If h(x) = ln[x cos x] then

29. If h(x) = ln[x cos x] then

Final Exam 319

30. 3 If g(x) = [x 53 + 4x] then (a)

31. Suppose that a steel ball is dropped from the top of a tall building. It takes the ball 7 seconds to hit the ground. How tall is the building?

(a) 824 feet (b) 720 feet (c) 550 feet (d) 652 feet (e) 784 feet

32. The position in feet of a moving vehicle is given by 8t 2 − 6t + 142. What is the acceleration of the vehicle at time t = 5 seconds?

(a)

12 ft/sec 2 (b)

8 ft/sec 2 (c)

−10 ft/sec 2 (d)

20 ft/sec 2 (e)

16 ft/sec 2

33. 3 Let f (x) = x 2 − 5x + 3x − 6. Then the graph of f is (a) concave up on (−3, ∞) and concave down on (−∞, −3)

(b) concave up on (5, ∞) and concave down on (−∞, 5) (c) concave up on (5/3, ∞) and concave down on (−∞, 5/3) (d) concave up on (3/5, ∞) and concave down on (−∞, 3/5) (e) concave up on (−∞, 5/3) and concave down on (5/3, ∞)

320 Final Exam

34. 3 Let g(x) = x 2 + x − 10x + 2. Then the graph of f is

(a) increasing on (−∞, −10/3) and decreasing on (−10/3, ∞) (b) increasing on (−∞, 1) and (10, ∞) and decreasing on (1, 10) (c) increasing on (−∞, −10/3) and (1, ∞) and decreasing on

( −10/3, 1) (d) increasing on (−10/3, ∞) and decreasing on (−∞, −10/3) (e) increasing on (−∞, −10) and (1, ∞) and decreasing on

3 Find all local maxima and minima of the function h(x) = −(4/3)x 2 +5x − 4x + 8.

(a) local minimum at x = 1/2, local maximum at x = 2 (b) local minimum at x = 1/2, local maximum at x = 1 (c) local minimum at x = −1, local maximum at x = 2 (d) local minimum at x = 1, local maximum at x = 3 (e) local minimum at x = 1/2, local maximum at x = 1/4

36. Find all local and global maxima and minima of the function h(x) = x +

2 sin x on the interval [0, 2π]. (a) local minimum at 4π/3, local maximum at 2π/3, global minimum

at 0, global maximum at 2π (b) local minimum at 2π/3, local maximum at 4π/3, global minimum

at 0, global maximum at 2π (c) local minimum at 2π , local maximum at 0, global minimum at 4π/3,

global maximum at 2π/3 (d) local minimum at 2π/3, local maximum at 2π , global minimum at

4π/3, global maximum at 0 (e) local minimum at 0, local maximum at 2π/3, global minimum at

4π/3, global maximum at 2π

Find all local and global maxima and minima of the function f (x) = x 3 + x 2 − x + 1. (a) local minimum at −1, local maximum at 1/3

(b) local minimum at 1, local maximum at −1/3 (c) local minimum at 1, local maximum at −1 (d) local minimum at 1/3, local maximum at −1 (e) local minimum at −1, local maximum at 1

38. A cylindrical tank is to be constructed to hold 100 cubic feet of liquid. The sides of the tank will be constructed of material costing $1 per

Final Exam 321

square foot, and the circular top and bottom of material costing $2 per square foot. What dimensions will result in the most economical tank?

3 height = 3 25/π , radius = 4 · 25/π (c)

height = 5 1/3 , radius = π (d) height = 4, radius = 1

(e)

height = 4 · 25/π , radius = 25/π

39. A pigpen is to be made in the shape of a rectangle. It is to hold 100 square feet. The fence for the north and south sides costs $8 per running foot, and the fence for the east and west sides costs $10 per running foot. What shape will result in the most economical pen?

(a) north/south = 4

5, east/west = 5 5

(b) north/south = 5

5, east/west = 4 5

(c) north/south = 4

4, east/west = 5 4

(d) north/south = 5

4, east/west = 4 4

(e) north/south =

5, east/west = 4

40. A spherical balloon is losing air at the rate of 2 cubic inches per minute. When the radius is 12 inches, at what rate is the radius changing?

