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. 11
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= ’2 and x = 4
(c) x
= 0 and x = 2
(d) x
= 2 and x = 2.1
(e) x
The limit expression that represents the derivative of f (x) = x 2 + x at
26.
c = 3 is
[(3 + h)2 + (3 + h)] ’ [32 + 3]
(a) lim
h
h’0
[(3 + 2h)2 + (3 + h)] ’ [32 + 3]
(b) lim
h
h’0
[(3 + h)2 + (3 + h)] ’ [32 + 3]
(c) lim
h2
h’0
[(3 + h)2 + (3 + 2h)] ’ [32 + 3]
(d) lim
h
h’0



Y
[(3 + h)2 + (3 + h)] ’ [32 + 4]
(e) lim
FL
h
h’0
x’3
If f (x) = 2
27. then
AM

x +x
1
(a) f (x) =
2x + 1
TE



x2 ’ x
(b) f (x) =
x’3
(c) f (x) = (x ’ 3) · (x 2 + x)
’x 2 + 6x + 3
(d) f (x) =
(x 2 + x)2
x 2 + 6x ’ 3
(e) f (x) =
x2 + x
If g(x) = x · sin x 2 then
28.
(x) = sin x 2
(a) f
(x) = 2x 2 sin x 2
(b) f
(x) = x 3 sin x 2
(c) f
(x) = x cos x 2
(d) f
(x) = sin x 2 + 2x 2 cos x 2
(e) f
If h(x) = ln[x cos x] then
29.
319
Final Exam

1
h (x) =
(a)
x cos x
x sin x
h (x) =
(b)
x cos x
cos x ’ x sin x
h (x) =
(c)
x cos x
h (x) = x · sin x · ln x
(d)
x cos x
h (x) =
(e)
sin x
If g(x) = [x 3 + 4x]53 then
30.
g (x) = 53 · [x 3 + 4x]52
(a)
g (x) = 53 · [x 3 + 4x]52 · (3x 2 + 4)
(b)
g (x) = (3x 2 + 4) · 53x 3
(c)
g (x) = x 3 · 4x
(d)
x 3 + 4x
(e) g (x) = 2
2x + 1
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
The position in feet of a moving vehicle is given by 8t 2 ’ 6t + 142. What
32.
is the acceleration of the vehicle at time t = 5 seconds?
12 ft/sec2
(a)
8 ft/sec2
(b)
’10 ft/sec2
(c)
20 ft/sec2
(d)
16 ft/sec2
(e)
Let f (x) = x 3 ’ 5x 2 + 3x ’ 6. Then the graph of f is
33.
concave up on (’3, ∞) and concave down on (’∞, ’3)
(a)
concave up on (5, ∞) and concave down on (’∞, 5)
(b)
concave up on (5/3, ∞) and concave down on (’∞, 5/3)
(c)
concave up on (3/5, ∞) and concave down on (’∞, 3/5)
(d)
concave up on (’∞, 5/3) and concave down on (5/3, ∞)
(e)
320 Final Exam

7
Let g(x) = x 3 + x 2 ’ 10x + 2. Then the graph of f is
34.
2
(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
(’10, 1)
Find all local maxima and minima of the function h(x) = ’(4/3)x 3 +5x 2 ’
35.
4x + 8.
= 1/2, local maximum at x = 2
(a) local minimum at x
= 1/2, local maximum at x = 1
(b) local minimum at x
= ’1, local maximum at x = 2
(c) local minimum at x
= 1, local maximum at x = 3
(d) local minimum at x
= 1/2, local maximum at x = 1/4
(e) local minimum at x
Find all local and global maxima and minima of the function h(x) = x +
36.
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 +
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
325
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
327
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
(b)
ln x
x 2 ln x
(c)
ex
x 2 ln x
2 1
+ ’1 ·
(d)
ex
x ln x
x
2 1
’ ’1
(e)
x ln x
x
The derivative of f (x) = Sin’1 (x · ln x) is
68.
1 + ln x
(a)
x 2 · ln2 x
1
(b)
1 ’ x 2 · ln2 x
328 Final Exam

ln x
(c)
1 ’ x 2 · ln2 x
1 + ln x
(d)
1 ’ x 2 · ln2 x
1 + ln x

(e)
1 ’ x2
The value of the derivative of Tan’1 (ex · cos x) is
69.
ex
(a)
1 + e2x cos2 x
ex sin x
(b)
1 + e2x cos2 x
ex cos x
(c)
1 + e2x cos2 x


