009B Sample Final 3 - Revision history

MathAdmin: Created page with "'''This is a sample, and is meant to represent the material usually covered in Math 9B for the final. An actual test may or may not be similar.''' '''Click on the''' '''
2017-04-10T16:51:48Z

Created page with "'''This is a sample, and is meant to represent the material usually covered in Math 9B for the final. An actual test may or may not be similar.''' '''Click on the''' '''<span..."

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

'''Click on the''' '''<span class="biglink" style="color:darkblue;"> boxed problem numbers </span> to go to a solution.'''
<div class="noautonum">TOC</div>

== [[009B_Sample Final 3,_Problem_1|<span class="biglink"><span style="font-size:80%"> Problem 1 </span></span>]] ==
<span class="exam">Divide the interval  <math style="vertical-align: -5px">[-1,1]</math>  into four subintervals of equal length  <math style="vertical-align: -14px">\frac{1}{2}</math>  and compute the left-endpoint Riemann sum of  <math style="vertical-align: -5px">y=1-x^2.</math>

== [[009B_Sample Final 3,_Problem_2|<span class="biglink"><span style="font-size:80%"> Problem 2 </span>]] ==
<span class="exam"> Evaluate the following integrals.

<span class="exam">(a)  <math>\int_0^{\frac{\sqrt{3}}{4}} \frac{1}{1+16x^2}~dx</math>

<span class="exam">(b)  <math>\int \frac{x^2}{(1+x^3)^2}~dx</math>

<span class="exam">(c)  <math>\int_1^e \frac{\cos(\ln(x))}{x}~dx</math>

== [[009B_Sample Final 3,_Problem_3|<span class="biglink"><span style="font-size:80%"> Problem 3 </span>]] ==
<span class="exam">The population density of trout in a stream is

::<math>\rho(x)=|-x^2+6x+16|</math>

<span class="exam">where  <math style="vertical-align: -5px">\rho</math>  is measured in trout per mile and  <math style="vertical-align: 0px">x</math>  is measured in miles.  <math style="vertical-align: 0px">x</math>  runs from 0 to 12.

<span class="exam">(a) Graph  <math style="vertical-align: -5px">\rho(x)</math>  and find the minimum and maximum.

<span class="exam">(b) Find the total number of trout in the stream.

== [[009B_Sample Final 3,_Problem_4|<span class="biglink"><span style="font-size:80%"> Problem 4 </span>]] ==
<span class="exam"> Find the volume of the solid obtained by rotating about the  <math>x</math>-axis the region bounded by  <math style="vertical-align: -4px">y=\sqrt{1-x^2}</math>  and  <math>y=0.</math>

== [[009B_Sample Final 3,_Problem_5|<span class="biglink"><span style="font-size:80%"> Problem 5 </span>]] ==
<span class="exam"> Find the following integrals.

<span class="exam">(a)  <math>\int x\cos(x)~dx</math>

<span class="exam">(b)  <math>\int \sin^3(x)\cos^2(x)~dx</math>

== [[009B_Sample Final 3,_Problem_6|<span class="biglink"><span style="font-size:80%"> Problem 6 </span>]] ==
<span class="exam"> Find the following integrals

<span class="exam">(a)  <math>\int \frac{3x-1}{2x^2-x}~dx</math>

<span class="exam">(b)  <math>\int \frac{\sqrt{x+1}}{x}~dx</math>

== [[009B_Sample Final 3,_Problem_7|<span class="biglink"><span style="font-size:80%"> Problem 7 </span>]] ==

<span class="exam">Does the following integral converge or diverge? Prove your answer!

::<math>\int_1^\infty \frac{\sin^2(x)}{x^3}~dx</math>