S1.qti

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      <qtimetadatafield><fieldlabel>bank_title</fieldlabel><fieldentry>Differential Equations -- S1</fieldentry></qtimetadatafield>
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  <item ident="S1-0579" title="S1 | Phase Planes | ver. 0579"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -2 \, x + 2 \, y - 6" alt="x'= -2 \, x + 2 \, y - 6" title="x'= -2 \, x + 2 \, y - 6" data-latex="x'= -2 \, x + 2 \, y - 6"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -4 \, x - 3 \, y + 23" alt="y'= -4 \, x - 3 \, y + 23" title="y'= -4 \, x - 3 \, y + 23" data-latex="y'= -4 \, x - 3 \, y + 23"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-2%20%5C,%20x%20+%202%20%5C,%20y%20-%206" alt="x'= -2 \, x + 2 \, y - 6" title="x'= -2 \, x + 2 \, y - 6" data-latex="x'= -2 \, x + 2 \, y - 6"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-4%20%5C,%20x%20-%203%20%5C,%20y%20+%2023" alt="y'= -4 \, x - 3 \, y + 23" title="y'= -4 \, x - 3 \, y + 23" data-latex="y'= -4 \, x - 3 \, y + 23"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(2, 5\right)" alt="\left(2, 5\right)" title="\left(2, 5\right)" data-latex="\left(2, 5\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a positive slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a negative slope. </p><p> The right region has a southwest arrow system, the top region has a southeast arrow system, the left region has a northeast arrow system, and the bottom region has a northwest arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(2,%205%5Cright)" alt="\left(2, 5\right)" title="\left(2, 5\right)" data-latex="\left(2, 5\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a positive slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a negative slope. &lt;/p&gt;
  &lt;p&gt; The right region has a southwest arrow system, the top region has a southeast arrow system, the left region has a northeast arrow system, and the bottom region has a northwest arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-2611" title="S1 | Phase Planes | ver. 2611"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -2 \, x - 3 \, y + 2" alt="x'= -2 \, x - 3 \, y + 2" title="x'= -2 \, x - 3 \, y + 2" data-latex="x'= -2 \, x - 3 \, y + 2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= x - 3 \, y - 10" alt="y'= x - 3 \, y - 10" title="y'= x - 3 \, y - 10" data-latex="y'= x - 3 \, y - 10"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-2%20%5C,%20x%20-%203%20%5C,%20y%20+%202" alt="x'= -2 \, x - 3 \, y + 2" title="x'= -2 \, x - 3 \, y + 2" data-latex="x'= -2 \, x - 3 \, y + 2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20x%20-%203%20%5C,%20y%20-%2010" alt="y'= x - 3 \, y - 10" title="y'= x - 3 \, y - 10" data-latex="y'= x - 3 \, y - 10"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(4, -2\right)" alt="\left(4, -2\right)" title="\left(4, -2\right)" data-latex="\left(4, -2\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(4,%20-2%5Cright)" alt="\left(4, -2\right)" title="\left(4, -2\right)" data-latex="\left(4, -2\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-2928" title="S1 | Phase Planes | ver. 2928"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 3 \, x - 2 \, y - 15" alt="x'= 3 \, x - 2 \, y - 15" title="x'= 3 \, x - 2 \, y - 15" data-latex="x'= 3 \, x - 2 \, y - 15"/></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'= 5 \, x + 3 \, y - 6" alt="y'= 5 \, x + 3 \, y - 6" title="y'= 5 \, x + 3 \, y - 6" data-latex="y'= 5 \, x + 3 \, y - 6"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%203%20%5C,%20x%20-%202%20%5C,%20y%20-%2015" alt="x'= 3 \, x - 2 \, y - 15" title="x'= 3 \, x - 2 \, y - 15" data-latex="x'= 3 \, x - 2 \, y - 15"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%205%20%5C,%20x%20+%203%20%5C,%20y%20-%206" alt="y'= 5 \, x + 3 \, y - 6" title="y'= 5 \, x + 3 \, y - 6" data-latex="y'= 5 \, x + 3 \, y - 6"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(3, -3\right)" alt="\left(3, -3\right)" title="\left(3, -3\right)" data-latex="\left(3, -3\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a positive slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a negative slope. </p><p> The right region has a northeast arrow system, the top region has a northwest arrow system, the left region has a southwest arrow system, and the bottom region has a southeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(3,%20-3%5Cright)" alt="\left(3, -3\right)" title="\left(3, -3\right)" data-latex="\left(3, -3\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a positive slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a negative slope. &lt;/p&gt;
  &lt;p&gt; The right region has a northeast arrow system, the top region has a northwest arrow system, the left region has a southwest arrow system, and the bottom region has a southeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-6995" title="S1 | Phase Planes | ver. 6995"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -2 \, x - 2 \, y" alt="x'= -2 \, x - 2 \, y" title="x'= -2 \, x - 2 \, y" data-latex="x'= -2 \, x - 2 \, y"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -4 \, x + 2 \, y + 18" alt="y'= -4 \, x + 2 \, y + 18" title="y'= -4 \, x + 2 \, y + 18" data-latex="y'= -4 \, x + 2 \, y + 18"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-2%20%5C,%20x%20-%202%20%5C,%20y" alt="x'= -2 \, x - 2 \, y" title="x'= -2 \, x - 2 \, y" data-latex="x'= -2 \, x - 2 \, y"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-4%20%5C,%20x%20+%202%20%5C,%20y%20+%2018" alt="y'= -4 \, x + 2 \, y + 18" title="y'= -4 \, x + 2 \, y + 18" data-latex="y'= -4 \, x + 2 \, y + 18"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(3, -3\right)" alt="\left(3, -3\right)" title="\left(3, -3\right)" data-latex="\left(3, -3\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(3,%20-3%5Cright)" alt="\left(3, -3\right)" title="\left(3, -3\right)" data-latex="\left(3, -3\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-8080" title="S1 | Phase Planes | ver. 8080"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -5 \, x + 4 \, y + 3" alt="x'= -5 \, x + 4 \, y + 3" title="x'= -5 \, x + 4 \, y + 3" data-latex="x'= -5 \, x + 4 \, y + 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'= -x - 5 \, y + 18" alt="y'= -x - 5 \, y + 18" title="y'= -x - 5 \, y + 18" data-latex="y'= -x - 5 \, y + 18"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-5%20%5C,%20x%20+%204%20%5C,%20y%20+%203" alt="x'= -5 \, x + 4 \, y + 3" title="x'= -5 \, x + 4 \, y + 3" data-latex="x'= -5 \, x + 4 \, y + 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?y'=%20-x%20-%205%20%5C,%20y%20+%2018" alt="y'= -x - 5 \, y + 18" title="y'= -x - 5 \, y + 18" data-latex="y'= -x - 5 \, y + 18"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(3, 3\right)" alt="\left(3, 3\right)" title="\left(3, 3\right)" data-latex="\left(3, 3\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a positive slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a negative slope. </p><p> The right region has a southwest arrow system, the top region has a southeast arrow system, the left region has a northeast arrow system, and the bottom region has a northwest arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(3,%203%5Cright)" alt="\left(3, 3\right)" title="\left(3, 3\right)" data-latex="\left(3, 3\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a positive slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a negative slope. &lt;/p&gt;
  &lt;p&gt; The right region has a southwest arrow system, the top region has a southeast arrow system, the left region has a northeast arrow system, and the bottom region has a northwest arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-8563" title="S1 | Phase Planes | ver. 8563"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -5 \, x + 2 \, y - 2" alt="x'= -5 \, x + 2 \, y - 2" title="x'= -5 \, x + 2 \, y - 2" data-latex="x'= -5 \, x + 2 \, y - 2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -4 \, x - y - 12" alt="y'= -4 \, x - y - 12" title="y'= -4 \, x - y - 12" data-latex="y'= -4 \, x - y - 12"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-5%20%5C,%20x%20+%202%20%5C,%20y%20-%202" alt="x'= -5 \, x + 2 \, y - 2" title="x'= -5 \, x + 2 \, y - 2" data-latex="x'= -5 \, x + 2 \, y - 2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-4%20%5C,%20x%20-%20y%20-%2012" alt="y'= -4 \, x - y - 12" title="y'= -4 \, x - y - 12" data-latex="y'= -4 \, x - y - 12"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(-2, -4\right)" alt="\left(-2, -4\right)" title="\left(-2, -4\right)" data-latex="\left(-2, -4\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a positive slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a negative slope. </p><p> The right region has a southwest arrow system, the top region has a southeast arrow system, the left region has a northeast arrow system, and the bottom region has a northwest arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(-2,%20-4%5Cright)" alt="\left(-2, -4\right)" title="\left(-2, -4\right)" data-latex="\left(-2, -4\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a positive slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a negative slope. &lt;/p&gt;
  &lt;p&gt; The right region has a southwest arrow system, the top region has a southeast arrow system, the left region has a northeast arrow system, and the bottom region has a northwest arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-5670" title="S1 | Phase Planes | ver. 5670"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 3 \, x - 4 \, y - 11" alt="x'= 3 \, x - 4 \, y - 11" title="x'= 3 \, x - 4 \, y - 11" data-latex="x'= 3 \, x - 4 \, y - 11"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 2 \, x + 5 \, y + 31" alt="y'= 2 \, x + 5 \, y + 31" title="y'= 2 \, x + 5 \, y + 31" data-latex="y'= 2 \, x + 5 \, y + 31"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%203%20%5C,%20x%20-%204%20%5C,%20y%20-%2011" alt="x'= 3 \, x - 4 \, y - 11" title="x'= 3 \, x - 4 \, y - 11" data-latex="x'= 3 \, x - 4 \, y - 11"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%202%20%5C,%20x%20+%205%20%5C,%20y%20+%2031" alt="y'= 2 \, x + 5 \, y + 31" title="y'= 2 \, x + 5 \, y + 31" data-latex="y'= 2 \, x + 5 \, y + 31"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(-3, -5\right)" alt="\left(-3, -5\right)" title="\left(-3, -5\right)" data-latex="\left(-3, -5\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a positive slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a negative slope. </p><p> The right region has a northeast arrow system, the top region has a northwest arrow system, the left region has a southwest arrow system, and the bottom region has a southeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(-3,%20-5%5Cright)" alt="\left(-3, -5\right)" title="\left(-3, -5\right)" data-latex="\left(-3, -5\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a positive slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a negative slope. &lt;/p&gt;
  &lt;p&gt; The right region has a northeast arrow system, the top region has a northwest arrow system, the left region has a southwest arrow system, and the bottom region has a southeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-7357" title="S1 | Phase Planes | ver. 7357"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 4 \, x + 4 \, y + 4" alt="x'= 4 \, x + 4 \, y + 4" title="x'= 4 \, x + 4 \, y + 4" data-latex="x'= 4 \, x + 4 \, y + 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'= 4 \, x - 5 \, y + 13" alt="y'= 4 \, x - 5 \, y + 13" title="y'= 4 \, x - 5 \, y + 13" data-latex="y'= 4 \, x - 5 \, y + 13"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%204%20%5C,%20x%20+%204%20%5C,%20y%20+%204" alt="x'= 4 \, x + 4 \, y + 4" title="x'= 4 \, x + 4 \, y + 4" data-latex="x'= 4 \, x + 4 \, y + 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?y'=%204%20%5C,%20x%20-%205%20%5C,%20y%20+%2013" alt="y'= 4 \, x - 5 \, y + 13" title="y'= 4 \, x - 5 \, y + 13" data-latex="y'= 4 \, x - 5 \, y + 13"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(-2, 1\right)" alt="\left(-2, 1\right)" title="\left(-2, 1\right)" data-latex="\left(-2, 1\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a northeast arrow system, the top region has a southeast arrow system, the left region has a southwest arrow system, and the bottom region has a northwest arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(-2,%201%5Cright)" alt="\left(-2, 1\right)" title="\left(-2, 1\right)" data-latex="\left(-2, 1\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a northeast arrow system, the top region has a southeast arrow system, the left region has a southwest arrow system, and the bottom region has a northwest arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-1892" title="S1 | Phase Planes | ver. 1892"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= x - y - 2" alt="x'= x - y - 2" title="x'= x - y - 2" data-latex="x'= x - y - 2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -2 \, x - y + 7" alt="y'= -2 \, x - y + 7" title="y'= -2 \, x - y + 7" data-latex="y'= -2 \, x - y + 7"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20x%20-%20y%20-%202" alt="x'= x - y - 2" title="x'= x - y - 2" data-latex="x'= x - y - 2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-2%20%5C,%20x%20-%20y%20+%207" alt="y'= -2 \, x - y + 7" title="y'= -2 \, x - y + 7" data-latex="y'= -2 \, x - y + 7"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(3, 1\right)" alt="\left(3, 1\right)" title="\left(3, 1\right)" data-latex="\left(3, 1\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a positive slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a negative slope. </p><p> The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(3,%201%5Cright)" alt="\left(3, 1\right)" title="\left(3, 1\right)" data-latex="\left(3, 1\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a positive slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a negative slope. &lt;/p&gt;
  &lt;p&gt; The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-7404" title="S1 | Phase Planes | ver. 7404"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -2 \, x - 3 \, y + 3" alt="x'= -2 \, x - 3 \, y + 3" title="x'= -2 \, x - 3 \, y + 3" data-latex="x'= -2 \, x - 3 \, y + 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'= x - 5 \, y - 8" alt="y'= x - 5 \, y - 8" title="y'= x - 5 \, y - 8" data-latex="y'= x - 5 \, y - 8"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-2%20%5C,%20x%20-%203%20%5C,%20y%20+%203" alt="x'= -2 \, x - 3 \, y + 3" title="x'= -2 \, x - 3 \, y + 3" data-latex="x'= -2 \, x - 3 \, y + 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?y'=%20x%20-%205%20%5C,%20y%20-%208" alt="y'= x - 5 \, y - 8" title="y'= x - 5 \, y - 8" data-latex="y'= x - 5 \, y - 8"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(3, -1\right)" alt="\left(3, -1\right)" title="\left(3, -1\right)" data-latex="\left(3, -1\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(3,%20-1%5Cright)" alt="\left(3, -1\right)" title="\left(3, -1\right)" data-latex="\left(3, -1\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-3539" title="S1 | Phase Planes | ver. 3539"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -5 \, x - 3 \, y + 14" alt="x'= -5 \, x - 3 \, y + 14" title="x'= -5 \, x - 3 \, y + 14" data-latex="x'= -5 \, x - 3 \, y + 14"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 4 \, x - 3 \, y + 5" alt="y'= 4 \, x - 3 \, y + 5" title="y'= 4 \, x - 3 \, y + 5" data-latex="y'= 4 \, x - 3 \, y + 5"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-5%20%5C,%20x%20-%203%20%5C,%20y%20+%2014" alt="x'= -5 \, x - 3 \, y + 14" title="x'= -5 \, x - 3 \, y + 14" data-latex="x'= -5 \, x - 3 \, y + 14"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%204%20%5C,%20x%20-%203%20%5C,%20y%20+%205" alt="y'= 4 \, x - 3 \, y + 5" title="y'= 4 \, x - 3 \, y + 5" data-latex="y'= 4 \, x - 3 \, y + 5"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(1, 3\right)" alt="\left(1, 3\right)" title="\left(1, 3\right)" data-latex="\left(1, 3\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(1,%203%5Cright)" alt="\left(1, 3\right)" title="\left(1, 3\right)" data-latex="\left(1, 3\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-6255" title="S1 | Phase Planes | ver. 6255"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 5 \, x - 4 \, y - 32" alt="x'= 5 \, x - 4 \, y - 32" title="x'= 5 \, x - 4 \, y - 32" data-latex="x'= 5 \, x - 4 \, y - 32"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -2 \, x - y + 5" alt="y'= -2 \, x - y + 5" title="y'= -2 \, x - y + 5" data-latex="y'= -2 \, x - y + 5"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%205%20%5C,%20x%20-%204%20%5C,%20y%20-%2032" alt="x'= 5 \, x - 4 \, y - 32" title="x'= 5 \, x - 4 \, y - 32" data-latex="x'= 5 \, x - 4 \, y - 32"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-2%20%5C,%20x%20-%20y%20+%205" alt="y'= -2 \, x - y + 5" title="y'= -2 \, x - y + 5" data-latex="y'= -2 \, x - y + 5"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(4, -3\right)" alt="\left(4, -3\right)" title="\left(4, -3\right)" data-latex="\left(4, -3\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a positive slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a negative slope. </p><p> The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(4,%20-3%5Cright)" alt="\left(4, -3\right)" title="\left(4, -3\right)" data-latex="\left(4, -3\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a positive slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a negative slope. &lt;/p&gt;
