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Addressing #558 by putting sample problem in example environment.
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@jbunn3
jbunn3 committed Jan 16, 2025
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75 changes: 38 additions & 37 deletions source/calculus/source/09-PS/05.ptx
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<subsection>
<title>Sample Problem</title>
<introduction>
<p>
Here you are tasked with approximating the value of <m>\cos(1)</m>.
</p>
</introduction>
<task>
<statement>
<p>
Calculate the 4th degree Taylor polynomial for <m>f(x)=\cos x</m> centered at <m>\pi</m>,
then use it to approximate the value of <m>\cos(1)</m> to three decimal places.
</p>
</statement>
</task>

<task>
<statement>
<p>
Apply Taylor's Theorem to find an upper bound for the error in this approximation.
</p>
</statement>
</task>
<example>
<introduction>
<p>
Here you are tasked with approximating the value of <m>\cos(1)</m>.
</p>
</introduction>
<task>
<statement>
<p>
Calculate the 4th degree Taylor polynomial for <m>f(x)=\cos x</m> centered at <m>\pi</m>,
then use it to approximate the value of <m>\cos(1)</m> to three decimal places.
</p>
</statement>
</task>

<task>
<statement>
<p>
Use technology to calculate <m>|R_4(1)|</m>. Is the error within the upper bound found
in part (b)?
</p>
</statement>
</task>
<task>
<statement>
<p>
Apply Taylor's Theorem to find an upper bound for the error in this approximation.
</p>
</statement>
</task>

<task>
<statement>
<p>
Explain whether the approximation error <m>|R_{k}(1)|</m> increases or decreases as
<m>k\rightarrow\infty</m>.
</p>
</statement>
</task>
<task>
<statement>
<p>
Use technology to calculate <m>|R_4(1)|</m>. Is the error within the upper bound found
in part (b)?
</p>
</statement>
</task>

<task>
<statement>
<p>
Explain whether the approximation error <m>|R_{k}(1)|</m> increases or decreases as
<m>k\rightarrow\infty</m>.
</p>
</statement>
</task>
</example>

</subsection>

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