0010.qti

<item ident="S1-3539" title="S1 | Phase Planes | ver. 3539">
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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 - 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">
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        <response_label ident="answer1" rshuffle="No"/>
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            <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;

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