009C Sample Final 2, Problem 5 - Revision history

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Created page with "<span class="exam"> Find the Taylor Polynomials of order 0, 1, 2, 3 generated by <math style="vertical-align: -5px">f(x)=\cos(x)</math> at <math style="verti..."

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<span class="exam"> Find the Taylor Polynomials of order 0, 1, 2, 3 generated by  <math style="vertical-align: -5px">f(x)=\cos(x)</math>  at  <math style="vertical-align: -14px">x=\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>
|}

'''Solution:'''

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 1:  
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|Let  <math style="vertical-align: -14px">a=\frac{\pi}{4}.</math>
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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="11" 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 x </math></td>
<td align = "center"><math> \frac{\sqrt{2}}{2}</math></td>
<td align = "center"><math> \frac{\sqrt{2}}{2}</math></td>
</tr>
<tr>
<td align = "center"><math>1</math></td>
<td align = "center"><math> -\sin x</math></td>
<td align = "center"><math> -\frac{\sqrt{2}}{2} </math></td>
<td align = "center"><math> -\frac{\sqrt{2}}{2} </math></td>
</tr>
<tr>
<td align = "center"><math>2</math></td>
<td align = "center"><math> -\cos x </math></td>
<td align = "center"><math> -\frac{\sqrt{2}}{2} </math></td>
<td align = "center"><math> -\frac{\sqrt{2}}{4} </math></td>
</tr>
<tr>
<td align = "center"><math>3</math></td>
<td align = "center"><math> \sin x </math></td>
<td align = "center"><math> \frac{\sqrt{2}}{2} </math></td>
<td align = "center"><math> \frac{\sqrt{2}}{12}</math></td>
</tr>
</table>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 2:  
|-
|Let  <math style="vertical-align: -4px">T_n</math>  be the Taylor polynomial of order  <math>n.</math>
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|Since   <math style="vertical-align: -14px">c_n=\frac{f^{(n)}(a)}{n!},</math>  we have
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| 
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|       <math>T_0=\frac{\sqrt{2}}{2}</math>
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|       <math>T_1=\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}\bigg(x-\frac{\pi}{4}\bigg)</math>
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|       <math>T_2=\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}\bigg(x-\frac{\pi}{4}\bigg)-\frac{\sqrt{2}}{4}\bigg(x-\frac{\pi}{4}\bigg)^2</math>
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|       <math>T_3=\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}\bigg(x-\frac{\pi}{4}\bigg)-\frac{\sqrt{2}}{4}\bigg(x-\frac{\pi}{4}\bigg)^2+\frac{\sqrt{2}}{12}\bigg(x-\frac{\pi}{4}\bigg)^3.</math>
|-
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Final Answer:  
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|       Let  <math style="vertical-align: -4px">T_n</math>  be the Taylor polynomial of order  <math>n.</math>
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| 
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|       <math>T_0=\frac{\sqrt{2}}{2}</math>
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|       <math>T_1=\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}\bigg(x-\frac{\pi}{4}\bigg)</math>
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|       <math>T_2=\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}\bigg(x-\frac{\pi}{4}\bigg)-\frac{\sqrt{2}}{4}\bigg(x-\frac{\pi}{4}\bigg)^2</math>
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|       <math>T_3=\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}\bigg(x-\frac{\pi}{4}\bigg)-\frac{\sqrt{2}}{4}\bigg(x-\frac{\pi}{4}\bigg)^2+\frac{\sqrt{2}}{12}\bigg(x-\frac{\pi}{4}\bigg)^3</math>
|}
[[009C_Sample_Final_2|'''<u>Return to Sample Exam</u>''']]