MATH 141 EXAM I, SAMPLE D SAMPLE EXAM

1. Which of the following functions is one-to-one on the given domain?

a) f(x) = sinx on (0, π)

b) f(x) = secx on (−π

2,π

2)

c) f(x) =1

xon [−1, 0) ∪ (0, 1]

d) f(x) =1

x2on [−1, 0) ∪ (0, 1]

e) f(x) = |x| on (−∞,∞)

2. Let f(x) = x3 + x+ 3. Find (f−1)′(5).

a) 76

b)1

76

c)1

8

d)1

4

e) 4

3. Find the slope of tangent line to the curve y =lnx

1 + lnxat the point(

e,1

2

).

a)1

2

b)1

2e

c)1

4

d)1

4e

e) − ln1

2

4. Find the derivative of y.

y = sin−1(e−x

)

a) cos−1(e−x

)b) ex cos ex

c)e−x

√1− e−x

d)−e−x

√1− e−2x

e)e−x√1− x2

1

MATH 141 EXAM I, SAMPLE D SAMPLE EXAM

5. Evaluate the limit.

limx→1+

e2

1−x

a) e2

b) ∞

c) 0

d) −∞

e) e

6. Find the derivative of y.

y = xtan−1 x

a) y′ = xtan−1 x

(lnx

x2 + 1+

tan−1 x

x

)

b) y′ = xtan−1 x

(x

x2 + 1+ tan−1 x

)

c) y′ = xtan−1 x

(x

x2 + 1

)

d) y′ = xtan−1 x

(1

x2 + 1

)e) y′ = (tan−1 x)x(tan

−1 x−1)

7. Evaluate the integral. ∫ 16

2

dx

2x√lnx

a) ln 2

b) 2 ln 2

c) 2e

d) 4−√2

e)√ln 2

8. Evaluate the integral. ∫ ln 9

0eθ√eθ − 1 dθ

a) e2

b) 32√2

c)32

3

√2

d) 16√2

e) 9

2

MATH 141 EXAM I, SAMPLE D SAMPLE EXAM

9. Evaluate the integral. ∫ π4

0sin3 2θ cos2 2θ dθ

a)1

15

b)−112

c)1

3

d)2

5

e)−215

10. Evaluate the integral.∫dx

√9x2 − 4

(Note : x >2

3)

a)1

3ln∣∣∣3x+

√9x2 − 4

∣∣∣+ C

b)3

2ln∣∣∣sin−1 3x+

√9x2 − 4

∣∣∣+ C

c)

∣∣∣ln√9x2 − 4∣∣∣

18x+ C

d) sec−1(3x) + C

e)1

2tan−1(3x)− ln(9x2 − 4) + C

11. Find the area of the region bounded by the curves y = lnx, y = 0,and x = e.

a) e− 1

b) 1

c) e2 + 1

d) e2

e) e2 − 1

12. Evaluate the integral. ∫5x+ 2

x2 + xdx

a) ln∣∣∣x5 (x2 + x

)2∣∣∣+ C

b) ln∣∣∣x2 (x+ 1)3

∣∣∣+ C

c) ln∣∣∣x3 (x+ 1)2

∣∣∣+ C

d) ln∣∣∣x (x2 + 1

)2∣∣∣+ C

e) ln |x (x+ 1)|+ C

3

MATH 141 EXAM I, SAMPLE D SAMPLE EXAM

For problems 13 to 18 (each worth 2 points), determine if the state-

ment is True or False. Be sure to code your scantron sheet asfollows:

A↔ if the statement is True,

B↔ if the statement is False.

13.d

dx(5x) = x5x−1

14.d

dx(ln 5) =

1

5

15.d

dx(aloga x) = 1

16.

∫4

x3 + 3dx = 4 ln

∣∣x3 + 1∣∣+ C

17. sin−1(sin 2π) = 2π

18. sin(cos−1 x

2

)=√

4− x2

For questions 19 to 21 show your work for full credit. An answer

with no work shown will receive no credit.

19. (8 pts.) Evaluate. ∫tan−1 x dx

4

MATH 141 EXAM I, SAMPLE D SAMPLE EXAM

20. (10 pts.) Evaluate. ∫ √16 + x2

x4dx

21. (10 pts.) Evaluate.∫3x2 − 3x+ 5

(x− 1)(x2 + 4)dx

EXAM I- Key

1. C, 2. D, 3. D, 4. D, 5. C, 6. A, 7. E, 8. C, 9. A, 10.A, 11. B, 12. B, 13. B, 14. B, 15. A, 16. B, 17. B, 18. B

19. x tan−1 x −1

2ln |1 + x2| + C, 20.

−(16 + x2)1/2

48x3 + C, 21.

ln |x− 1|+ ln |x2 + 4| −1

2tan−1 1

2+ C

5