Smith R., Minton R. Calculus
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MHDQ256-APP-B MHDQ256-Smith-v1.cls January 12, 2011 7:22
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APPENDIX B
..
Answers to Odd-Numbered Exercises A-73
33. undamped: y(t) = cos3t + 2t sin3t;
damped: y(t) =
e
−t/20
cos
√
3599
20
t
−
2399
√
3599
sin
√
3599
20
t
+ 40sin 3t
40
⫺100
100
undamped
damped
t
u
10
0
37. (a) u(t) = 5.12194 sin2t − 4.87804 sin2.1t
(b) u =
1
4
sin2t −
t
2
cos2t
(c) The solution in part (b) has resonance.
39.
1
√
ω
4
− 7.99ω
2
+ 16
41. y(t) = 1050 − 990e
−0.1t
− 98t
Exercises 16.3, page 1096
1.
Q(t) =−1.4549 ×10
−6
e
−443.65t
+ 1.1455 ×10
−5
e
−56.35t
;
I(t) = 6.4547 × 10
−4
e
−443.65t
− 6.4549 ×10
−4
e
−56.35t
3. Q(t) = 10
−6
cos707.11t; I (t) =−7.0711 ×10
−4
sin707.11t;
amplitude is 10
−6
, phase shift is
π
2
5. Q(t) = e
−2t
−
27
170
cos6t +
26
255
sin6t
+
27
170
cos2t +
3
85
sin2t;
I(t) = e
−2t
79
85
cos6t +
191
255
sin6t
+
6
85
cos2t −
27
85
sin2t
7. u(t) =−
1
2020
cos2t +
1
202
sin2t; amplitude ≈ 4.975 × 10
−3
;
phase shift ≈−0.100
9. Choose δ = tan
−1
2ω
ω
2
− 5
and A
2
as given in the exercise.
11. g(ω) =
1
ω
4
−
196
25
ω
2
+ 16
; for maximum gain, ω =
7
√
2
5
⫺1 ⫺0.5 0 0.5 1 1.5 2
⫺150
⫺100
⫺50
50
20 log
10
g(
ω
)
log
10
ω
13. g(ω) =
1
ω
4
−
199
25
ω
2
+ 16
; for maximum gain, ω =
199
50
0 0.5 1 1.5 2
50
⫺1 ⫺0.5
⫺150
⫺100
⫺50
20 log
10
g(
ω
)
log
10
ω
15. g(ω) =
ω
2
√
ω
4
− 7ω
2
+ 16
; for maximum gain, ω =
4
√
14
7
23. θ(t) = 0.2cos(7
√
2t); amplitude = 0.2; period =
π
√
2
7
; if the length
of the pendulum is doubled, the amplitude remains the same but the
period increases by a factor of
√
2
25. θ(t) =
1
70
√
2
sin(7
√
2t); amplitude =
1
70
√
2
; period =
π
√
2
7
27. amplitude ≈ 0.1367 rad; period =
2π
3
29. y = e
−4t
(c
1
cos
√
2516t + c
2
sin
√
2516t)
33. yes 35. yes 37. α>8.99 39.
2π
w
< 4; no
Exercises 16.4, page 1106
1. a
n+2
=
−2
n + 1
a
n
; a
0
∞
k=0
(−1)
k
2
2k
k!
(2k)!
x
2k
+ a
1
∞
k=0
(−1)
k
k!
x
2k+1
3. a
n+2
=
1
n + 2
a
n
; a
0
∞
k=0
x
2k
k!2
k
+ a
1
∞
k=0
2
k
k!x
2k+1
(2k + 1)!
5. a
n+2
=
n
(n + 2)(n +1)
a
n
for
n = 1, 3, 5,...; a
0
+ a
1
∞
k=0
x
2k+1
(2k + 1)k!2
k
7. a
n+3
=
n
(n + 2)(n +3)
a
n
for
n = 1, 4, 7,...; a
0
+ a
1
∞
k=0
x
3k+1
(3k + 1)k!3
k
9. a
0
∞
k=0
(x − 1)
2k
2
k
k!
