<< . .

. 40
( : 41)



. . >>

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)
This page intentionally left blank.




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

<< . .

. 40
( : 41)



. . >>