MathAdmin: Created page with " Find the Taylor polynomial of degree 4 of $f(x)=\cos^2x$ at $a=\frac{\pi}{4}$ . {| class="mw-collapsible..."

2016-04-19T01:24:21Z

Created page with "<span class="exam"> Find the Taylor polynomial of degree 4 of <math style="vertical-align: -5px">f(x)=\cos^2x</math> at <math>a=\frac{\pi}{4}</math>. {| class="mw-collapsible..."

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<span class="exam"> Find the Taylor polynomial of degree 4 of <math style="vertical-align: -5px">f(x)=\cos^2x</math> at <math>a=\frac{\pi}{4}</math>.

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Foundations:  
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|The Taylor polynomial of <math style="vertical-align: -5px">f(x)</math> at <math style="vertical-align: -1px">a</math> is
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::<math>\sum_{n=0}^{\infty}c_n(x-a)^n</math> where <math style="vertical-align: -14px">c_n=\frac{f^{(n)}(a)}{n!}.</math>
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'''Solution:'''

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 1:  
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|First, we make a table to find the coefficients of the Taylor polynomial.
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<table border="1" cellspacing="0" cellpadding="6" align = "center">
<tr>
<td align = "center"><math> n</math></td>
<td align = "center"><math> f^{(n)}(x) </math></td>
<td align = "center"><math> f^{(n)}(a) </math></td>
<td align = "center"><math> \frac{f^{(n)}(a)}{n!} </math></td>
</tr>
<tr>
<td align = "center"><math>0</math></td>
<td align = "center"><math> \cos^2x </math></td>
<td align = "center"><math> \frac{1}{2}</math></td>
<td align = "center"><math> \frac{1}{2}</math></td>
</tr>
<tr>
<td align = "center"><math>1</math></td>
<td align = "center"><math> -2\cos x\sin x</math></td>
<td align = "center"><math> -1 </math></td>
<td align = "center"><math> -1 </math></td>
</tr>
<tr>
<td align = "center"><math>2</math></td>
<td align = "center"><math> 2\sin^2x-2\cos^2x </math></td>
<td align = "center"><math> 0 </math></td>
<td align = "center"><math> 0 </math></td>
</tr>
<tr>
<td align = "center"><math>3</math></td>
<td align = "center"><math> 8\sin x\cos x </math></td>
<td align = "center"><math> 4 </math></td>
<td align = "center"><math> \frac{2}{3}</math></td>
</tr>
<tr>
<td align = "center"><math>4</math></td>
<td align = "center"><math> 8\cos^2x-8\sin^2x </math></td>
<td align = "center"><math> 0 </math></td>
<td align = "center"><math> 0 </math></td>
</tr>
</table>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 2:  
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|Since <math style="vertical-align: -14px">c_n=\frac{f^{(n)}(a)}{n!},</math> the Taylor polynomial of degree 4 of <math style="vertical-align: -5px">f(x)=\cos^2x</math> is
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::<math>T_4(x)=\frac{1}{2}+-1\bigg(x-\frac{\pi}{4}\bigg)+\frac{2}{3}\bigg(x-\frac{\pi}{4}\bigg)^3.</math>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Final Answer:  
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|   <math>\frac{1}{2}+-1\bigg(x-\frac{\pi}{4}\bigg)+\frac{2}{3}\bigg(x-\frac{\pi}{4}\bigg)^3</math>
|}
[[009C_Sample_Final_1|'''<u>Return to Sample Exam</u>''']]

009C Sample Final 1, Problem 6 - Revision history

MathAdmin: Created page with " Find the Taylor polynomial of degree 4 of f(x)=\cos^2x at a=\frac{\pi}{4}. {| class="mw-collapsible..."

MathAdmin: Created page with " Find the Taylor polynomial of degree 4 of $f(x)=\cos^2x$ at $a=\frac{\pi}{4}$ . {| class="mw-collapsible..."