+ a
1
∞
k=0
2
k
k!(x − 1)
2k+1
(2k + 1)!
11.
∞
n=0
a
n
(x − 1)
n
where a
0
and a
1
are chosen freely, a
2
=
a
0
2
, and
a
n+2
=
a
n−1
+ a
n
(n + 1)(n +2)
for n ≥ 1
13. 5
∞
k=0
(−1)
k
x
2k
k!
+ 7
∞
k=0
(−1)
k+1
2
2k
k!x
2k+1
(2k + 1)!
15. −3
∞
k=0
(x − 1)
2k
2
k
k!
+ 12
∞
k=0
2
k
k!(x − 1)
2k+1
(2k + 1)!
17. ∞ 19. ∞ 21.
∞
k=0
(−1)
k
x
2k
2
2k
(k!)
2
23. ∞
25. y
(4)
(x) =−2xyy
+ (x −4)y
+ 2y
; y
(4)
(0) =−10;
y
(5)
(x) =−2xyy
(4)
+ (x −6)y
+ 3y
; y
(5)
(0) =−72;
P
5
(x) = 2 − 5x −2x
3
−
5
12
x
4
−
3
5
x
5
27. P
5
(x) =−2 + x −
x
2
2
−
x
3
3
+
5x
4
24
+
x
5
60
29. P
5
(x) = 4(x − π ) − 2π(x − π )
2
+
2π
2
− 2
3
(x − π)
3
−
π
3
− 3π −2
6
(x − π)
4
+
4π
4
− 24π
2
− 20π +12
120
(x − π)
5
Chapter 16 Review Exercises, page 1107
1. y(t) = c
1
e
−4t
+ c
2
e
3t
3. y(t) = e
−t/2
c
1
cos
√
11
2
t
+ c
2
sin
√
11
2
t
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MHDQ256-APP-B MHDQ256-Smith-v1.cls January 12, 2011 7:22
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CONFIRMING PAGES
A-74 APPENDIX B
..
Answers to Odd-Numbered Exercises
5. y(t) = c
1
e
−2t
+ c
2
e
3t
+
1
5
te
3t
−
1
6
t
2
+
1
18
t −
25
108
7. y(t) = 2e
−4t
+ 3e
2t
9. y(t) = sin 2t + cost
11. u(t) =
1
6
cos(4
√
6t)
0 0.5 1 1.5 2
⫺0.2
⫺0.1
0.1
0.2
t
u
13. Q(t) =
−499 +20,000
√
1595
10,000
e
(−400+10
√
1595)t
+
1 −40
√
1595
20
e
(−400−10
√
1595)t
I(t) = 63801e
(−400−10
√
1595)t
− 49.9279e
(−400+10
√
1595)t
15. u(t) = te
−t
(A cos 2t + B sin2t) +C
3
t
3
+C
2
t
2
+C
1
t + C
0
+ D cos2t + E sin2t
17. u(t) =−
160
13289
cos2t +
2300
13289
sin2t
19. a
n+2
=
2
n + 1
a
n
; a
0
∞
k=0
2
2k
k!x
2k
(2k)!
+ a
1
∞
k=0
x
2k+1
k!
21.
∞
n=0
a
n
x
n
, where a
0
and a
1
are freely chosen and
a
n+2
= 2
a
n
n + 1
+
a
n+1
n + 2
for n ≥ 0
23. 4
∞
k=0
2
2k
k!x
2k
(2k)!
+ 2
∞
k=0
x
2k+1
k!