Y
ex (cos x ’ sin x)
FL
(d)
1 + cos2 x
ex (cos x ’ sin x)
AM

(e)
1 + e2x cos2 x
70. The value of the integral log3 x dx is
x
TE



(a) x ln x ’ +C
ln 3
(b) x log3 x ’ x + C
x
(c) x log3 x ’ +C
ln 3
x
(d) x log3 x ’ + C
3
x
(e) x ln x ’ + C
3
1 x2
· x dx is
71. The value of the integral 05
4
(a)
ln 5
ln 5
(b)
2
ln 5
(c)
ln 2
329
Final Exam

2
(d)
ln 5
ln 2
(e)
ln 5
x · 2x dx is
72. The value of the integral
x · 2x 2x
’ 2 +C
(a)
ln 2 ln 2
x·2 x 2x
+ 2 +C
(b)
ln 2 ln 2
x·2 x 2x
’ +C
(c)
ln 2 ln 2
2x
’ 2x + C
(d)
ln 2
2x
x
’ +C
(e)
ln 2 ln 2
73. A petri dish contains 7,000 bacteria at 10:00 a.m. and 10,000 bacteria at
1:00 p.m. How many bacteria will there be at 4:00 p.m.?
(a) 700000
(b) 10000
(c) 100000
(d) 10000/7
(e) 100000/7
74. There are 5 grams of a radioactive substance present at noon on January 1,
2005. At noon on January 1 of 2009 there are 3 grams present. When will
there be just 2 grams present?
= 5.127, or in early February of 2010
(a) t
= 7.712, or in mid-August of 2012
(b) t
= 7.175, or in early March of 2012
(c) t
= 6.135, or in early February of 2011
(d) t
= 6.712, or in mid-August of 2011
(e) t
75. If $8000 is placed in a savings account with 6% interest compounded
continuously, then how large is the account after ten years?
(a) 13331.46
(b) 11067.35
(c) 14771.05
(d) 13220.12
(e) 14576.95
330 Final Exam

76. A wealthy uncle wishes to ¬x an endowment for his favorite nephew.
He wants the fund to pay the young fellow $1,000,000 in cash on the
day of his thirtieth birthday. The endowment is set up on the day of the
nephew™s birth and is locked in at 8% interest compounded continuously.
How much principle should be put into the account to yield the necessary
payoff?
(a) 88,553.04
(b) 90,717.95
(c) 92,769.23
(d) 91,445.12
(e) 90,551.98

The values of Sin’1 1/2 and Tan’1 3 are
77.
(a) π/4 and π/3
(b) π/3 and π/2
(c) π/2 and π/3
(d) π/6 and π/3
(e) π/3 and π/6
dx
78. The value of the integral dx is
4 + x2
1 x
Tan’1 +C
(a)
2 2
1 x
Tan’1 +C
(b)
2 4
1 x
Tan’1 +C
(c)
4 2
x2
1
Tan’1 +C
(d)
2 2
1 2
Tan’1 +C
(e)
2 x
ex dx

79. The value of the integral dx
1 ’ e2x
Cos’1 ex + C
(a)
Sin’1 e2x + C
(b)
Sin’1 e’x + C
(c)
Cos’1 e’2x + C
(d)
Sin’1 ex + C
(e)
331
Final Exam