  &lt;p&gt; The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-7382" title="S1 | Phase Planes | ver. 7382"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -4 \, x - 5 \, y + 9" alt="x'= -4 \, x - 5 \, y + 9" title="x'= -4 \, x - 5 \, y + 9" data-latex="x'= -4 \, x - 5 \, y + 9"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= x - 2 \, y + 1" alt="y'= x - 2 \, y + 1" title="y'= x - 2 \, y + 1" data-latex="y'= x - 2 \, y + 1"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-4%20%5C,%20x%20-%205%20%5C,%20y%20+%209" alt="x'= -4 \, x - 5 \, y + 9" title="x'= -4 \, x - 5 \, y + 9" data-latex="x'= -4 \, x - 5 \, y + 9"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20x%20-%202%20%5C,%20y%20+%201" alt="y'= x - 2 \, y + 1" title="y'= x - 2 \, y + 1" data-latex="y'= x - 2 \, y + 1"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(1, 1\right)" alt="\left(1, 1\right)" title="\left(1, 1\right)" data-latex="\left(1, 1\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(1,%201%5Cright)" alt="\left(1, 1\right)" title="\left(1, 1\right)" data-latex="\left(1, 1\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-5724" title="S1 | Phase Planes | ver. 5724"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 2 \, x - 5 \, y - 6" alt="x'= 2 \, x - 5 \, y - 6" title="x'= 2 \, x - 5 \, y - 6" data-latex="x'= 2 \, x - 5 \, y - 6"/></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'= 5 \, x + 2 \, y + 14" alt="y'= 5 \, x + 2 \, y + 14" title="y'= 5 \, x + 2 \, y + 14" data-latex="y'= 5 \, x + 2 \, y + 14"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%202%20%5C,%20x%20-%205%20%5C,%20y%20-%206" alt="x'= 2 \, x - 5 \, y - 6" title="x'= 2 \, x - 5 \, y - 6" data-latex="x'= 2 \, x - 5 \, y - 6"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%205%20%5C,%20x%20+%202%20%5C,%20y%20+%2014" alt="y'= 5 \, x + 2 \, y + 14" title="y'= 5 \, x + 2 \, y + 14" data-latex="y'= 5 \, x + 2 \, y + 14"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(-2, -2\right)" alt="\left(-2, -2\right)" title="\left(-2, -2\right)" data-latex="\left(-2, -2\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a positive slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a negative slope. </p><p> The right region has a northeast arrow system, the top region has a northwest arrow system, the left region has a southwest arrow system, and the bottom region has a southeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(-2,%20-2%5Cright)" alt="\left(-2, -2\right)" title="\left(-2, -2\right)" data-latex="\left(-2, -2\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a positive slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a negative slope. &lt;/p&gt;
  &lt;p&gt; The right region has a northeast arrow system, the top region has a northwest arrow system, the left region has a southwest arrow system, and the bottom region has a southeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-0609" title="S1 | Phase Planes | ver. 0609"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -4 \, x - 4 \, y - 12" alt="x'= -4 \, x - 4 \, y - 12" title="x'= -4 \, x - 4 \, y - 12" data-latex="x'= -4 \, x - 4 \, y - 12"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 3 \, x - 2 \, y + 9" alt="y'= 3 \, x - 2 \, y + 9" title="y'= 3 \, x - 2 \, y + 9" data-latex="y'= 3 \, x - 2 \, y + 9"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-4%20%5C,%20x%20-%204%20%5C,%20y%20-%2012" alt="x'= -4 \, x - 4 \, y - 12" title="x'= -4 \, x - 4 \, y - 12" data-latex="x'= -4 \, x - 4 \, y - 12"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%203%20%5C,%20x%20-%202%20%5C,%20y%20+%209" alt="y'= 3 \, x - 2 \, y + 9" title="y'= 3 \, x - 2 \, y + 9" data-latex="y'= 3 \, x - 2 \, y + 9"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(-3, 0\right)" alt="\left(-3, 0\right)" title="\left(-3, 0\right)" data-latex="\left(-3, 0\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(-3,%200%5Cright)" alt="\left(-3, 0\right)" title="\left(-3, 0\right)" data-latex="\left(-3, 0\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-9339" title="S1 | Phase Planes | ver. 9339"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -x - y + 2" alt="x'= -x - y + 2" title="x'= -x - y + 2" data-latex="x'= -x - y + 2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 2 \, x - 4 \, y + 8" alt="y'= 2 \, x - 4 \, y + 8" title="y'= 2 \, x - 4 \, y + 8" data-latex="y'= 2 \, x - 4 \, y + 8"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-x%20-%20y%20+%202" alt="x'= -x - y + 2" title="x'= -x - y + 2" data-latex="x'= -x - y + 2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%202%20%5C,%20x%20-%204%20%5C,%20y%20+%208" alt="y'= 2 \, x - 4 \, y + 8" title="y'= 2 \, x - 4 \, y + 8" data-latex="y'= 2 \, x - 4 \, y + 8"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(0, 2\right)" alt="\left(0, 2\right)" title="\left(0, 2\right)" data-latex="\left(0, 2\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(0,%202%5Cright)" alt="\left(0, 2\right)" title="\left(0, 2\right)" data-latex="\left(0, 2\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-9899" title="S1 | Phase Planes | ver. 9899"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -2 \, x - y - 4" alt="x'= -2 \, x - y - 4" title="x'= -2 \, x - y - 4" data-latex="x'= -2 \, x - y - 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'= 3 \, x - 3 \, y + 15" alt="y'= 3 \, x - 3 \, y + 15" title="y'= 3 \, x - 3 \, y + 15" data-latex="y'= 3 \, x - 3 \, y + 15"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-2%20%5C,%20x%20-%20y%20-%204" alt="x'= -2 \, x - y - 4" title="x'= -2 \, x - y - 4" data-latex="x'= -2 \, x - y - 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?y'=%203%20%5C,%20x%20-%203%20%5C,%20y%20+%2015" alt="y'= 3 \, x - 3 \, y + 15" title="y'= 3 \, x - 3 \, y + 15" data-latex="y'= 3 \, x - 3 \, y + 15"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(-3, 2\right)" alt="\left(-3, 2\right)" title="\left(-3, 2\right)" data-latex="\left(-3, 2\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(-3,%202%5Cright)" alt="\left(-3, 2\right)" title="\left(-3, 2\right)" data-latex="\left(-3, 2\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-9518" title="S1 | Phase Planes | ver. 9518"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -2 \, x - 4 \, y - 6" alt="x'= -2 \, x - 4 \, y - 6" title="x'= -2 \, x - 4 \, y - 6" data-latex="x'= -2 \, x - 4 \, y - 6"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -4 \, x + y + 15" alt="y'= -4 \, x + y + 15" title="y'= -4 \, x + y + 15" data-latex="y'= -4 \, x + y + 15"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-2%20%5C,%20x%20-%204%20%5C,%20y%20-%206" alt="x'= -2 \, x - 4 \, y - 6" title="x'= -2 \, x - 4 \, y - 6" data-latex="x'= -2 \, x - 4 \, y - 6"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-4%20%5C,%20x%20+%20y%20+%2015" alt="y'= -4 \, x + y + 15" title="y'= -4 \, x + y + 15" data-latex="y'= -4 \, x + y + 15"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(3, -3\right)" alt="\left(3, -3\right)" title="\left(3, -3\right)" data-latex="\left(3, -3\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(3,%20-3%5Cright)" alt="\left(3, -3\right)" title="\left(3, -3\right)" data-latex="\left(3, -3\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-6012" title="S1 | Phase Planes | ver. 6012"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -x - 5 \, y - 1" alt="x'= -x - 5 \, y - 1" title="x'= -x - 5 \, y - 1" data-latex="x'= -x - 5 \, y - 1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -3 \, x + y + 13" alt="y'= -3 \, x + y + 13" title="y'= -3 \, x + y + 13" data-latex="y'= -3 \, x + y + 13"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-x%20-%205%20%5C,%20y%20-%201" alt="x'= -x - 5 \, y - 1" title="x'= -x - 5 \, y - 1" data-latex="x'= -x - 5 \, y - 1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-3%20%5C,%20x%20+%20y%20+%2013" alt="y'= -3 \, x + y + 13" title="y'= -3 \, x + y + 13" data-latex="y'= -3 \, x + y + 13"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(4, -1\right)" alt="\left(4, -1\right)" title="\left(4, -1\right)" data-latex="\left(4, -1\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(4,%20-1%5Cright)" alt="\left(4, -1\right)" title="\left(4, -1\right)" data-latex="\left(4, -1\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-4696" title="S1 | Phase Planes | ver. 4696"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -2 \, x - 2 \, y - 2" alt="x'= -2 \, x - 2 \, y - 2" title="x'= -2 \, x - 2 \, y - 2" data-latex="x'= -2 \, x - 2 \, y - 2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= x - 2 \, y + 4" alt="y'= x - 2 \, y + 4" title="y'= x - 2 \, y + 4" data-latex="y'= x - 2 \, y + 4"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-2%20%5C,%20x%20-%202%20%5C,%20y%20-%202" alt="x'= -2 \, x - 2 \, y - 2" title="x'= -2 \, x - 2 \, y - 2" data-latex="x'= -2 \, x - 2 \, y - 2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20x%20-%202%20%5C,%20y%20+%204" alt="y'= x - 2 \, y + 4" title="y'= x - 2 \, y + 4" data-latex="y'= x - 2 \, y + 4"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(-2, 1\right)" alt="\left(-2, 1\right)" title="\left(-2, 1\right)" data-latex="\left(-2, 1\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(-2,%201%5Cright)" alt="\left(-2, 1\right)" title="\left(-2, 1\right)" data-latex="\left(-2, 1\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-1319" title="S1 | Phase Planes | ver. 1319"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 5 \, x - 2 \, y - 16" alt="x'= 5 \, x - 2 \, y - 16" title="x'= 5 \, x - 2 \, y - 16" data-latex="x'= 5 \, x - 2 \, y - 16"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -2 \, x - 5 \, y + 18" alt="y'= -2 \, x - 5 \, y + 18" title="y'= -2 \, x - 5 \, y + 18" data-latex="y'= -2 \, x - 5 \, y + 18"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%205%20%5C,%20x%20-%202%20%5C,%20y%20-%2016" alt="x'= 5 \, x - 2 \, y - 16" title="x'= 5 \, x - 2 \, y - 16" data-latex="x'= 5 \, x - 2 \, y - 16"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-2%20%5C,%20x%20-%205%20%5C,%20y%20+%2018" alt="y'= -2 \, x - 5 \, y + 18" title="y'= -2 \, x - 5 \, y + 18" data-latex="y'= -2 \, x - 5 \, y + 18"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(4, 2\right)" alt="\left(4, 2\right)" title="\left(4, 2\right)" data-latex="\left(4, 2\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a positive slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a negative slope. </p><p> The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(4,%202%5Cright)" alt="\left(4, 2\right)" title="\left(4, 2\right)" data-latex="\left(4, 2\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a positive slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a negative slope. &lt;/p&gt;
  &lt;p&gt; The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-3957" title="S1 | Phase Planes | ver. 3957"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -4 \, x - 3 \, y - 22" alt="x'= -4 \, x - 3 \, y - 22" title="x'= -4 \, x - 3 \, y - 22" data-latex="x'= -4 \, x - 3 \, y - 22"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 2 \, x - 4 \, y" alt="y'= 2 \, x - 4 \, y" title="y'= 2 \, x - 4 \, y" data-latex="y'= 2 \, x - 4 \, y"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-4%20%5C,%20x%20-%203%20%5C,%20y%20-%2022" alt="x'= -4 \, x - 3 \, y - 22" title="x'= -4 \, x - 3 \, y - 22" data-latex="x'= -4 \, x - 3 \, y - 22"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%202%20%5C,%20x%20-%204%20%5C,%20y" alt="y'= 2 \, x - 4 \, y" title="y'= 2 \, x - 4 \, y" data-latex="y'= 2 \, x - 4 \, y"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(-4, -2\right)" alt="\left(-4, -2\right)" title="\left(-4, -2\right)" data-latex="\left(-4, -2\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(-4,%20-2%5Cright)" alt="\left(-4, -2\right)" title="\left(-4, -2\right)" data-latex="\left(-4, -2\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-4166" title="S1 | Phase Planes | ver. 4166"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 5 \, x - 5 \, y - 35" alt="x'= 5 \, x - 5 \, y - 35" title="x'= 5 \, x - 5 \, y - 35" data-latex="x'= 5 \, x - 5 \, y - 35"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -3 \, x - 2 \, y + 1" alt="y'= -3 \, x - 2 \, y + 1" title="y'= -3 \, x - 2 \, y + 1" data-latex="y'= -3 \, x - 2 \, y + 1"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%205%20%5C,%20x%20-%205%20%5C,%20y%20-%2035" alt="x'= 5 \, x - 5 \, y - 35" title="x'= 5 \, x - 5 \, y - 35" data-latex="x'= 5 \, x - 5 \, y - 35"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-3%20%5C,%20x%20-%202%20%5C,%20y%20+%201" alt="y'= -3 \, x - 2 \, y + 1" title="y'= -3 \, x - 2 \, y + 1" data-latex="y'= -3 \, x - 2 \, y + 1"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(3, -4\right)" alt="\left(3, -4\right)" title="\left(3, -4\right)" data-latex="\left(3, -4\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a positive slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a negative slope. </p><p> The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(3,%20-4%5Cright)" alt="\left(3, -4\right)" title="\left(3, -4\right)" data-latex="\left(3, -4\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a positive slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a negative slope. &lt;/p&gt;
  &lt;p&gt; The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-3178" title="S1 | Phase Planes | ver. 3178"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -4 \, x - 5 \, y - 37" alt="x'= -4 \, x - 5 \, y - 37" title="x'= -4 \, x - 5 \, y - 37" data-latex="x'= -4 \, x - 5 \, y - 37"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -2 \, x + 2 \, y + 4" alt="y'= -2 \, x + 2 \, y + 4" title="y'= -2 \, x + 2 \, y + 4" data-latex="y'= -2 \, x + 2 \, y + 4"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-4%20%5C,%20x%20-%205%20%5C,%20y%20-%2037" alt="x'= -4 \, x - 5 \, y - 37" title="x'= -4 \, x - 5 \, y - 37" data-latex="x'= -4 \, x - 5 \, y - 37"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-2%20%5C,%20x%20+%202%20%5C,%20y%20+%204" alt="y'= -2 \, x + 2 \, y + 4" title="y'= -2 \, x + 2 \, y + 4" data-latex="y'= -2 \, x + 2 \, y + 4"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(-3, -5\right)" alt="\left(-3, -5\right)" title="\left(-3, -5\right)" data-latex="\left(-3, -5\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(-3,%20-5%5Cright)" alt="\left(-3, -5\right)" title="\left(-3, -5\right)" data-latex="\left(-3, -5\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-8305" title="S1 | Phase Planes | ver. 8305"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -5 \, x - 2 \, y - 8" alt="x'= -5 \, x - 2 \, y - 8" title="x'= -5 \, x - 2 \, y - 8" data-latex="x'= -5 \, x - 2 \, y - 8"/></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'= 5 \, x - 2 \, y + 12" alt="y'= 5 \, x - 2 \, y + 12" title="y'= 5 \, x - 2 \, y + 12" data-latex="y'= 5 \, x - 2 \, y + 12"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-5%20%5C,%20x%20-%202%20%5C,%20y%20-%208" alt="x'= -5 \, x - 2 \, y - 8" title="x'= -5 \, x - 2 \, y - 8" data-latex="x'= -5 \, x - 2 \, y - 8"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%205%20%5C,%20x%20-%202%20%5C,%20y%20+%2012" alt="y'= 5 \, x - 2 \, y + 12" title="y'= 5 \, x - 2 \, y + 12" data-latex="y'= 5 \, x - 2 \, y + 12"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(-2, 1\right)" alt="\left(-2, 1\right)" title="\left(-2, 1\right)" data-latex="\left(-2, 1\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(-2,%201%5Cright)" alt="\left(-2, 1\right)" title="\left(-2, 1\right)" data-latex="\left(-2, 1\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-0143" title="S1 | Phase Planes | ver. 0143"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -5 \, x + 5 \, y - 30" alt="x'= -5 \, x + 5 \, y - 30" title="x'= -5 \, x + 5 \, y - 30" data-latex="x'= -5 \, x + 5 \, y - 30"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -4 \, x - 2 \, y" alt="y'= -4 \, x - 2 \, y" title="y'= -4 \, x - 2 \, y" data-latex="y'= -4 \, x - 2 \, y"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-5%20%5C,%20x%20+%205%20%5C,%20y%20-%2030" alt="x'= -5 \, x + 5 \, y - 30" title="x'= -5 \, x + 5 \, y - 30" data-latex="x'= -5 \, x + 5 \, y - 30"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-4%20%5C,%20x%20-%202%20%5C,%20y" alt="y'= -4 \, x - 2 \, y" title="y'= -4 \, x - 2 \, y" data-latex="y'= -4 \, x - 2 \, y"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(-2, 4\right)" alt="\left(-2, 4\right)" title="\left(-2, 4\right)" data-latex="\left(-2, 4\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a positive slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a negative slope. </p><p> The right region has a southwest arrow system, the top region has a southeast arrow system, the left region has a northeast arrow system, and the bottom region has a northwest arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(-2,%204%5Cright)" alt="\left(-2, 4\right)" title="\left(-2, 4\right)" data-latex="\left(-2, 4\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a positive slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a negative slope. &lt;/p&gt;
