} y( 2 )= -1" data-latex="{y'} = \sin\left(t + {y}\right) \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 4 )\approx -0.35" alt="y_p( 4 )\approx -0.35" title="y_p( 4 )\approx -0.35" data-latex="y_p( 4 )\approx -0.35"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%204%20)%5Capprox%20-0.35" alt="y_p( 4 )\approx -0.35" title="y_p( 4 )\approx -0.35" data-latex="y_p( 4 )\approx -0.35"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-4205" title="F1 | Direction fields for first-order ODEs | ver. 4205"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(t*y/9-t/3, (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" alt="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= -1" alt="t= -1" title="t= -1" data-latex="t= -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'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( 1 )= 0" alt="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( 1 )= 0" title="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( 1 )= 0" data-latex="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( 1 )= 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(t*y/9-t/3, (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20%7By%7D%20t%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t" alt="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%20-1" alt="t= -1" title="t= -1" data-latex="t= -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?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20%7By%7D%20t%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20%5Chspace%7B2em%7D%20y(%201%20)=%200" alt="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( 1 )= 0" title="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( 1 )= 0" data-latex="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( -1 )\approx 0.00" alt="y_p( -1 )\approx 0.00" title="y_p( -1 )\approx 0.00" data-latex="y_p( -1 )\approx 0.00"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%20-1%20)%5Capprox%200.00" alt="y_p( -1 )\approx 0.00" title="y_p( -1 )\approx 0.00" data-latex="y_p( -1 )\approx 0.00"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-0889" title="F1 | Direction fields for first-order ODEs | ver. 0889"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(t*y/9+t/3, (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" alt="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= -3" alt="t= -3" title="t= -3" data-latex="t= -3"/>. </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'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( -1 )= 1" alt="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( -1 )= 1" title="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( -1 )= 1" data-latex="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( -1 )= 1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(t*y/9+t/3, (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20%7By%7D%20t%20+%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t" alt="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%20-3" alt="t= -3" title="t= -3" data-latex="t= -3"&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?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20%7By%7D%20t%20+%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20%5Chspace%7B2em%7D%20y(%20-1%20)=%201" alt="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( -1 )= 1" title="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( -1 )= 1" data-latex="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( -3 )\approx 1.0" alt="y_p( -3 )\approx 1.0" title="y_p( -3 )\approx 1.0" data-latex="y_p( -3 )\approx 1.0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%20-3%20)%5Capprox%201.0" alt="y_p( -3 )\approx 1.0" title="y_p( -3 )\approx 1.0" data-latex="y_p( -3 )\approx 1.0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-5189" title="F1 | Direction fields for first-order ODEs | ver. 5189"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(cos(t+y), (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \cos\left({y} + t\right)" alt="{y'} = \cos\left({y} + t\right)" title="{y'} = \cos\left({y} + t\right)" data-latex="{y'} = \cos\left({y} + t\right)"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 0" alt="t= 0" title="t= 0" data-latex="t= 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'} = \cos\left({y} + t\right) \hspace{2em} y( 2 )= -1" alt="{y'} = \cos\left({y} + t\right) \hspace{2em} y( 2 )= -1" title="{y'} = \cos\left({y} + t\right) \hspace{2em} y( 2 )= -1" data-latex="{y'} = \cos\left({y} + t\right) \hspace{2em} y( 2 )= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(cos(t+y), (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Ccos%5Cleft(%7By%7D%20+%20t%5Cright)" alt="{y'} = \cos\left({y} + t\right)" title="{y'} = \cos\left({y} + t\right)" data-latex="{y'} = \cos\left({y} + t\right)"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%200" alt="t= 0" title="t= 0" data-latex="t= 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?%7By'%7D%20=%20%5Ccos%5Cleft(%7By%7D%20+%20t%5Cright)%20%5Chspace%7B2em%7D%20y(%202%20)=%20-1" alt="{y'} = \cos\left({y} + t\right) \hspace{2em} y( 2 )= -1" title="{y'} = \cos\left({y} + t\right) \hspace{2em} y( 2 )= -1" data-latex="{y'} = \cos\left({y} + t\right) \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 0 )\approx -1.0" alt="y_p( 0 )\approx -1.0" title="y_p( 0 )\approx -1.0" data-latex="y_p( 0 )\approx -1.0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%200%20)%5Capprox%20-1.0" alt="y_p( 0 )\approx -1.0" title="y_p( 0 )\approx -1.0" data-latex="y_p( 0 )\approx -1.0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-5694" title="F1 | Direction fields for first-order ODEs | ver. 5694"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(cos(y/2), (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \cos\left(\frac{1}{2} \, {y}\right)" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right)"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= -3" alt="t= -3" title="t= -3" data-latex="t= -3"/>. </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'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -1 )= 2" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -1 )= 2" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -1 )= 2" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -1 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(cos(y/2), (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Ccos%5Cleft(%5Cfrac%7B1%7D%7B2%7D%20%5C,%20%7By%7D%5Cright)" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right)"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%20-3" alt="t= -3" title="t= -3" data-latex="t= -3"&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?%7By'%7D%20=%20%5Ccos%5Cleft(%5Cfrac%7B1%7D%7B2%7D%20%5C,%20%7By%7D%5Cright)%20%5Chspace%7B2em%7D%20y(%20-1%20)=%202" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -1 )= 2" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -1 )= 2" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( -3 )\approx 2.0" alt="y_p( -3 )\approx 2.0" title="y_p( -3 )\approx 2.0" data-latex="y_p( -3 )\approx 2.0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%20-3%20)%5Capprox%202.0" alt="y_p( -3 )\approx 2.0" title="y_p( -3 )\approx 2.0" data-latex="y_p( -3 )\approx 2.0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-2239" title="F1 | Direction fields for first-order ODEs | ver. 2239"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(cos(y/2), (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \cos\left(\frac{1}{2} \, {y}\right)" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right)"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 1" alt="t= 1" title="t= 1" data-latex="t= 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'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -1 )= 0" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -1 )= 0" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -1 )= 0" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -1 )= 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(cos(y/2), (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Ccos%5Cleft(%5Cfrac%7B1%7D%7B2%7D%20%5C,%20%7By%7D%5Cright)" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right)"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%201" alt="t= 1" title="t= 1" data-latex="t= 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?%7By'%7D%20=%20%5Ccos%5Cleft(%5Cfrac%7B1%7D%7B2%7D%20%5C,%20%7By%7D%5Cright)%20%5Chspace%7B2em%7D%20y(%20-1%20)=%200" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -1 )= 0" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -1 )= 0" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 1 )\approx 1.7" alt="y_p( 1 )\approx 1.7" title="y_p( 1 )\approx 1.7" data-latex="y_p( 1 )\approx 1.7"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%201%20)%5Capprox%201.7" alt="y_p( 1 )\approx 1.7" title="y_p( 1 )\approx 1.7" data-latex="y_p( 1 )\approx 1.7"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-0038" title="F1 | Direction fields for first-order ODEs | ver. 0038"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(sin(t+y), (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \sin\left({y} + t\right)" alt="{y'} = \sin\left({y} + t\right)" title="{y'} = \sin\left({y} + t\right)" data-latex="{y'} = \sin\left({y} + t\right)"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= -4" alt="t= -4" title="t= -4" data-latex="t= -4"/>. </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'} = \sin\left({y} + t\right) \hspace{2em} y( -2 )= 1" alt="{y'} = \sin\left({y} + t\right) \hspace{2em} y( -2 )= 1" title="{y'} = \sin\left({y} + t\right) \hspace{2em} y( -2 )= 1" data-latex="{y'} = \sin\left({y} + t\right) \hspace{2em} y( -2 )= 1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(sin(t+y), (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Csin%5Cleft(%7By%7D%20+%20t%5Cright)" alt="{y'} = \sin\left({y} + t\right)" title="{y'} = \sin\left({y} + t\right)" data-latex="{y'} = \sin\left({y} + t\right)"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%20-4" alt="t= -4" title="t= -4" data-latex="t= -4"&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?%7By'%7D%20=%20%5Csin%5Cleft(%7By%7D%20+%20t%5Cright)%20%5Chspace%7B2em%7D%20y(%20-2%20)=%201" alt="{y'} = \sin\left({y} + t\right) \hspace{2em} y( -2 )= 1" title="{y'} = \sin\left({y} + t\right) \hspace{2em} y( -2 )= 1" data-latex="{y'} = \sin\left({y} + t\right) \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( -4 )\approx 1.0" alt="y_p( -4 )\approx 1.0" title="y_p( -4 )\approx 1.0" data-latex="y_p( -4 )\approx 1.0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%20-4%20)%5Capprox%201.0" alt="y_p( -4 )\approx 1.0" title="y_p( -4 )\approx 1.0" data-latex="y_p( -4 )\approx 1.0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-2833" title="F1 | Direction fields for first-order ODEs | ver. 2833"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(t*y/9-t/3, (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" alt="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 2" alt="t= 2" title="t= 2" data-latex="t= 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'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( 0 )= 2" alt="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( 0 )= 2" title="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( 0 )= 2" data-latex="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( 0 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(t*y/9-t/3, (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20%7By%7D%20t%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t" alt="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%202" alt="t= 2" title="t= 2" data-latex="t= 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?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20%7By%7D%20t%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20%5Chspace%7B2em%7D%20y(%200%20)=%202" alt="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( 0 )= 2" title="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( 0 )= 2" data-latex="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 2 )\approx 1.8" alt="y_p( 2 )\approx 1.8" title="y_p( 2 )\approx 1.8" data-latex="y_p( 2 )\approx 1.8"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%202%20)%5Capprox%201.8" alt="y_p( 2 )\approx 1.8" title="y_p( 2 )\approx 1.8" data-latex="y_p( 2 )\approx 1.8"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-5016" title="F1 | Direction fields for first-order ODEs | ver. 5016"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(t*y/9+t/3, (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" alt="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 3" alt="t= 3" title="t= 3" data-latex="t= 3"/>. </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'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( 1 )= 2" alt="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( 1 )= 2" title="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( 1 )= 2" data-latex="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( 1 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(t*y/9+t/3, (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20%7By%7D%20t%20+%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t" alt="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%203" alt="t= 3" title="t= 3" data-latex="t= 3"&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?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20%7By%7D%20t%20+%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20%5Chspace%7B2em%7D%20y(%201%20)=%202" alt="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( 1 )= 2" title="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( 1 )= 2" data-latex="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 3 )\approx 4.8" alt="y_p( 3 )\approx 4.8" title="y_p( 3 )\approx 4.8" data-latex="y_p( 3 )\approx 4.8"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%203%20)%5Capprox%204.8" alt="y_p( 3 )\approx 4.8" title="y_p( 3 )\approx 4.8" data-latex="y_p( 3 )\approx 4.8"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-6820" title="F1 | Direction fields for first-order ODEs | ver. 6820"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(t*y/9+t/3, (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" alt="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 0" alt="t= 0" title="t= 0" data-latex="t= 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'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( -2 )= 2" alt="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( -2 )= 2" title="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( -2 )= 2" data-latex="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( -2 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(t*y/9+t/3, (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20%7By%7D%20t%20+%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t" alt="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%200" alt="t= 0" title="t= 0" data-latex="t= 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?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20%7By%7D%20t%20+%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20%5Chspace%7B2em%7D%20y(%20-2%20)=%202" alt="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( -2 )= 2" title="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( -2 )= 2" data-latex="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 0 )\approx 1.0" alt="y_p( 0 )\approx 1.0" title="y_p( 0 )\approx 1.0" data-latex="y_p( 0 )\approx 1.0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%200%20)%5Capprox%201.0" alt="y_p( 0 )\approx 1.0" title="y_p( 0 )\approx 1.0" data-latex="y_p( 0 )\approx 1.0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-5147" title="F1 | Direction fields for first-order ODEs | ver. 5147"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(cos(y/2), (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \cos\left(\frac{1}{2} \, {y}\right)" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right)"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= -4" alt="t= -4" title="t= -4" data-latex="t= -4"/>. </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'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -2 )= 2" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -2 )= 2" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -2 )= 2" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -2 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(cos(y/2), (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Ccos%5Cleft(%5Cfrac%7B1%7D%7B2%7D%20%5C,%20%7By%7D%5Cright)" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right)"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%20-4" alt="t= -4" title="t= -4" data-latex="t= -4"&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?%7By'%7D%20=%20%5Ccos%5Cleft(%5Cfrac%7B1%7D%7B2%7D%20%5C,%20%7By%7D%5Cright)%20%5Chspace%7B2em%7D%20y(%20-2%20)=%202" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -2 )= 2" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -2 )= 2" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( -4 )\approx 2.0" alt="y_p( -4 )\approx 2.0" title="y_p( -4 )\approx 2.0" data-latex="y_p( -4 )\approx 2.0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%20-4%20)%5Capprox%202.0" alt="y_p( -4 )\approx 2.0" title="y_p( -4 )\approx 2.0" data-latex="y_p( -4 )\approx 2.0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-7330" title="F1 | Direction fields for first-order ODEs | ver. 7330"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(cos(t+y), (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \cos\left({y} + t\right)" alt="{y'} = \cos\left({y} + t\right)" title="{y'} = \cos\left({y} + t\right)" data-latex="{y'} = \cos\left({y} + t\right)"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 3" alt="t= 3" title="t= 3" data-latex="t= 3"/>. </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'} = \cos\left({y} + t\right) \hspace{2em} y( 1 )= 0" alt="{y'} = \cos\left({y} + t\right) \hspace{2em} y( 1 )= 0" title="{y'} = \cos\left({y} + t\right) \hspace{2em} y( 1 )= 0" data-latex="{y'} = \cos\left({y} + t\right) \hspace{2em} y( 1 )= 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(cos(t+y), (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Ccos%5Cleft(%7By%7D%20+%20t%5Cright)" alt="{y'} = \cos\left({y} + t\right)" title="{y'} = \cos\left({y} + t\right)" data-latex="{y'} = \cos\left({y} + t\right)"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%203" alt="t= 3" title="t= 3" data-latex="t= 3"&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?