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Subject Index
A
Absolute convergence, 571–577
Absolute extremum, 185, 882
approximation of, 192
finding, 882–883
on closed interval, 191
Absolute maximum, 185, 882
Absolute minimum, 185, 882
Absolute value
defined, 4
derivative of log of, 377
inequality with, 5
Acceleration, 132–133
angular, 774
normal component of, 790–794
tangential component of, 790–794
velocity from, 770
Acceleration vector, 770
Acid dissociation constant, 382
Addition principle, 479
Additive inverse, of vectors, 691, 700
Air resistance (drag), 356
Allosteric enzyme, 142
Alternating series
defined, 565
divergent, 567
harmonic, 565
sum of, 568–569
test, 565–566
Ambient temperature, 494
Ampere’s Law, 1066
Amplitude, altering, 30–31
Amplitude, of sine wave, 31
Angle(s)
between two vectors, 706
Angular acceleration, 774
Angular momentum, 775, 778
conservation of, 776, 925
Angular speed, 771
Angular velocity, 774
Antiderivatives
by trial and error, 292
changing variables for, 968–969
defined, 252
finding, of given function, 253
in computer algebra systems, 454
of vector-valued functions, 765
Approximate zero, 179
Approximation
linear
defined, 175
finding, 175–176
for linear interpolation, 177
of cube roots, 176
of sine function, 176
quadratic, 211
Arc length
defined, 345
estimation of, 51, 346
formula for, 346
in three dimensional space, 753–756
line integral with respect to, 990,
992
of plane curve, 643
of polar curve, 666
of power functions, 346
surface area and, 345–347, 641–647
Archimedean spiral, 653
Archimedes, 325
Arcsine. See Inverse sine function
Area, 266–270
approximating, with rectangles, 267
between curves, 315–321
between three curves, 318
between two curves, 316–318
computation of, 268
finding, with fundamental theorem,
285–286
in polar coordinates, 664, 927
of ellipse, 233, 639
of lima¸cons, 664
of parallelogram, 720
of region bounded by functions
of y, 320
of three-leaf rose, 663
signed, 275
surface
arc length and, 345–347, 641–647
calculation of, 934
computation of, 350
defined, 347
in polar coordinates, 935
numerical approximation of,
935–936
of parametric curves, 641–647
surface integrals for, 1038–1039
total, defined, 275
under curve
calculation of, 267
estimation of, 268
parametric, 638
with double integrals, 918
with Green’s Theorem, 1018
Area function, 272, 286
Arithmetic mean, 478, 896
Arnold, Vladimir, 353
Ascent, method of, 880–881
Asteroid, Trojan, 185
Asymptotes
horizontal
finding, 81
in graphing, 212
oblique, 83
slant, 83
vertical, 24
in graphing, 212, 215–216
Autocatalytic chemical reaction,
242–243
Automatic graphing window, 21
Average cost, 240
Average cost function, 211
Average linear density, 243
Average value, 1048
of function, 279–281
Average velocity, 112–113
Axes, of ellipse, 671
B
Bahill, Terry, 357
Ball, unit, 957
I-1
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I-2 Subject Index
Base(s), of exponential functions,
391–392
Bathtub curve, 421
Beats, 1087
Berkeley, George, 133
Bernoulli’s Theorem, 1064
derivation of, 1065
Bernoulli, Jacob, 645
Bessel function, 602–603
B´ezier curves, 641
Binomial series, 604–605
Binormal vector, 788
Bisections, method of, 74–75
Bode plot, 1096
Bohr, Niels, 708
Boltzmann integral, 478
Bombieri, Enrico, 951
Bound, of sequence, 539–540
Boundary point, 828
Boundary value problem, 1098
Boyle’s law, 238
Brachistochrone, 645
Brachistochrone problem, 644
Buffon needle problem, 486
C
Calculator(s). See Graphing calculators
Cantor set, 552
Cardioids, 654–655
Carrying capacity, 505
Cartesian plane, 6–7
CAS. See Computer algebra systems
