Stewart J. Calculus

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derivative(s) (continued)

as the slope of a tangent, 113

of a sum, 137

third, 130

of trigonometric functions, 148, 152

of a vector function, 860

Descartes, René, A11

descent of aircraft, determining

start of, 164

determinant, 822

Devil’s curve, 169

Difference Law of limits, 78

Difference Rule, 138

differentiable function, 126, 931

differential, 191, 932, 934

differential equation, 147, 276, 602, 605

autonomous, 611

Bernoulli, 643

ﬁrst-order, 616

general solution of, 605

homogeneous, 1147

linear, 638

linearly independent solutions, 1148

logistic, 723

nonhomogeneous, 1147, 1153

logistic, 628, 723

order of, 605

partial, 922

second-order, 605, 1147

separable, 616

solution of, 605

differentiation, 126

formulas for, 135, 144,

RP5

formulas for vector functions, 862

implicit, 164, 919, 941

logarithmic, 417, 428*

partial, 914, 919, 920

of a power series, 765

term by term, 765

of a vector function, 860

differentiation operator, 126

Direct Substitution Property, 80

direction angles, 817

direction cosines, 817

direction ﬁeld, 608, 609, 636

direction numbers, 831

directional derivative, 946, 947, 950

maximum value of, 951

of a temperature function, 946, 947

directrix, 691, 698

discontinuity, 97, 98

discontinuous function, 97, 98

discontinuous integrand, 547

disk method, 359

dispersion, 214

displacement, 115, 328

displacement vector, 806, 819

distance

between planes, 837

between point and line in space, 829

between point and plane, 829, 836

between points in a plane, A11

between points in space, 803

between real numbers, A7

distance formula, A12

in three dimensions, 803

distance problem, 296

divergence

of an improper integral, 545, 548

of an inﬁnite series, 724

of a sequence, 713

Test for, 728

of a vector ﬁeld, 1101

Divergence, Test for, 728

Divergence Theorem, 1135

divergent improper integral, 545, 548

divergent sequence, 713

divergent series, 724

division of power series, 781

DNA, helical shape of, 855

dog saddle, 904

domain of a function, 11, 891

Doppler effect, 945

dot product, 815

in component form, 815

properties of, 815

double integral, 987, 989

change of variable in, 1049, 1052

over general regions, 1001, 1002

Midpoint Rule for, 991

in polar coordinates, 1010, 1011

properties of, 994, 1006

over rectangles, 987

double Riemann sum, 990

double-angle formulas, A29

Douglas, Paul, 892

drumhead, vibration of, 760

dumpster design, minimizing

construction cost, 969

dye dilution method, 588

e (the number), 397, 424*

as a limit, 418, 444*

as a sum of an inﬁnite series, 775

eccentricity, 698

electric charge, 1016, 1032

electric circuit, 641

analysis of, 1165

electric ﬁeld (force per unit charge), 1067

electric force, 1067

electric ﬂux, 1125

elementary function, integrability of, 523

ellipse, 170, 692, 698

area, 527

directrix, 698

eccentricity, 698

foci, 692, 698

major axis, 693

polar equation, 700, 703

reﬂection property, 694

rotated, 170

vertices, 693

ellipsoid, 842, 844

elliptic paraboloid, 842, 844

empirical model, 25

end behavior of a function, 230, 242

endpoint extreme values, 205, 206

energy

conservation of, 1088

kinetic, 1088

potential, 1089

epicycloid, 666

equation(s)

