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08 04 2013 Exam | Calculus: Single Variable
https://class.coursera.org/calcsing-2012-001/quiz/attempt?quiz_id=502 1/12
Final ExamThe due date for this exam is Mon 15 Apr 2013 6:59 AM EEST +0300.
Question 1
What is the centroid of the region bounded by the curves and
?
Hint: draw a picture of this region as your first step.
Question 2
If satisfies the differential equation
and , then what is ?
y = x2
y = 3 − x2
( , ) = (0, 0)x y
( , ) = (0, )x y85
( , ) = ( , )x y76
32
( , ) = (− , )x y32
−−√ 3
2
−−√
( , ) = (0, )x y3√
2
( , ) = (0, 2)x y
( , ) = (0, )x y32
( , ) = (0, 3 )x y 6√
x(t)
= 2dx
dtet−x
x(0) = 0 x(1)
ln(e + 6)
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08 04 2013 Exam | Calculus: Single Variable
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Compute the arc-length of the graph of for .
Hint: your expression for the arclength element should admit a "miracle"
factorization that eliminates a certain square root.
Question 5
What is the volume of the solid generated by rotating about the -axis the
region defined by the inequalities:
1. , and
2. .
y = −x2
4ln x
21 ≤ x ≤ e2
dL
+ 11e6
12− 3e4
2− 11e4
12− 7e2
4+ 7e4
4+ 3e4
4e − 1
2+ 1e2
2
y
y ≥ 2x3
y ≤ 2x
2π
5π
332π
105224π
15
4π
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08 04 2013 Exam | Calculus: Single Variable
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Question 6
You approximate using the series:
If you use the first three terms as your approximation — that is, using terms up
to and including the term — then what is the bound on your error that
comes from observing that the series above is alternating?
4π
1516π
218π
15π
6
ln43
ln(1 + x) = (−1∑n=1
∞
)n+1 xn
n
x3 E
E <181
E <1
124
E <1
1215
E <1
243
E <1
120
E <124
E <1
108
E <1
324
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08 04 2013 Exam | Calculus: Single Variable
https://class.coursera.org/calcsing-2012-001/quiz/attempt?quiz_id=502 7/12
Question 11
Compute the expectation of the probability distribution function
where is the appropriate constant.
Question 12
Exactly four (4) of the following integrals converge. Which are they?
Note: you do not need to solve the integrals to complete this problem!
−45
E
ρ(x) = κ(x − 1 , 1 ≤ x ≤ 2)3/2
κ
E =107
E =2512
E =97
E =125
E =76
E =127
E =54
E =98
cos x dx∫ +∞
x=0
dx∫ +∞
x=1
3x + 2+ 2x + 1x2
dx+∞
−x
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08 04 2013 Exam | Calculus: Single Variable
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Question 13
Which of the following sequences is the second forward difference, , of the
sequence
Question 14
Which of the following statements about series convergence are true? Select all
dx∫ +∞
x=0e−x
(1 − cos ) dx∫ +∞
x=1
1x
∫ 1
x=−1
dx
csc x
∫ +∞
x=1
x dx
+ 1x3
ln x dx∫ 1
x=0
arctan x dx∫ +∞
x=0
aΔ2
a = (12, −1, 7, 8, 4, 1, 2, 0, 0, 5, 2, 3, 5, −1, 0, 0, 0, 0, …)
a = (21, −7, −5, 1, 4, −3, 2, 5, −8, 4, 1, −8, 7, −1, 0, 0, …)Δ2
a = (13, −8, −1, 4, 3, −1, 2, 0, −5, 3, −1, −2, 6, −1, 0, 0, …)Δ2
a = (−13, 8, 1, −4, −3, 1, −2, 0, 5, −3, 1, 2, −6, 1, 0, 0, …)Δ2
a = (−13, −8, 1, 4, 3, 0, 2, 5, 3, −1, 6, 2, −7, 1, 0, 0, …)Δ2
a = (21, −7, −5, 1, 4, −3, 2, 5, −8, 4, 1, −8, 7, −1, 0, 0, …)Δ2
a = (21, 7, 3, −1, 4, 3, −1, 5, 8, −4, 3, −9, 17, 1, 0, 0, …)Δ2
a = (8, 0, 2, 1, −5, −2, 3, 12, 0, 5, −17, 2, −5, 12, 0, 0, …)Δ2
a = (−8, 1, 4, −8, 5, 2, −3, 4, 1, −5, −7, 21, −25, 12, 0, 0, …)Δ2
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08 04 2013 Exam | Calculus: Single Variable
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that apply.
diverges by the -th term test.
converges conditionally by the ratio test.
converges conditionally.
converges conditionally.
converges absolutely by the ratio test.
diverges by the comparison test with .
converges conditionally.
diverges by the -th term test.
Question 15
Using your knowledge of Taylor series, evaluate the following infinite series:
∑n=5
∞ ln n
ln(ln n)n
∑n=1
∞n ⋅ 3n
(2n)!
(−1∑n=0
∞
)n
(−1∑n=0
∞
)n n
+ 1n2
∑n=1
∞n ⋅ 3n
(2n)!
∑n=5
∞ ln n
ln(ln n)∑n=5
∞ 1n
(−1∑n=3
∞
)n ln n
n3/4
∑n=3
∞ ln n
n3/4n
1 − + − ⋯ + (−1 + ⋯π2
9 ⋅ 2!π4
81 ⋅ 4!)n π2n
⋅ (2n)!32n
sinhπ
31
1 − π/3
coshπ
3
sinπ
3
cosπ
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08 04 2013 Exam | Calculus: Single Variable
https://class.coursera.org/calcsing-2012-001/quiz/attempt?quiz_id=502 10/12
This series does not converge.
Question 16
If and are related by the equation
find a formula for in terms of and .
Question 17
Which of the following is the Taylor expansion about of
cosπ
3
eπ/3
31 − π
x y
+ 5 y + = −2xy + 7,x3 x2 y 3
dy
dxx y
=dy
dx
3x2
5 + 3 + 2xx2 y 2
=dy
dx
3 − 10xy − 2yx2
5 − 3 − 2xx2 y 2
= −dy
dx
3 + 10xy + 2yx2
5 + 3 + 2xx2 y 2
= −dy
dx
3 + 2yx2
3 + 2xy 2
= −3 + 10xy + 2ydy
dxx2
=dy
dx
−1
5 + 3 + 2xx2 y 2
=dy
dx
3 − 10xy + 2yx2
5 − 3 + 2xx2 y 2
= −dy
dx
3 + 10xyx2
5 + 3x2 y 2
x = 0
f(x) = cos(2 ) dt∫ x2