MATH 140 FINAL EXAM SAMPLE C

1. Find the critical numbers of f(x) = x− 2 sinx on [0, 2π].

a) x = 0, 2π

b) x = π/12, 7π/12

c) x = π/6, 7π/6

d) x = π/3, 5π/3

e) x = 2π/3, 4π/3

2. If f(x) = 3√x+ 1 + cos(x

3

), find f ′′(0).

a) −1

3

b) −2

9

c) −1

9

d)1

3

e)1

9

3. Assume f is a continuous function on [1, 4] such that f(1) = 10,

f(3) = −2 and f(4) = 2. Which of the following statements mustbe true?

a) f has no zeros in [1, 4].

b) f has exactly one zero in [1, 4].

c) f has at least three zeros in [1, 4].

d) f has at least two zeros in [1, 4].

e) f has exactly two zeros in [1, 4].

4. Evaluate

∫x2

√1− 2x3

dx:

a) −√

1− 2x3

3+ C

b) −√x

3+ C

c) 2x3 − 1 + C

d) −√

1− 2x3

2+ C

e) −6

√1− 2x3

+ C

5. Find limx→1+

x2 − 3x− 4

x2 + 3x− 4.

a) 0

b) 1

c) −1

d) ∞

e) −∞

6. Find the horizontal and vertical asymptotes of f(x) =x+ 2

x2 − 2x− 3.

a) Horizontal asymptote y = 0, vertical asymptote x = 3.

b) Horizontal asymptote y = 1, vertical asymptote x = 3.

c) Horizontal asymptote y = 0, vertical asymptotes x = 3 and

x = −1.

d) No horizontal asymptotes, vertical asymptotes x = 3 and

x = −1.

e) Horizontal asymptote y = 1, vertical asymptote x = −3

7. Find the derivative f ′(x) of f(x) =

∫ x3

0

√t dt:

a)2x√x

3

b) 3x3√x

c)√x

d) x√x

e) x4√x

8. If xy2 = π + y, finddy

dx.

a)y2 − 1

1− 2xy

b)y2 + 1

1 + 2xy

c) 2xy = 1

d)y2

1− 2xy

e)1− 2xy

y2

1

MATH 140 FINAL EXAM SAMPLE C

9. Find

∫ π/2

π/4

cosx

sin2 xdx.

a)

√2

2− 1

b)√

2− 1

c) 1−√

2

2

d)

√2

2

e) 0

10. Evaluate

∫ 1

0x2(3 + x) dx.

a)−1

5

b) 1

c)1

5

d) 0

e)5

4

11. If f(x) =

x2 − 1 x < 1

3 x = 1

3x− 3 1 < x < 3

x2 x ≥ 3

, which one of the following state-

ments is true?

a) f has a removable discontinuity at x = 3.

b) f has a jump discontinuity at x = 1 and a removable dis-

continuity at x = 3.

c) f has a jump discontinuity at both x = 1 and x = 3.

d) f has a removable discontinuity at x = 1 and a jump dis-

continuity at x = 3.

e) f is continuous at x = 1 and x = 3.

12. If f(x) =√

cos(x2) + 1, find f ′(x).

a)√− sin(2x)

b)−x sin(x2)√cos(x2) + 1

c)x sinx√

cos(x2) + 1

d)1

2√−2x sin(x2)

e)

√− sin(2x)√

cos(x2) + 1

13. Evaluate limx→5

√x2 + 3−

√28

x− 5.

a)28

5

b) 0

c)5

2√

7

d) −∞

e) ∞

14. Find the interval(s) on which the function f(x) = x3 − 108x+ 10 isincreasing.

a) (−∞,−18), (18,∞)

b) (−∞, 6)

c) (−6, 6)

d) (−6,∞)

e) (−∞,−6), (6,∞)

15. If the sides of a square are growing at a constant rate of 5 cm/min,

how fast is the area increasing when the sides are 20 cm long?

a) 100 cm2/min

b) 200 cm2/min

c) 300 cm2/min

d) 400 cm2/min

e) 500 cm2/min

2

MATH 140 FINAL EXAM SAMPLE C

16. Find the area of the region enclosed by the curves y = |x| + 1 and

y = x2 − 1.

a)20

3

b) 10

c) 20

d)10

3

e) 6

17. Set up, but do not evaluate an integral for the volume of the solidgenerated by rotating the region bounded by y = cosx, y = 1 about

y = 4. 0 ≤ x ≤ 2π.

a) π

∫ 2π

0[(4 + cosx)2 − 9] dx

b) π

∫ 2π

0[(5− cos2 x)] dx

c) π

∫ 2π

0[(4− cosx)2 − 9] dx

d) 2π

∫ 2π

0[(4− x) cosx] dx

e) 2π

∫ 2π

0[(x− 4) cosx] dx

Questions 18 through 21 are true/false type. On your scantron

mark A for true, B for false. Each true/false question is worth3 points.

For questions 18 – 21, assume f(x) is a continuous function on[a, b].

18. The derivative f ′(x) must exist on (a, b).

a) True

b) False

19. If F (x) =

∫ x

af(t)dt, then F ′(x) is continuous on (a, b).

a) True

b) False

20. The function f has an absolute maximum on [a, b].

a) True

b) False

21. There is a point c in the interval (a, b) such that f ′(c) =f(b)− f(a)

b− a.

a) True

b) False

3

MATH 140 FINAL EXAM SAMPLE C

22. (12 pts total) A region R in the first quadrant is bounded by

y = 8− x2, y = 2x, and the y-axis.

a) (4 pts) Sketch the region R. You must find and label any points

of intersection.

b) (4pts) Write an integral expression for the volume when R isrevolved around the x-axis. (Do not evaluate the integral! Just set

it up.)

c) (4pts) Write an integral expression for the volume when R isrevolved around the y-axis. (Do not evaluate the integral! Just set

it up.)

23. (12 pts. total) Maximize the area of a rectangle inscribed in a

circle of radius 5.

a) (2 pts)Write the total area as a function.

b) (4 pts) Write the total area as a function of one variable.

c) (4 pts) Find the dimensions of the rectangle of maximum area

described above.

d) (2 pts) Justify that your answer is a maximum.

4

MATH 140 FINAL EXAM SAMPLE C

24. (12 pts.) A light on the ground shines on a wall 20 meters away.

If a man 2 meters tall walks from the light toward the wall at a rate

of 2.5 m/s, how fast is the length of his shadow decreasing when he’s10 meters away from the wall?

FINAL EXAM- FORM A

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

13. C 14. E 15. B 16. A 17. C 18. B 19. A 20. A 21. B

22a. intersection points: (0,0), (0,8), (2,4)

22b. V =

∫ 2

0π[(8− x2)2 − (2x)2] dx

22c. V =

∫ 2

02πx(8− x2 − 2x) dx

23a. A = xy

23b. A(x) = x√

100− x2

23c. x = y = 5√

2

23d. EVT works on Domain [0, 10]; or 1st der test; or 2nd der test

24. 1 m/s

5