446 0 INDEX
Difference of cubes, 5.84
Differentiable functions, 8.21, 8.25,
8.26, 8.30, 8.31, 8.43 to
8.47, 9.18, 9.45, 9.49, 42.96,
42.97
Differential equations:
Bernoulli's equation, 46.40 to
46.44
Bessel's equation, 38.70, 38.71,
46.81
exact, 46.25, 46.26
fourth-order, 46.76 to 46.78
integrating factors for, 46.27 to
46.33, 46.35, 46.36, 46.45
Legendre's equation, 46.80
nth order, 46.56
orthogonal trajectories and,
46.22 to 46.24
partial (see Partial differential
equations)
partial fraction decomposition
and, 46.22
power series solutions for, 38.49,
38.56, 38.66
predator-prey system, 46.70
Riccati equation, 46.43, 46.44
second-order homogeneous,
46.48 to 46.55, 46.72
second-order nonhomogeneous,
45.57 to 45.68
separable variables in, 19.88 to
19.94,46.1 to 46.21, 46.34,
46.37 to 46.39, 46.46, 46.47
substitutions in, 46.17 to 46.21,
46.34
third-order, 46.73 to 46.75, 46.79
variation of parameters and,
46.65 to 46.69
Differentials, approximation by
(see Approximation, by
differentials)
Differentiation:
implicit, 12.2, 12.11 to 12.22,
12.24, 12.34, 12.35, 12.38 to
12.48
implicit partial, 42.13 to 42.18
logarithmic, 23.23 to 23.26,
23.58, 23.67, 24.35 to 24.39
Direction:
of steepest ascent and descent,
43.2, 43.8
of vectors, 33.3, 33.13, 33.25
Direction angles, 40.32
Direction cosines of vectors, 40.27
to 40.29, 40.32
Directional derivatives, 43.1 to
43.13, 43.15 to 43.17
Discontinuities (see Continuity of
functions)
Discriminant, 30.13, 46.48
Disks, segments of, 29.43
Distance:
between lines, 40.113, 40.114,
43.31, 43.38
between planes, 40.91, 40.92
between points, 40.1 to 40.4
from points to lines, 33.9, 33.10,
33.34,40.14,40.20,40.24,
40.34, 40.77 to 40.79
from points to planes, 40.44,
40.88, 40.89
Divergence, 45.25 to 45.27, 45.30
to 45.32, 45.34, 45.35, 45.37,
45.39, 45.47 to 45.49
Domain:
of definition, 41.64 to 41.66
of functions, 5.1 to 5.19, 5.24 to
5.31, 5.34 to 5.37, 9.16,
27.21, 41.64 to 41.66
Doomsday equation, 26.31
Dot product of vectors, 33.4, 33.5,
33.9, 33.10,33.12, 33.18 to
33.20, 33.28 to 33.35, 33.37
to 33.41, 40.33 to 40.35,
40.39, 40.44, 40.45, 40.47,
40.49, 40.55 to 40.58, 40.60
to 40.67, 43.1 to 43.7, 43.10
to 43.12, 43.16, 43.17, 43.66,
45.9, 45.13 to 45.15, 45.35,
45.37,45.39,45.41,45.43,
45.45 to 45.49
Double integrals, 44.1 to 44.4, 44.6
to 44.46, 44.53 to 44.58,
44.74, 44.75, 44.79 to 44.82,
44.86, 44.87, 44.89, 44.90
Ellipses:
area of, 20.72,41.22
as contours (level curves), 41.37
equations of, 41.12
graphs of, 16.20, 16.41, 34.2,
34.5, 35.16
intersections of hyperbolas with,
12.43
parametric equations of, 34.2,
34.5,34.11
in polar coordinates, 35.16
rectangles inscribed in, 16.41
tangents to, 12.42
triangle perimeter and, 16.20
Ellipsoids:
boxes inscribed in, 43.42
equations of, 41.6, 41.79, 41.98
graphs of, 41.13
as level surfaces, 41.45
volume of, 41.22
Equiangular spirals, 35.71, 35.77,
35.101, 35.112
Equilateral triangles, 14.36, 16.20,
16.51, 16.60
