454 0 INDEX
Tangent vectors, 34.44 to 34.56,
34.59, 34.60, 34.65 to 34.72,
34.74 to 34.80, 34.89, 34.92,
42.118,42.119,45.1,45.3to
45.6
Tangential acceleration, 34.106 to
34.108
Taylor series:
Lagrange's remainder and, 39.17
to 39.20, 39.22 to 39.25
natural logarithms, 39.6
powers or roots, 39.4, 39.11,
39.37, 39.38, 39.47
trigonometric functions, 39.3,
39.16, 39.22, 39.45
Telescoping of infinite series, 37.10
Temperature scales, 3.71
Tetrahedrons, 22.23, 42.113
Toruses:
equations of, 41.94
volume of, 31.33
Tractrix (curve), 29.42
Trapezoidal rule for approximating
integrals, 20.66, 20.67,
20.70, 23.57
Trapezoids, 3.77
Triangle inequality, 2.18, 2.19,
2.35, 6.13, 33.29, 36.47,
36.49 to 36.51
Triangles:
altitudes of, 3.34, 3.41, 3.73
circles circumscribed about,
16.61
equilateral, 14.36, 16.20, 16.51,
16.60
isosceles, 14.46, 16.20, 16.55
law of cosines for, 14.33, 14.51,
33.41
medians of, 3.33, 3.43, 3.72, 33.23
midpoints of, 33.27
perimeter of, 16.19, 16.20, 16.55,
16.60, 16.61
perpendicular bisectors of, 3.35,
3.74
right (see Right triangles)
vertices of, 40.17
Trigonometric functions:
amplitude of, 10.13, 10.45
antiderivatives (indefinite
integrals) and, 19.11 to
19.13, 19.15, 19.16, 19.20 to
19.22, 19.31, 19.39, 19.40,
19.43, 19.45, 19.47, 19.48,
19.53, 19.55, 19.56, 19.60,
19.78 to 19.82, 19.98 to
19.100, 28.2, 28.5 to 28.12,
28.14,28.15,28.17, 28.18,
28.35 to 28.41, 28.44, 28.45,
28.49,29.1 to 29.16, 29.19 to
29.29, 29.34 to 29.37, 29.45
Trigonometric functions (Con/.):
approximation of, 18.23, 18.25,
18.26, 18.28, 18.30, 18.32,
38.55, 38.75, 38.77, 39.27,
39.29
defined, 10.8
definite integrals and, 20.10,
20.11,20.15, 20.20 to 20.22,
20.24, 20.39, 20.49, 20.60,
20.62, 20.63, 20.79 to 20.84,
28.32 to 28.34, 28.53, 29.17,
29.18,29.30,29.33,32.13,
32.31, 32.32, 32.44, 32.45,
32.51, 32.53, 32.59
derivatives of, 10.17 to 10.29,
10.36 to 10.43, 13.7, 13.15 to
13.17, 13.30to 13.34
extremaof, 13.7, 13.15 to 13.17,
13.30 to 13.34
frequency of, 10.12
graphs of, 10.11 to 10.13, 15.32
to 15.38, 15.51, 15.53,27.1,
27.3, 27.71
higher-order derivatives of, 12.8
to 12.10
implicit differentiation of, 12.16
to 12.18, 12.38 to 12.41,
12.47
inverse (see Inverse
trigonometric functions)
Laplace transform of, 32.59
limits of, 10.14 to 10.16, 10.30,
10.31, 10.44, 10.48
Maclaurin series and, 39.2, 39.8,
39.9, 39.15, 39.19, 39.23,
39.26, 39.27, 39.29, 39.36,
39.48
mean value theorem and, 11.34,
11.39, 11.40, 11.42
period of, 10.12, 10.13, 10.45
power series and, 38.58, 38.59,
38.63, 38.75 to 38.77, 38.79
to 38.83, 38.86, 38.89 to
38.91, 38.96
Rolle's theorem and, 11.47
Taylor series and, 39.3, 39.16,
39.22, 39.45
values of, 10.9, 10.10
wavelength of, 10.12
zeros of, 11.47
(See also Angles)
Trigonometric substitutions in
integrals, 29.3, 29.5, 29.19 to
29.21,29.23 to 29.27, 29.29,
29.30, 29.38 to 29.41, 29.43
to 29.45
Triple integrals, 44.5, 44.50 to