(CAS)
Catenary, 411, 413, 416–417
Cauchy Mean Value Theorem, 169
Cauchy, Augustin Louis, 87
Cauchy-Schwartz Inequality, 707, 712
Center of curvature, 789
Center of ellipse, 672
Center of hyperbola, 674
Center of mass, 365
of lamina, 920, 921–922
of solid, 945–946
triple integrals and, 944–946
Center of sphere, 702
Centripetal force, 771
Cesaro sum, 552
Chain rule, 142–145
Cobb-Douglas production function,
856–857
fundamental theorem and, 287–288
multiple, 144
use of, 856
with square root function, 144
Characteristic equation, 1075
Charge, in electric circuit, 1091
Chung, Fan, 144
Circle
curvature of, 782
osculating, 789
parametric equations of, 627–628
Circle of curvature, 789
Circular paraboloid, 737
Circulation, 1061
Closed curves, 1008
Closed disk, 828
Closed interval, 2
absolute extremum on, 191
continuity on, 72
Cobb-Douglas production function,
856–857
Coefficient(s)
drag, 525
of friction, 697
of polynomial, 13
of probability density function, 483
of restitution, 266, 358
second-order equations with
constant, 1074–1080
undetermined, method of, 1083
variable, differential equation with,
1100–1101
Colinear points, 9, 10
Combination of functions, 36
Commodities
complementary, 844
substitute, 844
Comparison test, 554–562
for convergent series, 559
for divergent integral, 475
for improper integrals, 474
for limits, 560
Complement of surface, 1062
Complementary commodities, 844
Completeness axiom, 541
Completing the square, 40–41, 423
Component functions, defined, 750
Component(s), of vector, 708–711
Componentwise, defined, 689
Composite function, continuity for, 72
Composition of functions, 36
continuity of, 830
finding, 37
identification of, 37
Compound interest, 496–498
continuous, 496
Computer algebra systems (CAS)
antiderivatives in, 454
integration with, 450–456
misinterpretation by, 454
shortcomings of, 453–454
Computer(s)
representation of real numbers on, 99
Concavity, 203–209
determination of, 205
down, 203
up, 203
Condition, initial, 493, 1077
Conditional convergence, 572
Conditional probabilities, 478
Conductivity, heat, 1041
Cone
elliptic, graph of, 738
equation for
in cylindrical coordinates, 949
in spherical coordinates, 956
frustrum of, 348
slant height in, 347
Conic sections
directrixes, effect of, 679
eccentricities, effect of, 678–679
in parametric equations, 668–675
in polar coordinates, 677–681
Conjugate diameter, 768
Connected region, definition of, 1004
Connes, Alain, 576
Conoid, Pl¨ucker’s, 955
Conservation of angular momentum,
776, 778, 925
Conservation of energy, 1013
Conservative vector field, 1003–1011
definition of, 1009
determination of, 1010, 1027
Constant
decay, 494
Euler’s, 554
gravitational, 794
growth, 492
spring, 1074
Constant force, work and, 361
Constant of integration, 253
Constant polynomial, 14
Constrained optimization, 887–894
Constraint, inequality, 890–891
Consumer surplus, 291
Continuity
defined, 828
intervals of, 72
limits and, 822–831
of composite functions, 72
of function
defined, 68
of rational function, 69, 71
of vector-valued function, 759–760
on a region, 829
Continuity equation, 1052
derivation of, 1064
Continuous compounding, 496
Continuous probability distributions,
481
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Subject Index I-3
Contour plot, 814
Contradiction, proof by, 8
Convergence
absolute, 571–577
comparison test for, 559
conditional, 572
interval of, 581
half-closed, 581
linear, 184
of Fourier series, 616–617
of geometric series, 547–548
of improper integrals, 469, 471
of infinite series, 546
of power series, 580
of sequence, 532–533