Escape velocity, 19.95, 19.96
Estimation (see Approximation)
Euler's constant ("/), 37.110
Euler's theorem for homogeneous
functions, 42.74 to 42.77
Even functions, 5.38, 5.40, 5.41,
5.48, 5.53, 5.55, 5.56, 5.90,
5.91, 5.93, 8.46, 8.47, 15.31,
15.36, 15.46, 20.50, 38.73,
38.93
Exact differential equations, 46.25,
46.26
Exponential functions:
absolute extrema of, 24.50, 24.57
to 24.62, 24.73
antiderivatives (indefinite
integrals) and, 24.17 to
24.29, 24.54, 28.1 to 28.3,
28.9, 28.13,28.20, 28.42,
28.43, 28.50
defined, 11.48
definite integrals and, 24.45 to
24.49, 24.55, 24.76, 24.92,
24.93, 32.6, 32.7, 32.12 to
32.14, 32.17 to 32.22, 32.38,
32.39, 32.58
derivatives of, 24.7 to 24.16,
24.30 to 24.39, 24.51,24.52,
24.64, 24.77, 24.88 to 24.90
graphs of, 24.57 to 24.63, 24.84
to 24.86, 24.91, 24.94, 25.56,
25.57
hyperbolic functions and, 24.95
to 24.111
Laplace transform of, 32.58
limits and, 24.74, 24.75, 24.78 to
24.81,24.83
Maclaurin series and, 39.1, 39.7,
39.18, 39.21, 39.24, 39.25,
39.32, 39.35, 39.44, 39.50
power series and, 38.39 to 38.42,
38.76, 38.78 to 38.80, 38.82,
38.97
properties of, 24.1 to 24.6, 24.35,
24.65 to 24.71
(See also Natural logarithms)
Exponential growth and decay:
bacterial growth, 26.2, 26.20,
26.27, 26.29, 26.37
compound interest, 26.8 to
26.14, 26.40 to 26.42
decay and growth constants for,
26.1
defined, 26.1
other applications, 26.3, 26.6,
26.22 to 26.24, 26.31,26.33
to 26.36, 26.39
population growth, 26.5, 26.7,
26.19, 26.25,26.26,26.28,
26.38
Difference of cubes, 5.84
Differentiable functions, 8.21, 8.25,
8.26, 8.30, 8.31, 8.43 to
8.47, 9.18, 9.45, 9.49, 42.96,
42.97
Differential equations:
Bernoulli's equation, 46.40 to
46.44
Bessel's equation, 38.70, 38.71,
46.81
exact, 46.25, 46.26
fourth-order, 46.76 to 46.78
integrating factors for, 46.27 to
46.33, 46.35, 46.36, 46.45
Legendre's equation, 46.80
nth order, 46.56
orthogonal trajectories and,
46.22 to 46.24
partial (see Partial differential
equations)
partial fraction decomposition
and, 46.22
power series solutions for, 38.49,
38.56, 38.66
predator-prey system, 46.70
Riccati equation, 46.43, 46.44
second-order homogeneous,
46.48 to 46.55, 46.72
second-order nonhomogeneous,
45.57 to 45.68
separable variables in, 19.88 to
19.94,46.1 to 46.21, 46.34,
46.37 to 46.39, 46.46, 46.47
substitutions in, 46.17 to 46.21,
46.34
third-order, 46.73 to 46.75, 46.79
variation of parameters and,
46.65 to 46.69
Differentials, approximation by
(see Approximation, by
differentials)
Differentiation:
implicit, 12.2, 12.11 to 12.22,
12.24, 12.34, 12.35, 12.38 to
12.48
implicit partial, 42.13 to 42.18
logarithmic, 23.23 to 23.26,
23.58, 23.67, 24.35 to 24.39
Direction:
of steepest ascent and descent,
43.2, 43.8
of vectors, 33.3, 33.13, 33.25
Direction angles, 40.32
Direction cosines of vectors, 40.27
to 40.29, 40.32
Directional derivatives, 43.1 to
43.13, 43.15 to 43.17
Discontinuities (see Continuity of
functions)
Discriminant, 30.13, 46.48
Disks, segments of, 29.43