2 x dx

80. The value of the integral dx is
2 x4 ’ 1
x
1
π
(a)
3
’π
(b)
4
’π
(c)
6
π
(d)
4
π
(e)
6
x 2 ln x dx is
81. The value of the integral

x3 x3
ln x ’ +C
(a)
3 9
x2 x2
ln x ’ +C
(b)
3 9
x3 x3
ln x ’ +C
(c)
2 6
x3 x3
ln x ’ +C
(d)
9 3
x5 x3
ln x ’ +C
(e)
3 6
1x
82. The value of the integral sin x dx is
0e
e · cos 1 ’ e · sin 1 + 1
(a)
e · sin 1 ’ e · cos 1 ’ 1
(b)
e · sin 1 ’ e · cos 1 + 1
(c)
e · sin 2 ’ e · cos 2 + 1
(d)
e · sin 2 + e · cos 2 ’ 1
(e)
x · e2x dx is
83. The value of the integral
xex ex
’ +C
(a)
2 4
xe2x e2x
’ +C
(b)
4 2
xex ex
’ +C
(c)
4 2
332 Final Exam

xe2x ex
’ +C
(d)
2 4
xe2x e2x
’ +C
(e)
2 4
dx
84. The value of the integral is
x(x + 1)
ln |x + 1| ’ ln |x| + C
(a)
ln |x ’ 1| ’ ln |x + 1| + C
(b)
ln |x| ’ ln |x + 1| + C
(c)
ln |x| ’ ln |x| + C
(d)
ln |x + 2| ’ ln |x + 1| + C
(e)
dx
85. The value of the integral is
x(x 2 + 4)
1 1
ln |x| ’ ln(x 2 + 2) + C
(a)
2 8
1 1
ln |x| ’ ln(x 2 + 4) + C
(b)
4 8
1 1
ln |x| ’ ln(x 2 + 4) + C
(c)
8 4
1 1
ln |x| ’ ln(x 2 + 1) + C
(d)
2 8
1 1
ln |x| ’ ln(x 2 + 4) + C
(e)
8 2
dx
86. The value of the integral is
(x ’ 1)2 (x + 1)
1 1 1
ln |x ’ 1| + ’ ln |x + 1| + C
(a)
x’1 2
2
’1 1 1
ln |x ’ 1| ’ + ln |x + 1| + C
(b)
(x ’ 1)2 2
2
’1 1 1
ln |x ’ 1| + + ln |x + 1| + C
(c)
x’1 4
2
’1 1 1
ln |x ’ 1| ’ + ln |x + 1| + C
(d)
x’1 2
2
’1 1/2 1
ln |x ’ 1| ’ + ln |x + 1| + C
(e)
x’1 4
4
2√
The value of the integral 1 x 1 + x 2 dx is
87.
333
Final Exam

’ 1 23/2
1 3/2
(a) 45 4
’ 1 33/2
1 3/2
(b) 34 3
’ 1 23/2
1 3/2
(c) 35 3
’ 1 33/2
1 3/2
(d) 54 5
’ 1 53/2
1 3/2
(e) 32 3
π/4 sin x cos x
88. The value of the integral dx
1 + cos2 x
0

3
(a) ln
5

5
(b) ln
2√
1 3
(c) ln
2√2
2
(d) ln
3
2
ln √
(e)
3
1x
sin(1 + ex ) dx is
89. The value of the integral 0e
’ sin(1 + e) + cos 2
(a)
’ sin(1 ’ e) + sin 2
(b)
’ cos(1 + e) + cos 2
(c)
cos(1 + e) ’ cos 2
(d)
cos(1 + e) + cos 2
(e)
π
sin4 x dx is
90. The value of the integral 0

(a)
8

(b)
8

(c)
6

(d)
10

(e)
5
334 Final Exam

π
sin2 x cos2 x dx is
91. The value of the integral 0
π
(a)
6
π
(b)
4
π
(c)
3
π
(d)
2
π
(e)
8
π/4
tan2 x dx
92. The value of the integral is
0
π
(a) 1 ’
3
π
(b) 2 ’
4
π
(c) 1 ’
2
π
(d) 1 ’
4
(e) 4 ’ π
93. A solid has base in the x-y plane that is the circle of radius 1 and center the
origin. The vertical slice parallel to the y-axis is a semi-circle. What is the
volume?

