Difference between revisions of "009B Sample Midterm 2"

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'''This is a sample, and is meant to represent the material usually covered in Math 9B for the midterm. An actual test may or may not be similar. Click on the'''  
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'''This is a sample, and is meant to represent the material usually covered in Math 9B for the midterm. An actual test may or may not be similar.'''  
  
'''<span class="biglink" style="color:darkblue;">&nbsp;boxed problem numbers&nbsp;</span> to go to a solution.'''
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'''Click on the <span class="biglink" style="color:darkblue;">&nbsp;boxed problem numbers&nbsp;</span> to go to a solution.'''
 
<div class="noautonum">__TOC__</div>
 
<div class="noautonum">__TOC__</div>
  
 
== [[009B_Sample Midterm 2,_Problem_1|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 1&nbsp;</span></span>]] ==
 
== [[009B_Sample Midterm 2,_Problem_1|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 1&nbsp;</span></span>]] ==
<span class="exam"> Consider the region <math>S</math> bounded by <math>x=1,x=5,y=\frac{1}{x^2}</math> and the <math>x</math>-axis.
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<span class="exam"> This problem has three parts:
  
::<span class="exam">a) Use four rectangles and a Riemann sum to approximate the area of the region <math>S</math>. Sketch the region <math>S</math> and the rectangles and indicate whether your rectangles overestimate or underestimate the area of <math>S</math>.
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<span class="exam">(a) State the both parts of the fundamental theorem of calculus.
::<span class="exam">b) Find an expression for the area of the region <math>S</math> as a limit. Do not evaluate the limit.
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 +
<span class="exam">(b) Compute &nbsp; <math style="vertical-align: -15px">\frac{d}{dx}\int_0^{\cos (x)}\sin (t)~dt</math>.
 +
 
 +
<span class="exam">(c) Evaluate &nbsp;<math style="vertical-align: -14px">\int_{0}^{\pi/4}\sec^2 x~dx</math>.
  
 
== [[009B_Sample Midterm 2,_Problem_2|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 2&nbsp;</span>]] ==
 
== [[009B_Sample Midterm 2,_Problem_2|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 2&nbsp;</span>]] ==
<span class="exam"> This problem has three parts:
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<span class="exam"> Evaluate
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 +
<span class="exam">(a) &nbsp; <math style="vertical-align: -14px">\int_1^2\bigg(2t+\frac{3}{t^2}\bigg)\bigg(4t^2-\frac{5}{t}\bigg)~dt</math>
  
::<span class="exam">a) State the fundamental theorem of calculus.
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<span class="exam">(b) &nbsp; <math style="vertical-align: -14px">\int_0^2 (x^3+x)\sqrt{x^4+2x^2+4}~dx</math>
::<span class="exam">b) Compute <math>\frac{d}{dx}\int_0^{\cos (x)}\sin (t)~dt</math>
 
::<span class="exam">c) Evaluate <math>\int_{0}^{\frac{\pi}{4}}\sec^2 x~dx</math>
 
  
 
== [[009B_Sample Midterm 2,_Problem_3|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 3&nbsp;</span>]] ==
 
== [[009B_Sample Midterm 2,_Problem_3|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 3&nbsp;</span>]] ==
<span class="exam"> Evaluate
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<span class="exam"> A particle moves along a straight line with velocity given by:
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 +
::<math>v(t)=-32t+200</math>
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 +
<span class="exam">feet per second. Determine the total distance traveled by the particle
  
::<span class="exam">a) <math>\int_1^2\bigg(2t+\frac{3}{t^2}\bigg)\bigg(4t^2-\frac{5}{t}\bigg)~dt</math>  
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<span class="exam">from time &nbsp;<math style="vertical-align: 0px">t=0</math>&nbsp; to time &nbsp;<math style="vertical-align: -1px">t=10.</math>
::<span class="exam">b) <math>\int_0^2 (x^3+x)\sqrt{x^4+2x^2+4}~dx</math>
 
  
 
== [[009B_Sample Midterm 2,_Problem_4|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 4&nbsp;</span>]] ==
 
== [[009B_Sample Midterm 2,_Problem_4|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 4&nbsp;</span>]] ==

Latest revision as of 18:28, 4 November 2017

This is a sample, and is meant to represent the material usually covered in Math 9B for the midterm. An actual test may or may not be similar.

Click on the  boxed problem numbers  to go to a solution.

 Problem 1 

This problem has three parts:

(a) State the both parts of the fundamental theorem of calculus.

(b) 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_0^{\cos (x)}\sin (t)~dt} .

(c) Evaluate  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}^{\pi/4}\sec^2 x~dx} .

 Problem 2 

Evaluate

(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 \int_1^2\bigg(2t+\frac{3}{t^2}\bigg)\bigg(4t^2-\frac{5}{t}\bigg)~dt}

(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 \int_0^2 (x^3+x)\sqrt{x^4+2x^2+4}~dx}

 Problem 3 

A particle moves along a straight line with velocity given by:

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 v(t)=-32t+200}

feet per second. Determine the total distance traveled by the particle

from time  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}   to time  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=10.}

 Problem 4 

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 e^{-2x}\sin (2x)~dx}

 Problem 5 

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 \tan^4 x ~dx}