Difference between revisions of "009B Sample Final 2, Problem 1"

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!Step 1:    
 
!Step 1:    
 
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|The Fundamental Theorem of Calculus Part 2 says that
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|-
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|&nbsp; &nbsp; &nbsp; &nbsp; <math>\int_0^1 \frac{d}{dx}(e^{\arctan(x)})~dx=F(1)-F(0)</math>
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|-
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|where &nbsp;<math>F(x)</math>&nbsp; is any antiderivative of &nbsp;<math>\frac{d}{dx}(e^{\arctan(x)}).</math>
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|-
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|Thus, we can take
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|-
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|&nbsp; &nbsp; &nbsp; &nbsp;<math>F(x)=e^{\arctan(x)}</math>
 
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|since then <math>F'(x)=\frac{d}{dx}(e^{\arctan(x)}).</math>
 
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!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
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|-
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|Now, we have
 
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|&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
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\displaystyle{\int_0^1 \frac{d}{dx}(e^{\arctan(x)})~dx} & = & \displaystyle{F(1)-F(0)}\\
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&&\\
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& = & \displaystyle{e^{\arctan(1}-e^{\arctan(0)}}\\
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&&\\
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& = & \displaystyle{e^{\frac{\pi}{4}}-e^0}\\
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&&\\
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& = & \displaystyle{e^{\frac{\pi}{4}}-1.}
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\end{array}</math>
 
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!Step 1: &nbsp;  
 
!Step 1: &nbsp;  
 
|-
 
|-
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|Using the Fundamental Theorem of Calculus Part 1 and the Chain Rule, we have
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|-
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|&nbsp; &nbsp; &nbsp; &nbsp;<math>\frac{d}{dx}\int_1^{\frac{1}{x}} \sin t~dt=\sin\bigg(\frac{1}{x}\bigg)\frac{d}{dx}\bigg(\frac{1}{x}\bigg).</math>
 
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|-
 
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!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
|-
 
|-
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|
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|Hence, we have
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|-
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|&nbsp; &nbsp; &nbsp; &nbsp;<math>\frac{d}{dx}\int_1^{\frac{1}{x}} \sin t~dt=\sin\bigg(\frac{1}{x}\bigg)\bigg(-\frac{1}{x^2}\bigg).</math>
 
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|&nbsp; &nbsp;'''(a)'''&nbsp; &nbsp; See above  
 
|&nbsp; &nbsp;'''(a)'''&nbsp; &nbsp; See above  
 
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|-
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|'''(b)'''  
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|&nbsp; &nbsp;'''(b)'''&nbsp; &nbsp; <math>e^{\frac{\pi}{4}}-1</math>
 
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|-
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|'''(c)'''  
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|&nbsp; &nbsp;'''(c)'''&nbsp; &nbsp; <math>\sin\bigg(\frac{1}{x}\bigg)\bigg(-\frac{1}{x^2}\bigg)</math>
 
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[[009B_Sample_Final_2|'''<u>Return to Sample Exam</u>''']]
 
[[009B_Sample_Final_2|'''<u>Return to Sample Exam</u>''']]

Revision as of 14:32, 4 March 2017

(a) State both parts of the Fundamental Theorem of Calculus.

(b) Evaluate the integral

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 \int_0^1 \frac{d}{dx} \bigg(e^{\tan^{-1}(x)}\bigg)dx}

(c) Compute

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{d}{dx}\int_1^{\frac{1}{x}} \sin t~dt}
Foundations:  

Solution:

(a)

Step 1:  
The Fundamental Theorem of Calculus has two parts.
The Fundamental Theorem of Calculus, Part 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 f}   be continuous on  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,b]}   and 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 F(x)=\int_a^x f(t)~dt.}
       Then,  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}   is a differentiable function on  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,b)}   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'(x)=f(x).}
Step 2:  
The Fundamental Theorem of Calculus, Part 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 f}   be continuous on  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,b]}   and 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 F}   be any antiderivative 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.}
       Then,  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 \int_a^b f(x)~dx=F(b)-F(a).}

(b)

Step 1:  
The Fundamental Theorem of Calculus Part 2 says that
        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 \int_0^1 \frac{d}{dx}(e^{\arctan(x)})~dx=F(1)-F(0)}
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 F(x)}   is any antiderivative 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 \frac{d}{dx}(e^{\arctan(x)}).}
Thus, we can take
       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^{\arctan(x)}}
since then 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)=\frac{d}{dx}(e^{\arctan(x)}).}
Step 2:  
Now, 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 \begin{array}{rcl} \displaystyle{\int_0^1 \frac{d}{dx}(e^{\arctan(x)})~dx} & = & \displaystyle{F(1)-F(0)}\\ &&\\ & = & \displaystyle{e^{\arctan(1}-e^{\arctan(0)}}\\ &&\\ & = & \displaystyle{e^{\frac{\pi}{4}}-e^0}\\ &&\\ & = & \displaystyle{e^{\frac{\pi}{4}}-1.} \end{array}}

(c)

Step 1:  
Using the Fundamental Theorem of Calculus Part 1 and the Chain Rule, 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 \frac{d}{dx}\int_1^{\frac{1}{x}} \sin t~dt=\sin\bigg(\frac{1}{x}\bigg)\frac{d}{dx}\bigg(\frac{1}{x}\bigg).}
Step 2:  
Hence, 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 \frac{d}{dx}\int_1^{\frac{1}{x}} \sin t~dt=\sin\bigg(\frac{1}{x}\bigg)\bigg(-\frac{1}{x^2}\bigg).}
Final Answer:  
   (a)    See above
   (b)    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 e^{\frac{\pi}{4}}-1}
   (c)    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\bigg(\frac{1}{x}\bigg)\bigg(-\frac{1}{x^2}\bigg)}

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