(a)
3

(b)
3
π
(c)
3

(d)
3
π
(e)
6
94. A solid has base in the x-y plane that is a square with center the origin and
vertices on the axes. The vertical slice parallel to the y-axis is an equilateral
triangle. What is the volume?

23
(a)
3
335
Final Exam

3
(b)
3

(c) 3

3+3
(d)

(e) 33
The planar region bounded by y = x 2 and y = x is rotated about the line
95.
y = ’1. What volume results?
11π
(a)
15

(b)
15

(c)
19

(d)
15

(e)
15

The planar region bounded by y = x and y =
96. x is rotated about the line
x = ’2. What volume results?

(a)
5

(b)
7

(c)
5

(d)
3
11π
(e)
5
97. A bird is ¬‚ying upward with a leaking bag of seaweed. The sack initially
weights 10 pounds. The bag loses 1/10 pound of liquid per minute, and the
bird increases its altitude by 100 feet per minute. How much work does the
bird perform in the ¬rst six minutes?
(a) 5660 foot-pounds
(b) 5500 foot-pounds
336 Final Exam

(c) 5800 foot-pounds
(d) 5820 foot-pounds
(e) 5810 foot-pounds
The average value of the function f (x) = sin x ’ x on the interval
98.
[0, π] is
3 π

(a)
4
π
2 π

(b)
3
π
2 π

(c)
2
π
4 π

(d)
4
π
1 π

(e)
2
π
= x 3,
99. The integral that equals the arc length of the curve y
1 ¤ x ¤ 4, is
4
1 + x 4 dx
(a)
1
4
1 + 9x 2 dx
(b)
1
4
1 + x 6 dx
(c)
1
4
1 + 4x 4 dx
(d)
1
4
1 + 9x 4 dx
(e)
1
1 dx

100. The Simpson™s Rule approximation to the integral dx
1 + x2
0
with k = 4 is
≈ 0.881
(a)
≈ 0.895
(b)
≈ 0.83
(c)
≈ 0.75
(d)
≈ 0.87
(e)
337
Final Exam

SOLUTIONS
1. (a), 2. (c), 3. (b), 4. (e), 5. (e), 6. (d), 7. (b),
8. (a), 9. (c), 10. (d), 11. (e), 12. (b), 13. (c), 14. (d),
15. (e), 16. (a), 17. (c), 18. (d), 19. (c), 20. (e), 21. (a),
22. (d), 23. (b), 24. (c), 25. (c), 26. (a), 27. (d), 28. (e),
29. (c), 30. (b), 31. (e), 32. (e), 33. (c), 34. (c), 35. (a),
36. (a), 37. (d), 38. (e), 39. (b), 40. (d), 41. (e), 42. (b),
43. (a), 44. (b), 45. (c), 46. (d), 47. (c), 48. (d), 49. (b),
50. (c), 51. (b), 52. (a), 53. (d), 54. (d), 55. (a), 56. (c),
57. (b), 58. (c), 59. (e), 60. (e), 61. (d), 62. (a), 63. (a),
64. (d), 65. (e), 66. (a), 67. (d), 68. (d), 69. (e), 70. (c),
71. (d), 72. (a), 73. (e), 74. (c), 75. (e), 76. (b), 77. (d),
78. (a), 79. (e), 80. (e), 81. (a), 82. (c), 83. (e), 84. (c),
85. (b), 86. (e), 87. (c), 88. (e), 89. (c), 90. (b), 91. (e),
92. (d), 93. (b), 94. (a), 95. (b), 96. (a), 97. (d), 98. (c),
99. (e), 100. (a)
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Y
FL
AM
TE
INDEX



