ax, derivatives of, 71–77
centripetal, 22
curves, 239
integrals, 119
addition
of complex numbers, 37
formulas, for cosine, 92
formulas, for sine, 92
of functions, 5
of matrices, 205
of vectors, 199–202
algebra, matrices and linear, 217–224
alternating series, 186
angles
double angle formula, 15
measurements, 11
polar, 36
rate of revolutions, 21
between vectors, 210
velocity, 21
antiderivatives, 117–120
applications
of exponential equations, 77–80
integrals, 157–161
approximating
areas, 127
linear approximations, 250–254
quadratic approximations, 250–254
slopes of curves, 109–112
arcs
of cycloid, 234
length of, 10–11
Arcsine, 19
Arctangent, 19
areas
approximating, 127
below X-axis, 131–134
under curves, 124–127
of cycloid, 235
derivatives, 60–63
integrals, 140–145
rectangles, 125
arithmetic
progressions, 180
series, 181
arrays, 203
associations, 3
errors, sums and, 128–131
asymptotes, 45
averages
flow, 299
slopes, 57
values, integrals, 131
axes, 148
base 10 logarithms, 9
base numbers, changing from one to the other, 10
binomial expressions, 194
Binomial Theorem, 194
boundary conditions, 324
calculating. See also formulas
derivatives, 72
functions, 109
integrals, 121
Cartesian coordinates, 33, 228, 296
Cauchy-Riemann equations, 287
centers
of mass, 159–161
of the series, 188
centripetal acceleration, 22
chain rule, 69
partial derivatives, 246–247
changing
coordinates, 152–157
circles, 10
areas of, 140
circular functions, 10
circular motion, 20–25
convergence, 193
on coordinate systems, 35
distance between points, 13
ellipses and, 42–44
length of, 229
motion of particles in, 231
particles moving around, 24
radius. See also radius
trigonometric functions, 11
circumference, 140, 229. See also circles
closed curves, 282
closeness of functions, 47
coefficients
constants, 308
columns, 147
multiplication, 207
common differences, 180
common integrals, 138–139
common ratios, 181
comparing integrals, 136
Comparison Tests, 123–124, 184–185
complex numbers, 36–38
complex planes, 38
components, 196–197
composite functions, 6
chain rule, 87
compound functions, 6
concave-down shapes, 94
concave-up shapes, 94
for infinite series, 183
initial, 324
conservative fields, 274
vectors, 276–282
coefficients, 308
derivatives, 63–64
functions, 66–67
integrals, 134–137
variables, 66–67
velocity, 61
constrained optimization, 263–265
continuable functions, 50
continuity, 50–52
continuous functions, 50–51
circles, 193
radius, 189
tests, 183–184
coordinate systems, 33–36
changing, 152–157
cylindrical, 149
polar, 156
rectangles, 224
spheres, 227
trigonometric functions, 10
vectors, 197
derivatives, 95
hyperbolic, 28
addition formulas for, 14
directions, 199
hyperbolic, 26
inverse, 98
derivatives, 95
hyperbolic, 28
critical points, 261
cross products, 211–215
cubes, 148
volume, 215
curl, 300–304
acceleration, 239
areas under, 124–127
closed, 282
continuity, 50
cosine, 24
flux, 288
length of, 139
level, 31
moving, 232
orientation of, 277
sections of, 140
sine, 24
slopes, approximating, 109–112
cycloid, 233–234
cylinders, 226
shells, volume of, 146
decay models, 77–80
definite integrals, 120–122, 127, 163, 277
definitions
of determinants, 215–217
of functions, 3–8
of limits, 47–50
of matrices, 202–205
partial derivatives, 244
of rate of change, 53–56
of relationships, 25–26
of vectors, 195–199
degrees, 10
polynomial functions, 28
delta (Δ)notation, 57–58
product rules, 83
denominators, 28
derivatives, 53
of ax, ex, and x, 71–77
antiderivatives, 117–120
areas, 60–63
chain rule, 86–90
constants, 63–64
cosecant, 95
cotangents, 95
delta (Δ) notation, 57–58
differentiating, 68–69
directional, 268–270
directional, gradients and, 255–259
distance, 60–63
first, 103–109
formulas, 64–66
functions, 69–71
hyperbolic functions, 99–100
inverse functions, 95–99
maximum, 103–109
minimum, 103–109
multiple, 82
of multiple products, 85
of natural logarithms, 76
