MATH 141, MATH 141E FINAL EXAM, VERSION A SAMPLE B

1. Differentiate y = sin−1(2x).

a)2x

√1− 2x

b)2x ln 2√

1 + 2x

c)1

√1 + 2x

d)2x ln 2√

1− 22x

e) (cos 2x)(2x ln 2)

2. Evaluate the integral

∫ 2

1x lnx dx.

a) 2(ln 2)2

b) 2 ln 2−3

4

c) ln 2−8

3

d) 1

e) 1− ln 2

3. Evaluate the integral

∫ π/2

0(cos3 x)

√sinx dx.

a)4

3

b)1

2

c)8

21

d)25

14

e)20

7

4. Evaluate the integral

∫5x− 3

(x+ 1)(x− 3)dx.

a) ln[(x+ 1)2|x− 3|3

]+ C

b) ln |(x+ 1)(x− 3)|+ C

c) ln

∣∣∣∣ (x+ 1)5

(x− 3)3

∣∣∣∣+ C

d) ln

∣∣∣∣x− 3

x+ 1

∣∣∣∣+ C

e) ln

∣∣∣∣ x+ 1

(x− 3)3

∣∣∣∣+ C

5. Evaluate the integral

∫x+ 1

x2 + 4dx.

a) ln |x|+1

4x+ C

b) tan−1(x

4

)+ C

c)1

2ln(x2 + 4) +

1

2tan−1

(x2

)+ C

d) ln(x2 + 4) +1

4tan−1

(x2

)+ C

e) (x+ 1) ln(x2 + 4) + C

1

MATH 141, MATH 141E FINAL EXAM, VERSION A SAMPLE B

6. Evaluate the integral

∫1

(1 + x2)3/2dx.

a)

√1 + x2

x2+ C

b) x√

1 + x2 + C

c) x2√

1 + x2 + C

d)x

(1 + x2)3/2+ C

e)x

√1 + x2

+ C

7. Evaluate the integral

∫ ∞4

e−√x

√x

dx if it exists.

a) −1

2e2

b) e4

c)1

e4

d)2

e2

e) The integral diverges.

8. Find the limit limx→0

(1 + 2x)1x .

a) 0

b) 1

c) e

d) e2

e) ∞

9. Determine whether the sequence

{sin(n2) + 2en

n2 + 4en

}is convergent or

divergent. If it converges, find the limit.

a) The sequence converges to 0.

b) The sequence converges to1

2.

c) The sequence converges to 1.

d) The sequence converges to3

2.

e) The sequence diverges.

10. Which one of the following statements is TRUE?

a)∞∑n=1

1

n4converges by the Integral Test.

b)

∞∑n=1

1

n4converges by the Ratio Test.

c)∞∑n=1

1

n4converges by the Root Test.

d)∞∑n=1

1

n4converges since lim

n→∞

1

n4= 0.

e)

∞∑n=1

1

n4diverges by the Integral Test.

2

MATH 141, MATH 141E FINAL EXAM, VERSION A SAMPLE B

11. Determine whether the series is convergent or divergent. If conver-

gent, find its sum.∞∑n=1

(4

2n−

5n−1

32n

)

a)41

4

b)15

4

c)15

14

d)31

8

e) The series diverges.

12. Which one of the following statements is FALSE?

a)

∞∑n=0

1

5nis a convergent p-series.

b)

∞∑n=0

1

5nconverges by the Ratio Test.

c)

∞∑n=0

1

5nconverges by the Root Test.

d)∞∑n=0

1

5nconverges by the Integral Test.

e)

∞∑n=0

1

5nis a convergent geometric series.

For Problems13-15, determine whether each series is absolutely con-

vergent, conditionally convergent, or divergent. Code on yourscantron sheet: A if the series is Absolutely Convergent, C if

it is Conditionally Convergent, D if it is Divergent.

13.∞∑

n=10

(−1)nlnn

n

14.

