0000.qti

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

</mattext>
    </material>
    <response_str ident="response1" rcardinality="Single">
      <render_fib>
        <response_label ident="answer1" rshuffle="No"/>
      </render_fib>
    </response_str>
  </presentation>
  <itemfeedback ident="general_fb">
    <flow_mat>
      <material>
        <mattextxml>
          <div class="exercise-answer">
            <h4>Partial Answer:</h4>
            <p style="text-align:center;">
              <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( -3 )\approx -2.0" alt="y_p( -3 )\approx -2.0" title="y_p( -3 )\approx -2.0" data-latex="y_p( -3 )\approx -2.0"/>
            </p>
          </div>
        </mattextxml>
        <mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p(%20-3%20)%5Capprox%20-2.0" alt="y_p( -3 )\approx -2.0" title="y_p( -3 )\approx -2.0" data-latex="y_p( -3 )\approx -2.0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

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