of a circle, A17

differential. See differential

equation

of an ellipse, 693, 700, A19

of a graph, A10, A16

heat conduction, 926

of a hyperbola, 694, 695, 700, A20

Laplace’s, 922, 1102

of a line, A12, A13, A14, A16

of a line in space, 830, 831, 832

linear, 834, A14

logistic, 604

logistic differential, 604, 636

Lotka-Volterra, 645

nth-degree, 167

of a parabola, 691, 700, A18

parametric, 667, 831, 854, 1106

of a plane, 834

point-slope, 18, A12

polar, 677, 700

predator-prey, 645

second-degree, A16

slope-intercept, A13

of a space curve, 854

of a sphere, 804

symmetric, 831

two-intercept form, A16

vector, 830

wave, 922

equilateral hyperbola, A21

equilibrium point, 646

equilibrium position, 155

equilibrium solution, 604, 645

equipotential curves, 904

equivalent vectors, 806

A128

||||

INDEX

error

in approximate integration, 557

percentage, 193

relative, 193

in Taylor approximation, 785

error bounds, 535, 539

error estimate

for alternating series, 748

for the Midpoint Rule, 557

for Simpson’s Rule, 561

for the Trapezoidal Rule, 557

escape velocity, 553

estimate of the sum of a series, 736, 744,

748, 753

Eudoxus, 3

Euler, Leonard, 734, 775

Euler’s constant, 740

Euler’s formula, A52

Euler’s Method, 611

even function, 19, 244

expected values, 1023

exponential decay, 627

exponential function(s), 33, 392, 438*

with base a, 392, 416, 438*

derivatives of, 396, 416, 438*

graphs of, 394, 399, 440*

integration of, 401, 435*, 441*,

779, 780

limits of, 395, 400, 432*

power series for, 772

properties of, 395, 432*, 438*

exponential graph, 399, 440*

exponential growth, 399, 627, 628

exponents, laws of, 395, 432*, 438*

extrapolation, 27

extreme value, 205

Extreme Value Theorem, 206, 964

family of functions, 50, 253

family of hypocycloids, 665

family of solutions, 604

fat circles, 168, 567

Fermat, Pierre, 4, 123, 207, 332

Fermat’s Principle, 266

Fermat’s Theorem, 207

Fibonacci, 722

Fibonacci sequence, 712

ﬁeld

conservative, 1068

electric, 1067

force, 1067

gradient, 955, 1067

gravitational, 1067

incompressible, 1102

irrotational, 1100

scalar, 1065

vector, 1063, 1064

velocity, 1063, 1066

First Derivative Test, 222

for Absolute Extreme Values, 259

ﬁrst octant, 801

ﬁrst-order linear differential equation, 638

ﬁrst-order optics, 790

ﬁxed point of a function, 111, 220

ﬂash bulb, current to, 62

ﬂow lines, 1069

ﬂuid ﬂow, 1066, 1102, 1124

ﬂux, 588, 588, 1123, 1125

ﬂux integral, 1123

FM synthesis, 253

focus

of a conic section, 698

of an ellipse, 692, 698

of a hyperbola, 694

of a parabola, 691

folium of Descartes, 165, 708

force, 370

centripetal, 888

constant, 819

exerted by ﬂuid, 575

resultant, 812

torque, 827

force ﬁeld, 1063, 1067

forced vibrations, 1164

Fourier, Joseph, 179

Fourier sequence, 503

four-leaved rose, 679

fractions (partial), 509, 510

Frenet-Serret formulas, 874

Fresnel, Augustin, 317

Fresnel function, 317

frustum

of a cone, 363

of a pyramid, 364

Fubini, Guido, 997

Fubini’s Theorem, 997, 1027

function(s), 11

absolute value, 17

Airy function, 764

algebraic, 31

arc length, 564, 866, 867

arrow diagram of, 12

average cost, 265

average value of, 374, 375, 593,

992, 1036

Bessel, 710, 760, 764

Cobb-Douglas production, 893,

923, 976

combinations of, 41

component, 853, 1064

composite, 41, 156, 911

constant, 135

continuity of, 97, 854, 910, 912

cost, 177, 261

cubic, 28

decreasing, 20

demand, 261

derivative of, 116

differentiability of, 126, 931

discontinuous, 97, 98

domain of, 11, 891

elementary, 523

even, 19, 244

exponential, 33, 392, 398

extreme values of, 205

family of, 50, 253

ﬁxed point of, 111, 220

Fresnel, 317

Gompertz, 636

gradient of, 949, 951

graph of, 12, 894

greatest integer, 83

harmonic, 922

Heaviside, 45, 70

homogeneous, 945

hyperbolic, 463

implicit, 164

increasing, 20

integrable, 989

inverse, 385, 386

inverse hyperbolic, 466, 467

inverse trigonometric, 454, 455

joint density, 1021, 1032

limit of, 66, 88, 907, 912

linear, 24, 894

logarithmic, 405, 421*, 442*

machine diagram of, 12

marginal cost, 118, 178, 261

marginal proﬁt, 262

marginal revenue, 261

maximum and minimum values of, 205,

958, 959

of n variables, 901

natural logarithmic, 406, 421*

nondifferentiable, 128

odd, 19, 244

one-to-one, 385

periodic, 244

piecewise deﬁned, 17

polynomial, 28, 910

position, 115

potential, 1068

power, 29, 142

probability density, 591, 1021

proﬁt, 262

quadratic, 28

ramp, 45

range of, 11, 891

INDEX

||||

A129

function(s) (continued)

