MATH 1080
Calculus of One Variable II
Test 1
Version A
Spring 2016
Sections 6.1 - 6.5, 7.1 - 7.3
Student’s Printed Name:
CUID:
Instructor:
Section:
Instructions: You are not permitted to use a calculator on any portion of this test. You are not
allowed to use any textbook, notes, cell phone, laptop, PDA, or any technology on either portion
of this test. All devices must be turned off while you are in the testing room.
During this test, any communication with any person (other than the instructor or a designated
proctor) in any form, including written, signed, verbal, or digital, is understood to be a violation
of academic integrity.
No part of this test may be removed from the examination room.
Read each question carefully. In order to receive full credit for the free response portion of the test,
you must:
1.
2.
3.
4.
Show legible and logical (relevant) justification that supports your final answer.
Use complete and correct mathematical notation.
Include proper units, if necessary.
Give exact numerical values whenever possible.
You have 90 minutes to complete the entire test.
On my honor, I have neither given nor received inappropriate or unauthorized information at any time before or during this test.
Student’s Signature:
Do not write below this line.
Free Response
Possible Points
Points Earned
Problem
1.
14
2.
14
3.
14
4.
10
5.
18
Free Response
70
Multiple Choice
30
Test Total
100
Version A - Page 1 of 12
MATH 1080
Calculus of One Variable II
Test 1
Version A
Spring 2016
Sections 6.1 - 6.5, 7.1 - 7.3
This page intentionally left blank.
Version A - Page 2 of 12
MATH 1080
Calculus of One Variable II
Test 1
Version A
Spring 2016
Sections 6.1 - 6.5, 7.1 - 7.3
Multiple Choice: There are 8 multiple choice questions. They do not all have the same
point value. Each question has one correct answer. The multiple choice problems will
count for 30% of the total grade. Use a number 2 pencil and bubble in the letter of
your response on the scantron sheet for problems 1 - 8. For your own record, also
circle your choice on your test since the scantron will not be returned to you. Only the
responses recorded on your scantron sheet will be graded. You are NOT permitted
to use a calculator on any portion of this test.
1. (4 pts.) Suppose f 00 is continuous and f and f 0 have the values given below. Evaluate
Z 3
xf 00 (x) dx.
1
f (x)
f 0 (x)
(a) 9
x=1
2
3
(b) 15
x=3
8
4
(c) 5
Z
2. (4 pts.) Evaluate the integral
x=2
5
1
(d) 3
cos2 (3x) dx.
(a)
1
1
x+
sin(6x) + C
2
12
(c)
1
1
x + sin(3x) + C
2
6
(b)
1
cos3 (3x) + C
9
(d)
1
1
x−
sin(6x) + C
2
12
Version A - Page 3 of 12
MATH 1080
Calculus of One Variable II
Test 1
Version A
Spring 2016
Sections 6.1 - 6.5, 7.1 - 7.3
Z p
3. (4 pts.) When evaluating
9 − 25x2 dx by trigonometric substitution, the expressions
√
9 − 25x2 and dx are replaced by
(a)
(b)
√
√
9 − 25x2 = 3 sin θ and dx =
3
cos θdθ
5
(c)
9 − 25x2 = 3 cos θ and dx =
3
sin θdθ
5
(d)
√
√
9 − 25x2 = 3 sin θ and dx =
3
sin θdθ
5
9 − 25x2 = 3 cos θ and dx =
3
cos θdθ
5
4. (2 pts.) To derive the formula for integration by parts, one would use:
(a) The Mean Value Theorem
(c) The Product Rule
(b) The Chain Rule
(d) None of these
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MATH 1080
Calculus of One Variable II
Test 1
Version A
Spring 2016
Sections 6.1 - 6.5, 7.1 - 7.3
5. (4 pts.) Which of the following integrals gives the volume of the solid whose base is bounded
by the graphs of y = x + 1 and y = x2 − 1 and whose cross sections perpendicular to the base
and parallel to the y-axis are isosceles triangles with height equal to twice the base? See figures
below.