  &lt;p&gt; The right region has a southwest arrow system, the top region has a southeast arrow system, the left region has a northeast arrow system, and the bottom region has a northwest arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-8408" title="S1 | Phase Planes | ver. 8408"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -4 \, x - y + 13" alt="x'= -4 \, x - y + 13" title="x'= -4 \, x - y + 13" data-latex="x'= -4 \, x - y + 13"/></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'= 5 \, x - 4 \, y + 10" alt="y'= 5 \, x - 4 \, y + 10" title="y'= 5 \, x - 4 \, y + 10" data-latex="y'= 5 \, x - 4 \, y + 10"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-4%20%5C,%20x%20-%20y%20+%2013" alt="x'= -4 \, x - y + 13" title="x'= -4 \, x - y + 13" data-latex="x'= -4 \, x - y + 13"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%205%20%5C,%20x%20-%204%20%5C,%20y%20+%2010" alt="y'= 5 \, x - 4 \, y + 10" title="y'= 5 \, x - 4 \, y + 10" data-latex="y'= 5 \, x - 4 \, y + 10"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(2, 5\right)" alt="\left(2, 5\right)" title="\left(2, 5\right)" data-latex="\left(2, 5\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(2,%205%5Cright)" alt="\left(2, 5\right)" title="\left(2, 5\right)" data-latex="\left(2, 5\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-9702" title="S1 | Phase Planes | ver. 9702"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -3 \, x - 3 \, y + 15" alt="x'= -3 \, x - 3 \, y + 15" title="x'= -3 \, x - 3 \, y + 15" data-latex="x'= -3 \, x - 3 \, y + 15"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= x - 5 \, y + 13" alt="y'= x - 5 \, y + 13" title="y'= x - 5 \, y + 13" data-latex="y'= x - 5 \, y + 13"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-3%20%5C,%20x%20-%203%20%5C,%20y%20+%2015" alt="x'= -3 \, x - 3 \, y + 15" title="x'= -3 \, x - 3 \, y + 15" data-latex="x'= -3 \, x - 3 \, y + 15"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20x%20-%205%20%5C,%20y%20+%2013" alt="y'= x - 5 \, y + 13" title="y'= x - 5 \, y + 13" data-latex="y'= x - 5 \, y + 13"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(2, 3\right)" alt="\left(2, 3\right)" title="\left(2, 3\right)" data-latex="\left(2, 3\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(2,%203%5Cright)" alt="\left(2, 3\right)" title="\left(2, 3\right)" data-latex="\left(2, 3\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-3231" title="S1 | Phase Planes | ver. 3231"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 2 \, x - 3 \, y + 7" alt="x'= 2 \, x - 3 \, y + 7" title="x'= 2 \, x - 3 \, y + 7" data-latex="x'= 2 \, x - 3 \, y + 7"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -4 \, x - 5 \, y + 41" alt="y'= -4 \, x - 5 \, y + 41" title="y'= -4 \, x - 5 \, y + 41" data-latex="y'= -4 \, x - 5 \, y + 41"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%202%20%5C,%20x%20-%203%20%5C,%20y%20+%207" alt="x'= 2 \, x - 3 \, y + 7" title="x'= 2 \, x - 3 \, y + 7" data-latex="x'= 2 \, x - 3 \, y + 7"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-4%20%5C,%20x%20-%205%20%5C,%20y%20+%2041" alt="y'= -4 \, x - 5 \, y + 41" title="y'= -4 \, x - 5 \, y + 41" data-latex="y'= -4 \, x - 5 \, y + 41"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(4, 5\right)" alt="\left(4, 5\right)" title="\left(4, 5\right)" data-latex="\left(4, 5\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a positive slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a negative slope. </p><p> The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(4,%205%5Cright)" alt="\left(4, 5\right)" title="\left(4, 5\right)" data-latex="\left(4, 5\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a positive slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a negative slope. &lt;/p&gt;
  &lt;p&gt; The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-5624" title="S1 | Phase Planes | ver. 5624"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -5 \, x - y + 2" alt="x'= -5 \, x - y + 2" title="x'= -5 \, x - y + 2" data-latex="x'= -5 \, x - y + 2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 3 \, x - 5 \, y + 10" alt="y'= 3 \, x - 5 \, y + 10" title="y'= 3 \, x - 5 \, y + 10" data-latex="y'= 3 \, x - 5 \, y + 10"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-5%20%5C,%20x%20-%20y%20+%202" alt="x'= -5 \, x - y + 2" title="x'= -5 \, x - y + 2" data-latex="x'= -5 \, x - y + 2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%203%20%5C,%20x%20-%205%20%5C,%20y%20+%2010" alt="y'= 3 \, x - 5 \, y + 10" title="y'= 3 \, x - 5 \, y + 10" data-latex="y'= 3 \, x - 5 \, y + 10"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(0, 2\right)" alt="\left(0, 2\right)" title="\left(0, 2\right)" data-latex="\left(0, 2\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(0,%202%5Cright)" alt="\left(0, 2\right)" title="\left(0, 2\right)" data-latex="\left(0, 2\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-6651" title="S1 | Phase Planes | ver. 6651"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 3 \, x - 4 \, y + 10" alt="x'= 3 \, x - 4 \, y + 10" title="x'= 3 \, x - 4 \, y + 10" data-latex="x'= 3 \, x - 4 \, y + 10"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -2 \, x - y + 8" alt="y'= -2 \, x - y + 8" title="y'= -2 \, x - y + 8" data-latex="y'= -2 \, x - y + 8"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%203%20%5C,%20x%20-%204%20%5C,%20y%20+%2010" alt="x'= 3 \, x - 4 \, y + 10" title="x'= 3 \, x - 4 \, y + 10" data-latex="x'= 3 \, x - 4 \, y + 10"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-2%20%5C,%20x%20-%20y%20+%208" alt="y'= -2 \, x - y + 8" title="y'= -2 \, x - y + 8" data-latex="y'= -2 \, x - y + 8"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(2, 4\right)" alt="\left(2, 4\right)" title="\left(2, 4\right)" data-latex="\left(2, 4\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a positive slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a negative slope. </p><p> The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(2,%204%5Cright)" alt="\left(2, 4\right)" title="\left(2, 4\right)" data-latex="\left(2, 4\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a positive slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a negative slope. &lt;/p&gt;
  &lt;p&gt; The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-5042" title="S1 | Phase Planes | ver. 5042"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -4 \, x - y - 20" alt="x'= -4 \, x - y - 20" title="x'= -4 \, x - y - 20" data-latex="x'= -4 \, x - y - 20"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 4 \, x - 4 \, y" alt="y'= 4 \, x - 4 \, y" title="y'= 4 \, x - 4 \, y" data-latex="y'= 4 \, x - 4 \, y"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-4%20%5C,%20x%20-%20y%20-%2020" alt="x'= -4 \, x - y - 20" title="x'= -4 \, x - y - 20" data-latex="x'= -4 \, x - y - 20"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%204%20%5C,%20x%20-%204%20%5C,%20y" alt="y'= 4 \, x - 4 \, y" title="y'= 4 \, x - 4 \, y" data-latex="y'= 4 \, x - 4 \, y"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(-4, -4\right)" alt="\left(-4, -4\right)" title="\left(-4, -4\right)" data-latex="\left(-4, -4\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(-4,%20-4%5Cright)" alt="\left(-4, -4\right)" title="\left(-4, -4\right)" data-latex="\left(-4, -4\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-6186" title="S1 | Phase Planes | ver. 6186"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -2 \, x - 5 \, y - 24" alt="x'= -2 \, x - 5 \, y - 24" title="x'= -2 \, x - 5 \, y - 24" data-latex="x'= -2 \, x - 5 \, y - 24"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -2 \, x + 5 \, y + 16" alt="y'= -2 \, x + 5 \, y + 16" title="y'= -2 \, x + 5 \, y + 16" data-latex="y'= -2 \, x + 5 \, y + 16"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-2%20%5C,%20x%20-%205%20%5C,%20y%20-%2024" alt="x'= -2 \, x - 5 \, y - 24" title="x'= -2 \, x - 5 \, y - 24" data-latex="x'= -2 \, x - 5 \, y - 24"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-2%20%5C,%20x%20+%205%20%5C,%20y%20+%2016" alt="y'= -2 \, x + 5 \, y + 16" title="y'= -2 \, x + 5 \, y + 16" data-latex="y'= -2 \, x + 5 \, y + 16"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(-2, -4\right)" alt="\left(-2, -4\right)" title="\left(-2, -4\right)" data-latex="\left(-2, -4\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(-2,%20-4%5Cright)" alt="\left(-2, -4\right)" title="\left(-2, -4\right)" data-latex="\left(-2, -4\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-7730" title="S1 | Phase Planes | ver. 7730"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -3 \, x + 5 \, y + 14" alt="x'= -3 \, x + 5 \, y + 14" title="x'= -3 \, x + 5 \, y + 14" data-latex="x'= -3 \, x + 5 \, y + 14"/></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'= -5 \, x - 3 \, y + 12" alt="y'= -5 \, x - 3 \, y + 12" title="y'= -5 \, x - 3 \, y + 12" data-latex="y'= -5 \, x - 3 \, y + 12"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-3%20%5C,%20x%20+%205%20%5C,%20y%20+%2014" alt="x'= -3 \, x + 5 \, y + 14" title="x'= -3 \, x + 5 \, y + 14" data-latex="x'= -3 \, x + 5 \, y + 14"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-5%20%5C,%20x%20-%203%20%5C,%20y%20+%2012" alt="y'= -5 \, x - 3 \, y + 12" title="y'= -5 \, x - 3 \, y + 12" data-latex="y'= -5 \, x - 3 \, y + 12"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(3, -1\right)" alt="\left(3, -1\right)" title="\left(3, -1\right)" data-latex="\left(3, -1\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a positive slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a negative slope. </p><p> The right region has a southwest arrow system, the top region has a southeast arrow system, the left region has a northeast arrow system, and the bottom region has a northwest arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(3,%20-1%5Cright)" alt="\left(3, -1\right)" title="\left(3, -1\right)" data-latex="\left(3, -1\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a positive slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a negative slope. &lt;/p&gt;
  &lt;p&gt; The right region has a southwest arrow system, the top region has a southeast arrow system, the left region has a northeast arrow system, and the bottom region has a northwest arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-6008" title="S1 | Phase Planes | ver. 6008"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= x - 2 \, y - 1" alt="x'= x - 2 \, y - 1" title="x'= x - 2 \, y - 1" data-latex="x'= x - 2 \, y - 1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= x + 3 \, y + 14" alt="y'= x + 3 \, y + 14" title="y'= x + 3 \, y + 14" data-latex="y'= x + 3 \, y + 14"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20x%20-%202%20%5C,%20y%20-%201" alt="x'= x - 2 \, y - 1" title="x'= x - 2 \, y - 1" data-latex="x'= x - 2 \, y - 1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20x%20+%203%20%5C,%20y%20+%2014" alt="y'= x + 3 \, y + 14" title="y'= x + 3 \, y + 14" data-latex="y'= x + 3 \, y + 14"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(-5, -3\right)" alt="\left(-5, -3\right)" title="\left(-5, -3\right)" data-latex="\left(-5, -3\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a positive slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a negative slope. </p><p> The right region has a northeast arrow system, the top region has a northwest arrow system, the left region has a southwest arrow system, and the bottom region has a southeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(-5,%20-3%5Cright)" alt="\left(-5, -3\right)" title="\left(-5, -3\right)" data-latex="\left(-5, -3\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a positive slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a negative slope. &lt;/p&gt;
  &lt;p&gt; The right region has a northeast arrow system, the top region has a northwest arrow system, the left region has a southwest arrow system, and the bottom region has a southeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-8563" title="S1 | Phase Planes | ver. 8563"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -5 \, x + 2 \, y - 2" alt="x'= -5 \, x + 2 \, y - 2" title="x'= -5 \, x + 2 \, y - 2" data-latex="x'= -5 \, x + 2 \, y - 2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -4 \, x - y - 12" alt="y'= -4 \, x - y - 12" title="y'= -4 \, x - y - 12" data-latex="y'= -4 \, x - y - 12"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-5%20%5C,%20x%20+%202%20%5C,%20y%20-%202" alt="x'= -5 \, x + 2 \, y - 2" title="x'= -5 \, x + 2 \, y - 2" data-latex="x'= -5 \, x + 2 \, y - 2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-4%20%5C,%20x%20-%20y%20-%2012" alt="y'= -4 \, x - y - 12" title="y'= -4 \, x - y - 12" data-latex="y'= -4 \, x - y - 12"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(-2, -4\right)" alt="\left(-2, -4\right)" title="\left(-2, -4\right)" data-latex="\left(-2, -4\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a positive slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a negative slope. </p><p> The right region has a southwest arrow system, the top region has a southeast arrow system, the left region has a northeast arrow system, and the bottom region has a northwest arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(-2,%20-4%5Cright)" alt="\left(-2, -4\right)" title="\left(-2, -4\right)" data-latex="\left(-2, -4\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a positive slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a negative slope. &lt;/p&gt;
