exam _ calculus_ single variable

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)



Question 3

What is the area between the curves and for

?

Question 4

ln(e + 6)

− 1e2

ln(2e − 1)

− 5e2

ln(3e + 2)

+ 5e2

ln(3e − 2)

ln(2e + 1)

y = −2x y =4

x(x + 4)1 ≤ x ≤ e

+ lne2

24

e − 1+ 1e2

4+ 1e2

3

+ lne2

31

1 + e

+ ln− 1e2

34

3 + 2e

+ lne2 54 + e

+ ln+ 1e2

23

2 + e

ln10

3 + e

y = −2



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



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



Question 7

Compute .

Question 8

Which of the following is ?

∫ sin x dxe2x

− (cos x + sin x) + C25

ex

− (2 cos x − sin x) + C15

e2x

(2 sin x − cos x) + C15

e2x

(2 cos x + sin x) + C25

e2x

(cos x − 2 sin x) + C25

e2x

(sin x − 2 cos x) + C25

e2x

− (cos x − 2 sin x) + C25

e2x

− (cos x − sin x) + C15

e2x

dx∫ 2

x=04 − x2− −−−−

√

π

4π

2√3

434π

π

23√

2



Question 9

Which of the following functions is in as ? Select all that

apply.

Question 10

Evaluate the limit

The limit does not exist.

6415

O( )x4 x → +∞

ln( )x3 x2

x4

ex

x5

x5

1000 ln x

( + 1x2 )2 3x−−√

x(x + 1)(x + 2)

cosh x

arctan x

limx→0

ln(cos(2x))

5x2

15

0

−25

−15

−25

45

Time remaining2:59:27



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



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



that apply.

diverges by the -th term test.

converges conditionally by the ratio test.

converges conditionally.


converges absolutely by the ratio test.

diverges by the comparison test with .


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



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



up to and including terms of order ?

Question 18

Which of the following is the interval of convergence of the series

?

f(x) = cos(2 ) dt∫ x

t=0t2

x10

f(x) = − + O( )12

x2 130

x8 x11

f(x) = x − + + O( )12

110

x5 1120

x9 x11

f(x) = x − + + O( )25

x5 227

x9 x11

f(x) = − + + O( )x2 16

x6 1120

x10 x11

f(x) = 1 − + + O( )12

x4 23

x8 x11

f(x) = x − + + O( )13

142

x5 3121

x9 x11

f(x) = − + O( )13

x3 142

x7 x11

f(x) = − + + O( )x2 16

x6 1120

x10 x11

(2x + 1∑n=1

∞n + 13n n2

)n

−2 < x < 1

−2 ≤ x < 1

−2 ≤ x ≤ 1

− ≤ x <32

32

− < x <32

32

− ≤ x <23

23

− ≤ x ≤16

56



In accordance with the Honor Code, I certify that my answers here are my

own work.

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