Difference between revisions of "009C Sample Final 3, Problem 5"

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!Step 1:    
 
!Step 1:    
 
|-
 
|-
−
|
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|We have
 +
|-
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|&nbsp; &nbsp; &nbsp; &nbsp;<math>f'(x)=\bigg(\frac{-1}{3}\bigg)e^{-\frac{1}{3}x},</math>
 +
|-
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|&nbsp; &nbsp; &nbsp; &nbsp;<math>f''(x)=\bigg(\frac{-1}{3}\bigg)^2 e^{-\frac{1}{3}x},</math>
 +
|-
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|and
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|-
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|&nbsp; &nbsp; &nbsp; &nbsp;<math>f^{(3)}(x)=\bigg(\frac{-1}{3}\bigg)^3e^{-\frac{1}{3}x}.</math>
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|-
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|If we compare these three equations, we notice a pattern.
 
|-
 
|-
−
|
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|We have
 
|-
 
|-
−
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|&nbsp; &nbsp; &nbsp; &nbsp;<math>f^{(n)}(x)=\bigg(\frac{-1}{3}\bigg)^3e^{-\frac{1}{3}x}.</math>
 
|}
 
|}
  
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!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
|-
 
|-
−
|
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|Since
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|-
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|&nbsp; &nbsp; &nbsp; &nbsp;<math>f'(x)=\bigg(\frac{-1}{3}\bigg)e^{-\frac{1}{3}x},</math>
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|-
 +
|we have
 
|-
 
|-
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|&nbsp; &nbsp; &nbsp; &nbsp;<math>f'(3)=\bigg(\frac{-1}{3}\bigg)e^{-1}.</math>
 
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!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
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|-
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|&nbsp;&nbsp; '''(a)'''  
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|&nbsp; &nbsp; '''(a)'''&nbsp; &nbsp; <math>f^{(n)}(x)=\bigg(\frac{-1}{3}\bigg)^3e^{-\frac{1}{3}x},~f'(3)=\bigg(\frac{-1}{3}\bigg)e^{-1}</math>
 
|-
 
|-
 
|&nbsp;&nbsp; '''(b)'''  
 
|&nbsp;&nbsp; '''(b)'''  
 
|}
 
|}
 
[[009C_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]
 
[[009C_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]

Revision as of 13:20, 5 March 2017

Consider the function

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)=e^{-\frac{1}{3}x}}

(a) Find a formula for the  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} th derivative  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)}   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}   and then find  

(b) Find the Taylor series for  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 x_0=3,}   i.e. write  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)}   in the form

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)=\sum_{n=0}^\infty a_n(x-3)^n.}
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:

(a)

Step 1:  
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 f'(x)=\bigg(\frac{-1}{3}\bigg)e^{-\frac{1}{3}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''(x)=\bigg(\frac{-1}{3}\bigg)^2 e^{-\frac{1}{3}x},}
and
       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^{(3)}(x)=\bigg(\frac{-1}{3}\bigg)^3e^{-\frac{1}{3}x}.}
If we compare these three equations, we notice a pattern.
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 f^{(n)}(x)=\bigg(\frac{-1}{3}\bigg)^3e^{-\frac{1}{3}x}.}
Step 2:  
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 f'(x)=\bigg(\frac{-1}{3}\bigg)e^{-\frac{1}{3}x},}
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 f'(3)=\bigg(\frac{-1}{3}\bigg)e^{-1}.}

(b)

Step 1:  
Step 2:  


Final Answer:  
    (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 f^{(n)}(x)=\bigg(\frac{-1}{3}\bigg)^3e^{-\frac{1}{3}x},~f'(3)=\bigg(\frac{-1}{3}\bigg)e^{-1}}
   (b)

Return to Sample Exam

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