  &lt;p&gt; The right region has a southwest arrow system, the top region has a southeast arrow system, the left region has a northeast arrow system, and the bottom region has a northwest arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-8884" title="S1 | Phase Planes | ver. 8884"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -3 \, x - 3 \, y - 18" alt="x'= -3 \, x - 3 \, y - 18" title="x'= -3 \, x - 3 \, y - 18" data-latex="x'= -3 \, x - 3 \, y - 18"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 2 \, x - 4 \, y - 18" alt="y'= 2 \, x - 4 \, y - 18" title="y'= 2 \, x - 4 \, y - 18" data-latex="y'= 2 \, x - 4 \, y - 18"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-3%20%5C,%20x%20-%203%20%5C,%20y%20-%2018" alt="x'= -3 \, x - 3 \, y - 18" title="x'= -3 \, x - 3 \, y - 18" data-latex="x'= -3 \, x - 3 \, y - 18"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%202%20%5C,%20x%20-%204%20%5C,%20y%20-%2018" alt="y'= 2 \, x - 4 \, y - 18" title="y'= 2 \, x - 4 \, y - 18" data-latex="y'= 2 \, x - 4 \, y - 18"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(-1, -5\right)" alt="\left(-1, -5\right)" title="\left(-1, -5\right)" data-latex="\left(-1, -5\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(-1,%20-5%5Cright)" alt="\left(-1, -5\right)" title="\left(-1, -5\right)" data-latex="\left(-1, -5\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-8625" title="S1 | Phase Planes | ver. 8625"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -5 \, x - 2 \, y + 11" alt="x'= -5 \, x - 2 \, y + 11" title="x'= -5 \, x - 2 \, y + 11" data-latex="x'= -5 \, x - 2 \, y + 11"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -x + 4 \, y - 11" alt="y'= -x + 4 \, y - 11" title="y'= -x + 4 \, y - 11" data-latex="y'= -x + 4 \, y - 11"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-5%20%5C,%20x%20-%202%20%5C,%20y%20+%2011" alt="x'= -5 \, x - 2 \, y + 11" title="x'= -5 \, x - 2 \, y + 11" data-latex="x'= -5 \, x - 2 \, y + 11"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-x%20+%204%20%5C,%20y%20-%2011" alt="y'= -x + 4 \, y - 11" title="y'= -x + 4 \, y - 11" data-latex="y'= -x + 4 \, y - 11"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(1, 3\right)" alt="\left(1, 3\right)" title="\left(1, 3\right)" data-latex="\left(1, 3\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(1,%203%5Cright)" alt="\left(1, 3\right)" title="\left(1, 3\right)" data-latex="\left(1, 3\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-4075" title="S1 | Phase Planes | ver. 4075"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= x + 5 \, y + 2" alt="x'= x + 5 \, y + 2" title="x'= x + 5 \, y + 2" data-latex="x'= x + 5 \, y + 2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 3 \, x - 2 \, y + 6" alt="y'= 3 \, x - 2 \, y + 6" title="y'= 3 \, x - 2 \, y + 6" data-latex="y'= 3 \, x - 2 \, y + 6"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20x%20+%205%20%5C,%20y%20+%202" alt="x'= x + 5 \, y + 2" title="x'= x + 5 \, y + 2" data-latex="x'= x + 5 \, y + 2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%203%20%5C,%20x%20-%202%20%5C,%20y%20+%206" alt="y'= 3 \, x - 2 \, y + 6" title="y'= 3 \, x - 2 \, y + 6" data-latex="y'= 3 \, x - 2 \, y + 6"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(-2, 0\right)" alt="\left(-2, 0\right)" title="\left(-2, 0\right)" data-latex="\left(-2, 0\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a northeast arrow system, the top region has a southeast arrow system, the left region has a southwest arrow system, and the bottom region has a northwest arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(-2,%200%5Cright)" alt="\left(-2, 0\right)" title="\left(-2, 0\right)" data-latex="\left(-2, 0\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a northeast arrow system, the top region has a southeast arrow system, the left region has a southwest arrow system, and the bottom region has a northwest arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-5557" title="S1 | Phase Planes | ver. 5557"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 5 \, x + 5 \, y - 15" alt="x'= 5 \, x + 5 \, y - 15" title="x'= 5 \, x + 5 \, y - 15" data-latex="x'= 5 \, x + 5 \, y - 15"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= x - 3 \, y - 11" alt="y'= x - 3 \, y - 11" title="y'= x - 3 \, y - 11" data-latex="y'= x - 3 \, y - 11"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%205%20%5C,%20x%20+%205%20%5C,%20y%20-%2015" alt="x'= 5 \, x + 5 \, y - 15" title="x'= 5 \, x + 5 \, y - 15" data-latex="x'= 5 \, x + 5 \, y - 15"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20x%20-%203%20%5C,%20y%20-%2011" alt="y'= x - 3 \, y - 11" title="y'= x - 3 \, y - 11" data-latex="y'= x - 3 \, y - 11"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(5, -2\right)" alt="\left(5, -2\right)" title="\left(5, -2\right)" data-latex="\left(5, -2\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a northeast arrow system, the top region has a southeast arrow system, the left region has a southwest arrow system, and the bottom region has a northwest arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(5,%20-2%5Cright)" alt="\left(5, -2\right)" title="\left(5, -2\right)" data-latex="\left(5, -2\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a northeast arrow system, the top region has a southeast arrow system, the left region has a southwest arrow system, and the bottom region has a northwest arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-4423" title="S1 | Phase Planes | ver. 4423"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 4 \, x + 3 \, y - 3" alt="x'= 4 \, x + 3 \, y - 3" title="x'= 4 \, x + 3 \, y - 3" data-latex="x'= 4 \, x + 3 \, y - 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'= 3 \, x - 4 \, y + 4" alt="y'= 3 \, x - 4 \, y + 4" title="y'= 3 \, x - 4 \, y + 4" data-latex="y'= 3 \, x - 4 \, y + 4"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%204%20%5C,%20x%20+%203%20%5C,%20y%20-%203" alt="x'= 4 \, x + 3 \, y - 3" title="x'= 4 \, x + 3 \, y - 3" data-latex="x'= 4 \, x + 3 \, y - 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?y'=%203%20%5C,%20x%20-%204%20%5C,%20y%20+%204" alt="y'= 3 \, x - 4 \, y + 4" title="y'= 3 \, x - 4 \, y + 4" data-latex="y'= 3 \, x - 4 \, y + 4"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(0, 1\right)" alt="\left(0, 1\right)" title="\left(0, 1\right)" data-latex="\left(0, 1\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a northeast arrow system, the top region has a southeast arrow system, the left region has a southwest arrow system, and the bottom region has a northwest arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(0,%201%5Cright)" alt="\left(0, 1\right)" title="\left(0, 1\right)" data-latex="\left(0, 1\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a northeast arrow system, the top region has a southeast arrow system, the left region has a southwest arrow system, and the bottom region has a northwest arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-6289" title="S1 | Phase Planes | ver. 6289"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -x - y" alt="x'= -x - y" title="x'= -x - y" data-latex="x'= -x - y"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 2 \, x - 3 \, y + 10" alt="y'= 2 \, x - 3 \, y + 10" title="y'= 2 \, x - 3 \, y + 10" data-latex="y'= 2 \, x - 3 \, y + 10"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-x%20-%20y" alt="x'= -x - y" title="x'= -x - y" data-latex="x'= -x - y"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%202%20%5C,%20x%20-%203%20%5C,%20y%20+%2010" alt="y'= 2 \, x - 3 \, y + 10" title="y'= 2 \, x - 3 \, y + 10" data-latex="y'= 2 \, x - 3 \, y + 10"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(-2, 2\right)" alt="\left(-2, 2\right)" title="\left(-2, 2\right)" data-latex="\left(-2, 2\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(-2,%202%5Cright)" alt="\left(-2, 2\right)" title="\left(-2, 2\right)" data-latex="\left(-2, 2\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-0638" title="S1 | Phase Planes | ver. 0638"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -2 \, x - y + 7" alt="x'= -2 \, x - y + 7" title="x'= -2 \, x - y + 7" data-latex="x'= -2 \, x - y + 7"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -3 \, x + y - 2" alt="y'= -3 \, x + y - 2" title="y'= -3 \, x + y - 2" data-latex="y'= -3 \, x + y - 2"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-2%20%5C,%20x%20-%20y%20+%207" alt="x'= -2 \, x - y + 7" title="x'= -2 \, x - y + 7" data-latex="x'= -2 \, x - y + 7"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-3%20%5C,%20x%20+%20y%20-%202" alt="y'= -3 \, x + y - 2" title="y'= -3 \, x + y - 2" data-latex="y'= -3 \, x + y - 2"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(1, 5\right)" alt="\left(1, 5\right)" title="\left(1, 5\right)" data-latex="\left(1, 5\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(1,%205%5Cright)" alt="\left(1, 5\right)" title="\left(1, 5\right)" data-latex="\left(1, 5\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-0697" title="S1 | Phase Planes | ver. 0697"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 5 \, x - y - 10" alt="x'= 5 \, x - y - 10" title="x'= 5 \, x - y - 10" data-latex="x'= 5 \, x - y - 10"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 2 \, x + 2 \, y + 8" alt="y'= 2 \, x + 2 \, y + 8" title="y'= 2 \, x + 2 \, y + 8" data-latex="y'= 2 \, x + 2 \, y + 8"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%205%20%5C,%20x%20-%20y%20-%2010" alt="x'= 5 \, x - y - 10" title="x'= 5 \, x - y - 10" data-latex="x'= 5 \, x - y - 10"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%202%20%5C,%20x%20+%202%20%5C,%20y%20+%208" alt="y'= 2 \, x + 2 \, y + 8" title="y'= 2 \, x + 2 \, y + 8" data-latex="y'= 2 \, x + 2 \, y + 8"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(1, -5\right)" alt="\left(1, -5\right)" title="\left(1, -5\right)" data-latex="\left(1, -5\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a positive slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a negative slope. </p><p> The right region has a northeast arrow system, the top region has a northwest arrow system, the left region has a southwest arrow system, and the bottom region has a southeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(1,%20-5%5Cright)" alt="\left(1, -5\right)" title="\left(1, -5\right)" data-latex="\left(1, -5\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a positive slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a negative slope. &lt;/p&gt;
  &lt;p&gt; The right region has a northeast arrow system, the top region has a northwest arrow system, the left region has a southwest arrow system, and the bottom region has a southeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-5647" title="S1 | Phase Planes | ver. 5647"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -4 \, x - y - 17" alt="x'= -4 \, x - y - 17" title="x'= -4 \, x - y - 17" data-latex="x'= -4 \, x - y - 17"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -2 \, x + 3 \, y - 19" alt="y'= -2 \, x + 3 \, y - 19" title="y'= -2 \, x + 3 \, y - 19" data-latex="y'= -2 \, x + 3 \, y - 19"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-4%20%5C,%20x%20-%20y%20-%2017" alt="x'= -4 \, x - y - 17" title="x'= -4 \, x - y - 17" data-latex="x'= -4 \, x - y - 17"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-2%20%5C,%20x%20+%203%20%5C,%20y%20-%2019" alt="y'= -2 \, x + 3 \, y - 19" title="y'= -2 \, x + 3 \, y - 19" data-latex="y'= -2 \, x + 3 \, y - 19"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(-5, 3\right)" alt="\left(-5, 3\right)" title="\left(-5, 3\right)" data-latex="\left(-5, 3\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(-5,%203%5Cright)" alt="\left(-5, 3\right)" title="\left(-5, 3\right)" data-latex="\left(-5, 3\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-0514" title="S1 | Phase Planes | ver. 0514"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 5 \, x + 3 \, y - 4" alt="x'= 5 \, x + 3 \, y - 4" title="x'= 5 \, x + 3 \, y - 4" data-latex="x'= 5 \, x + 3 \, y - 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'= x - 4 \, y - 10" alt="y'= x - 4 \, y - 10" title="y'= x - 4 \, y - 10" data-latex="y'= x - 4 \, y - 10"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%205%20%5C,%20x%20+%203%20%5C,%20y%20-%204" alt="x'= 5 \, x + 3 \, y - 4" title="x'= 5 \, x + 3 \, y - 4" data-latex="x'= 5 \, x + 3 \, y - 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?y'=%20x%20-%204%20%5C,%20y%20-%2010" alt="y'= x - 4 \, y - 10" title="y'= x - 4 \, y - 10" data-latex="y'= x - 4 \, y - 10"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(2, -2\right)" alt="\left(2, -2\right)" title="\left(2, -2\right)" data-latex="\left(2, -2\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a northeast arrow system, the top region has a southeast arrow system, the left region has a southwest arrow system, and the bottom region has a northwest arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(2,%20-2%5Cright)" alt="\left(2, -2\right)" title="\left(2, -2\right)" data-latex="\left(2, -2\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a northeast arrow system, the top region has a southeast arrow system, the left region has a southwest arrow system, and the bottom region has a northwest arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-3833" title="S1 | Phase Planes | ver. 3833"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 2 \, x + 3 \, y + 6" alt="x'= 2 \, x + 3 \, y + 6" title="x'= 2 \, x + 3 \, y + 6" data-latex="x'= 2 \, x + 3 \, y + 6"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 2 \, x - 5 \, y - 26" alt="y'= 2 \, x - 5 \, y - 26" title="y'= 2 \, x - 5 \, y - 26" data-latex="y'= 2 \, x - 5 \, y - 26"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%202%20%5C,%20x%20+%203%20%5C,%20y%20+%206" alt="x'= 2 \, x + 3 \, y + 6" title="x'= 2 \, x + 3 \, y + 6" data-latex="x'= 2 \, x + 3 \, y + 6"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%202%20%5C,%20x%20-%205%20%5C,%20y%20-%2026" alt="y'= 2 \, x - 5 \, y - 26" title="y'= 2 \, x - 5 \, y - 26" data-latex="y'= 2 \, x - 5 \, y - 26"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(3, -4\right)" alt="\left(3, -4\right)" title="\left(3, -4\right)" data-latex="\left(3, -4\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a northeast arrow system, the top region has a southeast arrow system, the left region has a southwest arrow system, and the bottom region has a northwest arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(3,%20-4%5Cright)" alt="\left(3, -4\right)" title="\left(3, -4\right)" data-latex="\left(3, -4\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a northeast arrow system, the top region has a southeast arrow system, the left region has a southwest arrow system, and the bottom region has a northwest arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-4373" title="S1 | Phase Planes | ver. 4373"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -2 \, x - 4 \, y + 6" alt="x'= -2 \, x - 4 \, y + 6" title="x'= -2 \, x - 4 \, y + 6" data-latex="x'= -2 \, x - 4 \, y + 6"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -2 \, x + 2 \, y" alt="y'= -2 \, x + 2 \, y" title="y'= -2 \, x + 2 \, y" data-latex="y'= -2 \, x + 2 \, y"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-2%20%5C,%20x%20-%204%20%5C,%20y%20+%206" alt="x'= -2 \, x - 4 \, y + 6" title="x'= -2 \, x - 4 \, y + 6" data-latex="x'= -2 \, x - 4 \, y + 6"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-2%20%5C,%20x%20+%202%20%5C,%20y" alt="y'= -2 \, x + 2 \, y" title="y'= -2 \, x + 2 \, y" data-latex="y'= -2 \, x + 2 \, y"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(1, 1\right)" alt="\left(1, 1\right)" title="\left(1, 1\right)" data-latex="\left(1, 1\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(1,%201%5Cright)" alt="\left(1, 1\right)" title="\left(1, 1\right)" data-latex="\left(1, 1\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-5658" title="S1 | Phase Planes | ver. 5658"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -3 \, x + 3 \, y + 12" alt="x'= -3 \, x + 3 \, y + 12" title="x'= -3 \, x + 3 \, y + 12" data-latex="x'= -3 \, x + 3 \, y + 12"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -3 \, x - 3 \, y + 6" alt="y'= -3 \, x - 3 \, y + 6" title="y'= -3 \, x - 3 \, y + 6" data-latex="y'= -3 \, x - 3 \, y + 6"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-3%20%5C,%20x%20+%203%20%5C,%20y%20+%2012" alt="x'= -3 \, x + 3 \, y + 12" title="x'= -3 \, x + 3 \, y + 12" data-latex="x'= -3 \, x + 3 \, y + 12"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-3%20%5C,%20x%20-%203%20%5C,%20y%20+%206" alt="y'= -3 \, x - 3 \, y + 6" title="y'= -3 \, x - 3 \, y + 6" data-latex="y'= -3 \, x - 3 \, y + 6"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(3, -1\right)" alt="\left(3, -1\right)" title="\left(3, -1\right)" data-latex="\left(3, -1\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a positive slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a negative slope. </p><p> The right region has a southwest arrow system, the top region has a southeast arrow system, the left region has a northeast arrow system, and the bottom region has a northwest arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(3,%20-1%5Cright)" alt="\left(3, -1\right)" title="\left(3, -1\right)" data-latex="\left(3, -1\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a positive slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a negative slope. &lt;/p&gt;
  &lt;p&gt; The right region has a southwest arrow system, the top region has a southeast arrow system, the left region has a northeast arrow system, and the bottom region has a northwest arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-7925" title="S1 | Phase Planes | ver. 7925"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -x - y + 1" alt="x'= -x - y + 1" title="x'= -x - y + 1" data-latex="x'= -x - y + 1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 2 \, x - 4 \, y + 16" alt="y'= 2 \, x - 4 \, y + 16" title="y'= 2 \, x - 4 \, y + 16" data-latex="y'= 2 \, x - 4 \, y + 16"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-x%20-%20y%20+%201" alt="x'= -x - y + 1" title="x'= -x - y + 1" data-latex="x'= -x - y + 1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%202%20%5C,%20x%20-%204%20%5C,%20y%20+%2016" alt="y'= 2 \, x - 4 \, y + 16" title="y'= 2 \, x - 4 \, y + 16" data-latex="y'= 2 \, x - 4 \, y + 16"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(-2, 3\right)" alt="\left(-2, 3\right)" title="\left(-2, 3\right)" data-latex="\left(-2, 3\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(-2,%203%5Cright)" alt="\left(-2, 3\right)" title="\left(-2, 3\right)" data-latex="\left(-2, 3\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-8786" title="S1 | Phase Planes | ver. 8786"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -2 \, x + 3 \, y + 15" alt="x'= -2 \, x + 3 \, y + 15" title="x'= -2 \, x + 3 \, y + 15" data-latex="x'= -2 \, x + 3 \, y + 15"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -x - 5 \, y - 25" alt="y'= -x - 5 \, y - 25" title="y'= -x - 5 \, y - 25" data-latex="y'= -x - 5 \, y - 25"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-2%20%5C,%20x%20+%203%20%5C,%20y%20+%2015" alt="x'= -2 \, x + 3 \, y + 15" title="x'= -2 \, x + 3 \, y + 15" data-latex="x'= -2 \, x + 3 \, y + 15"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-x%20-%205%20%5C,%20y%20-%2025" alt="y'= -x - 5 \, y - 25" title="y'= -x - 5 \, y - 25" data-latex="y'= -x - 5 \, y - 25"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(0, -5\right)" alt="\left(0, -5\right)" title="\left(0, -5\right)" data-latex="\left(0, -5\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a positive slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a negative slope. </p><p> The right region has a southwest arrow system, the top region has a southeast arrow system, the left region has a northeast arrow system, and the bottom region has a northwest arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(0,%20-5%5Cright)" alt="\left(0, -5\right)" title="\left(0, -5\right)" data-latex="\left(0, -5\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a positive slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a negative slope. &lt;/p&gt;
