009B Sample Final 2 - 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''' '''
2017-04-10T16:47:19Z

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..."

New page
'''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 2,_Problem_1|<span class="biglink"><span style="font-size:80%"> Problem 1 </span></span>]] ==
<span class="exam">(a) State '''both parts''' of the Fundamental Theorem of Calculus.

<span class="exam">(b) Evaluate the integral

::<math>\int_0^1 \frac{d}{dx} \bigg(e^{\tan^{-1}(x)}\bigg)dx</math>

<span class="exam">(c) Compute

::<math>\frac{d}{dx}\int_1^{\frac{1}{x}} \sin t~dt</math>

== [[009B_Sample Final 2,_Problem_2|<span class="biglink"><span style="font-size:80%"> Problem 2 </span>]] ==
<span class="exam"> Find the area of the region between the two curves  <math style="vertical-align: -4px">y=3x-x^2</math>  and  <math style="vertical-align: -4px">y=2x^3-x^2-5x.</math>

== [[009B_Sample Final 2,_Problem_3|<span class="biglink"><span style="font-size:80%"> Problem 3 </span>]] ==
<span class="exam">Find the volume of the solid obtained by rotating the region bounded by the curves  <math style="vertical-align: -4px">y=x</math>  and  <math style="vertical-align: -4px">y=x^2</math>  about the line  <math>y=2.</math>

== [[009B_Sample Final 2,_Problem_4|<span class="biglink"><span style="font-size:80%"> Problem 4 </span>]] ==
<span class="exam"> A city bordered on one side by a lake can be approximated by a semicircle of radius 7 miles, whose city center is on the shoreline. As we move away from the center along a radius the population density of the city can be approximated by:

::<math>\rho(x)=25000e^{-0.15x}</math>

<span class="exam">people per square mile. What is the population of the city?

== [[009B_Sample Final 2,_Problem_5|<span class="biglink"><span style="font-size:80%"> Problem 5 </span>]] ==
<span class="exam">(a) Find the area of the surface obtained by rotating the arc of the curve

::<math>y^3=x</math>

<span class="exam">between  <math style="vertical-align: -5px">(0,0)</math>  and  <math style="vertical-align: -5px">(1,1)</math>  about the  <math style="vertical-align: -4px">y</math>-axis.

<span class="exam">(b) Find the length of the arc

::<math>y=1+9x^{\frac{3}{2}}</math>

<span class="exam">between the points  <math style="vertical-align: -5px">(1,10)</math>  and  <math style="vertical-align: -5px">(4,73).</math>

== [[009B_Sample Final 2,_Problem_6|<span class="biglink"><span style="font-size:80%"> Problem 6 </span>]] ==
<span class="exam"> Evaluate the following integrals:

<span class="exam">(a)  <math>\int \frac{dx}{x^2\sqrt{x^2-16}}</math>

<span class="exam">(b)  <math>\int_{-\pi}^\pi \sin^3x\cos^3x~dx</math>

<span class="exam">(c)  <math>\int_0^1 \frac{x-3}{x^2+6x+5}~dx</math>

== [[009B_Sample Final 2,_Problem_7|<span class="biglink"><span style="font-size:80%"> Problem 7 </span>]] ==

<span class="exam">Evaluate the following integrals or show that they are divergent:

<span class="exam">(a)  <math>\int_1^\infty \frac{\ln x}{x^4}~dx</math>

<span class="exam">(b)  <math> \int_0^1 \frac{3\ln x}{\sqrt{x}}~dx</math>