Difference between revisions of "009C Sample Final 2, Problem 5"

From Grad Wiki
Jump to navigation Jump to search
Line 4: Line 4:
 
!Foundations:    
 
!Foundations:    
 
|-
 
|-
−
|
+
|The Taylor polynomial of  &nbsp; <math style="vertical-align: -5px">f(x)</math> &nbsp; at &nbsp; <math style="vertical-align: -1px">a</math> &nbsp; is
−
|-
 
−
|
 
−
|-
 
−
|
 
−
|-
 
−
|
 
 
|-
 
|-
 
|
 
|
 +
&nbsp; &nbsp; &nbsp; &nbsp;<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>
 
|}
 
|}
  
Line 21: Line 16:
 
!Step 1: &nbsp;  
 
!Step 1: &nbsp;  
 
|-
 
|-
−
|
+
|Let &nbsp;<math style="vertical-align: -14px">a=\frac{\pi}{4}.</math>
 
|-
 
|-
−
|
+
|First, we make a table to find the coefficients of the Taylor polynomial.
 
|-
 
|-
 
|
 
|
 +
<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 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>
 
|}
 
|}
  
Line 31: Line 58:
 
!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
|-
 
|-
−
|
+
|Let &nbsp;<math style="vertical-align: -4px">T_n</math>&nbsp; be the Taylor polynomial of order &nbsp;<math>n.</math>
 +
|-
 +
|Since &nbsp; <math style="vertical-align: -14px">c_n=\frac{f^{(n)}(a)}{n!},</math>&nbsp; we have
 +
|-
 +
|&nbsp; &nbsp; &nbsp; &nbsp;<math>T_0=\frac{\sqrt{2}}{2}</math>
 +
|-
 +
|&nbsp; &nbsp; &nbsp; &nbsp;<math>T_1=\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}\bigg(x-\frac{\pi}{4}\bigg)</math>
 +
|-
 +
|&nbsp; &nbsp; &nbsp; &nbsp;<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>
 +
|-
 +
|&nbsp; &nbsp; &nbsp; &nbsp;<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>
 
|-
 
|-
−
|
 
 
|}
 
|}
  
Line 40: Line 76:
 
!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
|-
 
|-
−
|&nbsp;&nbsp;
+
|&nbsp; &nbsp; &nbsp; &nbsp;Let &nbsp;<math style="vertical-align: -4px">T_n</math>&nbsp; be the Taylor polynomial of order &nbsp;<math>n.</math>
 +
|-
 +
|&nbsp; &nbsp; &nbsp; &nbsp;<math>T_0=\frac{\sqrt{2}}{2}</math>
 +
|-
 +
|&nbsp; &nbsp; &nbsp; &nbsp;<math>T_1=\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}\bigg(x-\frac{\pi}{4}\bigg)</math>
 +
|-
 +
|&nbsp; &nbsp; &nbsp; &nbsp;<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>
 +
|-
 +
|&nbsp; &nbsp; &nbsp; &nbsp;<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>''']]
 
[[009C_Sample_Final_2|'''<u>Return to Sample Exam</u>''']]

Revision as of 19:21, 4 March 2017

Find the Taylor Polynomials of order 0, 1, 2, 3 generated by    at  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=\frac{\pi}{4}.}

Foundations:  
The Taylor polynomial of   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)}   at   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a}   is

       Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=0}^{\infty}c_n(x-a)^n} where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_n=\frac{f^{(n)}(a)}{n!}.}


Solution:

Step 1:  
Let  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a=\frac{\pi}{4}.}
First, we make a table to find the coefficients of the Taylor polynomial.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{(n)}(x) } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{(n)}(a) } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{f^{(n)}(a)}{n!} }
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos x } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\sqrt{2}}{2}} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\sqrt{2}}{2}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\sin x} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{\sqrt{2}}{2} } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{\sqrt{2}}{2} }
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\cos x } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{\sqrt{2}}{2} } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{\sqrt{2}}{4} }
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin x } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\sqrt{2}}{2} } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\sqrt{2}}{12}}
Step 2:  
Let  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_n}   be the Taylor polynomial of order  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n.}
Since   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_n=\frac{f^{(n)}(a)}{n!},}   we have
       Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_0=\frac{\sqrt{2}}{2}}
       Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_1=\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}\bigg(x-\frac{\pi}{4}\bigg)}
       Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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}
       Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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.}


Final Answer:  
       Let  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_n}   be the Taylor polynomial of order  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n.}
       Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_0=\frac{\sqrt{2}}{2}}
       Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_1=\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}\bigg(x-\frac{\pi}{4}\bigg)}
       Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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}
       Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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}

Return to Sample Exam

Retrieved from "https://gradwiki.math.ucr.edu/index.php?title=009C_Sample_Final_2,_Problem_5&oldid=3065"