0005.qti

<item ident="S1-8563" title="S1 | Phase Planes | ver. 8563">
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        <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>
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  </presentation>
  <itemfeedback ident="general_fb">
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          <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;

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