%7By'%7D%20=%20%5Ccos%5Cleft(%7By%7D%20+%20t%5Cright)%20%5Chspace%7B2em%7D%20y(%201%20)=%200" alt="{y'} = \cos\left({y} + t\right) \hspace{2em} y( 1 )= 0" title="{y'} = \cos\left({y} + t\right) \hspace{2em} y( 1 )= 0" data-latex="{y'} = \cos\left({y} + t\right) \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 3 )\approx -0.61" alt="y_p( 3 )\approx -0.61" title="y_p( 3 )\approx -0.61" data-latex="y_p( 3 )\approx -0.61"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%203%20)%5Capprox%20-0.61" alt="y_p( 3 )\approx -0.61" title="y_p( 3 )\approx -0.61" data-latex="y_p( 3 )\approx -0.61"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-9391" title="F1 | Direction fields for first-order ODEs | ver. 9391"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(sin(y/2), (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \sin\left(\frac{1}{2} \, {y}\right)" alt="{y'} = \sin\left(\frac{1}{2} \, {y}\right)" title="{y'} = \sin\left(\frac{1}{2} \, {y}\right)" data-latex="{y'} = \sin\left(\frac{1}{2} \, {y}\right)"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 1" alt="t= 1" title="t= 1" data-latex="t= 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'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -1 )= 1" alt="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -1 )= 1" title="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -1 )= 1" data-latex="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -1 )= 1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(sin(y/2), (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Csin%5Cleft(%5Cfrac%7B1%7D%7B2%7D%20%5C,%20%7By%7D%5Cright)" alt="{y'} = \sin\left(\frac{1}{2} \, {y}\right)" title="{y'} = \sin\left(\frac{1}{2} \, {y}\right)" data-latex="{y'} = \sin\left(\frac{1}{2} \, {y}\right)"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%201" alt="t= 1" title="t= 1" data-latex="t= 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?%7By'%7D%20=%20%5Csin%5Cleft(%5Cfrac%7B1%7D%7B2%7D%20%5C,%20%7By%7D%5Cright)%20%5Chspace%7B2em%7D%20y(%20-1%20)=%201" alt="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -1 )= 1" title="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -1 )= 1" data-latex="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 1 )\approx 2.4" alt="y_p( 1 )\approx 2.4" title="y_p( 1 )\approx 2.4" data-latex="y_p( 1 )\approx 2.4"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%201%20)%5Capprox%202.4" alt="y_p( 1 )\approx 2.4" title="y_p( 1 )\approx 2.4" data-latex="y_p( 1 )\approx 2.4"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-7423" title="F1 | Direction fields for first-order ODEs | ver. 7423"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(sin(y/2), (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \sin\left(\frac{1}{2} \, {y}\right)" alt="{y'} = \sin\left(\frac{1}{2} \, {y}\right)" title="{y'} = \sin\left(\frac{1}{2} \, {y}\right)" data-latex="{y'} = \sin\left(\frac{1}{2} \, {y}\right)"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 1" alt="t= 1" title="t= 1" data-latex="t= 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'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -1 )= 1" alt="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -1 )= 1" title="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -1 )= 1" data-latex="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -1 )= 1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(sin(y/2), (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Csin%5Cleft(%5Cfrac%7B1%7D%7B2%7D%20%5C,%20%7By%7D%5Cright)" alt="{y'} = \sin\left(\frac{1}{2} \, {y}\right)" title="{y'} = \sin\left(\frac{1}{2} \, {y}\right)" data-latex="{y'} = \sin\left(\frac{1}{2} \, {y}\right)"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%201" alt="t= 1" title="t= 1" data-latex="t= 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?%7By'%7D%20=%20%5Csin%5Cleft(%5Cfrac%7B1%7D%7B2%7D%20%5C,%20%7By%7D%5Cright)%20%5Chspace%7B2em%7D%20y(%20-1%20)=%201" alt="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -1 )= 1" title="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -1 )= 1" data-latex="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 1 )\approx 2.4" alt="y_p( 1 )\approx 2.4" title="y_p( 1 )\approx 2.4" data-latex="y_p( 1 )\approx 2.4"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%201%20)%5Capprox%202.4" alt="y_p( 1 )\approx 2.4" title="y_p( 1 )\approx 2.4" data-latex="y_p( 1 )\approx 2.4"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-0744" title="F1 | Direction fields for first-order ODEs | ver. 0744"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(y/15-t/3, (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t" alt="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t" title="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= -4" alt="t= -4" title="t= -4" data-latex="t= -4"/>. </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'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( -2 )= 0" alt="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( -2 )= 0" title="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( -2 )= 0" data-latex="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( -2 )= 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(y/15-t/3, (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B15%7D%20%5C,%20%7By%7D%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t" alt="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t" title="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%20-4" alt="t= -4" title="t= -4" data-latex="t= -4"&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?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B15%7D%20%5C,%20%7By%7D%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20%5Chspace%7B2em%7D%20y(%20-2%20)=%200" alt="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( -2 )= 0" title="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( -2 )= 0" data-latex="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( -4 )\approx 0.00" alt="y_p( -4 )\approx 0.00" title="y_p( -4 )\approx 0.00" data-latex="y_p( -4 )\approx 0.00"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%20-4%20)%5Capprox%200.00" alt="y_p( -4 )\approx 0.00" title="y_p( -4 )\approx 0.00" data-latex="y_p( -4 )\approx 0.00"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-6371" title="F1 | Direction fields for first-order ODEs | 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>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(sin(y/2), (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \sin\left(\frac{1}{2} \, {y}\right)" alt="{y'} = \sin\left(\frac{1}{2} \, {y}\right)" title="{y'} = \sin\left(\frac{1}{2} \, {y}\right)" data-latex="{y'} = \sin\left(\frac{1}{2} \, {y}\right)"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 4" alt="t= 4" title="t= 4" data-latex="t= 4"/>. </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'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 2 )= 1" alt="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 2 )= 1" title="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 2 )= 1" data-latex="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 2 )= 1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(sin(y/2), (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Csin%5Cleft(%5Cfrac%7B1%7D%7B2%7D%20%5C,%20%7By%7D%5Cright)" alt="{y'} = \sin\left(\frac{1}{2} \, {y}\right)" title="{y'} = \sin\left(\frac{1}{2} \, {y}\right)" data-latex="{y'} = \sin\left(\frac{1}{2} \, {y}\right)"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%204" alt="t= 4" title="t= 4" data-latex="t= 4"&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?%7By'%7D%20=%20%5Csin%5Cleft(%5Cfrac%7B1%7D%7B2%7D%20%5C,%20%7By%7D%5Cright)%20%5Chspace%7B2em%7D%20y(%202%20)=%201" alt="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 2 )= 1" title="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 2 )= 1" data-latex="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 4 )\approx 2.4" alt="y_p( 4 )\approx 2.4" title="y_p( 4 )\approx 2.4" data-latex="y_p( 4 )\approx 2.4"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%204%20)%5Capprox%202.4" alt="y_p( 4 )\approx 2.4" title="y_p( 4 )\approx 2.4" data-latex="y_p( 4 )\approx 2.4"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-7439" title="F1 | Direction fields for first-order ODEs | ver. 7439"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(y/15-t/3, (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y}" alt="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y}" title="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y}" data-latex="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y}"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 1" alt="t= 1" title="t= 1" data-latex="t= 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'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y} \hspace{2em} y( -1 )= -1" alt="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y} \hspace{2em} y( -1 )= -1" title="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y} \hspace{2em} y( -1 )= -1" data-latex="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {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;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(y/15-t/3, (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20-%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20+%20%5Cfrac%7B1%7D%7B15%7D%20%5C,%20%7By%7D" alt="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y}" title="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y}" data-latex="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y}"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%201" alt="t= 1" title="t= 1" data-latex="t= 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?%7By'%7D%20=%20-%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20+%20%5Cfrac%7B1%7D%7B15%7D%20%5C,%20%7By%7D%20%5Chspace%7B2em%7D%20y(%20-1%20)=%20-1" alt="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y} \hspace{2em} y( -1 )= -1" title="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y} \hspace{2em} y( -1 )= -1" data-latex="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 1 )\approx -1.1" alt="y_p( 1 )\approx -1.1" title="y_p( 1 )\approx -1.1" data-latex="y_p( 1 )\approx -1.1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%201%20)%5Capprox%20-1.1" alt="y_p( 1 )\approx -1.1" title="y_p( 1 )\approx -1.1" data-latex="y_p( 1 )\approx -1.1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-4953" title="F1 | Direction fields for first-order ODEs | ver. 4953"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(cos(t+y), (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \cos\left({y} + t\right)" alt="{y'} = \cos\left({y} + t\right)" title="{y'} = \cos\left({y} + t\right)" data-latex="{y'} = \cos\left({y} + t\right)"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 2" alt="t= 2" title="t= 2" data-latex="t= 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'} = \cos\left({y} + t\right) \hspace{2em} y( 0 )= -2" alt="{y'} = \cos\left({y} + t\right) \hspace{2em} y( 0 )= -2" title="{y'} = \cos\left({y} + t\right) \hspace{2em} y( 0 )= -2" data-latex="{y'} = \cos\left({y} + t\right) \hspace{2em} y( 0 )= -2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(cos(t+y), (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Ccos%5Cleft(%7By%7D%20+%20t%5Cright)" alt="{y'} = \cos\left({y} + t\right)" title="{y'} = \cos\left({y} + t\right)" data-latex="{y'} = \cos\left({y} + t\right)"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%202" alt="t= 2" title="t= 2" data-latex="t= 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?%7By'%7D%20=%20%5Ccos%5Cleft(%7By%7D%20+%20t%5Cright)%20%5Chspace%7B2em%7D%20y(%200%20)=%20-2" alt="{y'} = \cos\left({y} + t\right) \hspace{2em} y( 0 )= -2" title="{y'} = \cos\left({y} + t\right) \hspace{2em} y( 0 )= -2" data-latex="{y'} = \cos\left({y} + t\right) \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 2 )\approx -1.2" alt="y_p( 2 )\approx -1.2" title="y_p( 2 )\approx -1.2" data-latex="y_p( 2 )\approx -1.2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%202%20)%5Capprox%20-1.2" alt="y_p( 2 )\approx -1.2" title="y_p( 2 )\approx -1.2" data-latex="y_p( 2 )\approx -1.2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-8321" title="F1 | Direction fields for first-order ODEs | ver. 8321"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(cos(t+y), (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \cos\left({y} + t\right)" alt="{y'} = \cos\left({y} + t\right)" title="{y'} = \cos\left({y} + t\right)" data-latex="{y'} = \cos\left({y} + t\right)"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= -4" alt="t= -4" title="t= -4" data-latex="t= -4"/>. </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'} = \cos\left({y} + t\right) \hspace{2em} y( -2 )= 0" alt="{y'} = \cos\left({y} + t\right) \hspace{2em} y( -2 )= 0" title="{y'} = \cos\left({y} + t\right) \hspace{2em} y( -2 )= 0" data-latex="{y'} = \cos\left({y} + t\right) \hspace{2em} y( -2 )= 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(cos(t+y), (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Ccos%5Cleft(%7By%7D%20+%20t%5Cright)" alt="{y'} = \cos\left({y} + t\right)" title="{y'} = \cos\left({y} + t\right)" data-latex="{y'} = \cos\left({y} + t\right)"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%20-4" alt="t= -4" title="t= -4" data-latex="t= -4"&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?%7By'%7D%20=%20%5Ccos%5Cleft(%7By%7D%20+%20t%5Cright)%20%5Chspace%7B2em%7D%20y(%20-2%20)=%200" alt="{y'} = \cos\left({y} + t\right) \hspace{2em} y( -2 )= 0" title="{y'} = \cos\left({y} + t\right) \hspace{2em} y( -2 )= 0" data-latex="{y'} = \cos\left({y} + t\right) \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( -4 )\approx 0.00" alt="y_p( -4 )\approx 0.00" title="y_p( -4 )\approx 0.00" data-latex="y_p( -4 )\approx 0.00"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%20-4%20)%5Capprox%200.00" alt="y_p( -4 )\approx 0.00" title="y_p( -4 )\approx 0.00" data-latex="y_p( -4 )\approx 0.00"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-3819" title="F1 | Direction fields for first-order ODEs | ver. 3819"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(cos(y/2), (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \cos\left(\frac{1}{2} \, {y}\right)" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right)"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= -4" alt="t= -4" title="t= -4" data-latex="t= -4"/>. </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'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -2 )= 1" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -2 )= 1" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -2 )= 1" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -2 )= 1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(cos(y/2), (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Ccos%5Cleft(%5Cfrac%7B1%7D%7B2%7D%20%5C,%20%7By%7D%5Cright)" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right)"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%20-4" alt="t= -4" title="t= -4" data-latex="t= -4"&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?%7By'%7D%20=%20%5Ccos%5Cleft(%5Cfrac%7B1%7D%7B2%7D%20%5C,%20%7By%7D%5Cright)%20%5Chspace%7B2em%7D%20y(%20-2%20)=%201" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -2 )= 1" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -2 )= 1" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( -4 )\approx 1.0" alt="y_p( -4 )\approx 1.0" title="y_p( -4 )\approx 1.0" data-latex="y_p( -4 )\approx 1.0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%20-4%20)%5Capprox%201.0" alt="y_p( -4 )\approx 1.0" title="y_p( -4 )\approx 1.0" data-latex="y_p( -4 )\approx 1.0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-2159" title="F1 | Direction fields for first-order ODEs | ver. 2159"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(cos(t+y), (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \cos\left(t + {y}\right)" alt="{y'} = \cos\left(t + {y}\right)" title="{y'} = \cos\left(t + {y}\right)" data-latex="{y'} = \cos\left(t + {y}\right)"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= -2" alt="t= -2" title="t= -2" data-latex="t= -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'} = \cos\left(t + {y}\right) \hspace{2em} y( 0 )= 2" alt="{y'} = \cos\left(t + {y}\right) \hspace{2em} y( 0 )= 2" title="{y'} = \cos\left(t + {y}\right) \hspace{2em} y( 0 )= 2" data-latex="{y'} = \cos\left(t + {y}\right) \hspace{2em} y( 0 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(cos(t+y), (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Ccos%5Cleft(t%20+%20%7By%7D%5Cright)" alt="{y'} = \cos\left(t + {y}\right)" title="{y'} = \cos\left(t + {y}\right)" data-latex="{y'} = \cos\left(t + {y}\right)"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%20-2" alt="t= -2" title="t= -2" data-latex="t= -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?%7By'%7D%20=%20%5Ccos%5Cleft(t%20+%20%7By%7D%5Cright)%20%5Chspace%7B2em%7D%20y(%200%20)=%202" alt="{y'} = \cos\left(t + {y}\right) \hspace{2em} y( 0 )= 2" title="{y'} = \cos\left(t + {y}\right) \hspace{2em} y( 0 )= 2" data-latex="{y'} = \cos\left(t + {y}\right) \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( -2 )\approx 2.0" alt="y_p( -2 )\approx 2.0" title="y_p( -2 )\approx 2.0" data-latex="y_p( -2 )\approx 2.0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%20-2%20)%5Capprox%202.0" alt="y_p( -2 )\approx 2.0" title="y_p( -2 )\approx 2.0" data-latex="y_p( -2 )\approx 2.0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-4730" title="F1 | Direction fields for first-order ODEs | ver. 4730"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(t*y/9+t/3, (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \frac{1}{9} \, t {y} + \frac{1}{3} \, t" alt="{y'} = \frac{1}{9} \, t {y} + \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, t {y} + \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, t {y} + \frac{1}{3} \, t"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= -2" alt="t= -2" title="t= -2" data-latex="t= -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'} = \frac{1}{9} \, t {y} + \frac{1}{3} \, t \hspace{2em} y( 0 )= 2" alt="{y'} = \frac{1}{9} \, t {y} + \frac{1}{3} \, t \hspace{2em} y( 0 )= 2" title="{y'} = \frac{1}{9} \, t {y} + \frac{1}{3} \, t \hspace{2em} y( 0 )= 2" data-latex="{y'} = \frac{1}{9} \, t {y} + \frac{1}{3} \, t \hspace{2em} y( 0 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(t*y/9+t/3, (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20t%20%7By%7D%20+%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t" alt="{y'} = \frac{1}{9} \, t {y} + \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, t {y} + \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, t {y} + \frac{1}{3} \, t"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%20-2" alt="t= -2" title="t= -2" data-latex="t= -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?