  &lt;p&gt; The right region has a southwest arrow system, the top region has a southeast arrow system, the left region has a northeast arrow system, and the bottom region has a northwest arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-1367" title="S1 | Phase Planes | ver. 1367"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 3 \, x - 2 \, y + 5" alt="x'= 3 \, x - 2 \, y + 5" title="x'= 3 \, x - 2 \, y + 5" data-latex="x'= 3 \, x - 2 \, y + 5"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -4 \, x - y - 14" alt="y'= -4 \, x - y - 14" title="y'= -4 \, x - y - 14" data-latex="y'= -4 \, x - y - 14"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%203%20%5C,%20x%20-%202%20%5C,%20y%20+%205" alt="x'= 3 \, x - 2 \, y + 5" title="x'= 3 \, x - 2 \, y + 5" data-latex="x'= 3 \, x - 2 \, y + 5"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-4%20%5C,%20x%20-%20y%20-%2014" alt="y'= -4 \, x - y - 14" title="y'= -4 \, x - y - 14" data-latex="y'= -4 \, x - y - 14"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(-3, -2\right)" alt="\left(-3, -2\right)" title="\left(-3, -2\right)" data-latex="\left(-3, -2\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a positive slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a negative slope. </p><p> The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(-3,%20-2%5Cright)" alt="\left(-3, -2\right)" title="\left(-3, -2\right)" data-latex="\left(-3, -2\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a positive slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a negative slope. &lt;/p&gt;
  &lt;p&gt; The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-2885" title="S1 | Phase Planes | ver. 2885"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 5 \, x - 3 \, y" alt="x'= 5 \, x - 3 \, y" title="x'= 5 \, x - 3 \, y" data-latex="x'= 5 \, x - 3 \, y"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -3 \, x - 5 \, y" alt="y'= -3 \, x - 5 \, y" title="y'= -3 \, x - 5 \, y" data-latex="y'= -3 \, x - 5 \, y"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%205%20%5C,%20x%20-%203%20%5C,%20y" alt="x'= 5 \, x - 3 \, y" title="x'= 5 \, x - 3 \, y" data-latex="x'= 5 \, x - 3 \, y"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-3%20%5C,%20x%20-%205%20%5C,%20y" alt="y'= -3 \, x - 5 \, y" title="y'= -3 \, x - 5 \, y" data-latex="y'= -3 \, x - 5 \, y"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(0, 0\right)" alt="\left(0, 0\right)" title="\left(0, 0\right)" data-latex="\left(0, 0\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a positive slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a negative slope. </p><p> The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(0,%200%5Cright)" alt="\left(0, 0\right)" title="\left(0, 0\right)" data-latex="\left(0, 0\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a positive slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a negative slope. &lt;/p&gt;
  &lt;p&gt; The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-6180" title="S1 | Phase Planes | ver. 6180"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -x - 3 \, y - 1" alt="x'= -x - 3 \, y - 1" title="x'= -x - 3 \, y - 1" data-latex="x'= -x - 3 \, y - 1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 3 \, x - 4 \, y - 23" alt="y'= 3 \, x - 4 \, y - 23" title="y'= 3 \, x - 4 \, y - 23" data-latex="y'= 3 \, x - 4 \, y - 23"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-x%20-%203%20%5C,%20y%20-%201" alt="x'= -x - 3 \, y - 1" title="x'= -x - 3 \, y - 1" data-latex="x'= -x - 3 \, y - 1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%203%20%5C,%20x%20-%204%20%5C,%20y%20-%2023" alt="y'= 3 \, x - 4 \, y - 23" title="y'= 3 \, x - 4 \, y - 23" data-latex="y'= 3 \, x - 4 \, y - 23"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(5, -2\right)" alt="\left(5, -2\right)" title="\left(5, -2\right)" data-latex="\left(5, -2\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(5,%20-2%5Cright)" alt="\left(5, -2\right)" title="\left(5, -2\right)" data-latex="\left(5, -2\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-5655" title="S1 | Phase Planes | ver. 5655"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= x - 5 \, y - 10" alt="x'= x - 5 \, y - 10" title="x'= x - 5 \, y - 10" data-latex="x'= x - 5 \, y - 10"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 2 \, x + 2 \, y - 8" alt="y'= 2 \, x + 2 \, y - 8" title="y'= 2 \, x + 2 \, y - 8" data-latex="y'= 2 \, x + 2 \, y - 8"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20x%20-%205%20%5C,%20y%20-%2010" alt="x'= x - 5 \, y - 10" title="x'= x - 5 \, y - 10" data-latex="x'= x - 5 \, y - 10"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%202%20%5C,%20x%20+%202%20%5C,%20y%20-%208" alt="y'= 2 \, x + 2 \, y - 8" title="y'= 2 \, x + 2 \, y - 8" data-latex="y'= 2 \, x + 2 \, y - 8"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(5, -1\right)" alt="\left(5, -1\right)" title="\left(5, -1\right)" data-latex="\left(5, -1\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a positive slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a negative slope. </p><p> The right region has a northeast arrow system, the top region has a northwest arrow system, the left region has a southwest arrow system, and the bottom region has a southeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(5,%20-1%5Cright)" alt="\left(5, -1\right)" title="\left(5, -1\right)" data-latex="\left(5, -1\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a positive slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a negative slope. &lt;/p&gt;
  &lt;p&gt; The right region has a northeast arrow system, the top region has a northwest arrow system, the left region has a southwest arrow system, and the bottom region has a southeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-4427" title="S1 | Phase Planes | ver. 4427"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -4 \, x - y - 1" alt="x'= -4 \, x - y - 1" title="x'= -4 \, x - y - 1" data-latex="x'= -4 \, x - y - 1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 2 \, x - 5 \, y - 27" alt="y'= 2 \, x - 5 \, y - 27" title="y'= 2 \, x - 5 \, y - 27" data-latex="y'= 2 \, x - 5 \, y - 27"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-4%20%5C,%20x%20-%20y%20-%201" alt="x'= -4 \, x - y - 1" title="x'= -4 \, x - y - 1" data-latex="x'= -4 \, x - y - 1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%202%20%5C,%20x%20-%205%20%5C,%20y%20-%2027" alt="y'= 2 \, x - 5 \, y - 27" title="y'= 2 \, x - 5 \, y - 27" data-latex="y'= 2 \, x - 5 \, y - 27"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(1, -5\right)" alt="\left(1, -5\right)" title="\left(1, -5\right)" data-latex="\left(1, -5\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(1,%20-5%5Cright)" alt="\left(1, -5\right)" title="\left(1, -5\right)" data-latex="\left(1, -5\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-4281" title="S1 | Phase Planes | ver. 4281"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -3 \, x - y - 3" alt="x'= -3 \, x - y - 3" title="x'= -3 \, x - y - 3" data-latex="x'= -3 \, x - y - 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'= -x + 3 \, y - 11" alt="y'= -x + 3 \, y - 11" title="y'= -x + 3 \, y - 11" data-latex="y'= -x + 3 \, y - 11"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-3%20%5C,%20x%20-%20y%20-%203" alt="x'= -3 \, x - y - 3" title="x'= -3 \, x - y - 3" data-latex="x'= -3 \, x - y - 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?y'=%20-x%20+%203%20%5C,%20y%20-%2011" alt="y'= -x + 3 \, y - 11" title="y'= -x + 3 \, y - 11" data-latex="y'= -x + 3 \, y - 11"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(-2, 3\right)" alt="\left(-2, 3\right)" title="\left(-2, 3\right)" data-latex="\left(-2, 3\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(-2,%203%5Cright)" alt="\left(-2, 3\right)" title="\left(-2, 3\right)" data-latex="\left(-2, 3\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-6345" title="S1 | Phase Planes | ver. 6345"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -3 \, x - y - 11" alt="x'= -3 \, x - y - 11" title="x'= -3 \, x - y - 11" data-latex="x'= -3 \, x - y - 11"/></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'= 5 \, x - 2 \, y" alt="y'= 5 \, x - 2 \, y" title="y'= 5 \, x - 2 \, y" data-latex="y'= 5 \, x - 2 \, y"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-3%20%5C,%20x%20-%20y%20-%2011" alt="x'= -3 \, x - y - 11" title="x'= -3 \, x - y - 11" data-latex="x'= -3 \, x - y - 11"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%205%20%5C,%20x%20-%202%20%5C,%20y" alt="y'= 5 \, x - 2 \, y" title="y'= 5 \, x - 2 \, y" data-latex="y'= 5 \, x - 2 \, y"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(-2, -5\right)" alt="\left(-2, -5\right)" title="\left(-2, -5\right)" data-latex="\left(-2, -5\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(-2,%20-5%5Cright)" alt="\left(-2, -5\right)" title="\left(-2, -5\right)" data-latex="\left(-2, -5\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-2997" title="S1 | Phase Planes | ver. 2997"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 5 \, x - y - 22" alt="x'= 5 \, x - y - 22" title="x'= 5 \, x - y - 22" data-latex="x'= 5 \, x - y - 22"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -2 \, x - 2 \, y + 16" alt="y'= -2 \, x - 2 \, y + 16" title="y'= -2 \, x - 2 \, y + 16" data-latex="y'= -2 \, x - 2 \, y + 16"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%205%20%5C,%20x%20-%20y%20-%2022" alt="x'= 5 \, x - y - 22" title="x'= 5 \, x - y - 22" data-latex="x'= 5 \, x - y - 22"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-2%20%5C,%20x%20-%202%20%5C,%20y%20+%2016" alt="y'= -2 \, x - 2 \, y + 16" title="y'= -2 \, x - 2 \, y + 16" data-latex="y'= -2 \, x - 2 \, y + 16"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(5, 3\right)" alt="\left(5, 3\right)" title="\left(5, 3\right)" data-latex="\left(5, 3\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a positive slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a negative slope. </p><p> The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(5,%203%5Cright)" alt="\left(5, 3\right)" title="\left(5, 3\right)" data-latex="\left(5, 3\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a positive slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a negative slope. &lt;/p&gt;
  &lt;p&gt; The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-6298" title="S1 | Phase Planes | ver. 6298"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -4 \, x - y + 2" alt="x'= -4 \, x - y + 2" title="x'= -4 \, x - y + 2" data-latex="x'= -4 \, x - y + 2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -2 \, x + 3 \, y - 6" alt="y'= -2 \, x + 3 \, y - 6" title="y'= -2 \, x + 3 \, y - 6" data-latex="y'= -2 \, x + 3 \, y - 6"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-4%20%5C,%20x%20-%20y%20+%202" alt="x'= -4 \, x - y + 2" title="x'= -4 \, x - y + 2" data-latex="x'= -4 \, x - y + 2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-2%20%5C,%20x%20+%203%20%5C,%20y%20-%206" alt="y'= -2 \, x + 3 \, y - 6" title="y'= -2 \, x + 3 \, y - 6" data-latex="y'= -2 \, x + 3 \, y - 6"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(0, 2\right)" alt="\left(0, 2\right)" title="\left(0, 2\right)" data-latex="\left(0, 2\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(0,%202%5Cright)" alt="\left(0, 2\right)" title="\left(0, 2\right)" data-latex="\left(0, 2\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-5916" title="S1 | Phase Planes | ver. 5916"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -x + y" alt="x'= -x + y" title="x'= -x + y" data-latex="x'= -x + y"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -x - y - 6" alt="y'= -x - y - 6" title="y'= -x - y - 6" data-latex="y'= -x - y - 6"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-x%20+%20y" alt="x'= -x + y" title="x'= -x + y" data-latex="x'= -x + y"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-x%20-%20y%20-%206" alt="y'= -x - y - 6" title="y'= -x - y - 6" data-latex="y'= -x - y - 6"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(-3, -3\right)" alt="\left(-3, -3\right)" title="\left(-3, -3\right)" data-latex="\left(-3, -3\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a positive slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a negative slope. </p><p> The right region has a southwest arrow system, the top region has a southeast arrow system, the left region has a northeast arrow system, and the bottom region has a northwest arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(-3,%20-3%5Cright)" alt="\left(-3, -3\right)" title="\left(-3, -3\right)" data-latex="\left(-3, -3\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a positive slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a negative slope. &lt;/p&gt;
  &lt;p&gt; The right region has a southwest arrow system, the top region has a southeast arrow system, the left region has a northeast arrow system, and the bottom region has a northwest arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-1304" title="S1 | Phase Planes | ver. 1304"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -2 \, x - 3 \, y + 2" alt="x'= -2 \, x - 3 \, y + 2" title="x'= -2 \, x - 3 \, y + 2" data-latex="x'= -2 \, x - 3 \, y + 2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 2 \, x - 3 \, y + 22" alt="y'= 2 \, x - 3 \, y + 22" title="y'= 2 \, x - 3 \, y + 22" data-latex="y'= 2 \, x - 3 \, y + 22"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-2%20%5C,%20x%20-%203%20%5C,%20y%20+%202" alt="x'= -2 \, x - 3 \, y + 2" title="x'= -2 \, x - 3 \, y + 2" data-latex="x'= -2 \, x - 3 \, y + 2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%202%20%5C,%20x%20-%203%20%5C,%20y%20+%2022" alt="y'= 2 \, x - 3 \, y + 22" title="y'= 2 \, x - 3 \, y + 22" data-latex="y'= 2 \, x - 3 \, y + 22"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(-5, 4\right)" alt="\left(-5, 4\right)" title="\left(-5, 4\right)" data-latex="\left(-5, 4\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(-5,%204%5Cright)" alt="\left(-5, 4\right)" title="\left(-5, 4\right)" data-latex="\left(-5, 4\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-5773" title="S1 | Phase Planes | ver. 5773"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -3 \, x - y + 3" alt="x'= -3 \, x - y + 3" title="x'= -3 \, x - y + 3" data-latex="x'= -3 \, x - y + 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'= -x + 3 \, y + 1" alt="y'= -x + 3 \, y + 1" title="y'= -x + 3 \, y + 1" data-latex="y'= -x + 3 \, y + 1"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-3%20%5C,%20x%20-%20y%20+%203" alt="x'= -3 \, x - y + 3" title="x'= -3 \, x - y + 3" data-latex="x'= -3 \, x - y + 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?y'=%20-x%20+%203%20%5C,%20y%20+%201" alt="y'= -x + 3 \, y + 1" title="y'= -x + 3 \, y + 1" data-latex="y'= -x + 3 \, y + 1"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(1, 0\right)" alt="\left(1, 0\right)" title="\left(1, 0\right)" data-latex="\left(1, 0\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(1,%200%5Cright)" alt="\left(1, 0\right)" title="\left(1, 0\right)" data-latex="\left(1, 0\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-2496" title="S1 | Phase Planes | ver. 2496"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -2 \, x - 4 \, y - 16" alt="x'= -2 \, x - 4 \, y - 16" title="x'= -2 \, x - 4 \, y - 16" data-latex="x'= -2 \, x - 4 \, y - 16"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -2 \, x + y + 9" alt="y'= -2 \, x + y + 9" title="y'= -2 \, x + y + 9" data-latex="y'= -2 \, x + y + 9"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-2%20%5C,%20x%20-%204%20%5C,%20y%20-%2016" alt="x'= -2 \, x - 4 \, y - 16" title="x'= -2 \, x - 4 \, y - 16" data-latex="x'= -2 \, x - 4 \, y - 16"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-2%20%5C,%20x%20+%20y%20+%209" alt="y'= -2 \, x + y + 9" title="y'= -2 \, x + y + 9" data-latex="y'= -2 \, x + y + 9"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(2, -5\right)" alt="\left(2, -5\right)" title="\left(2, -5\right)" data-latex="\left(2, -5\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(2,%20-5%5Cright)" alt="\left(2, -5\right)" title="\left(2, -5\right)" data-latex="\left(2, -5\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-4327" title="S1 | Phase Planes | ver. 4327"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -3 \, x + 2 \, y" alt="x'= -3 \, x + 2 \, y" title="x'= -3 \, x + 2 \, y" data-latex="x'= -3 \, x + 2 \, y"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -4 \, x - y" alt="y'= -4 \, x - y" title="y'= -4 \, x - y" data-latex="y'= -4 \, x - y"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-3%20%5C,%20x%20+%202%20%5C,%20y" alt="x'= -3 \, x + 2 \, y" title="x'= -3 \, x + 2 \, y" data-latex="x'= -3 \, x + 2 \, y"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-4%20%5C,%20x%20-%20y" alt="y'= -4 \, x - y" title="y'= -4 \, x - y" data-latex="y'= -4 \, x - y"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(0, 0\right)" alt="\left(0, 0\right)" title="\left(0, 0\right)" data-latex="\left(0, 0\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a positive slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a negative slope. </p><p> The right region has a southwest arrow system, the top region has a southeast arrow system, the left region has a northeast arrow system, and the bottom region has a northwest arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(0,%200%5Cright)" alt="\left(0, 0\right)" title="\left(0, 0\right)" data-latex="\left(0, 0\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a positive slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a negative slope. &lt;/p&gt;