acceleration as a second derivative, 77 concave down, 81
adjacent side of a triangle, 26 concave up, 81
angle, sketching, 21 cone, surface area of, 246
angles constant of integration, 100
in degree measure, 20 continuity, 64
in radian measure, 19, 21 measuring expected value, 64
antiderivative, concept of, 99 coordinates
antiderivatives, 94 in one dimension, 3
as organized guessing, 94 in two dimensions, 5
arc length, 240 cosecant function, 26
calculation of, 241 Cosine function, 182
area cosine function, principal, 182
between two curves, 116 cosine of an angle, 22
calculation of, 103 cotangent function, 28
examples of, 107 critical point, 87
function, 110 cubic, 16
of a rectangle, 103 cylindrical shells, method of, 229
positive, 114
signed, 111, 116
decreasing function, 81
area and volume, analysis of with improper
derivative, 66
integrals, 139
application of, 75
average value
as a rate of change, 76
comparison with minimum and maximum,
chain rule for, 71
238
importance of, 66
of a function, 237
of a logarithm, 72
average velocity, 67
of a power, 71
of a trigonometric function, 72
bacterial growth, 174
of an exponential, 72
product rule for, 71
Cartesian coordinates, 5
quotient rule for, 71
closed interval, 3
sum rule for, 71
composed functions, 40
derivatives, rules for calculating, 71
composition
differentiable, 66
not commutative, 41
differential equation
of functions, 40
for exponential decay, 174
compositions, recognizing, 41
compound interest, 178 for exponential growth, 174


339
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340 Index

improper integral
domain of a function, 31
convergence of, 134
divergence of, 135
element of a set, 30
incorrect analysis of, 137
endowment, growth of, 180
with in¬nite integrand, 134
Euler, Leonhard, 158
with interior singularity, 136
Euler™s constant, value of, 159
improper integrals, 132
Euler™s number e, 158
applications of, 143
exponential, 50
doubly in¬nite, 142
rules for, 51
over unbounded intervals, 140
exponential decay, 172
with in¬nite integrand, 133
exponential function, 154, 155
increasing function, 81
as inverse of the logarithm, 156
inde¬nite integral, 101
calculus properties of, 156
calculation of, 102
graph of, 155, 168
indeterminate forms, 123
properties of, 155
involving algebraic manipulation, 128
uniqueness of, 157
using algebraic manipulations to evaluate,
exponential growth, 172
131
exponentials
using common denominator to evaluate,
calculus with, 166
130
properties of, 164
using logarithm to evaluate, 128
rules for, 162
initial height, 96
with arbitrary bases, 160
initial velocity, 96
inside the parentheses, working, 40
falling bodies, 76, 94
instantaneous velocity, 66
examples of, 77
as derivative, 67
Fermat™s test, 87
integers, 2
function, 30
integral
speci¬ed by more than one formula, 32
as generalization of addition, 99
functions
linear properties of, 120
examples of, 31, 32
sign, 101, 106
with domain and range understood, 32
integrals
Fundamental Theorem of Calculus, 108
involving inverse trigonometric functions,
Justi¬cation for, 110
187
involving tangent, secant, etc., 213
Gauss, Carl Friedrich, 106 numerical methods for, 252
graph functions, using calculus to, 83 integrand, 106
graph of a function integration, rules for, 120
plotting, 35 integration by parts, 197, 198
point on, 33 choice of u and v, 199
graphs of trigonometric functions, 26 de¬nite integrals, 200
growth and decay, alternative model for, 177 limits of integration, 201
interest, continuous compounding of, 179
half-open interval, 3 intersection of sets, 30
Hooke™s Law, 235 inverse
horizontal line test for invertibility, 46 derivative of, 76
hydrostatic pressure, 247 restricting the domain to obtain, 44
calculation of, 248 rule for ¬nding, 42
341
Index