Distance:
between lines, 40.113, 40.114,
43.31, 43.38
between planes, 40.91, 40.92
between points, 40.1 to 40.4
from points to lines, 33.9, 33.10,
33.34,40.14,40.20,40.24,
40.34, 40.77 to 40.79
from points to planes, 40.44,
40.88, 40.89
Divergence, 45.25 to 45.27, 45.30
to 45.32, 45.34, 45.35, 45.37,
45.39, 45.47 to 45.49
Domain:
of definition, 41.64 to 41.66
of functions, 5.1 to 5.19, 5.24 to
5.31, 5.34 to 5.37, 9.16,
27.21, 41.64 to 41.66
Doomsday equation, 26.31
Dot product of vectors, 33.4, 33.5,
33.9, 33.10,33.12, 33.18 to
33.20, 33.28 to 33.35, 33.37
to 33.41, 40.33 to 40.35,
40.39, 40.44, 40.45, 40.47,
40.49, 40.55 to 40.58, 40.60
to 40.67, 43.1 to 43.7, 43.10
to 43.12, 43.16, 43.17, 43.66,
45.9, 45.13 to 45.15, 45.35,
45.37,45.39,45.41,45.43,
45.45 to 45.49
Double integrals, 44.1 to 44.4, 44.6
to 44.46, 44.53 to 44.58,
44.74, 44.75, 44.79 to 44.82,
44.86, 44.87, 44.89, 44.90
Ellipses:
area of, 20.72,41.22
as contours (level curves), 41.37
equations of, 41.12
graphs of, 16.20, 16.41, 34.2,
34.5, 35.16
intersections of hyperbolas with,
12.43
parametric equations of, 34.2,
34.5,34.11
in polar coordinates, 35.16
rectangles inscribed in, 16.41
tangents to, 12.42
triangle perimeter and, 16.20
Ellipsoids:
boxes inscribed in, 43.42
equations of, 41.6, 41.79, 41.98
graphs of, 41.13
as level surfaces, 41.45
volume of, 41.22
Equiangular spirals, 35.71, 35.77,
35.101, 35.112
Equilateral triangles, 14.36, 16.20,
16.51, 16.60
Escape velocity, 19.95, 19.96
Estimation (see Approximation)
Euler's constant ("/), 37.110
Euler's theorem for homogeneous
functions, 42.74 to 42.77
Even functions, 5.38, 5.40, 5.41,
5.48, 5.53, 5.55, 5.56, 5.90,
5.91, 5.93, 8.46, 8.47, 15.31,
15.36, 15.46, 20.50, 38.73,
38.93
Exact differential equations, 46.25,
46.26
Exponential functions:
absolute extrema of, 24.50, 24.57
to 24.62, 24.73
antiderivatives (indefinite
integrals) and, 24.17 to
24.29, 24.54, 28.1 to 28.3,
28.9, 28.13,28.20, 28.42,
28.43, 28.50
defined, 11.48
definite integrals and, 24.45 to
24.49, 24.55, 24.76, 24.92,
24.93, 32.6, 32.7, 32.12 to
32.14, 32.17 to 32.22, 32.38,
32.39, 32.58
derivatives of, 24.7 to 24.16,
24.30 to 24.39, 24.51,24.52,
24.64, 24.77, 24.88 to 24.90
graphs of, 24.57 to 24.63, 24.84
to 24.86, 24.91, 24.94, 25.56,
25.57
hyperbolic functions and, 24.95
to 24.111
Laplace transform of, 32.58
limits and, 24.74, 24.75, 24.78 to
24.81,24.83
Maclaurin series and, 39.1, 39.7,
39.18, 39.21, 39.24, 39.25,
39.32, 39.35, 39.44, 39.50
power series and, 38.39 to 38.42,
38.76, 38.78 to 38.80, 38.82,
38.97
properties of, 24.1 to 24.6, 24.35,
24.65 to 24.71
(See also Natural logarithms)
Exponential growth and decay:
bacterial growth, 26.2, 26.20,
26.27, 26.29, 26.37
compound interest, 26.8 to
26.14, 26.40 to 26.42
decay and growth constants for,
26.1
defined, 26.1
other applications, 26.3, 26.6,
26.22 to 26.24, 26.31,26.33
to 26.36, 26.39
population growth, 26.5, 26.7,
26.19, 26.25,26.26,26.28,
26.38