  &lt;p&gt; The right region has a southwest arrow system, the top region has a southeast arrow system, the left region has a northeast arrow system, and the bottom region has a northwest arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-6076" title="S1 | Phase Planes | ver. 6076"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 3 \, x + 5 \, y - 1" alt="x'= 3 \, x + 5 \, y - 1" title="x'= 3 \, x + 5 \, y - 1" data-latex="x'= 3 \, x + 5 \, y - 1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= x - y + 5" alt="y'= x - y + 5" title="y'= x - y + 5" data-latex="y'= x - y + 5"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%203%20%5C,%20x%20+%205%20%5C,%20y%20-%201" alt="x'= 3 \, x + 5 \, y - 1" title="x'= 3 \, x + 5 \, y - 1" data-latex="x'= 3 \, x + 5 \, y - 1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20x%20-%20y%20+%205" alt="y'= x - y + 5" title="y'= x - y + 5" data-latex="y'= x - y + 5"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(-3, 2\right)" alt="\left(-3, 2\right)" title="\left(-3, 2\right)" data-latex="\left(-3, 2\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a northeast arrow system, the top region has a southeast arrow system, the left region has a southwest arrow system, and the bottom region has a northwest arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(-3,%202%5Cright)" alt="\left(-3, 2\right)" title="\left(-3, 2\right)" data-latex="\left(-3, 2\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a northeast arrow system, the top region has a southeast arrow system, the left region has a southwest arrow system, and the bottom region has a northwest arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-3192" title="S1 | Phase Planes | ver. 3192"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 4 \, x - 3 \, y + 5" alt="x'= 4 \, x - 3 \, y + 5" title="x'= 4 \, x - 3 \, y + 5" data-latex="x'= 4 \, x - 3 \, y + 5"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= x + 5 \, y + 30" alt="y'= x + 5 \, y + 30" title="y'= x + 5 \, y + 30" data-latex="y'= x + 5 \, y + 30"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%204%20%5C,%20x%20-%203%20%5C,%20y%20+%205" alt="x'= 4 \, x - 3 \, y + 5" title="x'= 4 \, x - 3 \, y + 5" data-latex="x'= 4 \, x - 3 \, y + 5"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20x%20+%205%20%5C,%20y%20+%2030" alt="y'= x + 5 \, y + 30" title="y'= x + 5 \, y + 30" data-latex="y'= x + 5 \, y + 30"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(-5, -5\right)" alt="\left(-5, -5\right)" title="\left(-5, -5\right)" data-latex="\left(-5, -5\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a positive slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a negative slope. </p><p> The right region has a northeast arrow system, the top region has a northwest arrow system, the left region has a southwest arrow system, and the bottom region has a southeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(-5,%20-5%5Cright)" alt="\left(-5, -5\right)" title="\left(-5, -5\right)" data-latex="\left(-5, -5\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a positive slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a negative slope. &lt;/p&gt;
  &lt;p&gt; The right region has a northeast arrow system, the top region has a northwest arrow system, the left region has a southwest arrow system, and the bottom region has a southeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-9378" title="S1 | Phase Planes | ver. 9378"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -x - 5 \, y + 10" alt="x'= -x - 5 \, y + 10" title="x'= -x - 5 \, y + 10" data-latex="x'= -x - 5 \, y + 10"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -3 \, x + 4 \, y - 27" alt="y'= -3 \, x + 4 \, y - 27" title="y'= -3 \, x + 4 \, y - 27" data-latex="y'= -3 \, x + 4 \, y - 27"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-x%20-%205%20%5C,%20y%20+%2010" alt="x'= -x - 5 \, y + 10" title="x'= -x - 5 \, y + 10" data-latex="x'= -x - 5 \, y + 10"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-3%20%5C,%20x%20+%204%20%5C,%20y%20-%2027" alt="y'= -3 \, x + 4 \, y - 27" title="y'= -3 \, x + 4 \, y - 27" data-latex="y'= -3 \, x + 4 \, y - 27"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(-5, 3\right)" alt="\left(-5, 3\right)" title="\left(-5, 3\right)" data-latex="\left(-5, 3\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(-5,%203%5Cright)" alt="\left(-5, 3\right)" title="\left(-5, 3\right)" data-latex="\left(-5, 3\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-4512" title="S1 | Phase Planes | ver. 4512"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 4 \, x - 5 \, y - 10" alt="x'= 4 \, x - 5 \, y - 10" title="x'= 4 \, x - 5 \, y - 10" data-latex="x'= 4 \, x - 5 \, y - 10"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -4 \, x - 4 \, y + 28" alt="y'= -4 \, x - 4 \, y + 28" title="y'= -4 \, x - 4 \, y + 28" data-latex="y'= -4 \, x - 4 \, y + 28"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%204%20%5C,%20x%20-%205%20%5C,%20y%20-%2010" alt="x'= 4 \, x - 5 \, y - 10" title="x'= 4 \, x - 5 \, y - 10" data-latex="x'= 4 \, x - 5 \, y - 10"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-4%20%5C,%20x%20-%204%20%5C,%20y%20+%2028" alt="y'= -4 \, x - 4 \, y + 28" title="y'= -4 \, x - 4 \, y + 28" data-latex="y'= -4 \, x - 4 \, y + 28"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(5, 2\right)" alt="\left(5, 2\right)" title="\left(5, 2\right)" data-latex="\left(5, 2\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a positive slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a negative slope. </p><p> The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(5,%202%5Cright)" alt="\left(5, 2\right)" title="\left(5, 2\right)" data-latex="\left(5, 2\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a positive slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a negative slope. &lt;/p&gt;
  &lt;p&gt; The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-6404" title="S1 | Phase Planes | ver. 6404"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -4 \, x + y + 5" alt="x'= -4 \, x + y + 5" title="x'= -4 \, x + y + 5" data-latex="x'= -4 \, x + y + 5"/></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'= -5 \, x - y + 4" alt="y'= -5 \, x - y + 4" title="y'= -5 \, x - y + 4" data-latex="y'= -5 \, x - y + 4"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-4%20%5C,%20x%20+%20y%20+%205" alt="x'= -4 \, x + y + 5" title="x'= -4 \, x + y + 5" data-latex="x'= -4 \, x + y + 5"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-5%20%5C,%20x%20-%20y%20+%204" alt="y'= -5 \, x - y + 4" title="y'= -5 \, x - y + 4" data-latex="y'= -5 \, x - y + 4"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(1, -1\right)" alt="\left(1, -1\right)" title="\left(1, -1\right)" data-latex="\left(1, -1\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a positive slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a negative slope. </p><p> The right region has a southwest arrow system, the top region has a southeast arrow system, the left region has a northeast arrow system, and the bottom region has a northwest arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(1,%20-1%5Cright)" alt="\left(1, -1\right)" title="\left(1, -1\right)" data-latex="\left(1, -1\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a positive slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a negative slope. &lt;/p&gt;
  &lt;p&gt; The right region has a southwest arrow system, the top region has a southeast arrow system, the left region has a northeast arrow system, and the bottom region has a northwest arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-3325" title="S1 | Phase Planes | ver. 3325"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 2 \, x + 4 \, y - 12" alt="x'= 2 \, x + 4 \, y - 12" title="x'= 2 \, x + 4 \, y - 12" data-latex="x'= 2 \, x + 4 \, y - 12"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= x - 3 \, y + 9" alt="y'= x - 3 \, y + 9" title="y'= x - 3 \, y + 9" data-latex="y'= x - 3 \, y + 9"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%202%20%5C,%20x%20+%204%20%5C,%20y%20-%2012" alt="x'= 2 \, x + 4 \, y - 12" title="x'= 2 \, x + 4 \, y - 12" data-latex="x'= 2 \, x + 4 \, y - 12"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20x%20-%203%20%5C,%20y%20+%209" alt="y'= x - 3 \, y + 9" title="y'= x - 3 \, y + 9" data-latex="y'= x - 3 \, y + 9"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(0, 3\right)" alt="\left(0, 3\right)" title="\left(0, 3\right)" data-latex="\left(0, 3\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a northeast arrow system, the top region has a southeast arrow system, the left region has a southwest arrow system, and the bottom region has a northwest arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(0,%203%5Cright)" alt="\left(0, 3\right)" title="\left(0, 3\right)" data-latex="\left(0, 3\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a northeast arrow system, the top region has a southeast arrow system, the left region has a southwest arrow system, and the bottom region has a northwest arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-0855" title="S1 | Phase Planes | ver. 0855"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 5 \, x - 2 \, y - 29" alt="x'= 5 \, x - 2 \, y - 29" title="x'= 5 \, x - 2 \, y - 29" data-latex="x'= 5 \, x - 2 \, y - 29"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -4 \, x - 3 \, y + 14" alt="y'= -4 \, x - 3 \, y + 14" title="y'= -4 \, x - 3 \, y + 14" data-latex="y'= -4 \, x - 3 \, y + 14"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%205%20%5C,%20x%20-%202%20%5C,%20y%20-%2029" alt="x'= 5 \, x - 2 \, y - 29" title="x'= 5 \, x - 2 \, y - 29" data-latex="x'= 5 \, x - 2 \, y - 29"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-4%20%5C,%20x%20-%203%20%5C,%20y%20+%2014" alt="y'= -4 \, x - 3 \, y + 14" title="y'= -4 \, x - 3 \, y + 14" data-latex="y'= -4 \, x - 3 \, y + 14"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(5, -2\right)" alt="\left(5, -2\right)" title="\left(5, -2\right)" data-latex="\left(5, -2\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a positive slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a negative slope. </p><p> The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(5,%20-2%5Cright)" alt="\left(5, -2\right)" title="\left(5, -2\right)" data-latex="\left(5, -2\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a positive slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a negative slope. &lt;/p&gt;
  &lt;p&gt; The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-7549" title="S1 | Phase Planes | ver. 7549"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 3 \, x - 2 \, y + 13" alt="x'= 3 \, x - 2 \, y + 13" title="x'= 3 \, x - 2 \, y + 13" data-latex="x'= 3 \, x - 2 \, y + 13"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 2 \, x + 2 \, y + 12" alt="y'= 2 \, x + 2 \, y + 12" title="y'= 2 \, x + 2 \, y + 12" data-latex="y'= 2 \, x + 2 \, y + 12"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%203%20%5C,%20x%20-%202%20%5C,%20y%20+%2013" alt="x'= 3 \, x - 2 \, y + 13" title="x'= 3 \, x - 2 \, y + 13" data-latex="x'= 3 \, x - 2 \, y + 13"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%202%20%5C,%20x%20+%202%20%5C,%20y%20+%2012" alt="y'= 2 \, x + 2 \, y + 12" title="y'= 2 \, x + 2 \, y + 12" data-latex="y'= 2 \, x + 2 \, y + 12"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(-5, -1\right)" alt="\left(-5, -1\right)" title="\left(-5, -1\right)" data-latex="\left(-5, -1\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a positive slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a negative slope. </p><p> The right region has a northeast arrow system, the top region has a northwest arrow system, the left region has a southwest arrow system, and the bottom region has a southeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(-5,%20-1%5Cright)" alt="\left(-5, -1\right)" title="\left(-5, -1\right)" data-latex="\left(-5, -1\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a positive slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a negative slope. &lt;/p&gt;
  &lt;p&gt; The right region has a northeast arrow system, the top region has a northwest arrow system, the left region has a southwest arrow system, and the bottom region has a southeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-6660" title="S1 | Phase Planes | ver. 6660"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -3 \, x - 2 \, y" alt="x'= -3 \, x - 2 \, y" title="x'= -3 \, x - 2 \, y" data-latex="x'= -3 \, x - 2 \, y"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -4 \, x + 3 \, y + 17" alt="y'= -4 \, x + 3 \, y + 17" title="y'= -4 \, x + 3 \, y + 17" data-latex="y'= -4 \, x + 3 \, y + 17"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-3%20%5C,%20x%20-%202%20%5C,%20y" alt="x'= -3 \, x - 2 \, y" title="x'= -3 \, x - 2 \, y" data-latex="x'= -3 \, x - 2 \, y"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-4%20%5C,%20x%20+%203%20%5C,%20y%20+%2017" alt="y'= -4 \, x + 3 \, y + 17" title="y'= -4 \, x + 3 \, y + 17" data-latex="y'= -4 \, x + 3 \, y + 17"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(2, -3\right)" alt="\left(2, -3\right)" title="\left(2, -3\right)" data-latex="\left(2, -3\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(2,%20-3%5Cright)" alt="\left(2, -3\right)" title="\left(2, -3\right)" data-latex="\left(2, -3\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-5501" title="S1 | Phase Planes | ver. 5501"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -3 \, x - 5 \, y" alt="x'= -3 \, x - 5 \, y" title="x'= -3 \, x - 5 \, y" data-latex="x'= -3 \, x - 5 \, y"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 2 \, x - 5 \, y" alt="y'= 2 \, x - 5 \, y" title="y'= 2 \, x - 5 \, y" data-latex="y'= 2 \, x - 5 \, y"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-3%20%5C,%20x%20-%205%20%5C,%20y" alt="x'= -3 \, x - 5 \, y" title="x'= -3 \, x - 5 \, y" data-latex="x'= -3 \, x - 5 \, y"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%202%20%5C,%20x%20-%205%20%5C,%20y" alt="y'= 2 \, x - 5 \, y" title="y'= 2 \, x - 5 \, y" data-latex="y'= 2 \, x - 5 \, y"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(0, 0\right)" alt="\left(0, 0\right)" title="\left(0, 0\right)" data-latex="\left(0, 0\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(0,%200%5Cright)" alt="\left(0, 0\right)" title="\left(0, 0\right)" data-latex="\left(0, 0\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-7660" title="S1 | Phase Planes | ver. 7660"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 3 \, x + 2 \, y + 6" alt="x'= 3 \, x + 2 \, y + 6" title="x'= 3 \, x + 2 \, y + 6" data-latex="x'= 3 \, x + 2 \, y + 6"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= x - 5 \, y - 15" alt="y'= x - 5 \, y - 15" title="y'= x - 5 \, y - 15" data-latex="y'= x - 5 \, y - 15"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%203%20%5C,%20x%20+%202%20%5C,%20y%20+%206" alt="x'= 3 \, x + 2 \, y + 6" title="x'= 3 \, x + 2 \, y + 6" data-latex="x'= 3 \, x + 2 \, y + 6"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20x%20-%205%20%5C,%20y%20-%2015" alt="y'= x - 5 \, y - 15" title="y'= x - 5 \, y - 15" data-latex="y'= x - 5 \, y - 15"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(0, -3\right)" alt="\left(0, -3\right)" title="\left(0, -3\right)" data-latex="\left(0, -3\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a northeast arrow system, the top region has a southeast arrow system, the left region has a southwest arrow system, and the bottom region has a northwest arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(0,%20-3%5Cright)" alt="\left(0, -3\right)" title="\left(0, -3\right)" data-latex="\left(0, -3\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a northeast arrow system, the top region has a southeast arrow system, the left region has a southwest arrow system, and the bottom region has a northwest arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-3031" title="S1 | Phase Planes | ver. 3031"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 5 \, x - 2 \, y - 4" alt="x'= 5 \, x - 2 \, y - 4" title="x'= 5 \, x - 2 \, y - 4" data-latex="x'= 5 \, x - 2 \, y - 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'= x + 4 \, y - 14" alt="y'= x + 4 \, y - 14" title="y'= x + 4 \, y - 14" data-latex="y'= x + 4 \, y - 14"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%205%20%5C,%20x%20-%202%20%5C,%20y%20-%204" alt="x'= 5 \, x - 2 \, y - 4" title="x'= 5 \, x - 2 \, y - 4" data-latex="x'= 5 \, x - 2 \, y - 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?y'=%20x%20+%204%20%5C,%20y%20-%2014" alt="y'= x + 4 \, y - 14" title="y'= x + 4 \, y - 14" data-latex="y'= x + 4 \, y - 14"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(2, 3\right)" alt="\left(2, 3\right)" title="\left(2, 3\right)" data-latex="\left(2, 3\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a positive slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a negative slope. </p><p> The right region has a northeast arrow system, the top region has a northwest arrow system, the left region has a southwest arrow system, and the bottom region has a southeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(2,%203%5Cright)" alt="\left(2, 3\right)" title="\left(2, 3\right)" data-latex="\left(2, 3\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a positive slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a negative slope. &lt;/p&gt;