44.52, 44.59 to 44.73, 44.76
to 44.78, 44.83 to 44.85,
44.88, 44.91, 44.92
Unit tangent vector, 34.47 to 34.50,
34.81 to 34.93, 34.105 to
34.107
Unit vectors, 33.24, 34.67, 40.28,
40.31,40.33,40.40,40.54,
43.66
Vector convergence, 34.43
Vector functions:
chain rule for, 34.61
curl, 45.28 to 45.31, 45.33, 45.34,
45.36 to 45.38
divergence, 45.25 to 45.27, 45.30
to 45.32, 45.34, 45.35, 45.37,
45.39, 45.47 to 45.49
Gauss' theorem for, 45.49
gradient, 43.1 to 43.22, 43.56 to
43.64, 43.66, 43.67, 45.24,
45.32, 45.33, 45.35, 45.36,
45.39, 45.45
Green's theorem for, 45.47 to
45.49
Laplacian, 45.32
product rule for, 34.54, 34.57,
34.73, 34.92
quotient rule for, 34.73
Vector projection of vectors, 33.4,
33.20, 33.39, 40.33
Vectors:
acceleration, 34.51, 34.53, 34.56,
34.69, 34.71, 34.74 to 34.80,
34.83 to 34.86, 34.92, 34.106
to 34.108, 45.12, 45.16,
45.19,45.20,45.22,45.23
addition and subtraction of, 33.2,
33.3, 33.17, 33.21, 33.22,
40.30
angle between, 33.3, 33.13,
33.33, 33.35, 33.38, 33.40,
40.35, 40.37, 40.38
between two points, 40.25
binormal, 45.17
Cauchy's inequality for, 33.28,
33.29
components of, 33.4, 33.5
cross product of, 40.40 to 40.43,
40.45 to 40.68, 45.11,45.12,
45.17to45.20, 45.22, 45.23,
45.28, 45.36, 45.37, 45.39
defined, 33.1
direction cosines of, 40.27 to
40.29, 40.32
direction of, 33.3, 33.13, 33.25
distance from points to lines
with, 33.9, 33.10, 33.34
dot product of, 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,
Tangent vectors, 34.44 to 34.56,
34.59, 34.60, 34.65 to 34.72,
34.74 to 34.80, 34.89, 34.92,
42.118,42.119,45.1,45.3to
45.6
Tangential acceleration, 34.106 to
34.108
Taylor series:
Lagrange's remainder and, 39.17
to 39.20, 39.22 to 39.25
natural logarithms, 39.6
powers or roots, 39.4, 39.11,
39.37, 39.38, 39.47
trigonometric functions, 39.3,
39.16, 39.22, 39.45
Telescoping of infinite series, 37.10
Temperature scales, 3.71
Tetrahedrons, 22.23, 42.113
Toruses:
equations of, 41.94
volume of, 31.33
Tractrix (curve), 29.42
Trapezoidal rule for approximating
integrals, 20.66, 20.67,
20.70, 23.57
Trapezoids, 3.77
Triangle inequality, 2.18, 2.19,
2.35, 6.13, 33.29, 36.47,
36.49 to 36.51
Triangles:
altitudes of, 3.34, 3.41, 3.73
circles circumscribed about,
16.61
equilateral, 14.36, 16.20, 16.51,
16.60
isosceles, 14.46, 16.20, 16.55
law of cosines for, 14.33, 14.51,
33.41
medians of, 3.33, 3.43, 3.72, 33.23
midpoints of, 33.27
perimeter of, 16.19, 16.20, 16.55,
16.60, 16.61
perpendicular bisectors of, 3.35,
3.74
right (see Right triangles)
vertices of, 40.17
Trigonometric functions:
amplitude of, 10.13, 10.45
antiderivatives (indefinite
integrals) and, 19.11 to
19.13, 19.15, 19.16, 19.20 to
19.22, 19.31, 19.39, 19.40,
19.43, 19.45, 19.47, 19.48,
19.53, 19.55, 19.56, 19.60,
19.78 to 19.82, 19.98 to
19.100, 28.2, 28.5 to 28.12,
28.14,28.15,28.17, 28.18,
28.35 to 28.41, 28.44, 28.45,
28.49,29.1 to 29.16, 29.19 to