of Taylor series, 590
radius of, 581, 602–603
ratio test for, 571–577
root test for, 576
squeeze theorem and, 537–538
unusually slow convergence of, 181
Converse, 77
Cooling, Newton’s law of, 494
Coordinate systems
cylindrical, 948–953
cone equation in, 949
conversion to/from rectangular,
951–952
cylinder equation in, 948–949
defined, 948
triple integral in, 950–951
polar
arc length in, 666
area in, 664, 927
calculus and, 660–666
conic sections in, 677–681
conversion to/from rectangular,
649–650
defined, 649
double integrals in, 926–931
graph of, 652
horizontal tangent lines and, 661
intersections of, 665
plotting points in, 649
surface area in, 935
surface integral evaluation with,
1035
transformation with, 965
triple integrals with, 948
volume in, 928
rectangular
converting to/from cylindrical,
951–952
converting to/from polar, 649–650
converting to/from spherical,
956
defined, 649
right-handed, 697
spherical, 800, 956–960
cone equation in, 956
conversion to/from rectangular,
956
evaluation formula for, 970–971
surface integral evaluation with,
1038
triple integrals in, 957–960
volume with, 959–960
Cornu’s spiral, 648, 786
Cosecant function
hyperbolic, 412
Cosine function
combined with sine, 32
defined, 30
derivative of, 151
hyperbolic, 411
inverse, 414
integrand with even power of,
435
integrand with odd power of,
434
inverse
defined, 400
evaluation of, 400
parametric equations and, 627
power function in, 294
solving equations with, 29
Cost
average, 211, 240
marginal, 239
minimization of, 226–227
Cost functions, 211
Cotangent function
defined, 30
hyperbolic, 412
Coulomb force, 1013
Coupled system, 521
Coupon collector’s problem, 564
Criminal Geographic Targeting
algorithm, 388
Critical number
defined, 188
of rational function, 190
Critical point, 875, 878
Critical threshold, 512
Cross product, of vectors, 714–723
Cross-section, volume computation by,
326–327
Cube root, approximation of
linear, 176
Newton’s method, 180
Cubic polynomial, 14, 17
graph of, 23
Cumulative distribution function,
478
Curl, 1022–1029
of vector field
computing, 1023
definition of, 1023
interpretation of, 1024
Stokes Theorem and, 1053–1059
Curvature
center of, 789
circle of, 789
defined, 779, 781
of circle, 782
of helix, 783
of parabola, 784
of straight line, 782
radius of, 789
Curve(s)
arc length, 51
area between, 315–321
area between three, 318
area between two, 316–318
area under
calculation of, 267
estimation of, 268
parametric, 638
bathtub, 421
B´ezier, 641
closed, 1008
equiangular, 660
equipotential, 989
frequency response, 1092–1093,
1096
length of, 47–51
level, 814
directional derivatives and, 867
Lorentz, 272
orientation of
definition of, 990
negative, 1014
positive, 1014
parameterization of, 628, 780
piecewise-smooth, 993–995
plane
arc length of, 643
defined, 626
graph of, 626
parametric equations and,
625–630
unusual, 629
simple, 1014
sketching of, 212–221
slope of
estimating, 49
smooth, 763, 991
space, 750
thrust-time, 370
vector-valued
graph of, 750–751
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LT (Late Transcendental)
CONFIRMING PAGES
I-4 Subject Index
Cycloid, 640, 644
Cylinder
defined, 324, 734
in cylindrical coordinates,
948–949
volume of, 665
Cylindrical coordinates, 948–953
cone equation in, 949
conversion to/from rectangular
coordinates, 951–952
cylinder equation in, 948–949
defined, 948
triple integrals in, 950–951
Cylindrical shells method, 338–343
Cylindrical surface, graph of, 734
D
Damping force, 1074, 1094–1095
Daubechies, Ingrid, 610
de Branges, Louis, 270
Decay constant, 494
Decay problems, 491–496