negatives, 94
partial. See partial derivatives
points, 59
positive, 94
powers, 69–71
of products, 82–85
of quotients, 85–86
rate problem examples, 90–91
reciprocal functions, 89
secant, 95
slopes, 60–63
tangents, 95
variables, 66–67
vectors, 237
velocity, 60–63
zeroes, 104
determinants
definitions, 215–217
Jacobian, 154
diagrams, contour, 29–33, 274. See also graphs
differences, 80–81
common, 180
differentiability, 52–53
differential equations, 250–254, 307
first-order, 308–312
high-order linear, 314–317
Laplace transform method, 321–322
numerical methods for solving, 322–323
partial, 324–325
second-order linear, 312–314
series, 317–318
systems of, 319–321
differentiating, 60
chain rule, 86–90
derivatives, 68–69
explicit functions, 101–102
hyperbolic functions, 99–100
implicit functions, 101–102
matrices, 205
multivariable functions, 101
numerical methods, 111
quotients, 95
rules, 102–103
series, 189
sums, 80–81
trigonometric functions, 91–95
directional derivatives, 268–270
and gradients, 255–259
directions, 195
cosines, 199
of lines, 230
motion, 238
vectors, 199
discontinuous functions, 51
discontinuous integrands, 123
displacement vectors, 196, 210, 235, 279
distance, 210
center of mass, 159–161
derivatives, 60–63
horizontal, 233
integrals, 124
between points, 12
between points, circles, 13
between two points, 10
distributions, 161
series, 185
division
of complex numbers, 37
of functions, 5
domains, 180
sets, 4
trigonometric function values, 20
gradients, 258
double integrals, 136–138, 154
double-subscript notation, 204
dummy variables, 114
ex, derivatives of, 71–77
elements
in matrices, 203
in range sets, 3
ellipses and circles, 42–44
entries, matrices, 203
equal matrices, 204
equalpotentials, 275
equations, 3
for asymptotes, 45
Cauchy-Riemann, 287
Cramer’s rule, 215–216
differential. See differential equations
exponential, applications of, 77–80
Laplace’s, 287
linear, 218
for line normals, 60
of lines, 230
non-linear, 7
for parabolas, 39
planes, 226
for planes, 30
quadratic, 41
subsidiary, 321
equivalent vectors, 197
errors
associated errors, sums and, 128–131
trapezoid rules, 129
estimating
infinite series, 183
infinite series, sums of, 49
regions, 59
Euler’s formula, 193
Euler’s identity, 26
Euler’s method, 322
evaluating
derivatives, 63–64
integration, 162–177
partial derivatives, 243–246
even functions, 7, 27, 131–134
exact equations, 310
exact forms, 310
examples, rate problem, 90–91
expanding
series, 188–194
trigonometric functions, 38
explicit functions, differenti- ating, 101–102
exponential equations, appli- cations of, 77–80
exponential functions, 3, 8–10, 51, 71, 192
relationships, 25–26
expressions, 3
binomial, 194
polynomial functions, 28
extrema
local, 259
points, 104
factoring polynomials, 48
families
of functions, 119
of solutions, 308
fields
conservative, 276–282
divergence, 295
gradients, 274
lines, 271–276
slopes, 119
vectors, 271–276
finite arithmetic progressions, 180
finite sequences, 179
first derivatives, 103–109
first-order differential equations, 308–312
flow
averages, 299
fields, 275
lines, 276
flux, 287–295
foci, 43
focus points, 41
force, 159
dot products, 210
fields, 274
forms
exact, 310
solution, 308
formulas, 4
Euler’s, 193
integrals, 117–120
products, 83
progressions, 182
fractions, evaluating, 172–177
Frobenius method, 318
functions. See also fields
calculating, 109
chain rule, 86–90
circular, 10
circular motion, 20–25
closeness of, 47
complex numbers, 36–38
compound, 6
constants, 66–67
continuable, 50
coordinate systems, 33–36
derivatives, 69–71
discontinuous, 51
domain sets, 4
explicit, differentiating, 101–102
exponential, 8–10, 51, 71, 192
families of, 119
hyperbolic, 26–28
hyperbolic, differentiating, 99–100
implicit, differentiating, 101–102
integration, 134–136
inverse, 6
inverse, derivatives, 95–99
Lagrangian, 265
local extrema of, 104
logarithmic, 8–10
with multiple variables, 29–33
