0007.qti

<item ident="F1-6113" title="F1 | Direction fields for first-order ODEs | ver. 6113">
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          <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(cos(t+y), (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'} = \cos\left({y} + t\right)" alt="{y'} = \cos\left({y} + t\right)" title="{y'} = \cos\left({y} + t\right)" data-latex="{y'} = \cos\left({y} + t\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= 0" alt="t= 0" title="t= 0" data-latex="t= 0"/>. </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'} = \cos\left({y} + t\right) \hspace{2em} y( 2 )= -1" alt="{y'} = \cos\left({y} + t\right) \hspace{2em} y( 2 )= -1" title="{y'} = \cos\left({y} + t\right) \hspace{2em} y( 2 )= -1" data-latex="{y'} = \cos\left({y} + t\right) \hspace{2em} y( 2 )= -1"/>
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  &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(cos(t+y), (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%5Ccos%5Cleft(%7By%7D%20+%20t%5Cright)" alt="{y'} = \cos\left({y} + t\right)" title="{y'} = \cos\left({y} + t\right)" data-latex="{y'} = \cos\left({y} + t\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=%200" alt="t= 0" title="t= 0" data-latex="t= 0"&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%5Ccos%5Cleft(%7By%7D%20+%20t%5Cright)%20%5Chspace%7B2em%7D%20y(%202%20)=%20-1" alt="{y'} = \cos\left({y} + t\right) \hspace{2em} y( 2 )= -1" title="{y'} = \cos\left({y} + t\right) \hspace{2em} y( 2 )= -1" data-latex="{y'} = \cos\left({y} + t\right) \hspace{2em} y( 2 )= -1"&gt;
  &lt;/p&gt;
&lt;/div&gt;

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            <h4>Partial Answer:</h4>
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              <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y_p( 0 )\approx -1.0" alt="y_p( 0 )\approx -1.0" title="y_p( 0 )\approx -1.0" data-latex="y_p( 0 )\approx -1.0"/>
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  &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(%200%20)%5Capprox%20-1.0" alt="y_p( 0 )\approx -1.0" title="y_p( 0 )\approx -1.0" data-latex="y_p( 0 )\approx -1.0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

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