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20t%20%7By%7D%20+%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20%5Chspace%7B2em%7D%20y(%200%20)=%202" alt="{y'} = \frac{1}{9} \, t {y} + \frac{1}{3} \, t \hspace{2em} y( 0 )= 2" title="{y'} = \frac{1}{9} \, t {y} + \frac{1}{3} \, t \hspace{2em} y( 0 )= 2" data-latex="{y'} = \frac{1}{9} \, t {y} + \frac{1}{3} \, t \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( -2 )\approx 2.0" alt="y_p( -2 )\approx 2.0" title="y_p( -2 )\approx 2.0" data-latex="y_p( -2 )\approx 2.0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%20-2%20)%5Capprox%202.0" alt="y_p( -2 )\approx 2.0" title="y_p( -2 )\approx 2.0" data-latex="y_p( -2 )\approx 2.0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-4020" title="F1 | Direction fields for first-order ODEs | ver. 4020"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(y/15-t/3, (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y}" alt="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y}" title="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y}" data-latex="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y}"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 3" alt="t= 3" title="t= 3" data-latex="t= 3"/>. </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'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y} \hspace{2em} y( 1 )= 0" alt="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y} \hspace{2em} y( 1 )= 0" title="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y} \hspace{2em} y( 1 )= 0" data-latex="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y} \hspace{2em} y( 1 )= 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(y/15-t/3, (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20-%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20+%20%5Cfrac%7B1%7D%7B15%7D%20%5C,%20%7By%7D" alt="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y}" title="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y}" data-latex="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y}"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%203" alt="t= 3" title="t= 3" data-latex="t= 3"&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?%7By'%7D%20=%20-%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20+%20%5Cfrac%7B1%7D%7B15%7D%20%5C,%20%7By%7D%20%5Chspace%7B2em%7D%20y(%201%20)=%200" alt="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y} \hspace{2em} y( 1 )= 0" title="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y} \hspace{2em} y( 1 )= 0" data-latex="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y} \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 3 )\approx -1.4" alt="y_p( 3 )\approx -1.4" title="y_p( 3 )\approx -1.4" data-latex="y_p( 3 )\approx -1.4"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%203%20)%5Capprox%20-1.4" alt="y_p( 3 )\approx -1.4" title="y_p( 3 )\approx -1.4" data-latex="y_p( 3 )\approx -1.4"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-5401" title="F1 | Direction fields for first-order ODEs | ver. 5401"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(t*y/9+t/3, (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" alt="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 3" alt="t= 3" title="t= 3" data-latex="t= 3"/>. </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'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( 1 )= 2" alt="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( 1 )= 2" title="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( 1 )= 2" data-latex="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( 1 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(t*y/9+t/3, (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20%7By%7D%20t%20+%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t" alt="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%203" alt="t= 3" title="t= 3" data-latex="t= 3"&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?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20%7By%7D%20t%20+%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20%5Chspace%7B2em%7D%20y(%201%20)=%202" alt="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( 1 )= 2" title="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( 1 )= 2" data-latex="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 3 )\approx 4.8" alt="y_p( 3 )\approx 4.8" title="y_p( 3 )\approx 4.8" data-latex="y_p( 3 )\approx 4.8"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%203%20)%5Capprox%204.8" alt="y_p( 3 )\approx 4.8" title="y_p( 3 )\approx 4.8" data-latex="y_p( 3 )\approx 4.8"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-2690" title="F1 | Direction fields for first-order ODEs | ver. 2690"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(cos(y/2), (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \cos\left(\frac{1}{2} \, {y}\right)" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right)"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= -1" alt="t= -1" title="t= -1" data-latex="t= -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'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 1 )= 2" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 1 )= 2" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 1 )= 2" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 1 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(cos(y/2), (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Ccos%5Cleft(%5Cfrac%7B1%7D%7B2%7D%20%5C,%20%7By%7D%5Cright)" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right)"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%20-1" alt="t= -1" title="t= -1" data-latex="t= -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?%7By'%7D%20=%20%5Ccos%5Cleft(%5Cfrac%7B1%7D%7B2%7D%20%5C,%20%7By%7D%5Cright)%20%5Chspace%7B2em%7D%20y(%201%20)=%202" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 1 )= 2" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 1 )= 2" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( -1 )\approx 2.0" alt="y_p( -1 )\approx 2.0" title="y_p( -1 )\approx 2.0" data-latex="y_p( -1 )\approx 2.0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%20-1%20)%5Capprox%202.0" alt="y_p( -1 )\approx 2.0" title="y_p( -1 )\approx 2.0" data-latex="y_p( -1 )\approx 2.0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-5342" title="F1 | Direction fields for first-order ODEs | ver. 5342"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(cos(t+y), (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \cos\left(t + {y}\right)" alt="{y'} = \cos\left(t + {y}\right)" title="{y'} = \cos\left(t + {y}\right)" data-latex="{y'} = \cos\left(t + {y}\right)"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 0" alt="t= 0" title="t= 0" data-latex="t= 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'} = \cos\left(t + {y}\right) \hspace{2em} y( -2 )= -2" alt="{y'} = \cos\left(t + {y}\right) \hspace{2em} y( -2 )= -2" title="{y'} = \cos\left(t + {y}\right) \hspace{2em} y( -2 )= -2" data-latex="{y'} = \cos\left(t + {y}\right) \hspace{2em} y( -2 )= -2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(cos(t+y), (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Ccos%5Cleft(t%20+%20%7By%7D%5Cright)" alt="{y'} = \cos\left(t + {y}\right)" title="{y'} = \cos\left(t + {y}\right)" data-latex="{y'} = \cos\left(t + {y}\right)"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%200" alt="t= 0" title="t= 0" data-latex="t= 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?%7By'%7D%20=%20%5Ccos%5Cleft(t%20+%20%7By%7D%5Cright)%20%5Chspace%7B2em%7D%20y(%20-2%20)=%20-2" alt="{y'} = \cos\left(t + {y}\right) \hspace{2em} y( -2 )= -2" title="{y'} = \cos\left(t + {y}\right) \hspace{2em} y( -2 )= -2" data-latex="{y'} = \cos\left(t + {y}\right) \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 0 )\approx -3.6" alt="y_p( 0 )\approx -3.6" title="y_p( 0 )\approx -3.6" data-latex="y_p( 0 )\approx -3.6"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%200%20)%5Capprox%20-3.6" alt="y_p( 0 )\approx -3.6" title="y_p( 0 )\approx -3.6" data-latex="y_p( 0 )\approx -3.6"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-9719" title="F1 | Direction fields for first-order ODEs | ver. 9719"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(cos(t+y), (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \cos\left(t + {y}\right)" alt="{y'} = \cos\left(t + {y}\right)" title="{y'} = \cos\left(t + {y}\right)" data-latex="{y'} = \cos\left(t + {y}\right)"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= -1" alt="t= -1" title="t= -1" data-latex="t= -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'} = \cos\left(t + {y}\right) \hspace{2em} y( 1 )= 0" alt="{y'} = \cos\left(t + {y}\right) \hspace{2em} y( 1 )= 0" title="{y'} = \cos\left(t + {y}\right) \hspace{2em} y( 1 )= 0" data-latex="{y'} = \cos\left(t + {y}\right) \hspace{2em} y( 1 )= 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(cos(t+y), (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Ccos%5Cleft(t%20+%20%7By%7D%5Cright)" alt="{y'} = \cos\left(t + {y}\right)" title="{y'} = \cos\left(t + {y}\right)" data-latex="{y'} = \cos\left(t + {y}\right)"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%20-1" alt="t= -1" title="t= -1" data-latex="t= -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?%7By'%7D%20=%20%5Ccos%5Cleft(t%20+%20%7By%7D%5Cright)%20%5Chspace%7B2em%7D%20y(%201%20)=%200" alt="{y'} = \cos\left(t + {y}\right) \hspace{2em} y( 1 )= 0" title="{y'} = \cos\left(t + {y}\right) \hspace{2em} y( 1 )= 0" data-latex="{y'} = \cos\left(t + {y}\right) \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( -1 )\approx 0.00" alt="y_p( -1 )\approx 0.00" title="y_p( -1 )\approx 0.00" data-latex="y_p( -1 )\approx 0.00"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%20-1%20)%5Capprox%200.00" alt="y_p( -1 )\approx 0.00" title="y_p( -1 )\approx 0.00" data-latex="y_p( -1 )\approx 0.00"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-4408" title="F1 | Direction fields for first-order ODEs | ver. 4408"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(t*y/9-t/3, (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \frac{1}{9} \, t {y} - \frac{1}{3} \, t" alt="{y'} = \frac{1}{9} \, t {y} - \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, t {y} - \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, t {y} - \frac{1}{3} \, t"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 2" alt="t= 2" title="t= 2" data-latex="t= 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'} = \frac{1}{9} \, t {y} - \frac{1}{3} \, t \hspace{2em} y( 0 )= -2" alt="{y'} = \frac{1}{9} \, t {y} - \frac{1}{3} \, t \hspace{2em} y( 0 )= -2" title="{y'} = \frac{1}{9} \, t {y} - \frac{1}{3} \, t \hspace{2em} y( 0 )= -2" data-latex="{y'} = \frac{1}{9} \, t {y} - \frac{1}{3} \, t \hspace{2em} y( 0 )= -2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(t*y/9-t/3, (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20t%20%7By%7D%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t" alt="{y'} = \frac{1}{9} \, t {y} - \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, t {y} - \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, t {y} - \frac{1}{3} \, t"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%202" alt="t= 2" title="t= 2" data-latex="t= 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?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20t%20%7By%7D%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20%5Chspace%7B2em%7D%20y(%200%20)=%20-2" alt="{y'} = \frac{1}{9} \, t {y} - \frac{1}{3} \, t \hspace{2em} y( 0 )= -2" title="{y'} = \frac{1}{9} \, t {y} - \frac{1}{3} \, t \hspace{2em} y( 0 )= -2" data-latex="{y'} = \frac{1}{9} \, t {y} - \frac{1}{3} \, t \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 2 )\approx -3.2" alt="y_p( 2 )\approx -3.2" title="y_p( 2 )\approx -3.2" data-latex="y_p( 2 )\approx -3.2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%202%20)%5Capprox%20-3.2" alt="y_p( 2 )\approx -3.2" title="y_p( 2 )\approx -3.2" data-latex="y_p( 2 )\approx -3.2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-9171" title="F1 | Direction fields for first-order ODEs | ver. 9171"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(cos(y/2), (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \cos\left(\frac{1}{2} \, {y}\right)" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right)"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 4" alt="t= 4" title="t= 4" data-latex="t= 4"/>. </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'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 2 )= -1" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 2 )= -1" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 2 )= -1" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 2 )= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(cos(y/2), (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Ccos%5Cleft(%5Cfrac%7B1%7D%7B2%7D%20%5C,%20%7By%7D%5Cright)" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right)"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%204" alt="t= 4" title="t= 4" data-latex="t= 4"&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?%7By'%7D%20=%20%5Ccos%5Cleft(%5Cfrac%7B1%7D%7B2%7D%20%5C,%20%7By%7D%5Cright)%20%5Chspace%7B2em%7D%20y(%202%20)=%20-1" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 2 )= -1" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 2 )= -1" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 4 )\approx 0.92" alt="y_p( 4 )\approx 0.92" title="y_p( 4 )\approx 0.92" data-latex="y_p( 4 )\approx 0.92"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%204%20)%5Capprox%200.92" alt="y_p( 4 )\approx 0.92" title="y_p( 4 )\approx 0.92" data-latex="y_p( 4 )\approx 0.92"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-3417" title="F1 | Direction fields for first-order ODEs | ver. 3417"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(cos(t+y), (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \cos\left({y} + t\right)" alt="{y'} = \cos\left({y} + t\right)" title="{y'} = \cos\left({y} + t\right)" data-latex="{y'} = \cos\left({y} + t\right)"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 0" alt="t= 0" title="t= 0" data-latex="t= 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'} = \cos\left({y} + t\right) \hspace{2em} y( -2 )= -1" alt="{y'} = \cos\left({y} + t\right) \hspace{2em} y( -2 )= -1" title="{y'} = \cos\left({y} + t\right) \hspace{2em} y( -2 )= -1" data-latex="{y'} = \cos\left({y} + t\right) \hspace{2em} y( -2 )= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(cos(t+y), (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Ccos%5Cleft(%7By%7D%20+%20t%5Cright)" alt="{y'} = \cos\left({y} + t\right)" title="{y'} = \cos\left({y} + t\right)" data-latex="{y'} = \cos\left({y} + t\right)"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%200" alt="t= 0" title="t= 0" data-latex="t= 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?%7By'%7D%20=%20%5Ccos%5Cleft(%7By%7D%20+%20t%5Cright)%20%5Chspace%7B2em%7D%20y(%20-2%20)=%20-1" alt="{y'} = \cos\left({y} + t\right) \hspace{2em} y( -2 )= -1" title="{y'} = \cos\left({y} + t\right) \hspace{2em} y( -2 )= -1" data-latex="{y'} = \cos\left({y} + t\right) \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 0 )\approx -3.0" alt="y_p( 0 )\approx -3.0" title="y_p( 0 )\approx -3.0" data-latex="y_p( 0 )\approx -3.0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%200%20)%5Capprox%20-3.0" alt="y_p( 0 )\approx -3.0" title="y_p( 0 )\approx -3.0" data-latex="y_p( 0 )\approx -3.0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-1112" title="F1 | Direction fields for first-order ODEs | ver. 1112"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(cos(y/2), (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \cos\left(\frac{1}{2} \, {y}\right)" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right)"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= -1" alt="t= -1" title="t= -1" data-latex="t= -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'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 1 )= 0" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 1 )= 0" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 1 )= 0" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 1 )= 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(cos(y/2), (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Ccos%5Cleft(%5Cfrac%7B1%7D%7B2%7D%20%5C,%20%7By%7D%5Cright)" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right)"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%20-1" alt="t= -1" title="t= -1" data-latex="t= -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?%7By'%7D%20=%20%5Ccos%5Cleft(%5Cfrac%7B1%7D%7B2%7D%20%5C,%20%7By%7D%5Cright)%20%5Chspace%7B2em%7D%20y(%201%20)=%200" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 1 )= 0" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 1 )= 0" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( -1 )\approx 0.00" alt="y_p( -1 )\approx 0.00" title="y_p( -1 )\approx 0.00" data-latex="y_p( -1 )\approx 0.00"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%20-1%20)%5Capprox%200.00" alt="y_p( -1 )\approx 0.00" title="y_p( -1 )\approx 0.00" data-latex="y_p( -1 )\approx 0.00"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-5131" title="F1 | Direction fields for first-order ODEs | ver. 5131"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(cos(y/2), (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \cos\left(\frac{1}{2} \, {y}\right)" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right)"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= -1" alt="t= -1" title="t= -1" data-latex="t= -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'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 1 )= -2" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 1 )= -2" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 1 )= -2" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 1 )= -2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(cos(y/2), (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Ccos%5Cleft(%5Cfrac%7B1%7D%7B2%7D%20%5C,%20%7By%7D%5Cright)" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right)"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%20-1" alt="t= -1" title="t= -1" data-latex="t= -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?