rational, 31, 509, 910

reciprocal, 31

reﬂected, 38

representation as a power

series, 764

representations of, 11, 13

revenue, 261

root, 30

of several variables, 891, 900

shifted, 37

sine integral, 322

step, 18

stretched, 38

of three variables, 900

of two variables, 891

transcendental, 34

transformation of, 37, 38

translation of, 37

trigonometric, 32, A26

value of, 11

vector, 853

vector-valued, 853

Fundamental Theorem of Calculus, 313,

315, 318, 320

higher-dimensional versions, 1141

for line integrals, 1082

for vector functions, 864

G (gravitational constant), 181, 374

Gabriel’s horn, 573

Galileo, 661, 669

Galois, Evariste, 167

Gause, G. F., 632

Gauss, Karl Friedrich, 1135, A35

Gaussian optics, 790

Gauss’s Law, 1126

Gauss’s Theorem, 1135

geometric series, 724

geometry of a tetrahedron, 830

Gompertz function, 636

gradient, 949, 951

gradient vector, 949, 951

interpretations of, 955, 956

gradient vector ﬁeld, 955, 1067

graph(s)

of an equation, A10, A16

of exponential functions, 394,

399, 440*

of a function, 12

of a function of two variables, 894

of logarithmic functions, 405, 408,

424*, 442*, 443*

of a parametric curve, 658, 1119

polar, 677

of power functions, 30,

RP3

of a sequence, 716

of trignometric functions, A30, RP2

graphing calculator, 46, 250, 660, 682

graphing device. See computer algebra system

gravitation law, 180, 374

gravitational acceleration, 370

gravitational ﬁeld, 1067

great circle, 1047

greatest integer function, 83

Green, George, 1092, 1134

Green’s identities, 1105

Green’s Theorem, 1091, 1134

vector forms, 1102, 1103

Gregory, James, 768, 772

Gregory’s series, 768

grid curve, 1107

ground speed, 814

growth, law of natural, 447, 627

growth rate, 175

relative, 448

half-angle formulas, A29

half-life, 449

half-space, 900

hare-lynx system, 648

harmonic function, 922

harmonic series, 727

harmonic series, alternating, 747

heat conduction equation, 926

heat conductivity, 1126

heat ﬂow, 1126

heat index, 914

Heaviside, Oliver, 70

Heaviside function, 45, 70

Hecht, Eugene, 191, 790

height of a rocket, 495

helix, 854

higher derivatives, 129

higher partial derivatives, 920

homogeneous differential equation, 1147

homogeneous function, 945

Hooke’s Law, 371, 1161

horizontal asymptote, 232

horizontal line, A13

Horizontal Line Test, 385

horizontal plane, equation of, 802

Hubble Space Telescope, 210

Huygens, Christiaan, 661

hydrostatic pressure and force, 575

hydro-turbine optimization, 979

hyperbola, 169, 694, 698, A20

asymptotes, 694, A20

branches, 694, A20

directrix, 698

eccentricity, 698

equation, 694, A20

equilateral, A21

foci, 694, 698

polar equation, 700

reﬂection property, 698

vertices, 694

hyperbolic function(s), 463

derivatives, 463, 465

inverse, 466, 467

hyperbolic identities, 464

hyperbolic paraboloid, 843, 844

hyperbolic substitution, 507

hypocycloid, 665, 844, 846

hypersphere, 1036

hypocycloid, 665

i (imaginary number), A46

i (standard basis vector), 810

I/D Test, 221

ideal gas law, 182, 927

image of a point, 1049

image of a region, 1049

implicit differentiation, 164, 919, 941

implicit function, 164

Implicit Function Theorem, 942

improper integral, 544

impulse of a force, 637

incompressible velocity ﬁeld, 1102

increasing function, 20

increasing sequence, 717

Increasing/Decreasing Test, 221

increment, 117, 934

indeﬁnite integrals, 324

table of, 325

independence of path, 1083

independent random variable, 1022

independent variable, 11, 891, 939

indeterminate difference, 470, 471

indeterminate forms of limits, 470, 471

indeterminate power, 476

indeterminate product, 474

index of summation, A34

inequalities, rules for, A4

inertia (moment of ), 1019, 1032, 1081

inﬁnite discontinuity, 98

inﬁnite interval, 544

inﬁnite limit, 72, 94, 236

inﬁnite sequence. See sequence

inﬁnite series. See series

inﬂection point, 225

initial condition, 606

initial point

of a parametric curve, 658

of a vector, 806

initial-value problem, 606, 1151

inner product, 815

instantaneous rate of change, 64, 117, 171

A130

||||

INDEX

instantaneous rate of growth, 175

instantaneous rate of reaction, 174

instantaneous velocity, 64, 115, 171

integer, A2

integrable function, 989

integral(s)