29.29, 29.34 to 29.37, 29.45
Trigonometric functions (Con/.):
approximation of, 18.23, 18.25,
18.26, 18.28, 18.30, 18.32,
38.55, 38.75, 38.77, 39.27,
39.29
defined, 10.8
definite integrals and, 20.10,
20.11,20.15, 20.20 to 20.22,
20.24, 20.39, 20.49, 20.60,
20.62, 20.63, 20.79 to 20.84,
28.32 to 28.34, 28.53, 29.17,
29.18,29.30,29.33,32.13,
32.31, 32.32, 32.44, 32.45,
32.51, 32.53, 32.59
derivatives of, 10.17 to 10.29,
10.36 to 10.43, 13.7, 13.15 to
13.17, 13.30to 13.34
extremaof, 13.7, 13.15 to 13.17,
13.30 to 13.34
frequency of, 10.12
graphs of, 10.11 to 10.13, 15.32
to 15.38, 15.51, 15.53,27.1,
27.3, 27.71
higher-order derivatives of, 12.8
to 12.10
implicit differentiation of, 12.16
to 12.18, 12.38 to 12.41,
12.47
inverse (see Inverse
trigonometric functions)
Laplace transform of, 32.59
limits of, 10.14 to 10.16, 10.30,
10.31, 10.44, 10.48
Maclaurin series and, 39.2, 39.8,
39.9, 39.15, 39.19, 39.23,
39.26, 39.27, 39.29, 39.36,
39.48
mean value theorem and, 11.34,
11.39, 11.40, 11.42
period of, 10.12, 10.13, 10.45
power series and, 38.58, 38.59,
38.63, 38.75 to 38.77, 38.79
to 38.83, 38.86, 38.89 to
38.91, 38.96
Rolle's theorem and, 11.47
Taylor series and, 39.3, 39.16,
39.22, 39.45
values of, 10.9, 10.10
wavelength of, 10.12
zeros of, 11.47
(See also Angles)
Trigonometric substitutions in
integrals, 29.3, 29.5, 29.19 to
29.21,29.23 to 29.27, 29.29,
29.30, 29.38 to 29.41, 29.43
to 29.45
Triple integrals, 44.5, 44.50 to
44.52, 44.59 to 44.73, 44.76
to 44.78, 44.83 to 44.85,
44.88, 44.91, 44.92
Unit tangent vector, 34.47 to 34.50,
34.81 to 34.93, 34.105 to
34.107
Unit vectors, 33.24, 34.67, 40.28,
40.31,40.33,40.40,40.54,
43.66
Vector convergence, 34.43
Vector functions:
chain rule for, 34.61
curl, 45.28 to 45.31, 45.33, 45.34,
45.36 to 45.38
divergence, 45.25 to 45.27, 45.30
to 45.32, 45.34, 45.35, 45.37,
45.39, 45.47 to 45.49
Gauss' theorem for, 45.49
gradient, 43.1 to 43.22, 43.56 to
43.64, 43.66, 43.67, 45.24,
45.32, 45.33, 45.35, 45.36,
45.39, 45.45
Green's theorem for, 45.47 to
45.49
Laplacian, 45.32
product rule for, 34.54, 34.57,
34.73, 34.92
quotient rule for, 34.73
Vector projection of vectors, 33.4,
33.20, 33.39, 40.33
Vectors:
acceleration, 34.51, 34.53, 34.56,
34.69, 34.71, 34.74 to 34.80,
34.83 to 34.86, 34.92, 34.106
to 34.108, 45.12, 45.16,
45.19,45.20,45.22,45.23
addition and subtraction of, 33.2,
33.3, 33.17, 33.21, 33.22,
40.30
angle between, 33.3, 33.13,
33.33, 33.35, 33.38, 33.40,
40.35, 40.37, 40.38
between two points, 40.25
binormal, 45.17
Cauchy's inequality for, 33.28,
33.29
components of, 33.4, 33.5
cross product of, 40.40 to 40.43,
40.45 to 40.68, 45.11,45.12,
45.17to45.20, 45.22, 45.23,
45.28, 45.36, 45.37, 45.39
defined, 33.1
direction cosines of, 40.27 to
40.29, 40.32
direction of, 33.3, 33.13, 33.25
distance from points to lines
with, 33.9, 33.10, 33.34
dot product of, 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,