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(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 +
37.
x 2 ’ x + 1.
local minimum at ’1, local maximum at 1/3
(a)
local minimum at 1, local maximum at ’1/3
(b)
local minimum at 1, local maximum at ’1
(c)
local minimum at 1/3, local maximum at ’1
(d)
local minimum at ’1, local maximum at 1
(e)
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
321
Final Exam

square foot, and the circular top and bottom of material costing $2
per square foot. What dimensions will result in the most economical
tank?
√ √
(a) height = 4 · 3 π/25, radius = 3 π/25
√ √
(b) height = 3 25/π , radius = 4 · 3 25/π
(c) height = 51/3 , radius = π 1/3
(d) height = 4, radius = 1
√ √
(e) height = 4 · 3 25/π , radius = 3 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
’1 in./min
(b)
’2 in./min
(c)
’1/[144π] in./min
(d)
’1/[288π] in./min
(e)
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
= 10 and
1.5 inches per minute. How is the area changing when
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
The set of antiderivates of x 2 ’ cos x + 4x is
43.
x3
’ sin x + 2x 2 + C
(a)
3
(b) x 3 + cos x + x 2 + C
x3
’ sin x + x 2 + C
(c)
4
(d) x 2 + x + 1 + C
x3
’ cos x ’ 2x 2 + C
(e)
2
ln x
+ x dx equals
44. The inde¬nite integral
x
ln x 2 + ln2 x + C
(a)
ln2 x x2
+ +C
(b)
2 2
1
ln x + +C
(c)
ln x
x · ln x + C
(d)
x 2 · ln x 2 + C
(e)

2x cos x 2 dx equals
45. The inde¬nite integral

[cos x]2 + C
(a)
cos x 2 + C
(b)
sin x 2 + C
(c)
[sin x]2 + C
(d)
sin x · cos x
(e)
The area between the curve y = ’x 4 + 3x 2 + 4 and the x-axis is
46.
(a) 20
(b) 18
323
Final Exam

(c) 10
96
(d)
5
79
(e)
5
The area between the curve y = sin 2x + 1/2 and the x-axis for 0 ¤ x ¤
47.
2π is
√ π
(a) 2 3 ’
3
√ π
(b) ’2 3 +
3
√ π
(c) 2 3 +
3

(d) √3 + π
3’π
(e)
The area between the curve y = x 3 ’ 9x 2 + 26x ’ 24 and the x-axis is
48.
(a) 3/4
(b) 2/5
(c) 2/3
(d) 1/2
(e) 1/3
The area between the curves y = x 2 + x + 1 and y = ’x 2 ’ x + 13 is
49.
122
(a)
3
125
(b)
3
111
(c)
3
119
(d)
3
97
(e)
3
The area between the curves y = x 2 ’ x and y = 2x + 4 is
50.
117
(a)
6
111
(b)
6
324 Final Exam

125
(c)
6
119
(d)
6
121
(e)
12
5 5 3
= 7 and = 2 then =
51. If 1 f (x) dx 3 f (x) dx 1 f (x) dx
(a) 4
(b) 5
(c) 6
(d) 7
(e) 3
x2
If F (x) = ln t dt then F (x) =
52. x
(4x ’ 1) · ln x
(a)
x2 ’ x
(b)
ln x 2 ’ ln x
(c)
ln(x 2 ’ x)
(d)
1 1

(e)
x2 x
cos 2x ’ 1
53. Using l™Hôpital™s Rule, the limit lim equals
x2
x’0
(a) 1
(b) 0
’4
(c)
’2
(d)
(e) 4
x2
54. Using l™Hôpital™s Rule, the limit lim 3x equals
x’+∞ e
(a) ’1
(b) 1
(c) ’∞
(d) 0
(e) +∞

x
55. The limit lim x equals
x’0
(a) 1
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Final Exam

’1
(b)
(c) 0
+∞
(d)
(e) 2
√ √
x + 1 ’ 3 x equals
3
56. The limit lim
x’+∞
(a) 2
(b) 1
(c) 0
’2
(d)
’1
(e)
4 1

57. The improper integral dx equals
x’1
1

3’1
(a)

2( 3 ’ 1)
(b)

2( 3 + 1)
(c)

3+1
(d)

(e) 3
∞ x
58. The improper integral dx equals
1 + x4
1
π
(a)
3
π
(b)
2
π
(c)
8

(d)
3

(e)
4
The area under the curve y = x ’4 , above the x-axis, and from 3 to +∞, is
59.
2
(a)
79
1
(b)
79
326 Final Exam

2
(c)
97
2
(d)
81
1
(e)
81
The value of log2 (1/16) ’ log3 (1/27) is
60.
(a) 2
(b) 3
(c) 4
(d) 1
’1
(e)
log2 27
61. The value of is
log2 3
’1
(a)
(b) 2
(c) 0
(d) 3
’3
(e)
The graph of y = ln[1/x 2 ], x = 0, is
62.
concave up for all x = 0
(a)
concave down for all x = 0
(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
The graph of y = e’1/x , |x| > 2, is
2
63.
(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
d
64. The derivative log3 (cos x) equals
dx
sin x cos x
(a)
ln 3
ln 3 · sin x

(b)
cos x
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Final Exam

cos x

(c)
ln 3 · sin x
sin x
(d) ’
ln 3 · cos x
ln 3 · cos x
(e) ’
sin x
d x ln x
65. The derivative 3 equals
dx
(a) ln 3 · [x ln x]
(b) (x ln x) · 3x ln x’1
(c) 3x ln x
(d) ln 3 · [1 + ln x]
(e) ln 3 · [1 + ln x] · 3x·ln x
2
The value of the limit limh’0 (1 + h2 )1/ h is
66.
(a) e
e’1
(b)
(c) 1/e
e2
(d)
(e) 1
x 2 ln x
67. Using logarithmic differentiation, the value of the derivative is
ex
ln x
(a)
ex
x2

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