approximations to, 306

change of variables in, 334, 1047,

1052, 1055

comparison properties of, 307

conversion to cylindrical

coordinates, 1038

conversion to polar coordinates, 1011

conversion to spherical coordinates, 1043

deﬁnite, 300, 863, 987

derivative of, 334

double, 987, 989. See also double integral

evaluating, 303

improper, 544

indeﬁnite, 324

iterated, 995, 996

line, 1070. See also line integral

patterns in, 530

properties of, 307

surface, 1117, 1123

of symmetric functions, 337

table of, 520,

RP6–10

trigonometric, 496, 498, 499

triple, 1026. See also triple integral

units for, 329

Integral Test, 733, 735

integrand, 300

discontinuous, 547

integration, 300

approximate, 531

by computer algebra system, 527

of exponential functions, 401, 435*, 441*

formulas, 488, 520,

RP6–10

indeﬁnite, 324

limits of, 300

numerical, 531

partial, 996

by partial fractions, 509

by parts, 489, 492

of a power series, 765

reversing order of, 998, 1006

term by term, 765

of a vector function, 863

intercept(s), 244, A19

Intermediate Value Theorem, 104

intermediate variable, 939

interpolation, 27

intersection of planes, 835

intersection of polar graphs, area of, 687

intersection of sets, A3

intersection of three cylinders, 1041

interval, A3

interval of convergence, 761

inverse function(s), 40

inverse transformation, 1049

irrational number, A2

irrotational vector ﬁeld, 1100

isobars, 890, 897

isothermal compressibility, 175

isothermals, 897, 904

iterated integral, 995, 996

j (standard basis vector), 810

Jacobi, Carl, 1051

Jacobian of a transformation, 1051, 1055

jerk, 131

joint density function, 1021, 1032

joule, 370

jump discontinuity, 98

k (standard basis vector), 810

Kampyle of Eudoxus, 169

Kepler, Johannes, 880, 884

Kepler’s Laws, 880, 884

kinetic energy, 1088

Kirchhoff’s Laws, 1165

Kondo, Shigeru, 775

Lagrange, Joseph, 216, 217, 971

Lagrange multiplier, 970, 971

lamina, 579, 1016

Laplace, Pierre, 922, 1102

Laplace operator, 1102

Laplace’s equation, 922, 1102

lattice point, 203

Law of Conservation of Angular

Momentum, 884

Law of Conservation of Energy, 1089

law of cosines, A33

law of gravitation, 180, 374

law of laminar ﬂow, 176, 587

least squares method, 27, 968

least upper bound, 718

left-hand derivative, 134

left-hand limit, 71, 91

Leibniz, Gottfried Wilhelm, 4, 126, 332,

616, 784

Leibniz notation, 126

lemniscate, 169

length

of a curve, 561

of a line segment, A7, A12

of a parametric curve, 669

of a polar curve, 688

of a space curve, 866

of a vector, 809

level curve(s), 890, 896

of barometric pressure, 890

of temperatures, 897

level surface, 901

tangent plane to, 953

l’Hospital, Marquis de, 472, 481, 482

l’Hospital’s Rule, 472, 481, 482

origins of, 481, 482

libration point, 274

limaçon, 683

Limit Comparison Test, 743

Limit Laws, 77, 78

Limit Laws for sequences, 714

limit(s), 3, 66

calculating, 77, 233, 234

of exponential functions, 395,

400, 432*

of a function, 66, 88

of a function of three variables, 912

of a function of two variables, 907

inﬁnite, 72, 94, 236

at inﬁnity, 230, 231, 236

of integration, 300

left-hand, 71, 91

one-sided, 70, 91

precise deﬁnitions, 88, 91, 94, 237, 238,

240

properties of, 77, 83

right-hand, 71, 91

of a sequence, 6, 291, 713

involving sine and cosine

functions, 150, 151

of a vector function, 853

Limit Comparison Test, 743

Limit Laws

for functions of two variables, 909

for sequences, 714

linear approximation, 189, 930, 934

linear combination, 1147

linear density, 173

linear differential equation, 638, 1147

