Index

A

a^x, derivatives of, 71–77

acceleration, 54, 81, 230–241

centripetal, 22

curves, 239

integrals, 119

vectors, 236, 241

addition

of complex numbers, 37

formulas, 10, 14

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

of parallelograms, 213, 291

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

velocity, 54, 57

axes, 148

B

base 10 logarithms, 9

base numbers, changing from one to the other, 10

binomial expressions, 194

Binomial Theorem, 194

boundary conditions, 324

C

calculating. See also formulas

derivatives, 72

determinants, 157, 212

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

variables, 116, 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

matrices of, 203, 216

columns, 147

multiplication, 207

vectors, 196, 220

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

conditions, 49, 308

for infinite series, 183

initial, 324

conservative fields, 274

vectors, 276–282

constants, 8, 117

coefficients, 308

derivatives, 63–64

functions, 66–67

integrals, 134–137

integration, 118, 121

variables, 66–67

velocity, 61

constrained optimization, 263–265

continuable functions, 50

continuity, 50–52

continuous functions, 50–51

contour diagrams, 29–33, 274

convergence, 49, 122

circles, 193

intervals, 189, 192

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

cosecant, 10, 17

derivatives, 95

hyperbolic, 28

cosine, 7, 10, 17, 51

addition formulas for, 14

curves, 24, 92

directions, 199

hyperbolic, 26

inverse, 98

cotangents, 10, 17

derivatives, 95

hyperbolic, 28

Cramer’s rule, 126, 215, 220

critical points, 261

cross products, 211–215

cubes, 148

volume, 215

curl, 300–304

curves, 8, 92, 224–229

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

coordinates, 34, 149

shells, volume of, 146

D

decay models, 77–80

definite integrals, 120–122, 127, 163, 277

definitions

of derivatives, 53–56, 72

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

dependent variables, 29, 243

derivatives, 53

of a^x, e^x, and x, 71–77

antiderivatives, 117–120

areas, 60–63

chain rule, 86–90

constants, 63–64

cosecant, 95

cotangents, 95

definitions of, 53–56, 72

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

second, 81–82, 103–109

slopes, 60–63

tangents, 95

variables, 66–67

vectors, 237

velocity, 60–63

zeroes, 104

determinants

calculating, 157, 212

definitions, 215–217

Jacobian, 154

diagrams, contour, 29–33, 274. See also graphs

differences, 80–81

common, 180

quotients, 7, 325

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

divergence, 49, 183, 295–300

series, 185

division

of complex numbers, 37

of functions, 5

domains, 180

sets, 4

trigonometric function values, 20

dot products, 207, 208–211

gradients, 258

double angle formulas, 10, 15

double integrals, 136–138, 154

double-subscript notation, 204

dummy variables, 114

E

e^x, 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

Newton’s method for, 109, 111

non-linear, 7

for parabolas, 39

parameters, 224–229, 232

planes, 226

for planes, 30

quadratic, 41

subsidiary, 321

tangent lines, 59, 109

equivalent vectors, 197

errors

associated errors, sums and, 128–131

trapezoid rules, 129

estimating

infinite series, 183

infinite series, sums of, 49

integrals, 114, 128–131

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

F

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

addition, 10, 14

derivatives, 64–66, 68–69

double angle, 10, 15

Euler’s, 193

integrals, 117–120

products, 83

progressions, 182

Pythagorean, 10, 12

subtraction, 10, 14

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

composite, 6, 87

compound, 6

constants, 66–67

continuable, 50

continuous, 50, 51

coordinate systems, 33–36

derivatives, 69–71

discontinuous, 51

domain sets, 4

even, 7, 27, 131–134

explicit, differentiating, 101–102

exponential, 8–10, 51, 71, 192

families of, 119

graphs, 4, 70

hyperbolic, 26–28

hyperbolic, differentiating, 99–100

implicit, differentiating, 101–102

integration, 134–136

inverse, 6

inverse, derivatives, 95–99

Lagrangian, 265

linear, 6, 63–64

local extrema of, 104

logarithmic, 8–10

with multiple variables, 29–33

multivariable, differentiating, 101

non-linear, 6–7

odd, 7, 8, 27, 131–134

one-variable, 30, 260

polynomial, 28, 50, 80–81

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

two-variable, 30, 141

types of, 3–8

Fundamental Theorem of Calculus, 120–122

G

Gauss elimination, 220

Gauss-Jordon elimination, 220

Gauss’s Divergence Theorem, 298

generalized Ratio Tests, 186

general solutions, 308

geometry

progressions, 181

series, 182, 184, 193

globals, 104

gradients, 268–270

directional derivatives, and, 255–259

fields, 274

graphs

continuous functions, 51

functions, 4, 70

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

Green’s Theorem, 277, 282–287

growth models, 77–80

H

harmonic motion, 20, 22

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

I

