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

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:::::<math>\frac{d}{dx}\bigg(\int_{-1}^{x} \sin(t^2)2tdt\bigg)</math>.
 
:::::<math>\frac{d}{dx}\bigg(\int_{-1}^{x} \sin(t^2)2tdt\bigg)</math>.
  
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<span class="exam">a) Compute <math>f(x)=\int_{-1}^{x} \sin(t^2)2tdt</math>.
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<span class="exam">a) Compute <math>f(x)=\int_{-1}^{x} \sin(t^2)2t~dt</math>.
  
 
<span class="exam">b) Find <math>f'(x)</math>.
 
<span class="exam">b) Find <math>f'(x)</math>.
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<span class="exam">c) State the fundamental theorem of calculus.
 
<span class="exam">c) State the fundamental theorem of calculus.
  
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<span class="exam">d) Use the fundamental theorem of calculus to compute <math>\frac{d}{dx}\bigg(\int_{-1}^{x} \sin(t^2)2tdt\bigg)</math> without first computing the integral.
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<span class="exam">d) Use the fundamental theorem of calculus to compute <math>\frac{d}{dx}\bigg(\int_{-1}^{x} \sin(t^2)2t~dt\bigg)</math> without first computing the integral.
  
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Foundations: &nbsp;  
 
!Foundations: &nbsp;  
 
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|<math>u</math>-substitution
 
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!Step 1: &nbsp;  
 
!Step 1: &nbsp;  
 
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|We proceed using <math>u</math>-substitution. Let <math>u=t^2</math>. Then, <math>du=2tdt</math>.
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|Since this is a definite integral, we need to change the bounds of integration.
 
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|Plugging in our values into the equation <math>u=t^2</math>, we get <math>u_1=(-1)^2=1</math> and <math>u_2=x^2</math>.
 
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!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
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||So, we have
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|<math>f(x)=\int_{-1}^{x} \sin(t^2)2t~dt=\int_{1}^{x^2} \sin(u)du=\left.-\cos(u)\right|_{1}^{x^2}=-\cos(x^2)+\cos(1)</math>.
 
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!Step 1: &nbsp;  
 
!Step 1: &nbsp;  
 
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|From part (a), we have <math>f(x)=-\cos(x^2)+\cos(1)</math>.
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!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
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|If we take the derivative, we get <math>f'(x)=\sin(x^2)2x</math>.
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
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!Step 3: &nbsp;
 
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!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
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|'''(a)'''
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|'''(a)''' <math>f(x)=-\cos(x^2)+\cos(1)</math>
 
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|'''(b)'''  
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|'''(b)''' <math>f'(x)=\sin(x^2)2x</math>
 
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|'''(c)'''
 
|'''(c)'''

Revision as of 14:23, 2 February 2016

We would like to evaluate

.

a) 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 f(x)=\int_{-1}^{x} \sin(t^2)2t~dt} .

b) Find 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)} .

c) State the fundamental theorem of calculus.

d) Use the fundamental theorem of calculus to 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}\bigg(\int_{-1}^{x} \sin(t^2)2t~dt\bigg)} without first computing the integral.

Foundations:  
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 u} -substitution

Solution:

(a)

Step 1:  
We proceed using 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 u} -substitution. 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 u=t^2} . 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 du=2tdt} .
Since this is a definite integral, we need to change the bounds of integration.
Plugging in our values into the equation 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 u=t^2} , we get 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 u_1=(-1)^2=1} 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 u_2=x^2} .
Step 2:  
So, 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)=\int_{-1}^{x} \sin(t^2)2t~dt=\int_{1}^{x^2} \sin(u)du=\left.-\cos(u)\right|_{1}^{x^2}=-\cos(x^2)+\cos(1)} .

(b)

Step 1:  
From part (a), 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)=-\cos(x^2)+\cos(1)} .
Step 2:  
If we take the derivative, we get 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)=\sin(x^2)2x} .

(c)

Step 1:  
Step 2:  

(d)

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(x)=-\cos(x^2)+\cos(1)}
(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 f'(x)=\sin(x^2)2x}
(c)
(d)

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