<exercise masterit-seed="0688" masterit-slug="F1" masterit-name="Direction fields for first-order ODEs">
<statement>
<p>
Use <url href="https://sagecell.sagemath.org/"/> to run the SageMath code
<c>t,y = var('t y'); plot_slope_field(cos(y/2), (t,-5,5), (y,-5,5))</c>
producing the direction field for the ODE <m> {y'} = \cos\left(\frac{1}{2} \, {y}\right) </m>.
</p>
<p>
Let <m>y_p</m> be the solution to the following IVP.
Explain how to use its direction field to approximate the
value of <m>y_p</m> at <m>t= 1 </m>.
</p>
<me> {y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em}
y( -1 )= 0 </me>
</statement>
<answer>
<me>y_p( 1 )\approx 1.7 </me>
</answer>
</exercise>