Decreasing function
defined, 195
Decreasing sequence, 538
Definite integral
approximation of, with Midpoint
Rule, 274
computing exactly, 285
defined, 273, 903
integration by parts for, 430–431
of vector-valued function, 766
signed area and, 275–276
substitution in, 295–296
Taylor series for approximating,
601
with variable upper limit, 286
Degree of polynomial, 13
Demand
elasticity of, 241–242, 245
relative change in, 241
Density
linear, 243
mass density, 243
weight, 947
Density plot, 814, 816
Derivative Test
First, 199
Second, 877
Derivative(s)
at a point
unspecified, 118–119
chain rule for, 142–145
computation of, 127–133, 151
defined, 118
directional, 864–871
computation of, 865
defined, 864, 869
finding, 866
level curves and, 867
general rules, 130–131
given, finding functions with, 167
gradient, 864–871
higher order, 131–132
notation, alternative, 121–123
numeric approximation of, 122
of cosine function, 151
of exponential functions, 393–396
of log of absolute value, 377
of sum, 131
of trigonometric functions, 147–152
inverse, 406
of vector-valued function, 761
partial, 833–840
applications, 836–837, 839
computing of, 835–836
defined, 835
from table of data, 839–840
higher-order, 837
mixed second-order, 837
of three variable functions,
838–839
power rule, 127–133
product rule, 135–140, 151
quotient rule, 135–140
rewriting functions for, 131
second, 131
implicit, 158
third, 131
undefined, 188, 189
zero, function with, at local
maximum, 187
Determinant, 715
Deviation, standard, 486
Diaconis, Persi, 483
Diameter, conjugate, 768
Diameter, of section, 1032
Difference quotients, 110, 127
Difference, indefinite integral of, 256
Differentiable functions
defined, 118
Differential, 175
total, 850
Differential equations
as initial value problems, 503
defined, 252, 259, 492
equilibrium solutions, 505
family of solutions for, 503
first-order ordinary, 501
flow lines with, 983
general solution for, 492, 1075
higher-order, 1081
homogeneous, 1082
initial condition in, 503
linear ordinary, 989
modeling with, 491–498
nonhomogeneous, 1082–1088
power series solutions of, 1098–1106
second-order
applications, 1090–1095
as first-order systems, 525
with constant coefficients,
1074–1080
separable, 501–507
systems of, 521–525
with variable coefficients, solving,
1100–1101
Differential geometry, 788
Differential operator, 122
Differentiation
implicit, 155–160, 860
numerical, 122–123
of exponential functions, 395
of logarithms, 377, 380–381,
381–382
of power series, 582–583
of vector-valued functions, 761
term-by-term, 582
Diffusivity, thermal, 862
Dimensionless variables, 858
Dipole
electric, 586
electrostatic field of, 986–987
Dirac delta, 298
Direct linear transformation, 972
Direction field, 510, 512
Direction of maximum increase,
869–870
Direction vectors, 711
Directional derivative, 864–871
computation of, 865
defined, 864, 869
finding, 866
level curves and, 867
Directixes, effects of, 679
Directrix, of parabola, 668
Discontinuity
defined, 68, 828
removable, defined, 70
Discontinuous integrand, 278,
467–470
Discrete probability distributions, 481
Discriminant
defined, 877
local extrema and, 877–878
Disk
closed, 828
open, 828
Displacement vector, 710
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LT (Late Transcendental)
CONFIRMING PAGES
Subject Index I-5
Distance
from point, 720
minimization of, 226–227
minimum, 888
Distance formula, 6–7, 699
Divergence
comparison test for, 475, 559
kth term test for, 549–550
of alternating series, 567
of geometric series, 548–549
of improper integrals, 468, 471
of infinite series, 546
of sequence, 534–535
