(e)

π

sin4 x dx is

90. The value of the integral 0

5π

(a)

8

3π

(b)

8

5π

(c)

6

3π

(d)

10

2π

(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?

4π

(a)

3

2π

(b)

3

π

(c)

3

8π

(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

7π

(b)

15

7π

(c)

19

8π

(d)

15

2π

(e)

15

√

The planar region bounded by y = x and y =

96. x is rotated about the line

x = ’2. What volume results?

4π

(a)

5

4π

(b)

7

9π

(c)

5

4π

(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

Copyright 2003 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use.

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