  &lt;p&gt; The right region has a northeast arrow system, the top region has a northwest arrow system, the left region has a southwest arrow system, and the bottom region has a southeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-0367" title="S1 | Phase Planes | 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>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -4 \, x - 5 \, y - 5" alt="x'= -4 \, x - 5 \, y - 5" title="x'= -4 \, x - 5 \, y - 5" data-latex="x'= -4 \, x - 5 \, y - 5"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 4 \, x - 3 \, y + 29" alt="y'= 4 \, x - 3 \, y + 29" title="y'= 4 \, x - 3 \, y + 29" data-latex="y'= 4 \, x - 3 \, y + 29"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-4%20%5C,%20x%20-%205%20%5C,%20y%20-%205" alt="x'= -4 \, x - 5 \, y - 5" title="x'= -4 \, x - 5 \, y - 5" data-latex="x'= -4 \, x - 5 \, y - 5"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%204%20%5C,%20x%20-%203%20%5C,%20y%20+%2029" alt="y'= 4 \, x - 3 \, y + 29" title="y'= 4 \, x - 3 \, y + 29" data-latex="y'= 4 \, x - 3 \, y + 29"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(-5, 3\right)" alt="\left(-5, 3\right)" title="\left(-5, 3\right)" data-latex="\left(-5, 3\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(-5,%203%5Cright)" alt="\left(-5, 3\right)" title="\left(-5, 3\right)" data-latex="\left(-5, 3\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-0861" title="S1 | Phase Planes | ver. 0861"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -x - 3 \, y + 8" alt="x'= -x - 3 \, y + 8" title="x'= -x - 3 \, y + 8" data-latex="x'= -x - 3 \, y + 8"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -x + 3 \, y + 2" alt="y'= -x + 3 \, y + 2" title="y'= -x + 3 \, y + 2" data-latex="y'= -x + 3 \, y + 2"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-x%20-%203%20%5C,%20y%20+%208" alt="x'= -x - 3 \, y + 8" title="x'= -x - 3 \, y + 8" data-latex="x'= -x - 3 \, y + 8"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-x%20+%203%20%5C,%20y%20+%202" alt="y'= -x + 3 \, y + 2" title="y'= -x + 3 \, y + 2" data-latex="y'= -x + 3 \, y + 2"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(5, 1\right)" alt="\left(5, 1\right)" title="\left(5, 1\right)" data-latex="\left(5, 1\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(5,%201%5Cright)" alt="\left(5, 1\right)" title="\left(5, 1\right)" data-latex="\left(5, 1\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-4931" title="S1 | Phase Planes | ver. 4931"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 3 \, x - 3 \, y - 21" alt="x'= 3 \, x - 3 \, y - 21" title="x'= 3 \, x - 3 \, y - 21" data-latex="x'= 3 \, x - 3 \, y - 21"/></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'= -5 \, x - y + 17" alt="y'= -5 \, x - y + 17" title="y'= -5 \, x - y + 17" data-latex="y'= -5 \, x - y + 17"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%203%20%5C,%20x%20-%203%20%5C,%20y%20-%2021" alt="x'= 3 \, x - 3 \, y - 21" title="x'= 3 \, x - 3 \, y - 21" data-latex="x'= 3 \, x - 3 \, y - 21"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-5%20%5C,%20x%20-%20y%20+%2017" alt="y'= -5 \, x - y + 17" title="y'= -5 \, x - y + 17" data-latex="y'= -5 \, x - y + 17"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(4, -3\right)" alt="\left(4, -3\right)" title="\left(4, -3\right)" data-latex="\left(4, -3\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a positive slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a negative slope. </p><p> The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(4,%20-3%5Cright)" alt="\left(4, -3\right)" title="\left(4, -3\right)" data-latex="\left(4, -3\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a positive slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a negative slope. &lt;/p&gt;
  &lt;p&gt; The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-8552" title="S1 | Phase Planes | ver. 8552"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -3 \, x - 5 \, y + 25" alt="x'= -3 \, x - 5 \, y + 25" title="x'= -3 \, x - 5 \, y + 25" data-latex="x'= -3 \, x - 5 \, y + 25"/></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'= -5 \, x + 4 \, y + 17" alt="y'= -5 \, x + 4 \, y + 17" title="y'= -5 \, x + 4 \, y + 17" data-latex="y'= -5 \, x + 4 \, y + 17"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-3%20%5C,%20x%20-%205%20%5C,%20y%20+%2025" alt="x'= -3 \, x - 5 \, y + 25" title="x'= -3 \, x - 5 \, y + 25" data-latex="x'= -3 \, x - 5 \, y + 25"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-5%20%5C,%20x%20+%204%20%5C,%20y%20+%2017" alt="y'= -5 \, x + 4 \, y + 17" title="y'= -5 \, x + 4 \, y + 17" data-latex="y'= -5 \, x + 4 \, y + 17"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(5, 2\right)" alt="\left(5, 2\right)" title="\left(5, 2\right)" data-latex="\left(5, 2\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(5,%202%5Cright)" alt="\left(5, 2\right)" title="\left(5, 2\right)" data-latex="\left(5, 2\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-1120" title="S1 | Phase Planes | ver. 1120"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -5 \, x - 5 \, y + 35" alt="x'= -5 \, x - 5 \, y + 35" title="x'= -5 \, x - 5 \, y + 35" data-latex="x'= -5 \, x - 5 \, y + 35"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -4 \, x + 2 \, y + 16" alt="y'= -4 \, x + 2 \, y + 16" title="y'= -4 \, x + 2 \, y + 16" data-latex="y'= -4 \, x + 2 \, y + 16"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-5%20%5C,%20x%20-%205%20%5C,%20y%20+%2035" alt="x'= -5 \, x - 5 \, y + 35" title="x'= -5 \, x - 5 \, y + 35" data-latex="x'= -5 \, x - 5 \, y + 35"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-4%20%5C,%20x%20+%202%20%5C,%20y%20+%2016" alt="y'= -4 \, x + 2 \, y + 16" title="y'= -4 \, x + 2 \, y + 16" data-latex="y'= -4 \, x + 2 \, y + 16"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(5, 2\right)" alt="\left(5, 2\right)" title="\left(5, 2\right)" data-latex="\left(5, 2\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(5,%202%5Cright)" alt="\left(5, 2\right)" title="\left(5, 2\right)" data-latex="\left(5, 2\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-9600" title="S1 | Phase Planes | ver. 9600"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 5 \, x - 2 \, y + 17" alt="x'= 5 \, x - 2 \, y + 17" title="x'= 5 \, x - 2 \, y + 17" data-latex="x'= 5 \, x - 2 \, y + 17"/></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'= -5 \, x - 5 \, y - 10" alt="y'= -5 \, x - 5 \, y - 10" title="y'= -5 \, x - 5 \, y - 10" data-latex="y'= -5 \, x - 5 \, y - 10"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%205%20%5C,%20x%20-%202%20%5C,%20y%20+%2017" alt="x'= 5 \, x - 2 \, y + 17" title="x'= 5 \, x - 2 \, y + 17" data-latex="x'= 5 \, x - 2 \, y + 17"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-5%20%5C,%20x%20-%205%20%5C,%20y%20-%2010" alt="y'= -5 \, x - 5 \, y - 10" title="y'= -5 \, x - 5 \, y - 10" data-latex="y'= -5 \, x - 5 \, y - 10"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(-3, 1\right)" alt="\left(-3, 1\right)" title="\left(-3, 1\right)" data-latex="\left(-3, 1\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a positive slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a negative slope. </p><p> The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(-3,%201%5Cright)" alt="\left(-3, 1\right)" title="\left(-3, 1\right)" data-latex="\left(-3, 1\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a positive slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a negative slope. &lt;/p&gt;
  &lt;p&gt; The right region has a southeast arrow system, the top region has a southwest arrow system, the left region has a northwest arrow system, and the bottom region has a northeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-0950" title="S1 | Phase Planes | ver. 0950"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 2 \, x + y - 11" alt="x'= 2 \, x + y - 11" title="x'= 2 \, x + y - 11" data-latex="x'= 2 \, x + y - 11"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= x - 3 \, y + 12" alt="y'= x - 3 \, y + 12" title="y'= x - 3 \, y + 12" data-latex="y'= x - 3 \, y + 12"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%202%20%5C,%20x%20+%20y%20-%2011" alt="x'= 2 \, x + y - 11" title="x'= 2 \, x + y - 11" data-latex="x'= 2 \, x + y - 11"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20x%20-%203%20%5C,%20y%20+%2012" alt="y'= x - 3 \, y + 12" title="y'= x - 3 \, y + 12" data-latex="y'= x - 3 \, y + 12"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(3, 5\right)" alt="\left(3, 5\right)" title="\left(3, 5\right)" data-latex="\left(3, 5\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a northeast arrow system, the top region has a southeast arrow system, the left region has a southwest arrow system, and the bottom region has a northwest arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(3,%205%5Cright)" alt="\left(3, 5\right)" title="\left(3, 5\right)" data-latex="\left(3, 5\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a northeast arrow system, the top region has a southeast arrow system, the left region has a southwest arrow system, and the bottom region has a northwest arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-8023" title="S1 | Phase Planes | ver. 8023"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -5 \, x - 5 \, y + 15" alt="x'= -5 \, x - 5 \, y + 15" title="x'= -5 \, x - 5 \, y + 15" data-latex="x'= -5 \, x - 5 \, y + 15"/></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'= 5 \, x - 5 \, y + 25" alt="y'= 5 \, x - 5 \, y + 25" title="y'= 5 \, x - 5 \, y + 25" data-latex="y'= 5 \, x - 5 \, y + 25"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-5%20%5C,%20x%20-%205%20%5C,%20y%20+%2015" alt="x'= -5 \, x - 5 \, y + 15" title="x'= -5 \, x - 5 \, y + 15" data-latex="x'= -5 \, x - 5 \, y + 15"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%205%20%5C,%20x%20-%205%20%5C,%20y%20+%2025" alt="y'= 5 \, x - 5 \, y + 25" title="y'= 5 \, x - 5 \, y + 25" data-latex="y'= 5 \, x - 5 \, y + 25"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(-1, 4\right)" alt="\left(-1, 4\right)" title="\left(-1, 4\right)" data-latex="\left(-1, 4\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(-1,%204%5Cright)" alt="\left(-1, 4\right)" title="\left(-1, 4\right)" data-latex="\left(-1, 4\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-5093" title="S1 | Phase Planes | ver. 5093"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -x - 4 \, y + 17" alt="x'= -x - 4 \, y + 17" title="x'= -x - 4 \, y + 17" data-latex="x'= -x - 4 \, y + 17"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 4 \, x - 5 \, y + 37" alt="y'= 4 \, x - 5 \, y + 37" title="y'= 4 \, x - 5 \, y + 37" data-latex="y'= 4 \, x - 5 \, y + 37"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-x%20-%204%20%5C,%20y%20+%2017" alt="x'= -x - 4 \, y + 17" title="x'= -x - 4 \, y + 17" data-latex="x'= -x - 4 \, y + 17"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%204%20%5C,%20x%20-%205%20%5C,%20y%20+%2037" alt="y'= 4 \, x - 5 \, y + 37" title="y'= 4 \, x - 5 \, y + 37" data-latex="y'= 4 \, x - 5 \, y + 37"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(-3, 5\right)" alt="\left(-3, 5\right)" title="\left(-3, 5\right)" data-latex="\left(-3, 5\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(-3,%205%5Cright)" alt="\left(-3, 5\right)" title="\left(-3, 5\right)" data-latex="\left(-3, 5\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-4428" title="S1 | Phase Planes | ver. 4428"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -2 \, x - 2 \, y - 12" alt="x'= -2 \, x - 2 \, y - 12" title="x'= -2 \, x - 2 \, y - 12" data-latex="x'= -2 \, x - 2 \, y - 12"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= x - y - 2" alt="y'= x - y - 2" title="y'= x - y - 2" data-latex="y'= x - y - 2"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-2%20%5C,%20x%20-%202%20%5C,%20y%20-%2012" alt="x'= -2 \, x - 2 \, y - 12" title="x'= -2 \, x - 2 \, y - 12" data-latex="x'= -2 \, x - 2 \, y - 12"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20x%20-%20y%20-%202" alt="y'= x - y - 2" title="y'= x - y - 2" data-latex="y'= x - y - 2"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(-2, -4\right)" alt="\left(-2, -4\right)" title="\left(-2, -4\right)" data-latex="\left(-2, -4\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(-2,%20-4%5Cright)" alt="\left(-2, -4\right)" title="\left(-2, -4\right)" data-latex="\left(-2, -4\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-6887" title="S1 | Phase Planes | ver. 6887"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -5 \, x - y + 24" alt="x'= -5 \, x - y + 24" title="x'= -5 \, x - y + 24" data-latex="x'= -5 \, x - y + 24"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -3 \, x + y + 8" alt="y'= -3 \, x + y + 8" title="y'= -3 \, x + y + 8" data-latex="y'= -3 \, x + y + 8"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-5%20%5C,%20x%20-%20y%20+%2024" alt="x'= -5 \, x - y + 24" title="x'= -5 \, x - y + 24" data-latex="x'= -5 \, x - y + 24"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-3%20%5C,%20x%20+%20y%20+%208" alt="y'= -3 \, x + y + 8" title="y'= -3 \, x + y + 8" data-latex="y'= -3 \, x + y + 8"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(4, 4\right)" alt="\left(4, 4\right)" title="\left(4, 4\right)" data-latex="\left(4, 4\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(4,%204%5Cright)" alt="\left(4, 4\right)" title="\left(4, 4\right)" data-latex="\left(4, 4\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-7890" title="S1 | Phase Planes | ver. 7890"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -5 \, x + 3 \, y - 16" alt="x'= -5 \, x + 3 \, y - 16" title="x'= -5 \, x + 3 \, y - 16" data-latex="x'= -5 \, x + 3 \, y - 16"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -3 \, x - 2 \, y - 21" alt="y'= -3 \, x - 2 \, y - 21" title="y'= -3 \, x - 2 \, y - 21" data-latex="y'= -3 \, x - 2 \, y - 21"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-5%20%5C,%20x%20+%203%20%5C,%20y%20-%2016" alt="x'= -5 \, x + 3 \, y - 16" title="x'= -5 \, x + 3 \, y - 16" data-latex="x'= -5 \, x + 3 \, y - 16"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-3%20%5C,%20x%20-%202%20%5C,%20y%20-%2021" alt="y'= -3 \, x - 2 \, y - 21" title="y'= -3 \, x - 2 \, y - 21" data-latex="y'= -3 \, x - 2 \, y - 21"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(-5, -3\right)" alt="\left(-5, -3\right)" title="\left(-5, -3\right)" data-latex="\left(-5, -3\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a positive slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a negative slope. </p><p> The right region has a southwest arrow system, the top region has a southeast arrow system, the left region has a northeast arrow system, and the bottom region has a northwest arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(-5,%20-3%5Cright)" alt="\left(-5, -3\right)" title="\left(-5, -3\right)" data-latex="\left(-5, -3\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a positive slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a negative slope. &lt;/p&gt;