of vector field, 1022–1029
computing, 1025
definition of, 1025
interpretation of, 1025
ratio test for, 575
Divergence Theorem, 1044–1051
applications, 1047
definition of, 1044
proof of, 1045
proving general result with, 1048
Domain
in graph drawing, 212
of function, defined, 13
of three variable function, 810
of two variable function, 810
Domes, volume of, 327
Dot product, of vector, 704–711
Double integrals, 901–914
area with, 918
change of variables in, 967
defined, 906, 909
evaluation of, 910, 912
in polar coordinates, 926–931
irregular partitions and, 904
order in, 913
over general regions, 908–913
over rectangle, 903–907
volume with, 904–905, 918
Double Riemann sum, 904, 917
Doubling time, 493
Down concavity, 203
Drag coefficient, 525
E
e (irrational number)
as logarithmic base, 391
Taylor series for approximation of,
591
Eccentricity, 677, 678–679
Economic Order Quantity, 283, 291
Elasticity of demand, 241–242, 245
Electric dipole, 586
Electrical circuits, charge in, 1091
Electrical potential, 607
Electrostatic field, 986–987
Elementary polar regions, 926
Ellipse
area enclosed by, 233, 639
axes of, 671
center of, 672
defined, 671
equation of, 671, 672–673
features of, 672
parametric equations of, 627–628
vector-valued function defining, 751
vertices of, 671
Ellipsoid
equation for, 735
graph of, 735
inertia, 745
Elliptic cone, 738
Elliptical helix
defined, 752
vector-valued function defining,
751–752
Endpoints, 2
Energy
conservation of, 1013
kinetic, 361, 1013
potential, 361, 1013
Energy spectrum, 620
Enzyme, allosteric, 142
Epicycloid, 641
Equal-tempered tuning, 8
Equation(s)
calculator for solving, 26
characteristic, 1075
continuity, 1052
derivation of, 1064
converting between rectangular and
polar coordinates, 652
differential
as initial value problems, 503
defined, 252, 259, 492
equilibrium solutions, 505
family of solutions for, 503
first-order ordinary, 501
flow lines with, 983
general solution for, 492, 1075
higher-order, 1081
homogeneous, 1082
initial condition in, 503
linear ordinary, 989
modeling with, 491–498
nonhomogeneous, 1082–1088
power series solutions of,
1098–1106
second-order, 525, 1074–1080
applications, 1090–1095
separable, 501–507
systems of, 521–525
with variable coefficients, solving,
1100–1101
Euler, 1081
Euler’s, 1064
heat, 862, 1062, 1063
Hermite’s, 1107
impulse-momentum, 283, 364
Laplace, 989
linear ordinary differential, 989
linear, in three dimensions, 729
logistic, 244, 245, 505
Maxwell’s, 1065–1066
of cone, 956
of cylinder in cylindrical coordinates,
948–949
of ellipse, parametric, 671, 672–673
of ellipsoid, 735
of hyperbola, 674
of hyperboloid, 739
of lines, 9–18
of motion, 773–776
of parabola, 669, 670
of parallel line, 12
of perpendicular line, 12
of plane, 729
of sphere, 702
of tangent line, 110, 137
of tangent plane, 846–847
parametric, 457
calculus and, 634–639
circles defined by, 627–628
conic sections in, 668–675
defined, 625
ellipses defined by, 627–628
for intersecting surfaces, 755
for line segment, 628
for x-y equations, 629
parameter of, 626
plane curves and, 625–630
projectile motion and, 626–627
sine and, 627
slope of, 635
surface area with, 641–647
symmetric, of lines, 726
wave, 843
Equiangular curve, 660
Equilibrium position, of spring-mass
system, 1074
Equilibrium solutions, 505, 516–517
of systems of equations, 524
stable and unstable, 512, 522
Equipotential curves, 989
Equivalence point, 382
Error bounds, for numerical integration,
305–308
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LT (Late Transcendental)
CONFIRMING PAGES
I-6 Subject Index