multivariable, differentiating, 101
non-linear, 6–7
potential, 274
rational, 51
reciprocal, 89
relationships, 25–26
slope, 6
stream, 287
three-variable, 30
trigonometric, 10–20
trigonometric, derivatives, 95–99
trigonometric, differentiating, 91–95
trigonometric, inverse, 19
types of, 3–8
Fundamental Theorem of Calculus, 120–122
Gauss elimination, 220
Gauss-Jordon elimination, 220
Gauss’s Divergence Theorem, 298
generalized Ratio Tests, 186
general solutions, 308
geometry
progressions, 181
globals, 104
gradients, 268–270
directional derivatives, and, 255–259
fields, 274
graphs
continuous functions, 51
of inverse functions, 96
limits, 47
multiple variables, 29–30
pairs, 5
parabolas, 41
partial derivatives, 247–250
of trigonometric functions, 10, 16
gravitational fields, 274
gravity, 277
growth models, 77–80
Harmonic Series, 185
height, maximum, 233
Heun’s method, 323
high-order linear differential equations, 314–317
homogeneous systems, 218
horizontal distances, 233
horizontal fields, 272
hyperbolas, 44
hyperbolic functions, 26–28
differentiating, 99–100
hypotenuse, 33
identity matrices, 219
implicit functions, 101–102
improper integrals, 122–124
Improved Euler method, 323
increments of time, 54
indefinite integrals, 117–120, 135
independent variables, 29, 243
infinite geometric series, 182
infinite sequences, 179
sums of, 49
infinity, convergence, 122
initial value problems, 309
inner products, 208
instantaneous rate of change, 56
instantaneous velocity, 57
integrals, 113
acceleration, 119
applications, 157–161
areas, 140–145
areas under curves, 124–127
average values, 131
calculating, 121
common, 138–139
comparing, 136
definite, 120–122, 127, 163, 277
double, 154
evaluating, 162–177
formulas, 117–120
improper, 122–124
length, 139–140
lines, 276–282
multiple, 136–138
properties, 134–136
splitting, 123
sums and sigma (θ) notation, 114–117
surfaces, 287–295
triple, 154
volume, 145–152
Integral Test, 187
discontinuous, 123
functions, 134–136
limits of, 153
by parts, 162
intervals, 19
X-axis, 148
inverse cosine, 98
inverse functions, 6
derivatives, 95–99
inverse matrices, 219
inverse sine, 98
inverse trigonometric functions, 10, 19
iteration, Picard’s iteration method, 311
Jacobian determinants, 154
Jacobian J method, 153
J factor, 155
jumps, 51
Lagrange multipliers, 264
Lagrangian function, 265
Laplace’s equations, 287
Laplace transform method, 321–322
length, 195–196
of arcs, 10–11
of cycloid, 234
of integrals, 139–140
of quarter circles, 229
of vectors, 213
level curves, 31
limits
continuity, 50–52
definition of, 47–50
differentiability, 52–53
of integration, 153
switching, 135
linear algebra, matrices, 217–224
linear approximations, 250–254
linearity principles, 313
linearizing regions, 59
linear one-variable functions, 30
linear two-variable functions, 30
lines
contours, 31
curves. See curves
fields, 271–276
flow, 276
integrals, 276–282
motion in, 230
normals, 60
parametric equations for, 224
secant, 60
straight, 62
tangents, equations, 109
vertical line tests, 5
local extrema, 259
of functions, 104
local linearity, 109–112, 250–254
logarithms
natural, 75
Maclaurin series, 188–189, 191
magnitude, vectors, 195–196, 208, 237
major axis, 43
mapping
contours, 31
radius, 20
mass, center of, 159–161
matrices
of coefficients, 216
definitions, 202–205
identity, 219
inverse, 219
linear algebra, 217–224
multiplication, 205–208
transposing, 219
triangles, 220
maxima, 259–263
maximum derivatives, 103–109
maximum height, 233
measurements
of angles, 11
of Z-axis, 34
methods
Euler’s, 322
Frobenius, 318
Heun’s, 323
Jacobian J, 153
Laplace transform, 321–322
numerical, 109–112
numerical differentiation, 111
numerical for solving differential equations, 322–323
Picard’s iteration, 311
products, 324
Runge-Kutta method of fourth-order, 323
midpoint rule, 128
minima, 259–263
minimum derivatives, 103–109
models
decay, 77–80