(a) 1/[288π] in./min (b) −1 in./min (c) −2 in./min (d) −1/[144π] in./min (e) −1/[288π] in./min

41. Under heat, a rectangular plate is changing shape. The length is increasing by 0.5 inches per minute and the width is decreasing by

1.5 inches per minute. How is the area changing when ℓ = 10 and w = 5?

(a) The area is decreasing by 9.5 inches per minute. (b) The area is increasing by 13.5 inches per minute. (c) The area is decreasing by 10.5 inches per minute. (d) The area is increasing by 8.5 inches per minute. (e) The area is decreasing by 12.5 inches per minute.

322 Final Exam

42. An arrow is shot straight up into the air with initial velocity 50 ft/sec. After how long will it hit the ground?

(a)

12 seconds (b) 25/8 seconds (c) 25/4 seconds (d) 8/25 seconds (e)

8 seconds

43. The set of antiderivates of x 2 − cos x + 4x is

4 − sin x + x +C (d) x 2 +x+1+C

44. The indefinite integral

+ x dx equals

45. The indefinite integral 2x cos x 2 dx equals

(a)

[cos x] 2 +C

(b) cos x 2 +C (c) sin x 2 +C

(d)

[sin x] 2 +C (e) sin x · cos x

46. 4 The area between the curve y = −x 2 + 3x + 4 and the x-axis is (a) 20

(b) 18

Final Exam 323

47. The area between the curve y = sin 2x + 1/2 and the x-axis for 0 ≤ x ≤ 2π is

48. 3 The area between the curve y = x 2 − 9x + 26x − 24 and the x-axis is (a) 3/4

(b) 2/5 (c) 2/3 (d) 1/2 (e) 1/3

49. 2 The area between the curves y = x 2 + x + 1 and y = −x − x + 13 is 122

50. The area between the curves y = x 2 − x and y = 2x + 4 is 117

(a)

(b)

324 Final Exam

1 f (x) dx = 7 and 3 f (x) dx = 2 then 1 f (x) dx = (a) 4

(b) 5 (c) 6 (d) 7 (e) 3

x 52. 2 If F (x) =

x ln t dt then F ′ (x) = (a) ( 4x − 1) · ln x

(b) x 2 −x (c) ln x 2 − ln x

(d) ln(x 2 − x)

(e)

cos 2x − 1

53. Using l’Hôpital’s Rule, the limit lim

equals (a) 1

x →0

(b) 0 (c) −4 (d) −2 (e) 4

54. Using l’Hôpital’s Rule, the limit lim

55. The limit lim

x x equals

x →0

(a) 1

Final Exam 325

(b) −1 (c) 0 (d) +∞ (e) 2

56. The limit lim

57. The improper integral

dx equals

58. The improper integral

dx equals

59. The area under the curve y = x −4 , above the x-axis, and from 3 to +∞, is

2 (a)

1 (b)

326 Final Exam

60. The value of log 2 ( 1/16) − log 3 ( 1/27) is

(a) 2 (b) 3 (c) 4 (d) 1 (e) −1

log 2 27

61. The value of

62. The graph of y = ln[1/x 2 (a)

(b) (c) concave up for x < 0 and concave down for x > 0 (d) concave down for x < 0 and concave up for x > 0 (e) never concave up nor concave down

63. The graph of y = e −1/x 2 , |x| > 2, is

(a) concave up (b) concave down (c) concave up for x < 0 and concave down for x > 0 (d) concave down for x < 0 and concave up for x > 0 (e) never concave up nor concave down

64. The derivative

log 3 ( cos x) equals

dx sin x cos x

(a)

ln 3

ln 3 · sin x − cos x

(b)

Final Exam 327

65. The derivative

3 x ln x equals

66. The value of the limit lim h →0 2 ( 1+h ) 1/ h 2 is

(a) e (b) e −1 (c) 1/e (d) e 2 (e) 1

x 2 ln x

67. Using logarithmic differentiation, the value of the derivative

68. The derivative of f (x) = Sin −1 (x · ln x) is (a)

1 + ln x x 2 · ln 2 x

1 (b)

1−x 2 2 · ln x

328 Final Exam

ln x

(c)

2 2 1−x · ln x (d) 1 + ln x

1−x 2 2 · ln x

1 + ln x

(e) √ 1−x 2