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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>. | + | <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>. |
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| + | <hr> |
| − | !Foundations:
| + | [[009C Sample Final 1, Problem 6 Solution|'''<u>Solution</u>''']] |
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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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| − | |}
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| − | '''Solution:'''
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| + | [[009C Sample Final 1, Problem 6 Detailed Solution|'''<u>Detailed Solution</u>''']] |
| − | !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">
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| − | <tr>
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| − | <td align = "center"><math> n</math></td>
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| − | <td align = "center"><math> f^{(n)}(x) </math></td>
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| − | <td align = "center"><math> f^{(n)}(a) </math></td>
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| − | <td align = "center"><math> \frac{f^{(n)}(a)}{n!} </math></td>
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| − | </tr>
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| − | <tr>
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| − | <td align = "center"><math>0</math></td>
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| − | <td align = "center"><math> \cos^2x </math></td>
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| − | <td align = "center"><math> \frac{1}{2}</math></td>
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| − | <td align = "center"><math> \frac{1}{2}</math></td>
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| − | </tr>
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| − | <tr>
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| − | <td align = "center"><math>1</math></td>
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| − | <td align = "center"><math> -2\cos x\sin x</math></td>
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| − | <td align = "center"><math> -1 </math></td>
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| − | <td align = "center"><math> -1 </math></td>
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| − | </tr>
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| − | <tr>
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| − | <td align = "center"><math>2</math></td>
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| − | <td align = "center"><math> 2\sin^2x-2\cos^2x </math></td>
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| − | <td align = "center"><math> 0 </math></td>
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| − | <td align = "center"><math> 0 </math></td>
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| − | </tr>
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| − | <tr>
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| − | <td align = "center"><math>3</math></td>
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| − | <td align = "center"><math> 8\sin x\cos x </math></td>
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| − | <td align = "center"><math> 4 </math></td>
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| − | <td align = "center"><math> \frac{2}{3}</math></td>
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| − | </tr>
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| − | <tr>
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| − | <td align = "center"><math>4</math></td>
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| − | <td align = "center"><math> 8\cos^2x-8\sin^2x </math></td>
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| − | <td align = "center"><math> 0 </math></td>
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| − | <td align = "center"><math> 0 </math></td>
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| − | </tr>
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| − | </table>
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| − | |}
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Step 2:
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| − | |-
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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>
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| − | |}
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !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>
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| − | |}
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| | [[009C_Sample_Final_1|'''<u>Return to Sample Exam</u>''']] | | [[009C_Sample_Final_1|'''<u>Return to Sample Exam</u>''']] |