logarithm (contd.)
inverse cosecant, 189
properties of, 149
inverse cosine function, derivative of, 184
reciprocal law for, 150
inverse cosine, graph of, 182
to a base, 49, 148
inverse cotangent, 189
logarithm function
inverse function, graph of, 44
as inverse to exponential, 147
inverse of a function, 42
derivative of, 150
inverse secant, 189
logarithm functions, graph of, 168
inverse sine, graph of, 182
logarithmic derivative, 72
inverse sine function, derivative of, 184
logarithmic differentiation, 170
inverse tangent function, 185
logarithms
derivative of, 187
calculus with, 166
inverse trigonometric functions
properties of, 164
application of, 193
with arbitrary bases, 163
derivatives of, 76
graphs of, 190
key facts, 191 Maple, 256
inverses, some functions do not have, 43 Mathematica, 256
maxima and minima, applied, 88
maximum, derivative vanishing at, 77
Leibniz, Gottfried, 108
maximum/minimum problems, 86
l™Hôpital™s Rule, 123“127
minimum, derivative vanishing at, 87
limit
money, depreciation of, 144
as anticipated value rather than actual
value, 59 motion, 1
-δ de¬nition of, 57
informal de¬nition of, 57 natural logarithm as log to the base e, 163
non-existence of, 62 natural numbers, 1
rigorous de¬nition of, 57 Newton, Isaac, 108
uniqueness of, 62 non-repeating decimal expansion, 2
limits, 57 numerical approximation, 253
of integration, 106
one-sided, 60 open interval, 3
properties of, 61 opposite side of a triangle, 26
line
equation of, 13 parabola, 15, 18
key idea for ¬nding the equation of, 15 parallel lines have equal slopes, 12
point-slope form for, 13 partial fractions
two-point form for, 14 products of linear factors, 203
lines, graphs of, 7 quadratic factors, 206
loci in the plane, 15 repeated linear factors, 205
locus period of a trigonometric function, 25
of points, 39 perpendicular lines have negative reciprocal
plotting of, 7 slopes, 12
logarithm pinching theorem, 62
basic facts, 49 points in the plane, plotting, 5
formal de¬nition of, 148 points in the reals, plotting, 3
graph of, 151 polynomial functions, 147
natural, 49, 149 powers, derivatives of, 167
of the absolute value, 152 principal angle, associated, 25
342 Index

quotient, writing a product as, 128 Tangent function, 185
tangent function, 26
radioactive decay, 176 tangent line
range of a function, 31 calculation of, 69
rate of change and slope of tangent line, 70 slope of, 67
rates of change, 1 terminal point for an angle, 22
rational numbers, 2 transcendental functions, 147
real numbers, 2 trapezoid rule, 252, 254
reciprocals error in, 254
of linear functions, integrals of, 202 trigonometric expressions, integrals of, 210
of quadratic expressions, integrals of, 202, trigonometric functions
203 additional, 26
rectangles, method of, 253 fundamental identities, 29
related rates, 91 inverse, 180
repeating decimal expansion, 2 table of values, 28
Riemann sum, 104 trigonometric identities, useful,
rise over run, 10 210
trigonometry, 19
secant function, 26 classical formulation of, 25
set builder notation, 3
sets, 30 union of sets, 30
Simpson™s rule, 256, 257 unit circle, 19
error in, 257 u-substitution, 207
sine and cosine, fundamental properties of,
23 vertical line test for a function, 35
odd powers of, 211 volume
Sine function, 182 by slicing, 219
sine function, principal, 182 calculation of, 217
sine of an angle, 22 of solids of revolution, 224
slope
washers, method of, 225
of a line, 8
water
unde¬ned for vertical line, 12
pumping, 236
springs, 234
weight of, 249
substitution, method of, 207
work, 233
surface area, 243
calculation of, 234
calculation of, 245
ABOUT THE AUTHOR




Steven G. Krantz is the Chairman of the Mathematics Department at Washington
University in St. Louis. An award-winning teacher and author, Dr. Krantz has
written more than 30 books on mathematics, including a best-seller.




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