450 0 INDEX
Liebniz's formula for differentiable
functions, 42.96, 42.97
Limacons, 35.31, 35.38 to 35.40,
35.56, 35.59, 35.60
Limit comparison test for infinite
series, 37.42 to 37.47, 37.54,
37.76, 37.78, 37.82, 37.84,
37.86, 37.88 to 37.90, 37.96,
37.98, 37.100, 37.101, 37.108
Limits:
addition property of, 6.13
asymptotes and (see
Asymptotes)
cube roots in, 6.50, 6.51
defined, 6.1
exponential functions, 24.74,
24.75, 24.78 to 24.81, 24.83
infinite sequences, 36.1 to 36.53,
36.65
left-hand, 6.5, 6.33 to 6.36, 6.39
L'Hopital's rule for, 25.1 to
25.53, 32.4, 32.6 to 32.8,
32.20,32.36,36.15,36.20,
37.47,37.108,38.15,38.24,
41.56,41.57
multivariate functions, 41.48 to
41.63
natural logarithms, 22.43 to
22.45, 23.52, 23.71, 23.72
one-sided, 6.5, 6.33 to 6.36, 6.39
polynomials, 6.2 to 6.4, 6.6 to
6.8, 6.10, 6.11,6.12, 6.14 to
6.16, 6.17 to 6.19, 6.20 to
6.22, 6.28, 6.32, 6.37, 6.42,
6.43 to 6.49
product rule for, 36.50
right-hand, 6.5, 6.33 to 6.36,
6.39
square roots in, 6.9, 6.23 to 6.27,
6.29 to 6.31, 6.38, 6.40,
6.41,6.52
sum rule for, 36.49
trigonometric functions, 10.14 to
10.16, 10.30, 10.31, 10.44,
10.48
two-dimensional vector
functions, 34.43
Line integrals, 45.40 to 45.44
Line segments and lines:
as contours (level curves), 41.33,
41.39,41.41,41.43
curvature (K) of, 34.96
curvilinear motion and, 34.69,
34.70
distance between, 40.113,
40.114,43.31,43.38
distance from points to, 3.45,
3.47,4.28,4.30, 33.9,33.10,
33.34
families of, 3.39, 3.40
Line segments and lines (Cont.):
graphs of, 3.16, 3.36, 5.5, 5.12,
5.14, 5.15 to 5.18, 5.20,
5.23, 5.88, 6.33, 7.2, 7.4,
7.5, 7.8, 8.43, 10.46, 11.49,
16.45,34.9, 34.10, 35.10 to
35.12,35.19,35.20,40.21
intersections of, 3.44, 3.78, 3.79,
10.46, 10.47, 40.76
intersections with hyperbolas,
3.81
intersections with parabolas, 3.80
midpoints of, 3.33, 11.45, 20.91,
40.5
normal, 8.16, 8.19, 8.33, 8.36,
8.37,9.14,9.19, 10.23,
10.28, 34.27,42.117
parallel, 3.12 to 3.15, 3.23, 3.24,
3.66, 3.68, 8.23, 40.72 to
40.74
parametric equations of, 34.9,
34.10, 40.69 to 40.79, 40.85
perpendicular, 3.20 to 3.23, 3.25,
3.26, 3.31,3.69,40.75,40.77
perpendicular bisectors of, 3.35,
3.74
planes cut by, 40.99 to 40.102,
40.104,40.105
point-slope equations of, 3.2,
3.48 to 3.51, 10.22, 10.23,
10.25, 10.28
points above and below, 3.36 to
3.38
points on, 3.83
in polar coordinates, 35.10 to
35.12, 35.19, 35.20
rectangular equations of, 40.69,
40.71, 40.72, 40.97, 40.99,
40.101,40.113
reflections of, 5.97 to 5.99
slope-intercept equations of, 3.6
to 3.11, 3.20, 3.21,3.27, 3.29
to 3.31, 3.34, 3.35, 3.41,3.44,
3.46, 3.52 to 3.60, 8.13,8.16,
8.36,9.13,9.14
slope of, 3.1,3.5, 3.46, 3.61 to
3.65
tangent, 4.15, 4.19 to 4.22, 8.13 to
8.15,8.18,8.20,8.23,8.28,
8.34,8.35,8.38,8.42,9.13.
