007B Sample Final 1 - Revision history

Kayla Murray at 06:56, 3 December 2017

2017-12-03T06:56:30Z

Kayla Murray: /* Problem 7 */

2017-12-03T06:55:22Z

Problem 7

Kayla Murray: /* Problem 6 */

2017-12-03T06:50:41Z

Problem 6

Kayla Murray: /* Problem 3 */

2017-12-03T06:49:54Z

Problem 3

Kayla Murray: /* Problem 2 */

2017-12-03T06:45:45Z

Problem 2

Kayla Murray: Created page with "'''This is a sample, and is meant to represent the material usually covered in Math 7B for the final. An actual test may or may not be similar.''' '''Click on the''' '''
2017-12-03T06:45:13Z

Created page with "'''This is a sample, and is meant to represent the material usually covered in Math 7B 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 7B for the final. An actual test may or may not be similar.'''

'''Click on the''' ''' boxed problem numbers  to go to a solution.'''
<div class="noautonum">TOC</div>

== [[007B_Sample Final 3,_Problem_1| Problem 1 ]] ==
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>

== [[007B_Sample Final 3,_Problem_2| Problem 2 ]] ==
 Evaluate the following integrals.

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

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

(c)  <math>\int_1^e \frac{\cos(\ln(x))}{x}~dx</math>

== [[007B_Sample Final 3,_Problem_3| Problem 3 ]] ==
The population density of trout in a stream is

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

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.

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

(b) Find the total number of trout in the stream.

== [[007B_Sample Final 3,_Problem_4| Problem 4 ]] ==
 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>

== [[007B_Sample Final 3,_Problem_5| Problem 5 ]] ==
 Find the following integrals.

(a)  <math>\int x\cos(x)~dx</math>

(b)  <math>\int \sin^3(x)\cos^2(x)~dx</math>

== [[007B_Sample Final 3,_Problem_6| Problem 6 ]] ==
 Find the following integrals

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

(b)  <math>\int \frac{\sqrt{x+1}}{x}~dx</math>

== [[007B_Sample Final 3,_Problem_7| Problem 7 ]] ==

Does the following integral converge or diverge? Prove your answer!

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

@@ Line 4: / Line 4: @@
 <div class="noautonum">__TOC__</div>
-== [[007B_Sample Final 3,_Problem_1|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 1&nbsp;</span></span>]] ==
+== [[007B_Sample Final 1,_Problem_1|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 1&nbsp;</span></span>]] ==
 <span class="exam">Divide the interval &nbsp;<math style="vertical-align: -5px">[-1,1]</math>&nbsp; into four subintervals of equal length &nbsp;<math style="vertical-align: -14px">\frac{1}{2}</math>&nbsp; and compute the left-endpoint Riemann sum of &nbsp;<math style="vertical-align: -5px">y=1-x^2.</math>
-== [[007B_Sample Final 3,_Problem_2|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 2&nbsp;</span>]] ==
+== [[007B_Sample Final 1,_Problem_2|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 2&nbsp;</span>]] ==
 <span class="exam"> Evaluate the following integrals.
@@ Line 16: / Line 16: @@
 <span class="exam">(c) &nbsp;<math>\int_1^e \frac{\cos(\ln(x))}{x}~dx</math>
-== [[007B_Sample Final 3,_Problem_3|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 3&nbsp;</span>]] ==
+== [[007B_Sample Final 1,_Problem_3|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 3&nbsp;</span>]] ==
 <span class="exam">The rate &nbsp;<math style="vertical-align: 0px">r</math>&nbsp; at which people get sick during an epidemic of the flu can be approximated by
@@ Line 29: / Line 29: @@
 <span class="exam">(c) How many people get sick altogether?
-== [[007B_Sample Final 3,_Problem_4|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 4&nbsp;</span>]] ==
+== [[007B_Sample Final 1,_Problem_4|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 4&nbsp;</span>]] ==
 <span class="exam"> Find the volume of the solid obtained by rotating about the &nbsp;<math>x</math>-axis the region bounded by &nbsp;<math style="vertical-align: -4px">y=\sqrt{1-x^2}</math>&nbsp; and &nbsp;<math>y=0.</math>
-== [[007B_Sample Final 3,_Problem_5|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 5&nbsp;</span>]] ==
+== [[007B_Sample Final 1,_Problem_5|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 5&nbsp;</span>]] ==
 <span class="exam"> Find the following integrals.
@@ Line 39: / Line 39: @@
 <span class="exam">(b) &nbsp;<math>\int \sin^3(x)\cos^2(x)~dx</math>
-== [[007B_Sample Final 3,_Problem_6|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 6&nbsp;</span>]] ==
+== [[007B_Sample Final 1,_Problem_6|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 6&nbsp;</span>]] ==
 <span class="exam">Does the following integral converge or diverge? Prove your answer!
@@ Line 45: / Line 45: @@
 ::<math>\int_1^\infty \frac{\sin^2(x)}{x^3}~dx</math>
-== [[007B_Sample Final 3,_Problem_7|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 7&nbsp;</span>]] ==
+== [[007B_Sample Final 1,_Problem_7|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 7&nbsp;</span>]] ==
 <span class="exam">Solve the following differential equations:

@@ Line 47: / Line 47: @@
 == [[007B_Sample Final 3,_Problem_7|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 7&nbsp;</span>]] ==
-<span class="exam">Does the following integral converge or diverge? Prove your answer!
+<span class="exam">Solve the following differential equations:
-::<math>\int_1^\infty \frac{\sin^2(x)}{x^3}~dx</math>
+<span class="exam">(a) &nbsp;<math style="vertical-align: -13px">\frac{dy}{dx}=3y,</math>&nbsp; where &nbsp;<math style="vertical-align: -5px">y_0=2</math>&nbsp; for &nbsp;<math style="vertical-align: -3px">x_0=0</math>

@@ Line 40: / Line 40: @@
 == [[007B_Sample Final 3,_Problem_6|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 6&nbsp;</span>]] ==
-<span class="exam">(a) &nbsp;<math>\int \frac{3x-1}{2x^2-x}~dx</math>
+<span class="exam">Does the following integral converge or diverge? Prove your answer!
-<span class="exam">(b) &nbsp;<math>\int \frac{\sqrt{x+1}}{x}~dx</math>
+::<math>\int_1^\infty \frac{\sin^2(x)}{x^3}~dx</math>
 == [[007B_Sample Final 3,_Problem_7|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 7&nbsp;</span>]] ==

@@ Line 17: / Line 17: @@
 == [[007B_Sample Final 3,_Problem_3|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 3&nbsp;</span>]] ==
-<span class="exam">The population density of trout in a stream is
+<span class="exam">The rate &nbsp;<math style="vertical-align: 0px">r</math>&nbsp; at which people get sick during an epidemic of the flu can be approximated by
-::<math>\rho(x)=|-x^2+6x+16|</math>
+::<math>r=1600te^{-0.2t}</math>
-<span class="exam">where &nbsp;<math style="vertical-align: -5px">\rho</math>&nbsp; is measured in trout per mile and &nbsp;<math style="vertical-align: 0px">x</math>&nbsp; is measured in miles. &nbsp;<math style="vertical-align: 0px">x</math>&nbsp; runs from 0 to 12.
+<span class="exam">where &nbsp;<math style="vertical-align: 0px">r</math>&nbsp; is measured in people/day and &nbsp;<math style="vertical-align: 0px">t</math>&nbsp; is measured in days since the start of the epidemic.
-<span class="exam">(a) Graph &nbsp;<math style="vertical-align: -5px">\rho(x)</math>&nbsp; and find the minimum and maximum.
+<span class="exam">(a) Sketch a graph of &nbsp;<math style="vertical-align: 0px">r</math>&nbsp; as a function of &nbsp;<math style="vertical-align: 0px">t.</math>
-<span class="exam">(b) Find the total number of trout in the stream.
+<span class="exam">(b) When are people getting sick fastest?
 == [[007B_Sample Final 3,_Problem_4|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 4&nbsp;</span>]] ==

@@ Line 12: / Line 12: @@
 <span class="exam">(a) &nbsp;<math>\int_0^{\frac{\sqrt{3}}{4}} \frac{1}{1+16x^2}~dx</math>
-<span class="exam">(b) &nbsp;<math>\int \frac{x^2}{(1+x^3)^2}~dx</math>
+<span class="exam">(b) &nbsp;<math>\int \frac{\sqrt{x+1}}{x}~dx</math>
 <span class="exam">(c) &nbsp;<math>\int_1^e \frac{\cos(\ln(x))}{x}~dx</math>