Error estimate, for integral test, 557
Error function, 598
Escalante, Jaime, 432
Escape velocity, 86
Euler equation, 1081
Euler’s constant, 554
Euler’s equation, 1064
Euler’s formula, 598
Euler’s method, 510, 512, 983–984
Improved, 520
Euler, Leonhard, 510
Evaluation points, 269, 270
Evaluation Theorem, 991, 997, 1033
Expansion
Fourier series, 611–612, 614–615
of determinant, 715
Taylor series, 588
Exponent(s)
fractional, 191, 200, 255
negative, 254
Exponential decay law, 492
Exponential functions
base of, 391–392
defined, 391
derivatives of, 393–396
differentiation of, 395
integral of, 394
Extrapolation, 12
Extreme value theorem, 186
Extremum (extrema)
absolute, 185
approximation of, 192
finding, 882–883
on closed interval, 191
at undefined derivative, 189
defined, 22
for function with fractional
exponents, 191
local
approximation of, 200
defined, 22
discriminant in finding, 877–878
graph of, 197, 875–876
in functions of several variables,
874
of polynomial, 189
with first derivative test, 200
of functions of several variables,
874–883
with second derivative test, 207
F
Factor Theorem, 17
Factor, integrating, 989
Factorial, defined, 537
Factoring
finding limit by, 61
finding zeros by, 16
Family of solutions, to differential
equations, 503
Faraday’s Law of Induction, 1066
Fermat’s Theorem, 188
Fermat, Pierre de, 188
Feynman, Richard, 135
Fibonacci sequence, 1, 544
Field(s)
direction, 510, 522
electrostatic, 986–987
gradient
definition of, 984
finding, 984–985
potential function of, 985–986
inverse square, flux of,
1049–1050
magnetic, flux of, 1051, 1059
slope, 510
vector, 977–987
conservative, 1003–1011
determination of, 1027
curl of, 1022–1029
computing, 1023
definition of, 1023
interpretation of, 1024
definition of, 978
divergence of, 1022–1029
computing, 1025
defined, 1025
flux of, 1039–1041
graphing of, 978–979
incompressible, 1026
irrotational, 1024,
1027–1028
plotting, 978
potential function of, 1010
sink points, 1026
source points, 1026
source-free, 1026
with gradient, 1026
velocity, 981
flux of, 1061–1062
First component, of vector, 689
First Derivative Test, 199
First moment(s), 365, 944
First octant, 698
First-order ordinary differential
equations, 501
Fitzgerald, Ella, 1073
Fixed graphing window, 21
Fixed point, 42
Flow lines
definition of, 981
differential equations with, 983
Euler’s method in approximation of,
983–984
graphing of, 982
Flow, irrotational, 1058
Flux
definition of, 1040
of inverse square field, 1049–1050
of magnetic field, 1051, 1059
of vector field, 1039–1041
of velocity field, 1061–1062
FM. See Frequency modulation (FM)
Focus (foci)
of hyperbola, 673
of parabola, 668
Force(s)
centripetal, 771
constant, work and, 361
Coulomb, 1013
damping, 1074, 1094–1095
hydrostatic, 367
Magnus, 722, 723
resultant, 694
Four-leaf rose, 656
Fourier analysis, 620
Fourier convergence theorem, 616
Fourier cosine series, 620
Fourier series, 607–618
convergence of, 616–617
defined, 608
expansion of, 610, 611–612, 614–615
music synthesizers and, 617–618
Fourier sine series, 618, 620
Fourier’s law, 989
Fourier, Jean Baptiste Joseph, 608
Fraction(s)
inequality with, 4
partial, 442–448
Fractional exponent
extrema of function with, 191, 200
power rule with, 255
Freedman, Michael, 65
Frenet-Serret formulas, 798
Frequency
natural, 1087
of sine wave, 31
resonant, 1096
Frequency modulation (FM), 222
Frequency response curve, 1092–1093,
1096
Friction, coefficient of, 697
Frustrum of cone, 348
Fubini’s Theorem, 907, 927, 928
Fubini, Guido, 907
Function(s)
absolute extrema of, 186
antiderivative of given, 253
area, 272, 286