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Addresses #558 using example environment. #560

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siwelwerd merged 2 commits into from
Jan 21, 2025
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Addresses #558 using example environment. #560

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Addresses #558 using example environment.
jbunn3 Jan 19, 2025
e026770
Answers go outside of statements
siwelwerd Jan 21, 2025
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86 changes: 45 additions & 41 deletions source/calculus/source/09-PS/05.ptx
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Original file line number Diff line number Diff line change
Expand Up @@ -181,12 +181,12 @@
<p>
Calculate the derivatives <m>f'(x)</m>, <m>f''(x)</m>, <m>f'''(x)</m>, and <m>f^{(4)}(x)</m>.
</p>
</statement>
<answer>
<p>
<m>f'(x)=-1/x^2</m>, <m>f''(x)=2/x^3</m>, <m>f'''(x)=-6/x^4</m>, <m>f^{(4)}(x)=24/x^5</m>
</p>
</answer>
</statement>
</task>
<task>
<statement>
Expand Down Expand Up @@ -215,25 +215,27 @@
</p>
</li>
</ol>
</statement>
<answer>
<p>
B.
</p>
</answer>
</statement>

</task>

<task>
<statement>
<p>
Calculate <m>M_k</m> for each <m>k=1,2,3,4</m> using your results from part (b).
</p>
</statement>
<answer>
<p>
<m>M_1=1, M_2=2, M_3=6, M_4=24</m>
</p>
</answer>
</statement>

</task>

<task>
Expand All @@ -242,12 +244,13 @@
Use Taylor's Theorem to calculate <m>|R_k(1.5)|</m> for each <m>k=1,2,3,4</m>
to 3 decimal places. Use <m>a=1</m> as the center of the approximation.
</p>
</statement>
<answer>
<p>
<m>0.125, 0.042, 0.016, 0.006</m>
</p>
</answer>
</statement>

</task>

<task>
Expand Down Expand Up @@ -313,46 +316,47 @@

<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>

Expand Down