linear equation, A14

of a plane, 834

linear function, 24, 894

linearity of an integral, 994

linearization, 189, 930

linearly independent solutions, 1148

linear model, 24

linear regression, 27

line(s) in the plane, A12

equations of, A12, A13, A14

horizontal, A13

normal, 145, 146

parallel, A14

perpendicular, A14

secant, 4, 62

slope of, A120

tangent, 4, 62, 113

INDEX

||||

A131

line(s) in space

normal, 954

parametric equations of, 831

skew, 833

symmetric equations of, 831

tangent, 860

vector equation of, 830, 831

line integral, 1070

with respect to arc length, 1073

Fundamental Theorem for, 1082

for a plane curve, 1070

for a space curve, 1075

of vector ﬁelds, 1077, 1078

work deﬁned as, 1077

liquid force, 575, 576

Lissajous ﬁgure, 665

lithotripsy, 694

local maximum and minimum

values, 205, 959

logarithm(s), 34, 405, 421*, 438*

laws of, 405, 422*

natural, 406, 421*

notation for, 405, 443*

logarithmic differentiation, 417, 428*

logarithmic function(s), 34, 405,

421*, 442*

with base , 405

derivatives of, 412, 422*, 443*

graphs of, 405, 408, 424*,

442*, 443*

integration of, 417, 428*

limits of, 406, 408, 424*

properties of, 405, 422*

logistic difference equation, 723

logistic differential equation, 604, 628

logistic model, 636

logistic sequence, 723

LORAN system, 697

Lotka-Volterra equations, 645

machine diagram of a function, 12

Maclaurin, Colin, 772

Maclaurin series, 770, 772

table of, 779

magnitude of a vector, 809

major axis of ellipse, 693

marginal cost function, 118, 178, 261

marginal proﬁt function, 262

marginal productivity, 923

marginal propensity to consume or save, 731

marginal revenue function, 261

mass

of a lamina, 1016

of a solid, 1032

of a surface, 1119

of a wire, 1072

mass, center of. See center of mass

mathematical induction, 731

principle of, 55, 58, A36

mathematical model, 14, 24

Cobb-Douglas, for production

costs, 893, 923, 976

for vibration of membrane,

760

maximum and minimum values, 205,

958, 959

mean life of an atom, 553

mean of a probability density

function, 593

Mean Value Theorem, 214, 216

for double integrals, 1059

for integrals, 375

mean waiting time, 593

median of a probability density

function, 594

method of cylindrical shells, 365

method of exhaustion, 3, 80

method of Lagrange multipliers, 970,

971, 974

method of least squares, 27, 968

method of undetermined

coefﬁcients, 1154, 1158

midpoint formula, A16

Midpoint Rule, 306, 532

for double integrals, 991

error in using, 533

for triple integrals, 1034

mixing problems, 620

Möbius, August, 1121

Möbius strip, 1115, 1121

modeling, mathematical

with differential equations, 603

motion of a spring, 604

population growth, 396, 603, 627,

633, 636, 652

vibration of membrane, 760

model(s), mathematical, 24

comparison of natural growth vs.

logistic, 632

empirical, 25

exponential, 33

Gompertz function, 636

linear, 24

logarithmic, 34

polynomial, 28

power function, 29

predator-prey, 182, 645

rational function, 31

seasonal growth, 636

trigonometric, 32, 33

von Bertalanffy, 652

modulus, A47

moment

about an axis, 579, 1017

of inertia, 1019, 1032, 1081

of a lamina, 579, 1017

of a mass, 579

about a plane, 1032

polar, 1019

second, 1019

of a solid, 1031

of a system of particles, 579

momentum of an object, 637

monkey saddle, 904

monotonic sequence, 717

Monotonic Sequence Theorem, 719

motion in space, 874

motion of a spring, force affecting

restoring, 1161

damping, 1162

resonance, 1165

movie theater seating, 463

multiple integrals. See double integral;