∞∑n=1

(−1)n(lnn)n

n

15.∞∑n=1

(−1)nn

n3 + 25

3

MATH 141, MATH 141E FINAL EXAM, VERSION A SAMPLE B

16. Find the interval of convergence for the power series

∞∑n=1

(−1)n+1 1

n2n(x+

2)n.

a) [−4, 0]

b)

[−

5

2,−

3

2

]c) (−4, 0)

d)

(−

5

2,−

3

2

]e) (−4, 0]

17. Find the radius of convergence, R, for the power series

∞∑n=0

n!4n

(2n+ 1)!(x−

2)n.

a) R =1

2

b) R = 0

c) R = 2

d) R = 4

e) R =∞

18. Find the power series representation of f(x) =x

9 + x2for all x ∈

(−3, 3).

a)∞∑n=0

x2n+1

9n

b)

∞∑n=0

(−1)nx2n+1

9n+1

c)

∞∑n=0

(−1)nx2n

9

d)∞∑n=0

(−1)n9n−1x2n

e)∞∑n=0

9n−1x2n+1

19. For −1 < x < 1, find the sum of the power series

∞∑n=1

(−1)nnxn−1 = −1 + 2x− 3x2 + 4x3 − 5x4 + 6x5 − 7x6 + · · · .

a) −1

1 + x2

b)1

(1− x)2

c)1

1 + x

d) −1

(1 + x)2

e) −1

1− x2

20. Express the indefinite integral

∫x3 sin

(x2

)dx as a MacLaurin

series.

a) C +

∞∑n=0

(−1)nx2n+4

2n(2n+ 4)(2n)!

b) C +

∞∑n=0

(−1)nx2n+5

(2n+ 5)(2n+ 1)!

c) C +∞∑n=0

(−1)nx2n+1

(2n+ 1)(2n+ 1)!

d) C +∞∑n=0

(−1)nx2n+3

(2n+ 3)(2n)!

e) C +

∞∑n=0

(−1)nx2n+5

22n+1(2n+ 5)(2n+ 1)!

4

MATH 141, MATH 141E FINAL EXAM, VERSION A SAMPLE B

21. Find the 3rd-degree Taylor polynomial, T3, of f(x) = e(1+x2) about

a = 0.

a) T3(x) = e+e

2x+

e

4x2 +

e

8x3

b) T3(x) = e+e

2x+

e

8x2 +

e

48x3

c) T3(x) = 1 + x+1

2x2 +

1

6x3

d) T3(x) = e+ ex+e

2x2 +

e

3x3

e) T3(x) = e− ex+e

2x2 −

e

3x3

22. Use the Taylor’s inequality to estimate the accuracy of the approx-

imation f(x) ≈ T3(x) for the function f(x) = e(1+x2) at a = 0 for

−2 ≤ x ≤ 2.

a)e2

6

b)e2

12

c)e2

24

d)e2

48

e)e2

16

23. Find an equation of the tangent line to the parametric curve{x(t) = t2 − sin(2t)

y(t) = 3et/2

at the point when t = 0.

a) y =3

2x+ 3

b) y = −2

3x+ 3

c) y =4

3x+ 3

d) y = −1

3x+ 3

e) y = −3

4x+ 3

24. Find the Cartesian equation for the polar curve r = tan θ sec θ.

a) y2 = x

b) y = x2

c) y = x

d) y = x2 + y2

e) y =1

x2

25. Set up but do not evaluate an integral expression for the area

inside the curve r = 1 + sin θ and outside the curve r =3

2

a)

∫ π2

π6

[(1 + sin θ)2 −

9

4

]dθ

b)

∫ 5π6

π6

(sin θ −

1

2

)dθ

c)

∫ 2π3

π3

[(1 + sin θ)2 −

9

4

]dθ

d)

∫ 5π6

π6

(sin θ +

5

2

)dθ

e) 2

∫ π2

π6

[9

4− (1 + sin θ)2

]dθ

FINAL EXAM- VERSION A

1. D 2. B 3. C 4. A 5. C 6. E 7. D 8. D 9. B 10. A 11. B 12. A

16. E 17. E 18. B 19. D 20. E 21. B 22. C 23. E 24. B 25. A

5