Z
2
(a)
−1
Z
2
(b)
2
1
x + 1 − (x2 − 1) dx
2
2
π x + 1 − (x − 1) dx
2
Z
2
(c)
2
x + 1 − (x2 − 1) dx
−1
Z
2
(d)
2πx x + 1 − (x2 − 1) dx
−1
−1
6. (4 pts.) At what point does the function f (x) = 3x2 − 2 equal its average value on the interval
[0, 2]?
(a) 1
2
(b) √
3
(c) 2
(d)
√
2
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MATH 1080
Calculus of One Variable II
Test 1
Version A
Spring 2016
Sections 6.1 - 6.5, 7.1 - 7.3
7. (4 pts.) A force of 12 pounds is required to stretch a spring 0.5 feet from its natural position.
Assuming Hooke’s Law applies, how much work is done in stretching the spring 3 feet from its
natural position?
(a) 54 ft-lb
(b) 72 ft-lb
(c) 36 ft-lb
(d) 108 ft-lb
8. (4 pts.) Find the area of the region bounded by the curves y = x3 − x2 and y = −x2 + 4x for
−1 ≤ x ≤ 2.
(a)
23
4
(b)
7
4
(c)
9
4
(d)
55
4
Version A - Page 6 of 12
MATH 1080
Calculus of One Variable II
Test 1
Version A
Spring 2016
Sections 6.1 - 6.5, 7.1 - 7.3
Free Response. The Free Response questions will count for 70% of the total grade.
Read each question carefully. To receive full credit, you must show legible, logical, and
relevant justification which supports your final answer. Give answers as exact values.
You are NOT permitted to use a calculator or any other technology on any portion
of this test.
Z
1. (14 pts.) Evaluate the integral:
x2 e−3x dx.
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MATH 1080
Calculus of One Variable II
Test 1
Version A
Z
2. (14 pts.) Evaluate the integral:
√
x2
Spring 2016
Sections 6.1 - 6.5, 7.1 - 7.3
x
dx.
+ 4x + 8
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MATH 1080
Calculus of One Variable II
Test 1
Version A
Z
3. (14 pts.) Evaluate the integral:
0
π/4
Spring 2016
Sections 6.1 - 6.5, 7.1 - 7.3
tan3 x
dx.
sec2 x
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MATH 1080
Calculus of One Variable II
Test 1
Version A
Spring 2016
Sections 6.1 - 6.5, 7.1 - 7.3
4. (10 pts.) A 15-ft chain with a 100-pound load attached to it hangs from a rod that is 15 ft
above the ground. Set up, but do not evaluate or simplify, the integral that represents the
work done in winding up the entire chain (with the load attached) onto the rod if the chain
weighs 3 pounds per foot.
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MATH 1080
Calculus of One Variable II
Test 1
Version A
Spring 2016
Sections 6.1 - 6.5, 7.1 - 7.3
5. (18 pts.) Let R be the region bounded by y = e2x , y = 4, and the y-axis.
(a) Set up, but do not evaluate or simplify, the integral that gives the volume of the solid
obtained by rotating the region R around the y-axis using the disk/washer method.
(b) Set up, but do not evaluate or simplify, the integral that gives the volume of the solid
obtained by rotating the region R around the line y = −1 using the disk/washer method.
(c) Set up, but do not evaluate or simplify, the integral that gives the volume of the solid
obtained by rotating the region R around the line x = 2 using the shell method.
Version A - Page 11 of 12
MATH 1080
Calculus of One Variable II
Test 1
Version A
Spring 2016
Sections 6.1 - 6.5, 7.1 - 7.3
Scantron: Check to make sure your Scantron form meets the following criteria:
My Scantron:
• is bubbled with firm marks so that the form can be machine read;
• is not damaged and has no stray marks (the form can be machine read);
• has 8 bubbled in answers;
• has MATH 1080 and my Section number written at the top;
• has my Instructor’s last name written at the top;
• has Test No. 1 written at the top;
• has the correct test version written at the top and bubbled in below my XID;
• shows my correct XID both written and bubbled in.
**Bubble a zero for the leading C in your XID**.
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