%7By'%7D%20=%20%5Ccos%5Cleft(%5Cfrac%7B1%7D%7B2%7D%20%5C,%20%7By%7D%5Cright)%20%5Chspace%7B2em%7D%20y(%201%20)=%20-2" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 1 )= -2" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 1 )= -2" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( -1 )\approx -2.0" alt="y_p( -1 )\approx -2.0" title="y_p( -1 )\approx -2.0" data-latex="y_p( -1 )\approx -2.0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%20-1%20)%5Capprox%20-2.0" alt="y_p( -1 )\approx -2.0" title="y_p( -1 )\approx -2.0" data-latex="y_p( -1 )\approx -2.0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-6997" title="F1 | Direction fields for first-order ODEs | ver. 6997"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(t*y/9+t/3, (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" alt="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= -3" alt="t= -3" title="t= -3" data-latex="t= -3"/>. </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'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( -1 )= -1" alt="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( -1 )= -1" title="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( -1 )= -1" data-latex="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( -1 )= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(t*y/9+t/3, (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20%7By%7D%20t%20+%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t" alt="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%20-3" alt="t= -3" title="t= -3" data-latex="t= -3"&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?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20%7By%7D%20t%20+%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20%5Chspace%7B2em%7D%20y(%20-1%20)=%20-1" alt="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( -1 )= -1" title="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( -1 )= -1" data-latex="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( -3 )\approx -1.0" alt="y_p( -3 )\approx -1.0" title="y_p( -3 )\approx -1.0" data-latex="y_p( -3 )\approx -1.0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%20-3%20)%5Capprox%20-1.0" alt="y_p( -3 )\approx -1.0" title="y_p( -3 )\approx -1.0" data-latex="y_p( -3 )\approx -1.0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-3483" title="F1 | Direction fields for first-order ODEs | ver. 3483"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(t*y/9-t/3, (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" alt="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 1" alt="t= 1" title="t= 1" data-latex="t= 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'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( -1 )= -2" alt="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( -1 )= -2" title="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( -1 )= -2" data-latex="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( -1 )= -2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(t*y/9-t/3, (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20%7By%7D%20t%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t" alt="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%201" alt="t= 1" title="t= 1" data-latex="t= 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?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20%7By%7D%20t%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20%5Chspace%7B2em%7D%20y(%20-1%20)=%20-2" alt="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( -1 )= -2" title="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( -1 )= -2" data-latex="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 1 )\approx -2.0" alt="y_p( 1 )\approx -2.0" title="y_p( 1 )\approx -2.0" data-latex="y_p( 1 )\approx -2.0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%201%20)%5Capprox%20-2.0" alt="y_p( 1 )\approx -2.0" title="y_p( 1 )\approx -2.0" data-latex="y_p( 1 )\approx -2.0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-1879" title="F1 | Direction fields for first-order ODEs | ver. 1879"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(cos(t+y), (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \cos\left({y} + t\right)" alt="{y'} = \cos\left({y} + t\right)" title="{y'} = \cos\left({y} + t\right)" data-latex="{y'} = \cos\left({y} + t\right)"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 0" alt="t= 0" title="t= 0" data-latex="t= 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'} = \cos\left({y} + t\right) \hspace{2em} y( -2 )= 2" alt="{y'} = \cos\left({y} + t\right) \hspace{2em} y( -2 )= 2" title="{y'} = \cos\left({y} + t\right) \hspace{2em} y( -2 )= 2" data-latex="{y'} = \cos\left({y} + t\right) \hspace{2em} y( -2 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(cos(t+y), (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Ccos%5Cleft(%7By%7D%20+%20t%5Cright)" alt="{y'} = \cos\left({y} + t\right)" title="{y'} = \cos\left({y} + t\right)" data-latex="{y'} = \cos\left({y} + t\right)"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%200" alt="t= 0" title="t= 0" data-latex="t= 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?%7By'%7D%20=%20%5Ccos%5Cleft(%7By%7D%20+%20t%5Cright)%20%5Chspace%7B2em%7D%20y(%20-2%20)=%202" alt="{y'} = \cos\left({y} + t\right) \hspace{2em} y( -2 )= 2" title="{y'} = \cos\left({y} + t\right) \hspace{2em} y( -2 )= 2" data-latex="{y'} = \cos\left({y} + t\right) \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 0 )\approx 2.2" alt="y_p( 0 )\approx 2.2" title="y_p( 0 )\approx 2.2" data-latex="y_p( 0 )\approx 2.2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%200%20)%5Capprox%202.2" alt="y_p( 0 )\approx 2.2" title="y_p( 0 )\approx 2.2" data-latex="y_p( 0 )\approx 2.2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-0468" title="F1 | Direction fields for first-order ODEs | ver. 0468"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(t/3-y/15, (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t" alt="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t" title="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t" data-latex="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= -1" alt="t= -1" title="t= -1" data-latex="t= -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'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( 1 )= 0" alt="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( 1 )= 0" title="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( 1 )= 0" data-latex="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( 1 )= 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(t/3-y/15, (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20-%5Cfrac%7B1%7D%7B15%7D%20%5C,%20%7By%7D%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t" alt="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t" title="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t" data-latex="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%20-1" alt="t= -1" title="t= -1" data-latex="t= -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?%7By'%7D%20=%20-%5Cfrac%7B1%7D%7B15%7D%20%5C,%20%7By%7D%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20%5Chspace%7B2em%7D%20y(%201%20)=%200" alt="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( 1 )= 0" title="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( 1 )= 0" data-latex="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( -1 )\approx 0.00" alt="y_p( -1 )\approx 0.00" title="y_p( -1 )\approx 0.00" data-latex="y_p( -1 )\approx 0.00"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%20-1%20)%5Capprox%200.00" alt="y_p( -1 )\approx 0.00" title="y_p( -1 )\approx 0.00" data-latex="y_p( -1 )\approx 0.00"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-9775" title="F1 | Direction fields for first-order ODEs | ver. 9775"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(cos(y/2), (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \cos\left(\frac{1}{2} \, {y}\right)" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right)"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= -4" alt="t= -4" title="t= -4" data-latex="t= -4"/>. </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'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -2 )= -1" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -2 )= -1" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -2 )= -1" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -2 )= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(cos(y/2), (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Ccos%5Cleft(%5Cfrac%7B1%7D%7B2%7D%20%5C,%20%7By%7D%5Cright)" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right)"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%20-4" alt="t= -4" title="t= -4" data-latex="t= -4"&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?%7By'%7D%20=%20%5Ccos%5Cleft(%5Cfrac%7B1%7D%7B2%7D%20%5C,%20%7By%7D%5Cright)%20%5Chspace%7B2em%7D%20y(%20-2%20)=%20-1" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -2 )= -1" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -2 )= -1" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( -4 )\approx -1.0" alt="y_p( -4 )\approx -1.0" title="y_p( -4 )\approx -1.0" data-latex="y_p( -4 )\approx -1.0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%20-4%20)%5Capprox%20-1.0" alt="y_p( -4 )\approx -1.0" title="y_p( -4 )\approx -1.0" data-latex="y_p( -4 )\approx -1.0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-0703" title="F1 | Direction fields for first-order ODEs | ver. 0703"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(t*y/9-t/3, (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" alt="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 3" alt="t= 3" title="t= 3" data-latex="t= 3"/>. </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'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( 1 )= 1" alt="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( 1 )= 1" title="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( 1 )= 1" data-latex="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( 1 )= 1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(t*y/9-t/3, (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20%7By%7D%20t%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t" alt="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%203" alt="t= 3" title="t= 3" data-latex="t= 3"&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?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20%7By%7D%20t%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20%5Chspace%7B2em%7D%20y(%201%20)=%201" alt="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( 1 )= 1" title="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( 1 )= 1" data-latex="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 3 )\approx -0.12" alt="y_p( 3 )\approx -0.12" title="y_p( 3 )\approx -0.12" data-latex="y_p( 3 )\approx -0.12"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%203%20)%5Capprox%20-0.12" alt="y_p( 3 )\approx -0.12" title="y_p( 3 )\approx -0.12" data-latex="y_p( 3 )\approx -0.12"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-6981" title="F1 | Direction fields for first-order ODEs | ver. 6981"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(sin(y/2), (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \sin\left(\frac{1}{2} \, {y}\right)" alt="{y'} = \sin\left(\frac{1}{2} \, {y}\right)" title="{y'} = \sin\left(\frac{1}{2} \, {y}\right)" data-latex="{y'} = \sin\left(\frac{1}{2} \, {y}\right)"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 0" alt="t= 0" title="t= 0" data-latex="t= 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'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -2 )= -2" alt="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -2 )= -2" title="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -2 )= -2" data-latex="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -2 )= -2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(sin(y/2), (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Csin%5Cleft(%5Cfrac%7B1%7D%7B2%7D%20%5C,%20%7By%7D%5Cright)" alt="{y'} = \sin\left(\frac{1}{2} \, {y}\right)" title="{y'} = \sin\left(\frac{1}{2} \, {y}\right)" data-latex="{y'} = \sin\left(\frac{1}{2} \, {y}\right)"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%200" alt="t= 0" title="t= 0" data-latex="t= 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?%7By'%7D%20=%20%5Csin%5Cleft(%5Cfrac%7B1%7D%7B2%7D%20%5C,%20%7By%7D%5Cright)%20%5Chspace%7B2em%7D%20y(%20-2%20)=%20-2" alt="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -2 )= -2" title="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -2 )= -2" data-latex="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 0 )\approx -3.9" alt="y_p( 0 )\approx -3.9" title="y_p( 0 )\approx -3.9" data-latex="y_p( 0 )\approx -3.9"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%200%20)%5Capprox%20-3.9" alt="y_p( 0 )\approx -3.9" title="y_p( 0 )\approx -3.9" data-latex="y_p( 0 )\approx -3.9"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-6520" title="F1 | Direction fields for first-order ODEs | ver. 6520"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(y/15-t/3, (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t" alt="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t" title="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 1" alt="t= 1" title="t= 1" data-latex="t= 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'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( -1 )= -1" alt="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( -1 )= -1" title="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( -1 )= -1" data-latex="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( -1 )= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(y/15-t/3, (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B15%7D%20%5C,%20%7By%7D%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t" alt="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t" title="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%201" alt="t= 1" title="t= 1" data-latex="t= 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?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B15%7D%20%5C,%20%7By%7D%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20%5Chspace%7B2em%7D%20y(%20-1%20)=%20-1" alt="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( -1 )= -1" title="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( -1 )= -1" data-latex="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 1 )\approx -1.1" alt="y_p( 1 )\approx -1.1" title="y_p( 1 )\approx -1.1" data-latex="y_p( 1 )\approx -1.1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%201%20)%5Capprox%20-1.1" alt="y_p( 1 )\approx -1.1" title="y_p( 1 )\approx -1.1" data-latex="y_p( 1 )\approx -1.1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-7729" title="F1 | Direction fields for first-order ODEs | ver. 7729"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(y/15-t/3, (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t" alt="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t" title="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 4" alt="t= 4" title="t= 4" data-latex="t= 4"/>. </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'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( 2 )= 1" alt="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( 2 )= 1" title="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( 2 )= 1" data-latex="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( 2 )= 1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(y/15-t/3, (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B15%7D%20%5C,%20%7By%7D%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t" alt="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t" title="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%204" alt="t= 4" title="t= 4" data-latex="t= 4"&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?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B15%7D%20%5C,%20%7By%7D%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20%5Chspace%7B2em%7D%20y(%202%20)=%201" alt="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( 2 )= 1" title="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( 2 )= 1" data-latex="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 4 )\approx -0.98" alt="y_p( 4 )\approx -0.98" title="y_p( 4 )\approx -0.98" data-latex="y_p( 4 )\approx -0.98"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%204%20)%5Capprox%20-0.98" alt="y_p( 4 )\approx -0.98" title="y_p( 4 )\approx -0.98" data-latex="y_p( 4 )\approx -0.98"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-1105" title="F1 | Direction fields for first-order ODEs | ver. 1105"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(t*y/9+t/3, (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" alt="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= -4" alt="t= -4" title="t= -4" data-latex="t= -4"/>. </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'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( -2 )= -1" alt="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( -2 )= -1" title="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( -2 )= -1" data-latex="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( -2 )= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(t*y/9+t/3, (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20%7By%7D%20t%20+%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t" alt="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%20-4" alt="t= -4" title="t= -4" data-latex="t= -4"&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?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20%7By%7D%20t%20+%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20%5Chspace%7B2em%7D%20y(%20-2%20)=%20-1" alt="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( -2 )= -1" title="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( -2 )= -1" data-latex="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( -4 )\approx -1.0" alt="y_p( -4 )\approx -1.0" title="y_p( -4 )\approx -1.0" data-latex="y_p( -4 )\approx -1.0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%20-4%20)%5Capprox%20-1.0" alt="y_p( -4 )\approx -1.0" title="y_p( -4 )\approx -1.0" data-latex="y_p( -4 )\approx -1.0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-3667" title="F1 | Direction fields for first-order ODEs | ver. 3667"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(t/3-y/15, (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t" alt="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t" title="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t" data-latex="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= -4" alt="t= -4" title="t= -4" data-latex="t= -4"/>. </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'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( -2 )= 0" alt="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( -2 )= 0" title="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( -2 )= 0" data-latex="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( -2 )= 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(t/3-y/15, (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20-%5Cfrac%7B1%7D%7B15%7D%20%5C,%20%7By%7D%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t" alt="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t" title="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t" data-latex="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%20-4" alt="t= -4" title="t= -4" data-latex="t= -4"&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?