triple integral

multiplication, scalar, of vectors, 807, 809

multiplication of power series, 781

multiplier (Lagrange), 970, 971, 974

multiplier effect, 731

natural exponential function, 398, 400, 430*

derivatives of, 398, 433*, 434*

graph of, 399, 432*

power series for, 772

properties of, 400, 432*

natural growth law, 447, 627

natural logarithm function, 406, 421*

derivatives of, 413, 424*, 427*

limits of, 408, 424*

properties of, 405, 422*

-dimensional vector, 810

negative angle, A25

net area, 301

Net Change Theorem, 327

net investment ﬂow, 590

newton (unit of force), 370

Newton, Sir Isaac, 4, 9, 80, 123, 313,

332, 784, 880, 884

Newton’s Law of Cooling, 450

Newton’s Law of Gravitation, 180, 374,

880, 1066

Newton’s method, 269

Newton’s Second Law of Motion, 370, 876,

880, 1161

Nicomedes, 662

nondifferentiable function, 128

nonhomogeneous differential

equation, 1147, 1153

nonparallel planes, 835

A132

||||

INDEX

normal component of acceleration, 878

normal derivative, 1105

normal distribution, 595

normal line, 145, 146, 954

normal plane, 871

normal vector, 833, 870

nth-degree equation, ﬁnding roots of, 167

nth-degree Taylor polynomial, 195, 773

number

complex, A46

integer, A2

irrational, A2

rational, A2

real, A2

numerical integration, 531

octant, 801

odd function, 19, 244

one-sided limits, 70, 91

one-to-one function, 385

one-to-one transformation, 1049

open interval, A3

open region, 1084

optics

ﬁrst-order, 790

Gaussian, 790

third-order, 791

optimization problems, 205, 256

orbits of planets, 880, 884

order of a differential equation, 605

order of integration, reversed, 998, 1006

ordered pair, A10

ordered triple, 801

Oresme, Nicole, 728

orientation

of a curve, 1075, 1091

of a surface, 1122

oriented surface, 1121, 1122

origin, A2, A10

orthogonal curves, 170

orthogonal projection, 821

orthogonal surfaces, 958

orthogonal trajectory, 170, 619

orthogonal vectors, 817

osculating circle, 871

osculating plane, 871

Ostrogradsky, Mikhail, 1135

ovals of Cassini, 685

overdamped vibration, 1163

Pappus, Theorem of, 582

Pappus of Alexandria, 582

parabola, 691, 698, A18

axis, 691

directrix, 691, 698

equation, 691, 692

focus, 691, 698

polar equation, 700

reﬂection property, 203

vertex, 691

parabolic cylinder, 841

paraboloid, 842, 846

paradoxes of Zeno, 6, 7

parallel lines, A14

parallel planes, 835

parallel vectors, 807

parallelepiped, 354

volume of, 827

Parallelogram Law, 807, 822

parameter, 657, 831, 854

parametric curve, 657, 854

arc length of, 669

area under, 668

slope of tangent line to, 666

parametric equations, 657

of a line, 831

of a space curve, 854

of a surface, 1106

of a trajectory, 877

parametric surface, 1106

graph of, 1119

surface area of, 1111, 1112

surface integral over, 1117

tangent plane to, 1110

parametrization of a space curve, 856

with respect to arc length, 867

smooth, 867

paraxial rays, 191

partial derivative(s), 915, 916

of a function of more than three

variables, 919

interpretations of, 917

notations for, 916

as rates of change, 916

rules for ﬁnding, 916

second, 920

as slopes of tangent lines, 917

partial differential equation, 922

partial fractions, 509, 510

partial integration, 489, 490, 996

partial sum of a series, 724

particle, motion of, 874

parts, integration by, 489, 490

path, 1083

patterns in integrals, 530

pendulum, approximating the period

of, 191, 194

percentage error, 193

perihelion, 703

perilune, 697

period, 244

periodic function, 244

perpendicular lines, A14

perpendicular vectors, 817

phase plane, 646

phase portrait, 646

phase trajectory, 646

piecewise deﬁned function, 17

piecewise-smooth curve, 1071

Planck’s Law, 793

plane(s), 833

coordinate, 801

equation(s) of, 833, 834

horizontal, 802

normal, 871

osculating, 871

parallel, 835

tangent to a surface, 928, 953, 1110

vertical, 802

plane region of type I, 1002

plane region of type II, 1003

planetary motion, 880

point of inﬂection, 225

point(s) in space

coordinates of, 801

distance between, 803