9.19,9.23, 10.22, 10.23,
10.25, 10.28, 10.32 to 10.35,
10.47, 11.44, 12.14, 12.16,
12.18, 12.27, 12.28, 12.37,
12.42 to 12.46, 15.21, 15.28,
15.29, 19.66 to 19.70, 19.77,
19.83 to 19.86,21.46,24.56,
27.58,34.26,34.31,35.89to
35.92, 35.95, 35.98, 42.26 to
42.33,42.119,45.1,45.4,45.6
Line segments and lines (Cont.):
through two points, 3.16 to 3.19
vector representation of, 40.69,
40.71,40.72
vectors and, 33.7, 33.8
Logarithmic differentiation, 23.23
to 23.26, 23.58, 23.67, 24.35
to 24.39
Logarithmic functions (see Natural
logarithms)
Logarithmic spirals, 34.42
Maclaurin series:
exponential functions and, 39.1,
39.7, 39.18,39,21, 39.24,
39.25, 39.32, 39.35, 39.44,
39.50
higher-order derivatives and,
39.30 to 39.32, 39.43
hyperbolic functions and,
39.14
inverse trigonometric functions
and, 39.10, 39.13, 39.31
natural logarithms and, 39.5,
39.20, 39.28, 39.34, 39.40,
39.46, 39.50
powers or roots, 39.12, 39.33,
39.39, 39.42, 39.43
trigonometric functions and,
39.2, 39.8, 39.9, 39.15,
39.19, 39.23, 39.26, 39.27,
39.29, 39.36, 39.48
Magnitude of vectors, 33.3, 33.13
Mass:
center of (see Centroids)
by integration, 44.74 to 44.79
Mathematical induction, 2.18, 20.3,
20.65, 24.89, 24.90, 32.22,
43.20
Maxima and minima of functions
(see Extrema of functions)
Mean, arithmetic and geometric,
43.51
Mean value theorem:
for functions, 11.10 to 11.17,
11.30, 11.33to 11.36, 11.38
to 11.41, 11.43, 11.45, 11.46
for integrals, 20.34 to 20.37,
20.42
Medians of triangles, 3.33, 3.43,
3.72, 33.23
Method of Lagrange multipliers,
43.56 to 43.64, 43.66, 43.67
Method of partial fractions for
integrals, 30.1 to 30.33
Midpoints:
of lines, 3.33, 11.45, 20.91, 40.5
of triangles, 33.27
Minima and maxima of functions
(see Extrema of functions)
Liebniz's formula for differentiable
functions, 42.96, 42.97
Limacons, 35.31, 35.38 to 35.40,
35.56, 35.59, 35.60
Limit comparison test for infinite
series, 37.42 to 37.47, 37.54,
37.76, 37.78, 37.82, 37.84,
37.86, 37.88 to 37.90, 37.96,
37.98, 37.100, 37.101, 37.108
Limits:
addition property of, 6.13
asymptotes and (see
Asymptotes)
cube roots in, 6.50, 6.51
defined, 6.1
exponential functions, 24.74,
24.75, 24.78 to 24.81, 24.83
infinite sequences, 36.1 to 36.53,
36.65
left-hand, 6.5, 6.33 to 6.36, 6.39
L'Hopital's rule for, 25.1 to
25.53, 32.4, 32.6 to 32.8,
32.20,32.36,36.15,36.20,
37.47,37.108,38.15,38.24,
41.56,41.57
multivariate functions, 41.48 to
41.63
natural logarithms, 22.43 to
22.45, 23.52, 23.71, 23.72
one-sided, 6.5, 6.33 to 6.36, 6.39
polynomials, 6.2 to 6.4, 6.6 to
6.8, 6.10, 6.11,6.12, 6.14 to
6.16, 6.17 to 6.19, 6.20 to
6.22, 6.28, 6.32, 6.37, 6.42,
6.43 to 6.49
product rule for, 36.50
right-hand, 6.5, 6.33 to 6.36,
6.39
square roots in, 6.9, 6.23 to 6.27,
6.29 to 6.31, 6.38, 6.40,
6.41,6.52
sum rule for, 36.49
trigonometric functions, 10.14 to
10.16, 10.30, 10.31, 10.44,
10.48
two-dimensional vector
functions, 34.43
Line integrals, 45.40 to 45.44
Line segments and lines:
as contours (level curves), 41.33,
41.39,41.41,41.43
curvature (K) of, 34.96
curvilinear motion and, 34.69,
34.70
distance between, 40.113,
40.114,43.31,43.38