%7By'%7D%20=%20-%5Cfrac%7B1%7D%7B15%7D%20%5C,%20%7By%7D%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20%5Chspace%7B2em%7D%20y(%20-2%20)=%200" alt="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( -2 )= 0" title="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( -2 )= 0" data-latex="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( -4 )\approx 0.00" alt="y_p( -4 )\approx 0.00" title="y_p( -4 )\approx 0.00" data-latex="y_p( -4 )\approx 0.00"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%20-4%20)%5Capprox%200.00" alt="y_p( -4 )\approx 0.00" title="y_p( -4 )\approx 0.00" data-latex="y_p( -4 )\approx 0.00"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-1310" title="F1 | Direction fields for first-order ODEs | ver. 1310"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(y/15-t/3, (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y}" alt="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y}" title="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y}" data-latex="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y}"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 4" alt="t= 4" title="t= 4" data-latex="t= 4"/>. </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'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y} \hspace{2em} y( 2 )= 0" alt="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y} \hspace{2em} y( 2 )= 0" title="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y} \hspace{2em} y( 2 )= 0" data-latex="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y} \hspace{2em} y( 2 )= 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(y/15-t/3, (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20-%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20+%20%5Cfrac%7B1%7D%7B15%7D%20%5C,%20%7By%7D" alt="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y}" title="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y}" data-latex="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y}"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%204" alt="t= 4" title="t= 4" data-latex="t= 4"&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?%7By'%7D%20=%20-%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20+%20%5Cfrac%7B1%7D%7B15%7D%20%5C,%20%7By%7D%20%5Chspace%7B2em%7D%20y(%202%20)=%200" alt="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y} \hspace{2em} y( 2 )= 0" title="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y} \hspace{2em} y( 2 )= 0" data-latex="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y} \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 4 )\approx -2.1" alt="y_p( 4 )\approx -2.1" title="y_p( 4 )\approx -2.1" data-latex="y_p( 4 )\approx -2.1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%204%20)%5Capprox%20-2.1" alt="y_p( 4 )\approx -2.1" title="y_p( 4 )\approx -2.1" data-latex="y_p( 4 )\approx -2.1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-5269" title="F1 | Direction fields for first-order ODEs | ver. 5269"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(sin(t+y), (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \sin\left(t + {y}\right)" alt="{y'} = \sin\left(t + {y}\right)" title="{y'} = \sin\left(t + {y}\right)" data-latex="{y'} = \sin\left(t + {y}\right)"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 4" alt="t= 4" title="t= 4" data-latex="t= 4"/>. </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'} = \sin\left(t + {y}\right) \hspace{2em} y( 2 )= -1" alt="{y'} = \sin\left(t + {y}\right) \hspace{2em} y( 2 )= -1" title="{y'} = \sin\left(t + {y}\right) \hspace{2em} y( 2 )= -1" data-latex="{y'} = \sin\left(t + {y}\right) \hspace{2em} y( 2 )= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(sin(t+y), (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Csin%5Cleft(t%20+%20%7By%7D%5Cright)" alt="{y'} = \sin\left(t + {y}\right)" title="{y'} = \sin\left(t + {y}\right)" data-latex="{y'} = \sin\left(t + {y}\right)"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%204" alt="t= 4" title="t= 4" data-latex="t= 4"&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?%7By'%7D%20=%20%5Csin%5Cleft(t%20+%20%7By%7D%5Cright)%20%5Chspace%7B2em%7D%20y(%202%20)=%20-1" alt="{y'} = \sin\left(t + {y}\right) \hspace{2em} y( 2 )= -1" title="{y'} = \sin\left(t + {y}\right) \hspace{2em} y( 2 )= -1" data-latex="{y'} = \sin\left(t + {y}\right) \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 4 )\approx -0.35" alt="y_p( 4 )\approx -0.35" title="y_p( 4 )\approx -0.35" data-latex="y_p( 4 )\approx -0.35"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%204%20)%5Capprox%20-0.35" alt="y_p( 4 )\approx -0.35" title="y_p( 4 )\approx -0.35" data-latex="y_p( 4 )\approx -0.35"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-5276" title="F1 | Direction fields for first-order ODEs | ver. 5276"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(t/3-y/15, (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = -\frac{1}{3} \, t - \frac{1}{15} \, {y}" alt="{y'} = -\frac{1}{3} \, t - \frac{1}{15} \, {y}" title="{y'} = -\frac{1}{3} \, t - \frac{1}{15} \, {y}" data-latex="{y'} = -\frac{1}{3} \, t - \frac{1}{15} \, {y}"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 4" alt="t= 4" title="t= 4" data-latex="t= 4"/>. </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'} = -\frac{1}{3} \, t - \frac{1}{15} \, {y} \hspace{2em} y( 2 )= -2" alt="{y'} = -\frac{1}{3} \, t - \frac{1}{15} \, {y} \hspace{2em} y( 2 )= -2" title="{y'} = -\frac{1}{3} \, t - \frac{1}{15} \, {y} \hspace{2em} y( 2 )= -2" data-latex="{y'} = -\frac{1}{3} \, t - \frac{1}{15} \, {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;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(t/3-y/15, (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20-%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20-%20%5Cfrac%7B1%7D%7B15%7D%20%5C,%20%7By%7D" alt="{y'} = -\frac{1}{3} \, t - \frac{1}{15} \, {y}" title="{y'} = -\frac{1}{3} \, t - \frac{1}{15} \, {y}" data-latex="{y'} = -\frac{1}{3} \, t - \frac{1}{15} \, {y}"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%204" alt="t= 4" title="t= 4" data-latex="t= 4"&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?%7By'%7D%20=%20-%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20-%20%5Cfrac%7B1%7D%7B15%7D%20%5C,%20%7By%7D%20%5Chspace%7B2em%7D%20y(%202%20)=%20-2" alt="{y'} = -\frac{1}{3} \, t - \frac{1}{15} \, {y} \hspace{2em} y( 2 )= -2" title="{y'} = -\frac{1}{3} \, t - \frac{1}{15} \, {y} \hspace{2em} y( 2 )= -2" data-latex="{y'} = -\frac{1}{3} \, t - \frac{1}{15} \, {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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 4 )\approx -3.6" alt="y_p( 4 )\approx -3.6" title="y_p( 4 )\approx -3.6" data-latex="y_p( 4 )\approx -3.6"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%204%20)%5Capprox%20-3.6" alt="y_p( 4 )\approx -3.6" title="y_p( 4 )\approx -3.6" data-latex="y_p( 4 )\approx -3.6"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-7222" title="F1 | Direction fields for first-order ODEs | ver. 7222"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(sin(y/2), (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \sin\left(\frac{1}{2} \, {y}\right)" alt="{y'} = \sin\left(\frac{1}{2} \, {y}\right)" title="{y'} = \sin\left(\frac{1}{2} \, {y}\right)" data-latex="{y'} = \sin\left(\frac{1}{2} \, {y}\right)"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 2" alt="t= 2" title="t= 2" data-latex="t= 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'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 0 )= 2" alt="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 0 )= 2" title="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 0 )= 2" data-latex="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 0 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(sin(y/2), (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Csin%5Cleft(%5Cfrac%7B1%7D%7B2%7D%20%5C,%20%7By%7D%5Cright)" alt="{y'} = \sin\left(\frac{1}{2} \, {y}\right)" title="{y'} = \sin\left(\frac{1}{2} \, {y}\right)" data-latex="{y'} = \sin\left(\frac{1}{2} \, {y}\right)"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%202" alt="t= 2" title="t= 2" data-latex="t= 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?%7By'%7D%20=%20%5Csin%5Cleft(%5Cfrac%7B1%7D%7B2%7D%20%5C,%20%7By%7D%5Cright)%20%5Chspace%7B2em%7D%20y(%200%20)=%202" alt="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 0 )= 2" title="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 0 )= 2" data-latex="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 2 )\approx 3.9" alt="y_p( 2 )\approx 3.9" title="y_p( 2 )\approx 3.9" data-latex="y_p( 2 )\approx 3.9"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%202%20)%5Capprox%203.9" alt="y_p( 2 )\approx 3.9" title="y_p( 2 )\approx 3.9" data-latex="y_p( 2 )\approx 3.9"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-1607" title="F1 | Direction fields for first-order ODEs | ver. 1607"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(sin(y/2), (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \sin\left(\frac{1}{2} \, {y}\right)" alt="{y'} = \sin\left(\frac{1}{2} \, {y}\right)" title="{y'} = \sin\left(\frac{1}{2} \, {y}\right)" data-latex="{y'} = \sin\left(\frac{1}{2} \, {y}\right)"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 3" alt="t= 3" title="t= 3" data-latex="t= 3"/>. </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'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 1 )= -1" alt="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 1 )= -1" title="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 1 )= -1" data-latex="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 1 )= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(sin(y/2), (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Csin%5Cleft(%5Cfrac%7B1%7D%7B2%7D%20%5C,%20%7By%7D%5Cright)" alt="{y'} = \sin\left(\frac{1}{2} \, {y}\right)" title="{y'} = \sin\left(\frac{1}{2} \, {y}\right)" data-latex="{y'} = \sin\left(\frac{1}{2} \, {y}\right)"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%203" alt="t= 3" title="t= 3" data-latex="t= 3"&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?%7By'%7D%20=%20%5Csin%5Cleft(%5Cfrac%7B1%7D%7B2%7D%20%5C,%20%7By%7D%5Cright)%20%5Chspace%7B2em%7D%20y(%201%20)=%20-1" alt="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 1 )= -1" title="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 1 )= -1" data-latex="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 3 )\approx -2.4" alt="y_p( 3 )\approx -2.4" title="y_p( 3 )\approx -2.4" data-latex="y_p( 3 )\approx -2.4"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%203%20)%5Capprox%20-2.4" alt="y_p( 3 )\approx -2.4" title="y_p( 3 )\approx -2.4" data-latex="y_p( 3 )\approx -2.4"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-6819" title="F1 | Direction fields for first-order ODEs | ver. 6819"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(cos(y/2), (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \cos\left(\frac{1}{2} \, {y}\right)" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right)"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 1" alt="t= 1" title="t= 1" data-latex="t= 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'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -1 )= -2" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -1 )= -2" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -1 )= -2" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -1 )= -2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(cos(y/2), (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Ccos%5Cleft(%5Cfrac%7B1%7D%7B2%7D%20%5C,%20%7By%7D%5Cright)" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right)"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%201" alt="t= 1" title="t= 1" data-latex="t= 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?%7By'%7D%20=%20%5Ccos%5Cleft(%5Cfrac%7B1%7D%7B2%7D%20%5C,%20%7By%7D%5Cright)%20%5Chspace%7B2em%7D%20y(%20-1%20)=%20-2" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -1 )= -2" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -1 )= -2" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 1 )\approx -0.45" alt="y_p( 1 )\approx -0.45" title="y_p( 1 )\approx -0.45" data-latex="y_p( 1 )\approx -0.45"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%201%20)%5Capprox%20-0.45" alt="y_p( 1 )\approx -0.45" title="y_p( 1 )\approx -0.45" data-latex="y_p( 1 )\approx -0.45"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-8511" title="F1 | Direction fields for first-order ODEs | ver. 8511"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(cos(y/2), (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \cos\left(\frac{1}{2} \, {y}\right)" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right)"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 0" alt="t= 0" title="t= 0" data-latex="t= 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'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -2 )= -1" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -2 )= -1" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -2 )= -1" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -2 )= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(cos(y/2), (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Ccos%5Cleft(%5Cfrac%7B1%7D%7B2%7D%20%5C,%20%7By%7D%5Cright)" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right)"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%200" alt="t= 0" title="t= 0" data-latex="t= 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?%7By'%7D%20=%20%5Ccos%5Cleft(%5Cfrac%7B1%7D%7B2%7D%20%5C,%20%7By%7D%5Cright)%20%5Chspace%7B2em%7D%20y(%20-2%20)=%20-1" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -2 )= -1" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -2 )= -1" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 0 )\approx 0.92" alt="y_p( 0 )\approx 0.92" title="y_p( 0 )\approx 0.92" data-latex="y_p( 0 )\approx 0.92"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%200%20)%5Capprox%200.92" alt="y_p( 0 )\approx 0.92" title="y_p( 0 )\approx 0.92" data-latex="y_p( 0 )\approx 0.92"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-2167" title="F1 | Direction fields for first-order ODEs | ver. 2167"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(y/15-t/3, (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y}" alt="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y}" title="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y}" data-latex="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y}"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= -2" alt="t= -2" title="t= -2" data-latex="t= -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'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y} \hspace{2em} y( 0 )= 2" alt="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y} \hspace{2em} y( 0 )= 2" title="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y} \hspace{2em} y( 0 )= 2" data-latex="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {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;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(y/15-t/3, (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20-%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20+%20%5Cfrac%7B1%7D%7B15%7D%20%5C,%20%7By%7D" alt="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y}" title="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y}" data-latex="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y}"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%20-2" alt="t= -2" title="t= -2" data-latex="t= -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?%7By'%7D%20=%20-%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20+%20%5Cfrac%7B1%7D%7B15%7D%20%5C,%20%7By%7D%20%5Chspace%7B2em%7D%20y(%200%20)=%202" alt="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y} \hspace{2em} y( 0 )= 2" title="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y} \hspace{2em} y( 0 )= 2" data-latex="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( -2 )\approx 2.0" alt="y_p( -2 )\approx 2.0" title="y_p( -2 )\approx 2.0" data-latex="y_p( -2 )\approx 2.0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%20-2%20)%5Capprox%202.0" alt="y_p( -2 )\approx 2.0" title="y_p( -2 )\approx 2.0" data-latex="y_p( -2 )\approx 2.0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-7591" title="F1 | Direction fields for first-order ODEs | ver. 7591"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(t/3-y/15, (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t" alt="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t" title="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t" data-latex="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 4" alt="t= 4" title="t= 4" data-latex="t= 4"/>. </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'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( 2 )= 0" alt="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( 2 )= 0" title="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( 2 )= 0" data-latex="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( 2 )= 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(t/3-y/15, (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20-%5Cfrac%7B1%7D%7B15%7D%20%5C,%20%7By%7D%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t" alt="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t" title="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t" data-latex="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%204" alt="t= 4" title="t= 4" data-latex="t= 4"&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?