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

infinite series, 183–184, 188

sums of, 49

infinity, convergence, 122

initial conditions, 309, 324

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

estimating, 114, 128–131

evaluating, 162–177

formulas, 117–120

improper, 122–124

indefinite, 117–120, 135

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

integrands, 117, 122

discontinuous, 123

integration, 60, 161

constants, 118, 121

functions, 134–136

limits of, 153

by parts, 162

intervals, 19

convergence, 189, 192

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

J

Jacobian determinants, 154

Jacobian J method, 153

J factor, 155

jumps, 51

L

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

of sums, 114, 116

switching, 135

linear algebra, matrices, 217–224

linear approximations, 250–254

linear functions, 6, 63–64

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, 53, 58–60

tangents, equations, 109

vertical line tests, 5

local extrema, 259

of functions, 104

local linearity, 109–112, 250–254

logarithms

functions, 3, 8–10

natural, 75

M

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

of determinants, 216, 218

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

harmonic, 20, 22

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

N

natural logarithms, 9, 75

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

O

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

P

pairs, 4, 5

parabolas, 38–42

paralleledpipeds, volume of, 214

parallelograms

areas, 291

areas of, 213

parallel vectors, 209

parameters, 278

equations, 224–229, 232

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

polar coordinates, 33, 156

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

series, 188, 190, 318

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 formulas, 10, 12

Pythagorean Theorem, 33

Q

quadratic approximations, 250–254

quadratic equations, 41

quarter circles, length of, 229

quotients

derivatives, 85–86

differences, 7, 325

differentiating, 95

R

radians, 10, 12

radius, 12

of convergence, 189

mapping, 20

trigonometric functions, 10

ranges

sets, 3

trigonometric function values, 20

rates

of change, 53–56, 64

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

coordinate systems, 33, 224

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

Riemann sums, 126, 128

right-hand screw rule, 211

roots, 112

Root Test, 188

rotation fields, 273

rows

multiplication, 207

vectors, 196

rules, 3

chain, 69, 86–90

chain, partial derivatives, 246–247

Cramer’s, 215–216, 220

differentiating, 102–103

midpoint, 128

products, 82–85, 162

quotients, 85–86

right-hand screw, 211

Simpson’s, 130

trapezoids, 129

Runge-Kutta method, 323

S

saddle points, 263

saddle-shaped surfaces, 32

scalars, 195, 208–211, 267–268

gradients, 256

multiplication, 206

secant, 10, 17

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

sequences, 114, 179–182

series, 179–182

alternating, 186

differential equations, 317–318

expanding, 188–194

geometric, 193

Harmonic Series, 185

infinite, 183–184, 188

Maclaurin, 188–189, 191

powers, 188, 190, 318

Taylor, 188–189, 191, 253

tests, 187–188

sets

domain, 4

range, 3

sigma (θ) notation, 114–117

Simpson’s rule, 130

sine, 10, 17, 51

addition formulas for, 15

curves, 24, 92

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

volume, 147, 153

spiral of Archimedes, 35

splitting integrals, 123

squares, matrices, 203

Stoke’s Theorem, 277, 304–306

straight lines, 62

motion in, 230

stream functions, 287

strips, 127

subsidiary equations, 321

substitution integrals, 164–172

subtraction

complex numbers, 37

formulas, 10, 14

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

Riemann, 126, 128

and sigma (θ) notation, 114–117

of vectors, 200

superposition, 313

surfaces, 30, 224–229

integrals, 287–295

saddle-shaped, 32

switching limits, 135

symmetric matrices, 204

symmetry, axis of, 40

systems of differential equations, 319–321

T

tables, 4

tangents, 10

derivatives, 95

hyperbolic, 28

lines, 53, 58–60

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

U

units

tangent vectors, 238

vectors, 196–197, 210

V

values, 4

averages, integrals, 131

of cross products, 213

trigonometric functions, 20

variables, 8, 243

changing, 116, 152–157

constants, 66–67

derivatives, 66–67

dummy, 114

functions with multiple, 29–33

separation of, 325

two-variable functions, 141

vectors, 267–268

acceleration, 236, 241

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

position, 224–229, 237, 273

products, 211–215

velocity, 54, 230–241

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

W

waveforms, 12

wave patterns, 23

work, 157–159, 210

along a curve, 279

X

x, derivatives of, 71–77

X-axis, 4, 8

areas below, 131–134

derivatives, 71

integrals, 119

intervals, 148

XY-planes, 141

Y

Y-axis, 4, 8

Z

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