distance from points to, 3.45,
3.47,4.28,4.30, 33.9,33.10,
33.34
families of, 3.39, 3.40
Line segments and lines (Cont.):
graphs of, 3.16, 3.36, 5.5, 5.12,
5.14, 5.15 to 5.18, 5.20,
5.23, 5.88, 6.33, 7.2, 7.4,
7.5, 7.8, 8.43, 10.46, 11.49,
16.45,34.9, 34.10, 35.10 to
35.12,35.19,35.20,40.21
intersections of, 3.44, 3.78, 3.79,
10.46, 10.47, 40.76
intersections with hyperbolas,
3.81
intersections with parabolas, 3.80
midpoints of, 3.33, 11.45, 20.91,
40.5
normal, 8.16, 8.19, 8.33, 8.36,
8.37,9.14,9.19, 10.23,
10.28, 34.27,42.117
parallel, 3.12 to 3.15, 3.23, 3.24,
3.66, 3.68, 8.23, 40.72 to
40.74
parametric equations of, 34.9,
34.10, 40.69 to 40.79, 40.85
perpendicular, 3.20 to 3.23, 3.25,
3.26, 3.31,3.69,40.75,40.77
perpendicular bisectors of, 3.35,
3.74
planes cut by, 40.99 to 40.102,
40.104,40.105
point-slope equations of, 3.2,
3.48 to 3.51, 10.22, 10.23,
10.25, 10.28
points above and below, 3.36 to
3.38
points on, 3.83
in polar coordinates, 35.10 to
35.12, 35.19, 35.20
rectangular equations of, 40.69,
40.71, 40.72, 40.97, 40.99,
40.101,40.113
reflections of, 5.97 to 5.99
slope-intercept equations of, 3.6
to 3.11, 3.20, 3.21,3.27, 3.29
to 3.31, 3.34, 3.35, 3.41,3.44,
3.46, 3.52 to 3.60, 8.13,8.16,
8.36,9.13,9.14
slope of, 3.1,3.5, 3.46, 3.61 to
3.65
tangent, 4.15, 4.19 to 4.22, 8.13 to
8.15,8.18,8.20,8.23,8.28,
8.34,8.35,8.38,8.42,9.13.
9.19,9.23, 10.22, 10.23,
10.25, 10.28, 10.32 to 10.35,
10.47, 11.44, 12.14, 12.16,
12.18, 12.27, 12.28, 12.37,
12.42 to 12.46, 15.21, 15.28,
15.29, 19.66 to 19.70, 19.77,
19.83 to 19.86,21.46,24.56,
27.58,34.26,34.31,35.89to
35.92, 35.95, 35.98, 42.26 to
42.33,42.119,45.1,45.4,45.6
Line segments and lines (Cont.):
through two points, 3.16 to 3.19
vector representation of, 40.69,
40.71,40.72
vectors and, 33.7, 33.8
Logarithmic differentiation, 23.23
to 23.26, 23.58, 23.67, 24.35
to 24.39
Logarithmic functions (see Natural
logarithms)
Logarithmic spirals, 34.42
Maclaurin series:
exponential functions and, 39.1,
39.7, 39.18,39,21, 39.24,
39.25, 39.32, 39.35, 39.44,
39.50
higher-order derivatives and,
39.30 to 39.32, 39.43
hyperbolic functions and,
39.14
inverse trigonometric functions
and, 39.10, 39.13, 39.31
natural logarithms and, 39.5,
39.20, 39.28, 39.34, 39.40,
39.46, 39.50
powers or roots, 39.12, 39.33,
39.39, 39.42, 39.43
trigonometric functions and,
39.2, 39.8, 39.9, 39.15,
39.19, 39.23, 39.26, 39.27,
39.29, 39.36, 39.48
Magnitude of vectors, 33.3, 33.13
Mass:
center of (see Centroids)
by integration, 44.74 to 44.79
Mathematical induction, 2.18, 20.3,
20.65, 24.89, 24.90, 32.22,
43.20
Maxima and minima of functions
(see Extrema of functions)
Mean, arithmetic and geometric,
43.51
Mean value theorem:
for functions, 11.10 to 11.17,
11.30, 11.33to 11.36, 11.38
to 11.41, 11.43, 11.45, 11.46
for integrals, 20.34 to 20.37,
20.42
Medians of triangles, 3.33, 3.43,
3.72, 33.23
Method of Lagrange multipliers,
43.56 to 43.64, 43.66, 43.67
Method of partial fractions for
integrals, 30.1 to 30.33
Midpoints:
of lines, 3.33, 11.45, 20.91, 40.5
of triangles, 33.27
Minima and maxima of functions
(see Extrema of functions)