  &lt;p&gt; The right region has a southwest arrow system, the top region has a southeast arrow system, the left region has a northeast arrow system, and the bottom region has a northwest arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-7471" title="S1 | Phase Planes | ver. 7471"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -5 \, x - 2 \, y + 9" alt="x'= -5 \, x - 2 \, y + 9" title="x'= -5 \, x - 2 \, y + 9" data-latex="x'= -5 \, x - 2 \, y + 9"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 3 \, x - 5 \, y - 24" alt="y'= 3 \, x - 5 \, y - 24" title="y'= 3 \, x - 5 \, y - 24" data-latex="y'= 3 \, x - 5 \, y - 24"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-5%20%5C,%20x%20-%202%20%5C,%20y%20+%209" alt="x'= -5 \, x - 2 \, y + 9" title="x'= -5 \, x - 2 \, y + 9" data-latex="x'= -5 \, x - 2 \, y + 9"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%203%20%5C,%20x%20-%205%20%5C,%20y%20-%2024" alt="y'= 3 \, x - 5 \, y - 24" title="y'= 3 \, x - 5 \, y - 24" data-latex="y'= 3 \, x - 5 \, y - 24"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(3, -3\right)" alt="\left(3, -3\right)" title="\left(3, -3\right)" data-latex="\left(3, -3\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(3,%20-3%5Cright)" alt="\left(3, -3\right)" title="\left(3, -3\right)" data-latex="\left(3, -3\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-6016" title="S1 | Phase Planes | ver. 6016"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -3 \, x - y - 1" alt="x'= -3 \, x - y - 1" title="x'= -3 \, x - y - 1" data-latex="x'= -3 \, x - y - 1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= x - 3 \, y - 3" alt="y'= x - 3 \, y - 3" title="y'= x - 3 \, y - 3" data-latex="y'= x - 3 \, y - 3"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-3%20%5C,%20x%20-%20y%20-%201" alt="x'= -3 \, x - y - 1" title="x'= -3 \, x - y - 1" data-latex="x'= -3 \, x - y - 1"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20x%20-%203%20%5C,%20y%20-%203" alt="y'= x - 3 \, y - 3" title="y'= x - 3 \, y - 3" data-latex="y'= x - 3 \, y - 3"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(0, -1\right)" alt="\left(0, -1\right)" title="\left(0, -1\right)" data-latex="\left(0, -1\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(0,%20-1%5Cright)" alt="\left(0, -1\right)" title="\left(0, -1\right)" data-latex="\left(0, -1\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-0244" title="S1 | Phase Planes | ver. 0244"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -x + 3 \, y + 14" alt="x'= -x + 3 \, y + 14" title="x'= -x + 3 \, y + 14" data-latex="x'= -x + 3 \, y + 14"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -2 \, x - 2 \, y + 4" alt="y'= -2 \, x - 2 \, y + 4" title="y'= -2 \, x - 2 \, y + 4" data-latex="y'= -2 \, x - 2 \, y + 4"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-x%20+%203%20%5C,%20y%20+%2014" alt="x'= -x + 3 \, y + 14" title="x'= -x + 3 \, y + 14" data-latex="x'= -x + 3 \, y + 14"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-2%20%5C,%20x%20-%202%20%5C,%20y%20+%204" alt="y'= -2 \, x - 2 \, y + 4" title="y'= -2 \, x - 2 \, y + 4" data-latex="y'= -2 \, x - 2 \, y + 4"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(5, -3\right)" alt="\left(5, -3\right)" title="\left(5, -3\right)" data-latex="\left(5, -3\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a positive slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a negative slope. </p><p> The right region has a southwest arrow system, the top region has a southeast arrow system, the left region has a northeast arrow system, and the bottom region has a northwest arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(5,%20-3%5Cright)" alt="\left(5, -3\right)" title="\left(5, -3\right)" data-latex="\left(5, -3\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a positive slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a negative slope. &lt;/p&gt;
  &lt;p&gt; The right region has a southwest arrow system, the top region has a southeast arrow system, the left region has a northeast arrow system, and the bottom region has a northwest arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-4173" title="S1 | Phase Planes | ver. 4173"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -2 \, x - 2 \, y - 2" alt="x'= -2 \, x - 2 \, y - 2" title="x'= -2 \, x - 2 \, y - 2" data-latex="x'= -2 \, x - 2 \, y - 2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 4 \, x - 5 \, y - 41" alt="y'= 4 \, x - 5 \, y - 41" title="y'= 4 \, x - 5 \, y - 41" data-latex="y'= 4 \, x - 5 \, y - 41"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-2%20%5C,%20x%20-%202%20%5C,%20y%20-%202" alt="x'= -2 \, x - 2 \, y - 2" title="x'= -2 \, x - 2 \, y - 2" data-latex="x'= -2 \, x - 2 \, y - 2"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%204%20%5C,%20x%20-%205%20%5C,%20y%20-%2041" alt="y'= 4 \, x - 5 \, y - 41" title="y'= 4 \, x - 5 \, y - 41" data-latex="y'= 4 \, x - 5 \, y - 41"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(4, -5\right)" alt="\left(4, -5\right)" title="\left(4, -5\right)" data-latex="\left(4, -5\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(4,%20-5%5Cright)" alt="\left(4, -5\right)" title="\left(4, -5\right)" data-latex="\left(4, -5\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-6116" title="S1 | Phase Planes | ver. 6116"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -x - 4 \, y + 17" alt="x'= -x - 4 \, y + 17" title="x'= -x - 4 \, y + 17" data-latex="x'= -x - 4 \, y + 17"/></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'= -5 \, x + 3 \, y - 30" alt="y'= -5 \, x + 3 \, y - 30" title="y'= -5 \, x + 3 \, y - 30" data-latex="y'= -5 \, x + 3 \, y - 30"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-x%20-%204%20%5C,%20y%20+%2017" alt="x'= -x - 4 \, y + 17" title="x'= -x - 4 \, y + 17" data-latex="x'= -x - 4 \, y + 17"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-5%20%5C,%20x%20+%203%20%5C,%20y%20-%2030" alt="y'= -5 \, x + 3 \, y - 30" title="y'= -5 \, x + 3 \, y - 30" data-latex="y'= -5 \, x + 3 \, y - 30"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(-3, 5\right)" alt="\left(-3, 5\right)" title="\left(-3, 5\right)" data-latex="\left(-3, 5\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(-3,%205%5Cright)" alt="\left(-3, 5\right)" title="\left(-3, 5\right)" data-latex="\left(-3, 5\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-1912" title="S1 | Phase Planes | ver. 1912"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 5 \, x - 2 \, y - 25" alt="x'= 5 \, x - 2 \, y - 25" title="x'= 5 \, x - 2 \, y - 25" data-latex="x'= 5 \, x - 2 \, y - 25"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 3 \, x + 5 \, y + 16" alt="y'= 3 \, x + 5 \, y + 16" title="y'= 3 \, x + 5 \, y + 16" data-latex="y'= 3 \, x + 5 \, y + 16"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%205%20%5C,%20x%20-%202%20%5C,%20y%20-%2025" alt="x'= 5 \, x - 2 \, y - 25" title="x'= 5 \, x - 2 \, y - 25" data-latex="x'= 5 \, x - 2 \, y - 25"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%203%20%5C,%20x%20+%205%20%5C,%20y%20+%2016" alt="y'= 3 \, x + 5 \, y + 16" title="y'= 3 \, x + 5 \, y + 16" data-latex="y'= 3 \, x + 5 \, y + 16"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(3, -5\right)" alt="\left(3, -5\right)" title="\left(3, -5\right)" data-latex="\left(3, -5\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a positive slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a negative slope. </p><p> The right region has a northeast arrow system, the top region has a northwest arrow system, the left region has a southwest arrow system, and the bottom region has a southeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(3,%20-5%5Cright)" alt="\left(3, -5\right)" title="\left(3, -5\right)" data-latex="\left(3, -5\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a positive slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a negative slope. &lt;/p&gt;
  &lt;p&gt; The right region has a northeast arrow system, the top region has a northwest arrow system, the left region has a southwest arrow system, and the bottom region has a southeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-7242" title="S1 | Phase Planes | ver. 7242"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -3 \, x - 2 \, y - 10" alt="x'= -3 \, x - 2 \, y - 10" title="x'= -3 \, x - 2 \, y - 10" data-latex="x'= -3 \, x - 2 \, y - 10"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= 2 \, x - 3 \, y + 11" alt="y'= 2 \, x - 3 \, y + 11" title="y'= 2 \, x - 3 \, y + 11" data-latex="y'= 2 \, x - 3 \, y + 11"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-3%20%5C,%20x%20-%202%20%5C,%20y%20-%2010" alt="x'= -3 \, x - 2 \, y - 10" title="x'= -3 \, x - 2 \, y - 10" data-latex="x'= -3 \, x - 2 \, y - 10"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%202%20%5C,%20x%20-%203%20%5C,%20y%20+%2011" alt="y'= 2 \, x - 3 \, y + 11" title="y'= 2 \, x - 3 \, y + 11" data-latex="y'= 2 \, x - 3 \, y + 11"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(-4, 1\right)" alt="\left(-4, 1\right)" title="\left(-4, 1\right)" data-latex="\left(-4, 1\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(-4,%201%5Cright)" alt="\left(-4, 1\right)" title="\left(-4, 1\right)" data-latex="\left(-4, 1\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-3930" title="S1 | Phase Planes | ver. 3930"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= 3 \, x + 2 \, y + 7" alt="x'= 3 \, x + 2 \, y + 7" title="x'= 3 \, x + 2 \, y + 7" data-latex="x'= 3 \, x + 2 \, y + 7"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= x - y - 1" alt="y'= x - y - 1" title="y'= x - y - 1" data-latex="y'= x - y - 1"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%203%20%5C,%20x%20+%202%20%5C,%20y%20+%207" alt="x'= 3 \, x + 2 \, y + 7" title="x'= 3 \, x + 2 \, y + 7" data-latex="x'= 3 \, x + 2 \, y + 7"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20x%20-%20y%20-%201" alt="y'= x - y - 1" title="y'= x - y - 1" data-latex="y'= x - y - 1"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(-1, -2\right)" alt="\left(-1, -2\right)" title="\left(-1, -2\right)" data-latex="\left(-1, -2\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a northeast arrow system, the top region has a southeast arrow system, the left region has a southwest arrow system, and the bottom region has a northwest arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(-1,%20-2%5Cright)" alt="\left(-1, -2\right)" title="\left(-1, -2\right)" data-latex="\left(-1, -2\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a northeast arrow system, the top region has a southeast arrow system, the left region has a southwest arrow system, and the bottom region has a northwest arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-5423" title="S1 | Phase Planes | ver. 5423"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -4 \, x - 3 \, y + 24" alt="x'= -4 \, x - 3 \, y + 24" title="x'= -4 \, x - 3 \, y + 24" data-latex="x'= -4 \, x - 3 \, y + 24"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -2 \, x + y + 2" alt="y'= -2 \, x + y + 2" title="y'= -2 \, x + y + 2" data-latex="y'= -2 \, x + y + 2"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-4%20%5C,%20x%20-%203%20%5C,%20y%20+%2024" alt="x'= -4 \, x - 3 \, y + 24" title="x'= -4 \, x - 3 \, y + 24" data-latex="x'= -4 \, x - 3 \, y + 24"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-2%20%5C,%20x%20+%20y%20+%202" alt="y'= -2 \, x + y + 2" title="y'= -2 \, x + y + 2" data-latex="y'= -2 \, x + y + 2"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(3, 4\right)" alt="\left(3, 4\right)" title="\left(3, 4\right)" data-latex="\left(3, 4\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(3,%204%5Cright)" alt="\left(3, 4\right)" title="\left(3, 4\right)" data-latex="\left(3, 4\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-6309" title="S1 | Phase Planes | ver. 6309"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -3 \, x - 5 \, y + 30" alt="x'= -3 \, x - 5 \, y + 30" title="x'= -3 \, x - 5 \, y + 30" data-latex="x'= -3 \, x - 5 \, y + 30"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -x + 5 \, y - 10" alt="y'= -x + 5 \, y - 10" title="y'= -x + 5 \, y - 10" data-latex="y'= -x + 5 \, y - 10"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-3%20%5C,%20x%20-%205%20%5C,%20y%20+%2030" alt="x'= -3 \, x - 5 \, y + 30" title="x'= -3 \, x - 5 \, y + 30" data-latex="x'= -3 \, x - 5 \, y + 30"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-x%20+%205%20%5C,%20y%20-%2010" alt="y'= -x + 5 \, y - 10" title="y'= -x + 5 \, y - 10" data-latex="y'= -x + 5 \, y - 10"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(5, 3\right)" alt="\left(5, 3\right)" title="\left(5, 3\right)" data-latex="\left(5, 3\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a negative slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a positive slope. </p><p> The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(5,%203%5Cright)" alt="\left(5, 3\right)" title="\left(5, 3\right)" data-latex="\left(5, 3\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a negative slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a positive slope. &lt;/p&gt;
  &lt;p&gt; The right region has a southwest arrow system, the top region has a northwest arrow system, the left region has a northeast arrow system, and the bottom region has a southeast arrow system. &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="S1-3967" title="S1 | Phase Planes | ver. 3967"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>S1.</strong></p><p> Draw a phase plane for the following system of ODEs. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -3 \, x + y + 16" alt="x'= -3 \, x + y + 16" title="x'= -3 \, x + y + 16" data-latex="x'= -3 \, x + y + 16"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'= -3 \, x - 4 \, y - 4" alt="y'= -3 \, x - 4 \, y - 4" title="y'= -3 \, x - 4 \, y - 4" data-latex="y'= -3 \, x - 4 \, y - 4"/></p><p>Be sure to include and label the following features in your sketch:</p><ul><li>The isoclines <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/>.</li><li>Arrow systems describing trajectories of solutions.</li><li>The equillibrium of the system.</li></ul></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;S1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Draw a phase plane for the following system of ODEs. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=%20-3%20%5C,%20x%20+%20y%20+%2016" alt="x'= -3 \, x + y + 16" title="x'= -3 \, x + y + 16" data-latex="x'= -3 \, x + y + 16"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%20-3%20%5C,%20x%20-%204%20%5C,%20y%20-%204" alt="y'= -3 \, x - 4 \, y - 4" title="y'= -3 \, x - 4 \, y - 4" data-latex="y'= -3 \, x - 4 \, y - 4"&gt;
  &lt;/p&gt;
  &lt;p&gt;Be sure to include and label the following features in your sketch:&lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;The isoclines &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt;.&lt;/li&gt;
    &lt;li&gt;Arrow systems describing trajectories of solutions.&lt;/li&gt;
    &lt;li&gt;The equillibrium of the system.&lt;/li&gt;
  &lt;/ul&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> The isoclines cross at the equillibrium <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left(4, -4\right)" alt="\left(4, -4\right)" title="\left(4, -4\right)" data-latex="\left(4, -4\right)"/>, with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"/> having a positive slope and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"/> having a negative slope. </p><p> The right region has a southwest arrow system, the top region has a southeast arrow system, the left region has a northeast arrow system, and the bottom region has a northwest arrow system. </p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt; The isoclines cross at the equillibrium &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft(4,%20-4%5Cright)" alt="\left(4, -4\right)" title="\left(4, -4\right)" data-latex="\left(4, -4\right)"&gt;, with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'=0" alt="x'=0" title="x'=0" data-latex="x'=0"&gt; having a positive slope and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=0" alt="y'=0" title="y'=0" data-latex="y'=0"&gt; having a negative slope. &lt;/p&gt;
  &lt;p&gt; The right region has a southwest arrow system, the top region has a southeast arrow system, the left region has a northeast arrow system, and the bottom region has a northwest arrow system. &lt;/p&gt;
&lt;/div&gt;

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