0002.qti

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

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        <response_label ident="answer1" rshuffle="No"/>
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  <itemfeedback ident="general_fb">
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          <div class="exercise-answer">
            <h4>Partial Answer:</h4>
            <p style="text-align:center;">
              <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 2 )\approx -1.8" alt="y_p( 2 )\approx -1.8" title="y_p( 2 )\approx -1.8" data-latex="y_p( 2 )\approx -1.8"/>
            </p>
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        <mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p(%202%20)%5Capprox%20-1.8" alt="y_p( 2 )\approx -1.8" title="y_p( 2 )\approx -1.8" data-latex="y_p( 2 )\approx -1.8"&gt;
  &lt;/p&gt;
&lt;/div&gt;

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