growth, 77–80
motion, 230–241
circular, 20–25
directions, 238
oscillatory, 22
polar coordinates, 239
moving. See also motion
curves, 232
objects, 224
particles, 235
multiple derivatives, 82
multiple integrals, 136–138
multiple products, derivatives of, 85
multiple variables, functions with, 29–33
multiplication
complex numbers, 37
of functions, 5
matrices, 205–208
scalars, 206
vectors, 205–208
multipliers, Lagrange, 264
multivariable functions, differentiating, 101
derivatives of, 76
negatives
derivatives, 94
of vectors, 200
Newton’s method for equations, 109, 111
non-homogeneous systems, 218
non-linear
equations, 7
functions, 6–7
normals
line, 60
vectors, 238
notation
for delta (Δ), 57–58
for delta (Δ), product rules, 83
for derivatives, 53
for directional derivatives, 255
for double-subscript, 204
of gradients, 257
for integration, 162
for line integrals, 278
for matrices, 203
for multiple derivatives, 82
for sigma (Δ), 114–117
for vectors, 196
nth derivatives, 54
nth-order homogeneous differential equations, 315
nth-order homogeneous linear differential equations, 315
numbers
complex, 36–38
in range sets, 3
sequences, 114
numerical differentiation methods, 111
numerical methods, 109–112
for solving differential equations, 322–323
objects
moving, 224
volume of, 145
odd functions, 7, 8, 27, 131–134
one-variable functions, 30, 260
optimization, 259–263
constrained, 263–265
orientation, 277
oscillatory motion, 22
velocity, 23
outflow, 297
parabolas, 38–42
paralleledpipeds, volume of, 214
parallelograms
areas, 291
areas of, 213
parallel vectors, 209
parameters, 278
flux, 294
planes, 225
partial derivatives
chain rule, 246–247
evaluating, 243–246
graphs, 247–250
representation, 243–246
partial differential equations, 324–325
partial fractions, evaluating, 172–177
particles
motion in planes, 230
moving around circles, 24
speed of, 238
velocity, 235
parts, integration by, 162
perpendicular vectors, 209
perspective in contour diagrams, 32
Picard’s iteration method, 311
planes, 30
complex, 38
equations, 226
points, 225
tangents, 261
XY-planes, 141
plotting, spiral of Archimedes, 35
points
critical, 261
derivatives, 59
discontinuous functions, 51
distances between, 12–13
extrema, 104
focus, 41
planes, 225
saddle, 263
slopes, 57
trigonometric functions, 10
in vector fields, 272
vertex, 39
polar angles, 36
motion, 239
polynomial functions, 28, 50, 80–81
factoring, 48
integrals, 134–136
position vectors, 197, 224–229, 237, 273
positive derivatives, 94
potential functions, 274
powers
derivatives, 69–71
second, functions raised to, 85
variables raised to, 66–67
pressure, 157–159
probabilities, 161
products
cross, 211–215
derivatives of, 82–85
dot, 207–211
dot, gradients, 258
inner, 208
methods, 324
representations of, 117
rules, 162
scalars. See scalars
vectors, 211–215
progressions, 179–182
projectiles, 232
properties, 3
of addition, matrices and vectors, 205
of cross products, 215
of dot products, 211
of hyperbolic cosines, 27
of integrals, 134–136
of line integrals, 279
of multiplication of matrices, 205
of sums, 115
Pythagorean Theorem, 33
quadratic approximations, 250–254
quadratic equations, 41
quarter circles, length of, 229
quotients
derivatives, 85–86
differentiating, 95
radius, 12
of convergence, 189
mapping, 20
trigonometric functions, 10
ranges
sets, 3
trigonometric function values, 20
rates
problem examples, 90–91
rational functions, 51
ratios, 181
trigonometric functions, 10
Ratio Tests, 185–186
r-component, 34
reciprocal functions, 89
rectangles
areas, 125
regions, 141
volume, 149
regions
linearizing, 59
rectangles, 141
relations, 3
trigonometric formulas and, 16
relationships
between coordinate systems, 33
functions, 25–26
between trigonometric/exponential functions, 25–26
representations
graphs, 247–250
partial derivatives, 243–246
of products, 117
revolutions
angular rate of, 21
volume of, 146