%7By'%7D%20=%20-%5Cfrac%7B1%7D%7B15%7D%20%5C,%20%7By%7D%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20%5Chspace%7B2em%7D%20y(%202%20)=%200" alt="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( 2 )= 0" title="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( 2 )= 0" data-latex="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 4 )\approx -1.9" alt="y_p( 4 )\approx -1.9" title="y_p( 4 )\approx -1.9" data-latex="y_p( 4 )\approx -1.9"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%204%20)%5Capprox%20-1.9" alt="y_p( 4 )\approx -1.9" title="y_p( 4 )\approx -1.9" data-latex="y_p( 4 )\approx -1.9"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-6901" title="F1 | Direction fields for first-order ODEs | ver. 6901"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(y/15-t/3, (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y}" alt="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y}" title="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y}" data-latex="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y}"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= -3" alt="t= -3" title="t= -3" data-latex="t= -3"/>. </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'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y} \hspace{2em} y( -1 )= 1" alt="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y} \hspace{2em} y( -1 )= 1" title="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y} \hspace{2em} y( -1 )= 1" data-latex="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {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;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(y/15-t/3, (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20-%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20+%20%5Cfrac%7B1%7D%7B15%7D%20%5C,%20%7By%7D" alt="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y}" title="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y}" data-latex="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y}"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%20-3" alt="t= -3" title="t= -3" data-latex="t= -3"&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?%7By'%7D%20=%20-%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20+%20%5Cfrac%7B1%7D%7B15%7D%20%5C,%20%7By%7D%20%5Chspace%7B2em%7D%20y(%20-1%20)=%201" alt="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y} \hspace{2em} y( -1 )= 1" title="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y} \hspace{2em} y( -1 )= 1" data-latex="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( -3 )\approx 1.0" alt="y_p( -3 )\approx 1.0" title="y_p( -3 )\approx 1.0" data-latex="y_p( -3 )\approx 1.0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%20-3%20)%5Capprox%201.0" alt="y_p( -3 )\approx 1.0" title="y_p( -3 )\approx 1.0" data-latex="y_p( -3 )\approx 1.0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-2135" title="F1 | Direction fields for first-order ODEs | ver. 2135"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(t*y/9-t/3, (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" alt="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 1" alt="t= 1" title="t= 1" data-latex="t= 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'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( -1 )= 2" alt="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( -1 )= 2" title="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( -1 )= 2" data-latex="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( -1 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(t*y/9-t/3, (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20%7By%7D%20t%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t" alt="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%201" alt="t= 1" title="t= 1" data-latex="t= 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?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20%7By%7D%20t%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20%5Chspace%7B2em%7D%20y(%20-1%20)=%202" alt="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( -1 )= 2" title="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( -1 )= 2" data-latex="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 1 )\approx 2.0" alt="y_p( 1 )\approx 2.0" title="y_p( 1 )\approx 2.0" data-latex="y_p( 1 )\approx 2.0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%201%20)%5Capprox%202.0" alt="y_p( 1 )\approx 2.0" title="y_p( 1 )\approx 2.0" data-latex="y_p( 1 )\approx 2.0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-8570" title="F1 | Direction fields for first-order ODEs | ver. 8570"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(y/15-t/3, (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t" alt="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t" title="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= -4" alt="t= -4" title="t= -4" data-latex="t= -4"/>. </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'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( -2 )= -1" alt="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( -2 )= -1" title="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( -2 )= -1" data-latex="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( -2 )= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(y/15-t/3, (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B15%7D%20%5C,%20%7By%7D%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t" alt="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t" title="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%20-4" alt="t= -4" title="t= -4" data-latex="t= -4"&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?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B15%7D%20%5C,%20%7By%7D%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20%5Chspace%7B2em%7D%20y(%20-2%20)=%20-1" alt="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( -2 )= -1" title="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( -2 )= -1" data-latex="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( -4 )\approx -1.0" alt="y_p( -4 )\approx -1.0" title="y_p( -4 )\approx -1.0" data-latex="y_p( -4 )\approx -1.0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%20-4%20)%5Capprox%20-1.0" alt="y_p( -4 )\approx -1.0" title="y_p( -4 )\approx -1.0" data-latex="y_p( -4 )\approx -1.0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-9830" title="F1 | Direction fields for first-order ODEs | ver. 9830"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(sin(t+y), (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \sin\left(t + {y}\right)" alt="{y'} = \sin\left(t + {y}\right)" title="{y'} = \sin\left(t + {y}\right)" data-latex="{y'} = \sin\left(t + {y}\right)"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= -1" alt="t= -1" title="t= -1" data-latex="t= -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'} = \sin\left(t + {y}\right) \hspace{2em} y( 1 )= -2" alt="{y'} = \sin\left(t + {y}\right) \hspace{2em} y( 1 )= -2" title="{y'} = \sin\left(t + {y}\right) \hspace{2em} y( 1 )= -2" data-latex="{y'} = \sin\left(t + {y}\right) \hspace{2em} y( 1 )= -2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(sin(t+y), (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Csin%5Cleft(t%20+%20%7By%7D%5Cright)" alt="{y'} = \sin\left(t + {y}\right)" title="{y'} = \sin\left(t + {y}\right)" data-latex="{y'} = \sin\left(t + {y}\right)"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%20-1" alt="t= -1" title="t= -1" data-latex="t= -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?%7By'%7D%20=%20%5Csin%5Cleft(t%20+%20%7By%7D%5Cright)%20%5Chspace%7B2em%7D%20y(%201%20)=%20-2" alt="{y'} = \sin\left(t + {y}\right) \hspace{2em} y( 1 )= -2" title="{y'} = \sin\left(t + {y}\right) \hspace{2em} y( 1 )= -2" data-latex="{y'} = \sin\left(t + {y}\right) \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( -1 )\approx -2.0" alt="y_p( -1 )\approx -2.0" title="y_p( -1 )\approx -2.0" data-latex="y_p( -1 )\approx -2.0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%20-1%20)%5Capprox%20-2.0" alt="y_p( -1 )\approx -2.0" title="y_p( -1 )\approx -2.0" data-latex="y_p( -1 )\approx -2.0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-7207" title="F1 | Direction fields for first-order ODEs | ver. 7207"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(sin(t+y), (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \sin\left({y} + t\right)" alt="{y'} = \sin\left({y} + t\right)" title="{y'} = \sin\left({y} + t\right)" data-latex="{y'} = \sin\left({y} + t\right)"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= -1" alt="t= -1" title="t= -1" data-latex="t= -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'} = \sin\left({y} + t\right) \hspace{2em} y( 1 )= -1" alt="{y'} = \sin\left({y} + t\right) \hspace{2em} y( 1 )= -1" title="{y'} = \sin\left({y} + t\right) \hspace{2em} y( 1 )= -1" data-latex="{y'} = \sin\left({y} + t\right) \hspace{2em} y( 1 )= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(sin(t+y), (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Csin%5Cleft(%7By%7D%20+%20t%5Cright)" alt="{y'} = \sin\left({y} + t\right)" title="{y'} = \sin\left({y} + t\right)" data-latex="{y'} = \sin\left({y} + t\right)"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%20-1" alt="t= -1" title="t= -1" data-latex="t= -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?%7By'%7D%20=%20%5Csin%5Cleft(%7By%7D%20+%20t%5Cright)%20%5Chspace%7B2em%7D%20y(%201%20)=%20-1" alt="{y'} = \sin\left({y} + t\right) \hspace{2em} y( 1 )= -1" title="{y'} = \sin\left({y} + t\right) \hspace{2em} y( 1 )= -1" data-latex="{y'} = \sin\left({y} + t\right) \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( -1 )\approx -1.0" alt="y_p( -1 )\approx -1.0" title="y_p( -1 )\approx -1.0" data-latex="y_p( -1 )\approx -1.0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%20-1%20)%5Capprox%20-1.0" alt="y_p( -1 )\approx -1.0" title="y_p( -1 )\approx -1.0" data-latex="y_p( -1 )\approx -1.0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-0060" title="F1 | Direction fields for first-order ODEs | ver. 0060"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(t/3-y/15, (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t" alt="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t" title="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t" data-latex="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= -2" alt="t= -2" title="t= -2" data-latex="t= -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'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( 0 )= 2" alt="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( 0 )= 2" title="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( 0 )= 2" data-latex="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( 0 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(t/3-y/15, (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20-%5Cfrac%7B1%7D%7B15%7D%20%5C,%20%7By%7D%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t" alt="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t" title="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t" data-latex="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%20-2" alt="t= -2" title="t= -2" data-latex="t= -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?%7By'%7D%20=%20-%5Cfrac%7B1%7D%7B15%7D%20%5C,%20%7By%7D%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20%5Chspace%7B2em%7D%20y(%200%20)=%202" alt="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( 0 )= 2" title="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( 0 )= 2" data-latex="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( -2 )\approx 2.0" alt="y_p( -2 )\approx 2.0" title="y_p( -2 )\approx 2.0" data-latex="y_p( -2 )\approx 2.0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%20-2%20)%5Capprox%202.0" alt="y_p( -2 )\approx 2.0" title="y_p( -2 )\approx 2.0" data-latex="y_p( -2 )\approx 2.0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-7081" title="F1 | Direction fields for first-order ODEs | ver. 7081"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(cos(t+y), (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \cos\left({y} + t\right)" alt="{y'} = \cos\left({y} + t\right)" title="{y'} = \cos\left({y} + t\right)" data-latex="{y'} = \cos\left({y} + t\right)"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 1" alt="t= 1" title="t= 1" data-latex="t= 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'} = \cos\left({y} + t\right) \hspace{2em} y( -1 )= 2" alt="{y'} = \cos\left({y} + t\right) \hspace{2em} y( -1 )= 2" title="{y'} = \cos\left({y} + t\right) \hspace{2em} y( -1 )= 2" data-latex="{y'} = \cos\left({y} + t\right) \hspace{2em} y( -1 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(cos(t+y), (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Ccos%5Cleft(%7By%7D%20+%20t%5Cright)" alt="{y'} = \cos\left({y} + t\right)" title="{y'} = \cos\left({y} + t\right)" data-latex="{y'} = \cos\left({y} + t\right)"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%201" alt="t= 1" title="t= 1" data-latex="t= 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?%7By'%7D%20=%20%5Ccos%5Cleft(%7By%7D%20+%20t%5Cright)%20%5Chspace%7B2em%7D%20y(%20-1%20)=%202" alt="{y'} = \cos\left({y} + t\right) \hspace{2em} y( -1 )= 2" title="{y'} = \cos\left({y} + t\right) \hspace{2em} y( -1 )= 2" data-latex="{y'} = \cos\left({y} + t\right) \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 1 )\approx 1.4" alt="y_p( 1 )\approx 1.4" title="y_p( 1 )\approx 1.4" data-latex="y_p( 1 )\approx 1.4"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%201%20)%5Capprox%201.4" alt="y_p( 1 )\approx 1.4" title="y_p( 1 )\approx 1.4" data-latex="y_p( 1 )\approx 1.4"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-7721" title="F1 | Direction fields for first-order ODEs | ver. 7721"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(y/15-t/3, (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t" alt="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t" title="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= -4" alt="t= -4" title="t= -4" data-latex="t= -4"/>. </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'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( -2 )= -2" alt="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( -2 )= -2" title="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( -2 )= -2" data-latex="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( -2 )= -2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(y/15-t/3, (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B15%7D%20%5C,%20%7By%7D%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t" alt="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t" title="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%20-4" alt="t= -4" title="t= -4" data-latex="t= -4"&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?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B15%7D%20%5C,%20%7By%7D%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20%5Chspace%7B2em%7D%20y(%20-2%20)=%20-2" alt="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( -2 )= -2" title="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( -2 )= -2" data-latex="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( -4 )\approx -2.0" alt="y_p( -4 )\approx -2.0" title="y_p( -4 )\approx -2.0" data-latex="y_p( -4 )\approx -2.0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%20-4%20)%5Capprox%20-2.0" alt="y_p( -4 )\approx -2.0" title="y_p( -4 )\approx -2.0" data-latex="y_p( -4 )\approx -2.0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-4192" title="F1 | Direction fields for first-order ODEs | ver. 4192"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(cos(t+y), (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \cos\left({y} + t\right)" alt="{y'} = \cos\left({y} + t\right)" title="{y'} = \cos\left({y} + t\right)" data-latex="{y'} = \cos\left({y} + t\right)"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 2" alt="t= 2" title="t= 2" data-latex="t= 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'} = \cos\left({y} + t\right) \hspace{2em} y( 0 )= -2" alt="{y'} = \cos\left({y} + t\right) \hspace{2em} y( 0 )= -2" title="{y'} = \cos\left({y} + t\right) \hspace{2em} y( 0 )= -2" data-latex="{y'} = \cos\left({y} + t\right) \hspace{2em} y( 0 )= -2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(cos(t+y), (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Ccos%5Cleft(%7By%7D%20+%20t%5Cright)" alt="{y'} = \cos\left({y} + t\right)" title="{y'} = \cos\left({y} + t\right)" data-latex="{y'} = \cos\left({y} + t\right)"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%202" alt="t= 2" title="t= 2" data-latex="t= 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?%7By'%7D%20=%20%5Ccos%5Cleft(%7By%7D%20+%20t%5Cright)%20%5Chspace%7B2em%7D%20y(%200%20)=%20-2" alt="{y'} = \cos\left({y} + t\right) \hspace{2em} y( 0 )= -2" title="{y'} = \cos\left({y} + t\right) \hspace{2em} y( 0 )= -2" data-latex="{y'} = \cos\left({y} + t\right) \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 2 )\approx -1.2" alt="y_p( 2 )\approx -1.2" title="y_p( 2 )\approx -1.2" data-latex="y_p( 2 )\approx -1.2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%202%20)%5Capprox%20-1.2" alt="y_p( 2 )\approx -1.2" title="y_p( 2 )\approx -1.2" data-latex="y_p( 2 )\approx -1.2"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-1016" title="F1 | Direction fields for first-order ODEs | ver. 1016"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(sin(t+y), (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \sin\left({y} + t\right)" alt="{y'} = \sin\left({y} + t\right)" title="{y'} = \sin\left({y} + t\right)" data-latex="{y'} = \sin\left({y} + t\right)"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 0" alt="t= 0" title="t= 0" data-latex="t= 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'} = \sin\left({y} + t\right) \hspace{2em} y( -2 )= 1" alt="{y'} = \sin\left({y} + t\right) \hspace{2em} y( -2 )= 1" title="{y'} = \sin\left({y} + t\right) \hspace{2em} y( -2 )= 1" data-latex="{y'} = \sin\left({y} + t\right) \hspace{2em} y( -2 )= 1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(sin(t+y), (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Csin%5Cleft(%7By%7D%20+%20t%5Cright)" alt="{y'} = \sin\left({y} + t\right)" title="{y'} = \sin\left({y} + t\right)" data-latex="{y'} = \sin\left({y} + t\right)"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%200" alt="t= 0" title="t= 0" data-latex="t= 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?