1730winter-2022-final-exam
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1730Winter 2022 Final Exam
Differential Calculus (University of Windsor)
Studocu is not sponsored or endorsed by any college or university
1730Winter 2022 Final Exam
Differential Calculus (University of Windsor)
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UNIVERSITY OF WINDSOR
DEPARTMENT OF MATHEMATICS AND STATISTICS
Integral Calculus MATH 1730-01 & MATH 1730-02
Final Exam
Monday, April 25, 2022
Last name
(PRINT):
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:
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:
Section No.
:
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:
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This exam has 12 questions. You have 180 minutes (3 hours).
Read carefully and answer
all
questions.
Show all your work to receive full credit
.
Give
exact answers
not the decimal approximations.
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1.
(22)
Evaluate the following integrals.
(a)
integraldisplay
x
5
4
√
9 +
x
3
dx
(b)
integraldisplay
3
x
-
1
(
x
-
2)(
x
2
+ 1)
dx
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(c)
integraldisplay
x
tan
2
xdx
(d)
integraldisplay
1
x
2
√
x
2
-
64
dx
Page 3
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2.
(5)
Evaluate the following integral.
integraldisplay
π/
2
0
cos
2
x
sin
5
x dx
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3.
(6)
Evaluate the following
improper integral
if exists (converges).
integraldisplay
1
0
1
√
x
+
x
√
x
dx
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4.
(5)
Find the number
a
such that the line
x
=
a
bisects the area under the curve
y
=
4
x
2
, where 1
≤
x
≤
4
Note.
The line ‘bisects’ the region if the area of the portion of the region to the left of the line is
equal to the area of the portion of the region to the right of the line.
Page 6
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5.
(6)
The region
R
is bounded by the curves
x
=
y
6
and
x
= 64.
(a) Using cylindrical shell method, set up the integral that represents the volume when the region is
rotated about the line
y
= 2.
(b) Using washer method (also called disk method or cross section method), set up the integral that
represents the volume when the region is rotated about the line
x
= 64.
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6.
(6)
A 20m chain weighs 12kg and hangs from a ceiling. Find the work done in lifting the lower end of the
chain to the ceiling so that it’s level with the upper end.
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7.
(8)
Find the radius of convergence and the interval of convergence of the following power series.
∞
summationdisplay
n
=1
1
3
n
√
n
+ 1
(
x
+ 1)
n
Show your work.
Page 9
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8.
(4)
Determine whether the sequence
a
n
=
2
n
2
+(
-
1)
n
n
3
n
2
+1
converges or diverges. If it converges, find the limit.
Page 10
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9.
(20)
Determine whether the series is convergent or divergent.
Justify your answer with the details, i.e.
indicate the test that you are applying and show that the test works.
(a)
∞
summationdisplay
n
=1
n
2
+
n
5
n
5
+
n
-
3
(b)
∞
summationdisplay
n
=1
(
-
1)
n
e
1
/n
2
(c)
∞
summationdisplay
n
=1
√
2
n
n
!
(d)
∞
summationdisplay
n
=2
1
n
(ln
n
)
5
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Page 12
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10.
(8)
Determine whether the series
∞
summationdisplay
n
=1
(
-
1)
n
sin
parenleftbigg
1
√
n
parenrightbigg
is absolutely convergent, conditionally convergent, or divergent. Show your reasoning.
Page 13
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11.
(6)
(a) Find a power series representation for the function
f
(
x
) =
x
cos(
x
3
).
(b) Evaluate
integraldisplay
x
cos (
x
3
)d
x
as a power series. Show your work.
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12.
(4)
Find the exact value of the sum of the series
∞
summationdisplay
n
=0
(
-
1)
n
(2
n
+ 1)3
n
+
1
2
Page 15
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