right-hand screw rule, 211
roots, 112
Root Test, 188
rotation fields, 273
rows
multiplication, 207
vectors, 196
rules, 3
chain, partial derivatives, 246–247
differentiating, 102–103
midpoint, 128
quotients, 85–86
right-hand screw, 211
Simpson’s, 130
trapezoids, 129
Runge-Kutta method, 323
saddle points, 263
saddle-shaped surfaces, 32
scalars, 195, 208–211, 267–268
gradients, 256
multiplication, 206
derivatives, 95
hyperbolic, 28
lines, 60
second derivatives, 81–82, 103–109, 260
second-order linear differential equations, 312–314
second power, functions raised to, 85
sections of curves, 140
separable equations, 309
separation of variables, 325
series, 179–182
alternating, 186
differential equations, 317–318
expanding, 188–194
geometric, 193
Harmonic Series, 185
tests, 187–188
sets
domain, 4
range, 3
sigma (θ) notation, 114–117
Simpson’s rule, 130
addition formulas for, 15
hyperbolic, 27
inverse, 98
slopes, 24
wave patterns, 23
slicing surfaces, 31
slopes, 24
curves, approximating, 109–112
of cycloid, 234
derivatives, 60–63
directional derivatives, 258
fields, 119
functions, 6
points, 57
secant lines, 60
of tangent lines, 58–60
solutions, 308
specified conditions, 308
speed, 54
of particles, 238
spheres, 227
coordinates, 36
spiral of Archimedes, 35
splitting integrals, 123
squares, matrices, 203
straight lines, 62
motion in, 230
stream functions, 287
strips, 127
subsidiary equations, 321
substitution integrals, 164–172
subtraction
complex numbers, 37
of functions, 5
matrices, 205
vectors, 199–202
sums
of areas, 125
associated errors, 128–131
differentiating, 80–81
of infinite series, 49
integrals, 134–136
of integrals, 136
limits of, 116
properties of, 115
and sigma (θ) notation, 114–117
of vectors, 200
superposition, 313
integrals, 287–295
saddle-shaped, 32
switching limits, 135
symmetric matrices, 204
symmetry, axis of, 40
systems of differential equations, 319–321
tables, 4
tangents, 10
derivatives, 95
hyperbolic, 28
lines, equations, 109
planes, 261
vectors, 237
velocity, 22
Taylor series, 188–189, 191, 253
terms, 179
tests
Comparison Tests, 123–124, 184–185
convergence, 183–184
Integral Test, 187
Ratio Tests, 185–186
Root Test, 188
series, 187–188
vertical line, 5
three-dimensional space, 13
three-variable function, 30
time rate of change, 54
torque, 214
total volume, 149. See also volume
transposing matrices, 197, 204, 219
trapezoid rules, 129
triangles
matrices, 220
trigonometric functions, 10–11
wedge sections, 35
trigonometric functions, 3, 10–20
differentiating, 91–95
expansions of, 38
inverse, 19
inverse, derivatives, 95–99
relationships, 25–26
triple integrals, 154
turning directions, 238
two-variable functions, 30, 141
types
of functions, 3–8
of vectors, 196
units
tangent vectors, 238
values, 4
averages, integrals, 131
of cross products, 213
trigonometric functions, 20
constants, 66–67
derivatives, 66–67
dummy, 114
functions with multiple, 29–33
separation of, 325
two-variable functions, 141
vectors, 267–268
addition and subtraction, 199–202
columns, 220
conservative fields, 276–282
definitions, 195–199
displacement, 235
divergence, 295
fields, 271–276
multiplication, 205–208
normal, 238
products, 211–215
acceleration, 119
angular, 21
averages, 57
derivatives, 60–63
distances, increasing, 61
fields, 275
instantaneous, 57
integrals, 124
of oscillatory motion, 23
particles, 235
second derivatives, 81
tangents, 22
vectors, 196
vertical line tests, 5
vertices, 45
points, 39
volume
cubes, 215
integrals, 145–152
of paralleledpiped, 214
rectangles, 149
spheres, 153
waveforms, 12
wave patterns, 23
along a curve, 279
x, derivatives of, 71–77
areas below, 131–134
derivatives, 71
integrals, 119
intervals, 148
XY-planes, 141
Z-axis, 34
distance measurements, 227
flux, 289
integrals, 147
z-component, 34
zeroes
convergence, 49
denominators, 28
derivatives, 104
derivatives of constants, 64
divergence, 297
limits, 48
vectors, 197