%7By'%7D%20=%20%5Csin%5Cleft(%7By%7D%20+%20t%5Cright)%20%5Chspace%7B2em%7D%20y(%20-2%20)=%201" alt="{y'} = \sin\left({y} + t\right) \hspace{2em} y( -2 )= 1" title="{y'} = \sin\left({y} + t\right) \hspace{2em} y( -2 )= 1" data-latex="{y'} = \sin\left({y} + t\right) \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 0 )\approx -0.34" alt="y_p( 0 )\approx -0.34" title="y_p( 0 )\approx -0.34" data-latex="y_p( 0 )\approx -0.34"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%200%20)%5Capprox%20-0.34" alt="y_p( 0 )\approx -0.34" title="y_p( 0 )\approx -0.34" data-latex="y_p( 0 )\approx -0.34"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-7758" title="F1 | Direction fields for first-order ODEs | ver. 7758"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(t*y/9-t/3, (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \frac{1}{9} \, t {y} - \frac{1}{3} \, t" alt="{y'} = \frac{1}{9} \, t {y} - \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, t {y} - \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, t {y} - \frac{1}{3} \, t"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 0" alt="t= 0" title="t= 0" data-latex="t= 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'} = \frac{1}{9} \, t {y} - \frac{1}{3} \, t \hspace{2em} y( -2 )= 0" alt="{y'} = \frac{1}{9} \, t {y} - \frac{1}{3} \, t \hspace{2em} y( -2 )= 0" title="{y'} = \frac{1}{9} \, t {y} - \frac{1}{3} \, t \hspace{2em} y( -2 )= 0" data-latex="{y'} = \frac{1}{9} \, t {y} - \frac{1}{3} \, t \hspace{2em} y( -2 )= 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(t*y/9-t/3, (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20t%20%7By%7D%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t" alt="{y'} = \frac{1}{9} \, t {y} - \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, t {y} - \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, t {y} - \frac{1}{3} \, t"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%200" alt="t= 0" title="t= 0" data-latex="t= 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?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20t%20%7By%7D%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20%5Chspace%7B2em%7D%20y(%20-2%20)=%200" alt="{y'} = \frac{1}{9} \, t {y} - \frac{1}{3} \, t \hspace{2em} y( -2 )= 0" title="{y'} = \frac{1}{9} \, t {y} - \frac{1}{3} \, t \hspace{2em} y( -2 )= 0" data-latex="{y'} = \frac{1}{9} \, t {y} - \frac{1}{3} \, t \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 0 )\approx 0.60" alt="y_p( 0 )\approx 0.60" title="y_p( 0 )\approx 0.60" data-latex="y_p( 0 )\approx 0.60"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%200%20)%5Capprox%200.60" alt="y_p( 0 )\approx 0.60" title="y_p( 0 )\approx 0.60" data-latex="y_p( 0 )\approx 0.60"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-1934" title="F1 | Direction fields for first-order ODEs | ver. 1934"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(y/15-t/3, (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y}" alt="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y}" title="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y}" data-latex="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y}"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= -1" alt="t= -1" title="t= -1" data-latex="t= -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'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y} \hspace{2em} y( 1 )= -1" alt="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y} \hspace{2em} y( 1 )= -1" title="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y} \hspace{2em} y( 1 )= -1" data-latex="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {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;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(y/15-t/3, (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20-%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20+%20%5Cfrac%7B1%7D%7B15%7D%20%5C,%20%7By%7D" alt="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y}" title="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y}" data-latex="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y}"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%20-1" alt="t= -1" title="t= -1" data-latex="t= -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?%7By'%7D%20=%20-%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20+%20%5Cfrac%7B1%7D%7B15%7D%20%5C,%20%7By%7D%20%5Chspace%7B2em%7D%20y(%201%20)=%20-1" alt="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y} \hspace{2em} y( 1 )= -1" title="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y} \hspace{2em} y( 1 )= -1" data-latex="{y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( -1 )\approx -1.0" alt="y_p( -1 )\approx -1.0" title="y_p( -1 )\approx -1.0" data-latex="y_p( -1 )\approx -1.0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%20-1%20)%5Capprox%20-1.0" alt="y_p( -1 )\approx -1.0" title="y_p( -1 )\approx -1.0" data-latex="y_p( -1 )\approx -1.0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-7004" title="F1 | Direction fields for first-order ODEs | ver. 7004"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(cos(t+y), (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \cos\left(t + {y}\right)" alt="{y'} = \cos\left(t + {y}\right)" title="{y'} = \cos\left(t + {y}\right)" data-latex="{y'} = \cos\left(t + {y}\right)"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= -3" alt="t= -3" title="t= -3" data-latex="t= -3"/>. </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'} = \cos\left(t + {y}\right) \hspace{2em} y( -1 )= 1" alt="{y'} = \cos\left(t + {y}\right) \hspace{2em} y( -1 )= 1" title="{y'} = \cos\left(t + {y}\right) \hspace{2em} y( -1 )= 1" data-latex="{y'} = \cos\left(t + {y}\right) \hspace{2em} y( -1 )= 1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(cos(t+y), (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Ccos%5Cleft(t%20+%20%7By%7D%5Cright)" alt="{y'} = \cos\left(t + {y}\right)" title="{y'} = \cos\left(t + {y}\right)" data-latex="{y'} = \cos\left(t + {y}\right)"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%20-3" alt="t= -3" title="t= -3" data-latex="t= -3"&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?%7By'%7D%20=%20%5Ccos%5Cleft(t%20+%20%7By%7D%5Cright)%20%5Chspace%7B2em%7D%20y(%20-1%20)=%201" alt="{y'} = \cos\left(t + {y}\right) \hspace{2em} y( -1 )= 1" title="{y'} = \cos\left(t + {y}\right) \hspace{2em} y( -1 )= 1" data-latex="{y'} = \cos\left(t + {y}\right) \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( -3 )\approx 1.0" alt="y_p( -3 )\approx 1.0" title="y_p( -3 )\approx 1.0" data-latex="y_p( -3 )\approx 1.0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%20-3%20)%5Capprox%201.0" alt="y_p( -3 )\approx 1.0" title="y_p( -3 )\approx 1.0" data-latex="y_p( -3 )\approx 1.0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-2647" title="F1 | Direction fields for first-order ODEs | ver. 2647"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(t*y/9-t/3, (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" alt="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 1" alt="t= 1" title="t= 1" data-latex="t= 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'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( -1 )= 0" alt="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( -1 )= 0" title="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( -1 )= 0" data-latex="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( -1 )= 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(t*y/9-t/3, (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20%7By%7D%20t%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t" alt="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%201" alt="t= 1" title="t= 1" data-latex="t= 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?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20%7By%7D%20t%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20%5Chspace%7B2em%7D%20y(%20-1%20)=%200" alt="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( -1 )= 0" title="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( -1 )= 0" data-latex="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 1 )\approx 3.0 \times 10^{-9}" alt="y_p( 1 )\approx 3.0 \times 10^{-9}" title="y_p( 1 )\approx 3.0 \times 10^{-9}" data-latex="y_p( 1 )\approx 3.0 \times 10^{-9}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%201%20)%5Capprox%203.0%20%5Ctimes%2010%5E%7B-9%7D" alt="y_p( 1 )\approx 3.0 \times 10^{-9}" title="y_p( 1 )\approx 3.0 \times 10^{-9}" data-latex="y_p( 1 )\approx 3.0 \times 10^{-9}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-4171" title="F1 | Direction fields for first-order ODEs | ver. 4171"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(cos(y/2), (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \cos\left(\frac{1}{2} \, {y}\right)" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right)"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= -4" alt="t= -4" title="t= -4" data-latex="t= -4"/>. </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'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -2 )= 2" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -2 )= 2" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -2 )= 2" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -2 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(cos(y/2), (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Ccos%5Cleft(%5Cfrac%7B1%7D%7B2%7D%20%5C,%20%7By%7D%5Cright)" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right)"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%20-4" alt="t= -4" title="t= -4" data-latex="t= -4"&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?%7By'%7D%20=%20%5Ccos%5Cleft(%5Cfrac%7B1%7D%7B2%7D%20%5C,%20%7By%7D%5Cright)%20%5Chspace%7B2em%7D%20y(%20-2%20)=%202" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -2 )= 2" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -2 )= 2" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( -4 )\approx 2.0" alt="y_p( -4 )\approx 2.0" title="y_p( -4 )\approx 2.0" data-latex="y_p( -4 )\approx 2.0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%20-4%20)%5Capprox%202.0" alt="y_p( -4 )\approx 2.0" title="y_p( -4 )\approx 2.0" data-latex="y_p( -4 )\approx 2.0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-2327" title="F1 | Direction fields for first-order ODEs | ver. 2327"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(t*y/9-t/3, (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" alt="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 0" alt="t= 0" title="t= 0" data-latex="t= 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'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( 2 )= -2" alt="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( 2 )= -2" title="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( 2 )= -2" data-latex="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( 2 )= -2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(t*y/9-t/3, (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20%7By%7D%20t%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t" alt="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%200" alt="t= 0" title="t= 0" data-latex="t= 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?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20%7By%7D%20t%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20%5Chspace%7B2em%7D%20y(%202%20)=%20-2" alt="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( 2 )= -2" title="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( 2 )= -2" data-latex="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 0 )\approx -2.0" alt="y_p( 0 )\approx -2.0" title="y_p( 0 )\approx -2.0" data-latex="y_p( 0 )\approx -2.0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%200%20)%5Capprox%20-2.0" alt="y_p( 0 )\approx -2.0" title="y_p( 0 )\approx -2.0" data-latex="y_p( 0 )\approx -2.0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-4942" title="F1 | Direction fields for first-order ODEs | ver. 4942"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(cos(t+y), (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \cos\left(t + {y}\right)" alt="{y'} = \cos\left(t + {y}\right)" title="{y'} = \cos\left(t + {y}\right)" data-latex="{y'} = \cos\left(t + {y}\right)"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 2" alt="t= 2" title="t= 2" data-latex="t= 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'} = \cos\left(t + {y}\right) \hspace{2em} y( 0 )= 1" alt="{y'} = \cos\left(t + {y}\right) \hspace{2em} y( 0 )= 1" title="{y'} = \cos\left(t + {y}\right) \hspace{2em} y( 0 )= 1" data-latex="{y'} = \cos\left(t + {y}\right) \hspace{2em} y( 0 )= 1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(cos(t+y), (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Ccos%5Cleft(t%20+%20%7By%7D%5Cright)" alt="{y'} = \cos\left(t + {y}\right)" title="{y'} = \cos\left(t + {y}\right)" data-latex="{y'} = \cos\left(t + {y}\right)"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%202" alt="t= 2" title="t= 2" data-latex="t= 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?%7By'%7D%20=%20%5Ccos%5Cleft(t%20+%20%7By%7D%5Cright)%20%5Chspace%7B2em%7D%20y(%200%20)=%201" alt="{y'} = \cos\left(t + {y}\right) \hspace{2em} y( 0 )= 1" title="{y'} = \cos\left(t + {y}\right) \hspace{2em} y( 0 )= 1" data-latex="{y'} = \cos\left(t + {y}\right) \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 2 )\approx 0.39" alt="y_p( 2 )\approx 0.39" title="y_p( 2 )\approx 0.39" data-latex="y_p( 2 )\approx 0.39"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%202%20)%5Capprox%200.39" alt="y_p( 2 )\approx 0.39" title="y_p( 2 )\approx 0.39" data-latex="y_p( 2 )\approx 0.39"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-6432" title="F1 | Direction fields for first-order ODEs | ver. 6432"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(cos(t+y), (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \cos\left({y} + t\right)" alt="{y'} = \cos\left({y} + t\right)" title="{y'} = \cos\left({y} + t\right)" data-latex="{y'} = \cos\left({y} + t\right)"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 0" alt="t= 0" title="t= 0" data-latex="t= 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'} = \cos\left({y} + t\right) \hspace{2em} y( 2 )= -1" alt="{y'} = \cos\left({y} + t\right) \hspace{2em} y( 2 )= -1" title="{y'} = \cos\left({y} + t\right) \hspace{2em} y( 2 )= -1" data-latex="{y'} = \cos\left({y} + t\right) \hspace{2em} y( 2 )= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(cos(t+y), (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Ccos%5Cleft(%7By%7D%20+%20t%5Cright)" alt="{y'} = \cos\left({y} + t\right)" title="{y'} = \cos\left({y} + t\right)" data-latex="{y'} = \cos\left({y} + t\right)"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%200" alt="t= 0" title="t= 0" data-latex="t= 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?%7By'%7D%20=%20%5Ccos%5Cleft(%7By%7D%20+%20t%5Cright)%20%5Chspace%7B2em%7D%20y(%202%20)=%20-1" alt="{y'} = \cos\left({y} + t\right) \hspace{2em} y( 2 )= -1" title="{y'} = \cos\left({y} + t\right) \hspace{2em} y( 2 )= -1" data-latex="{y'} = \cos\left({y} + t\right) \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 0 )\approx -1.0" alt="y_p( 0 )\approx -1.0" title="y_p( 0 )\approx -1.0" data-latex="y_p( 0 )\approx -1.0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%200%20)%5Capprox%20-1.0" alt="y_p( 0 )\approx -1.0" title="y_p( 0 )\approx -1.0" data-latex="y_p( 0 )\approx -1.0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-7723" title="F1 | Direction fields for first-order ODEs | ver. 7723"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(t/3-y/15, (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t" alt="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t" title="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t" data-latex="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= -4" alt="t= -4" title="t= -4" data-latex="t= -4"/>. </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'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( -2 )= 1" alt="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( -2 )= 1" title="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( -2 )= 1" data-latex="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( -2 )= 1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(t/3-y/15, (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20-%5Cfrac%7B1%7D%7B15%7D%20%5C,%20%7By%7D%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t" alt="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t" title="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t" data-latex="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%20-4" alt="t= -4" title="t= -4" data-latex="t= -4"&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?%7By'%7D%20=%20-%5Cfrac%7B1%7D%7B15%7D%20%5C,%20%7By%7D%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20%5Chspace%7B2em%7D%20y(%20-2%20)=%201" alt="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( -2 )= 1" title="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( -2 )= 1" data-latex="{y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( -4 )\approx 1.0" alt="y_p( -4 )\approx 1.0" title="y_p( -4 )\approx 1.0" data-latex="y_p( -4 )\approx 1.0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%20-4%20)%5Capprox%201.0" alt="y_p( -4 )\approx 1.0" title="y_p( -4 )\approx 1.0" data-latex="y_p( -4 )\approx 1.0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-5233" title="F1 | Direction fields for first-order ODEs | ver. 5233"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(cos(y/2), (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \cos\left(\frac{1}{2} \, {y}\right)" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right)"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 4" alt="t= 4" title="t= 4" data-latex="t= 4"/>. </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'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 2 )= -1" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 2 )= -1" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 2 )= -1" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 2 )= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(cos(y/2), (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Ccos%5Cleft(%5Cfrac%7B1%7D%7B2%7D%20%5C,%20%7By%7D%5Cright)" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right)"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%204" alt="t= 4" title="t= 4" data-latex="t= 4"&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?%7By'%7D%20=%20%5Ccos%5Cleft(%5Cfrac%7B1%7D%7B2%7D%20%5C,%20%7By%7D%5Cright)%20%5Chspace%7B2em%7D%20y(%202%20)=%20-1" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 2 )= -1" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 2 )= -1" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 4 )\approx 0.92" alt="y_p( 4 )\approx 0.92" title="y_p( 4 )\approx 0.92" data-latex="y_p( 4 )\approx 0.92"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%204%20)%5Capprox%200.92" alt="y_p( 4 )\approx 0.92" title="y_p( 4 )\approx 0.92" data-latex="y_p( 4 )\approx 0.92"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-7073" title="F1 | Direction fields for first-order ODEs | ver. 7073"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(sin(y/2), (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \sin\left(\frac{1}{2} \, {y}\right)" alt="{y'} = \sin\left(\frac{1}{2} \, {y}\right)" title="{y'} = \sin\left(\frac{1}{2} \, {y}\right)" data-latex="{y'} = \sin\left(\frac{1}{2} \, {y}\right)"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 2" alt="t= 2" title="t= 2" data-latex="t= 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'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 0 )= 0" alt="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 0 )= 0" title="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 0 )= 0" data-latex="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 0 )= 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(sin(y/2), (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Csin%5Cleft(%5Cfrac%7B1%7D%7B2%7D%20%5C,%20%7By%7D%5Cright)" alt="{y'} = \sin\left(\frac{1}{2} \, {y}\right)" title="{y'} = \sin\left(\frac{1}{2} \, {y}\right)" data-latex="{y'} = \sin\left(\frac{1}{2} \, {y}\right)"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%202" alt="t= 2" title="t= 2" data-latex="t= 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?%7By'%7D%20=%20%5Csin%5Cleft(%5Cfrac%7B1%7D%7B2%7D%20%5C,%20%7By%7D%5Cright)%20%5Chspace%7B2em%7D%20y(%200%20)=%200" alt="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 0 )= 0" title="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 0 )= 0" data-latex="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 2 )\approx 0.00" alt="y_p( 2 )\approx 0.00" title="y_p( 2 )\approx 0.00" data-latex="y_p( 2 )\approx 0.00"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%202%20)%5Capprox%200.00" alt="y_p( 2 )\approx 0.00" title="y_p( 2 )\approx 0.00" data-latex="y_p( 2 )\approx 0.00"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-3716" title="F1 | Direction fields for first-order ODEs | ver. 3716"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(t*y/9+t/3, (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" alt="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 4" alt="t= 4" title="t= 4" data-latex="t= 4"/>. </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'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( 2 )= -2" alt="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( 2 )= -2" title="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( 2 )= -2" data-latex="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( 2 )= -2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(t*y/9+t/3, (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20%7By%7D%20t%20+%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t" alt="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%204" alt="t= 4" title="t= 4" data-latex="t= 4"&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?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20%7By%7D%20t%20+%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20%5Chspace%7B2em%7D%20y(%202%20)=%20-2" alt="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( 2 )= -2" title="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( 2 )= -2" data-latex="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 4 )\approx -1.1" alt="y_p( 4 )\approx -1.1" title="y_p( 4 )\approx -1.1" data-latex="y_p( 4 )\approx -1.1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%204%20)%5Capprox%20-1.1" alt="y_p( 4 )\approx -1.1" title="y_p( 4 )\approx -1.1" data-latex="y_p( 4 )\approx -1.1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-6412" title="F1 | Direction fields for first-order ODEs | ver. 6412"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(cos(y/2), (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \cos\left(\frac{1}{2} \, {y}\right)" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right)"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= -3" alt="t= -3" title="t= -3" data-latex="t= -3"/>. </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'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -1 )= 1" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -1 )= 1" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -1 )= 1" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -1 )= 1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(cos(y/2), (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Ccos%5Cleft(%5Cfrac%7B1%7D%7B2%7D%20%5C,%20%7By%7D%5Cright)" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right)" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right)"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%20-3" alt="t= -3" title="t= -3" data-latex="t= -3"&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?%7By'%7D%20=%20%5Ccos%5Cleft(%5Cfrac%7B1%7D%7B2%7D%20%5C,%20%7By%7D%5Cright)%20%5Chspace%7B2em%7D%20y(%20-1%20)=%201" alt="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -1 )= 1" title="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -1 )= 1" data-latex="{y'} = \cos\left(\frac{1}{2} \, {y}\right) \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( -3 )\approx 1.0" alt="y_p( -3 )\approx 1.0" title="y_p( -3 )\approx 1.0" data-latex="y_p( -3 )\approx 1.0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%20-3%20)%5Capprox%201.0" alt="y_p( -3 )\approx 1.0" title="y_p( -3 )\approx 1.0" data-latex="y_p( -3 )\approx 1.0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-9432" title="F1 | Direction fields for first-order ODEs | ver. 9432"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(sin(y/2), (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \sin\left(\frac{1}{2} \, {y}\right)" alt="{y'} = \sin\left(\frac{1}{2} \, {y}\right)" title="{y'} = \sin\left(\frac{1}{2} \, {y}\right)" data-latex="{y'} = \sin\left(\frac{1}{2} \, {y}\right)"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 0" alt="t= 0" title="t= 0" data-latex="t= 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'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -2 )= 2" alt="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -2 )= 2" title="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -2 )= 2" data-latex="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -2 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(sin(y/2), (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Csin%5Cleft(%5Cfrac%7B1%7D%7B2%7D%20%5C,%20%7By%7D%5Cright)" alt="{y'} = \sin\left(\frac{1}{2} \, {y}\right)" title="{y'} = \sin\left(\frac{1}{2} \, {y}\right)" data-latex="{y'} = \sin\left(\frac{1}{2} \, {y}\right)"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%200" alt="t= 0" title="t= 0" data-latex="t= 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?%7By'%7D%20=%20%5Csin%5Cleft(%5Cfrac%7B1%7D%7B2%7D%20%5C,%20%7By%7D%5Cright)%20%5Chspace%7B2em%7D%20y(%20-2%20)=%202" alt="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -2 )= 2" title="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -2 )= 2" data-latex="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 0 )\approx 3.9" alt="y_p( 0 )\approx 3.9" title="y_p( 0 )\approx 3.9" data-latex="y_p( 0 )\approx 3.9"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%200%20)%5Capprox%203.9" alt="y_p( 0 )\approx 3.9" title="y_p( 0 )\approx 3.9" data-latex="y_p( 0 )\approx 3.9"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-4969" title="F1 | Direction fields for first-order ODEs | ver. 4969"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(t*y/9+t/3, (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" alt="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= -4" alt="t= -4" title="t= -4" data-latex="t= -4"/>. </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'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( -2 )= 0" alt="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( -2 )= 0" title="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( -2 )= 0" data-latex="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( -2 )= 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(t*y/9+t/3, (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20%7By%7D%20t%20+%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t" alt="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%20-4" alt="t= -4" title="t= -4" data-latex="t= -4"&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?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20%7By%7D%20t%20+%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20%5Chspace%7B2em%7D%20y(%20-2%20)=%200" alt="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( -2 )= 0" title="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( -2 )= 0" data-latex="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( -4 )\approx 0.00" alt="y_p( -4 )\approx 0.00" title="y_p( -4 )\approx 0.00" data-latex="y_p( -4 )\approx 0.00"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%20-4%20)%5Capprox%200.00" alt="y_p( -4 )\approx 0.00" title="y_p( -4 )\approx 0.00" data-latex="y_p( -4 )\approx 0.00"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-6119" title="F1 | Direction fields for first-order ODEs | ver. 6119"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(t*y/9+t/3, (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" alt="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= -3" alt="t= -3" title="t= -3" data-latex="t= -3"/>. </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'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( -1 )= 2" alt="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( -1 )= 2" title="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( -1 )= 2" data-latex="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( -1 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(t*y/9+t/3, (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20%7By%7D%20t%20+%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t" alt="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%20-3" alt="t= -3" title="t= -3" data-latex="t= -3"&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?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20%7By%7D%20t%20+%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20%5Chspace%7B2em%7D%20y(%20-1%20)=%202" alt="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( -1 )= 2" title="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( -1 )= 2" data-latex="{y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( -3 )\approx 2.0" alt="y_p( -3 )\approx 2.0" title="y_p( -3 )\approx 2.0" data-latex="y_p( -3 )\approx 2.0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%20-3%20)%5Capprox%202.0" alt="y_p( -3 )\approx 2.0" title="y_p( -3 )\approx 2.0" data-latex="y_p( -3 )\approx 2.0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-5192" title="F1 | Direction fields for first-order ODEs | ver. 5192"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(y/15-t/3, (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t" alt="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t" title="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 4" alt="t= 4" title="t= 4" data-latex="t= 4"/>. </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'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( 2 )= -1" alt="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( 2 )= -1" title="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( 2 )= -1" data-latex="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( 2 )= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(y/15-t/3, (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B15%7D%20%5C,%20%7By%7D%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t" alt="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t" title="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%204" alt="t= 4" title="t= 4" data-latex="t= 4"&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?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B15%7D%20%5C,%20%7By%7D%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20%5Chspace%7B2em%7D%20y(%202%20)=%20-1" alt="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( 2 )= -1" title="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( 2 )= -1" data-latex="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 4 )\approx -3.3" alt="y_p( 4 )\approx -3.3" title="y_p( 4 )\approx -3.3" data-latex="y_p( 4 )\approx -3.3"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%204%20)%5Capprox%20-3.3" alt="y_p( 4 )\approx -3.3" title="y_p( 4 )\approx -3.3" data-latex="y_p( 4 )\approx -3.3"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-4244" title="F1 | Direction fields for first-order ODEs | ver. 4244"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(sin(y/2), (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \sin\left(\frac{1}{2} \, {y}\right)" alt="{y'} = \sin\left(\frac{1}{2} \, {y}\right)" title="{y'} = \sin\left(\frac{1}{2} \, {y}\right)" data-latex="{y'} = \sin\left(\frac{1}{2} \, {y}\right)"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 0" alt="t= 0" title="t= 0" data-latex="t= 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'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -2 )= 2" alt="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -2 )= 2" title="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -2 )= 2" data-latex="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -2 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(sin(y/2), (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Csin%5Cleft(%5Cfrac%7B1%7D%7B2%7D%20%5C,%20%7By%7D%5Cright)" alt="{y'} = \sin\left(\frac{1}{2} \, {y}\right)" title="{y'} = \sin\left(\frac{1}{2} \, {y}\right)" data-latex="{y'} = \sin\left(\frac{1}{2} \, {y}\right)"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%200" alt="t= 0" title="t= 0" data-latex="t= 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?%7By'%7D%20=%20%5Csin%5Cleft(%5Cfrac%7B1%7D%7B2%7D%20%5C,%20%7By%7D%5Cright)%20%5Chspace%7B2em%7D%20y(%20-2%20)=%202" alt="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -2 )= 2" title="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -2 )= 2" data-latex="{y'} = \sin\left(\frac{1}{2} \, {y}\right) \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 0 )\approx 3.9" alt="y_p( 0 )\approx 3.9" title="y_p( 0 )\approx 3.9" data-latex="y_p( 0 )\approx 3.9"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%200%20)%5Capprox%203.9" alt="y_p( 0 )\approx 3.9" title="y_p( 0 )\approx 3.9" data-latex="y_p( 0 )\approx 3.9"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-7977" title="F1 | Direction fields for first-order ODEs | ver. 7977"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(y/15-t/3, (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t" alt="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t" title="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= 3" alt="t= 3" title="t= 3" data-latex="t= 3"/>. </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'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( 1 )= -1" alt="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( 1 )= -1" title="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( 1 )= -1" data-latex="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( 1 )= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(y/15-t/3, (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B15%7D%20%5C,%20%7By%7D%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t" alt="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t" title="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%203" alt="t= 3" title="t= 3" data-latex="t= 3"&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?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B15%7D%20%5C,%20%7By%7D%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20%5Chspace%7B2em%7D%20y(%201%20)=%20-1" alt="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( 1 )= -1" title="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( 1 )= -1" data-latex="{y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 3 )\approx -2.6" alt="y_p( 3 )\approx -2.6" title="y_p( 3 )\approx -2.6" data-latex="y_p( 3 )\approx -2.6"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%203%20)%5Capprox%20-2.6" alt="y_p( 3 )\approx -2.6" title="y_p( 3 )\approx -2.6" data-latex="y_p( 3 )\approx -2.6"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F1-2707" title="F1 | Direction fields for first-order ODEs | ver. 2707"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F1.</strong></p><p> Use <a href="https://sagecell.sagemath.org/">https://sagecell.sagemath.org/</a> to run the SageMath code <code>t,y = var('t y'); plot_slope_field(t*y/9-t/3, (t,-5,5), (y,-5,5))</code> producing the direction field for the ODE <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" alt="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t"/>. </p><p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> be the solution to the following IVP. Explain how to use its direction field to approximate the value of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"/> at <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t= -4" alt="t= -4" title="t= -4" data-latex="t= -4"/>. </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'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( -2 )= 2" alt="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( -2 )= 2" title="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( -2 )= 2" data-latex="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( -2 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use &lt;a href="https://sagecell.sagemath.org/"&gt;https://sagecell.sagemath.org/&lt;/a&gt; to run the SageMath code &lt;code&gt;t,y = var('t y'); plot_slope_field(t*y/9-t/3, (t,-5,5), (y,-5,5))&lt;/code&gt; producing the direction field for the ODE &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20%7By%7D%20t%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t" alt="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" title="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t" data-latex="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t"&gt;. &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; be the solution to the following IVP. Explain how to use its direction field to approximate the value of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p" alt="y_p" title="y_p" data-latex="y_p"&gt; at &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t=%20-4" alt="t= -4" title="t= -4" data-latex="t= -4"&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?%7By'%7D%20=%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20%7By%7D%20t%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20t%20%5Chspace%7B2em%7D%20y(%20-2%20)=%202" alt="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( -2 )= 2" title="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( -2 )= 2" data-latex="{y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( -4 )\approx 2.0" alt="y_p( -4 )\approx 2.0" title="y_p( -4 )\approx 2.0" data-latex="y_p( -4 )\approx 2.0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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_p(%20-4%20)%5Capprox%202.0" alt="y_p( -4 )\approx 2.0" title="y_p( -4 )\approx 2.0" data-latex="y_p( -4 )\approx 2.0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

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

Resources

Company