projection of, 802

point-slope equation of a line, 18, A12

Poiseuille, Jean-Louis-Marie, 176

Poiseuille’s Laws, 194, 267, 588

polar axis, 675

polar coordinate system, 675

area in, 686

conic sections in, 698

conversion equations for Cartesian

coordinates, 676, 677

conversion of double integral

to, 1010, 1011

polar curve, 677

arc length of, 688

graph of, 677

symmetry in, 680

tangent line to, 680

polar equation, graph of, 677

polar equation of a conic, 700

polar form of a complex number, A48

polar graph, 677

polar moment of inertia, 1019

polar rectangle, 1010

polar region, area of, 686

pole, 675

polynomial function, 28

of two variables, 910

population growth, 627

of bacteria, 176, 627, 634

of insects, 519

models, 603

world, 396, 448, 449

INDEX

||||

A133

position function, 115

position vector, 809

positive angle, A25

positive orientation

of a boundary curve, 1129

of a closed curve, 1091

of a surface, 1122

potential, 556

potential energy, 1089

potential function, 1068

pound, 370

power consumption, approximation

of, 329

power function, 29, 142

Power Law of limits, 79

Power Rule, 136, 142

power series, 759

coefﬁcients of, 759

for cosine, 776

differentiation of, 765

division of, 781

for an exponenial function, 776

integration of, 765

interval of convergence, 761

multiplication of, 781

radius of convergence, 761

representations of functions as, 764

for sine, 776

predator, 644

predator-prey model, 182, 645

pressure exerted by a ﬂuid, 576

prey, 645

prime notation, 116, 137, 139, 140

principal square root of a complex

number, A47

principal unit normal vector, 870

principle of mathematical induction, 55,

58, A36

principle of superposition, 1156

probability, 1021

probability density function, 591, 1021

problem-solving principles, 54

producer surplus, 589

product

cross, 822. See also cross product

dot, 815. See also dot product

scalar, 815

scalar triple, 826

triple, 826

product formulas, A29

Product Law of limits, 78

Product Rule, 139

proﬁt function, 262

projectile, path of, 665, 877

projection, 802, 818, 819, 821

p-series, 735

quadrant, A11

quadratic approximation, 195, 969

quadratic function, 28

quadric surface(s), 841

cone, 844

cylinder, 841

ellipsoid, 842, 844

hyperboloid, 844, 846

paraboloid, 842, 846

table of graphs, 844

Quotient Law of limits, 78

Quotient Rule, 140

radian measure, 148, A24

radiation from stars, 793

radioactive decay, 449

radiocarbon dating, 454

radius of convergence, 761

radius of gyration, 1020

rainbow, formation and location of, 213

rainbow angle, 213

ramp function, 45

range of a function, 11, 891

rate of change

average, 117, 171

derivative as, 117

instantaneous, 64, 117, 171

rate of growth, 175

rate of reaction, 119, 174

rates, related, 182

rational function, 31, 910

integration of, 509

rational number, A2

rationalizing substitution, 517

Ratio Test, 752

Rayleigh-Jeans Law, 793

real line, A3

real number, A2

rearrangement of a series, 755

reciprocal function, 31

rectangular coordinate system, A11

conversion to cylindrical

coordinates, 1037

conversion to spherical coordinates, 1042

three-dimensional, 802

rectilinear motion, 277

recursion relation, 1170

reduction formula, 493

reﬂecting a function, 38

reﬂection property

of an ellipse, 694

of a hyperbola, 698

of a parabola, 203

region

connected, 1084

under a graph, 289, 295

open, 1084

plane, of type I or II, 1002, 1003

simple plane, 1092

simple solid, 1135

simply-connected, 1085

solid (of type 1, 2, or 3), 1027, 1029

between two graphs, 347

related rates, 182

relative error, 193

relative growth rate, 448, 628

relative maximum and minimum, 205

remainder estimates

for the Alternating Series, 748

for the Integral Test, 737

remainder of the Taylor series, 773

removable discontinuity, 98

representations of functions, 11, 13

as power series, 764

resonance, 1165

restoring force, 1161

resultant force, 812

revenue function, 261

reversing order of integration, 998, 1006

revolution, solid of, 359

revolution, surface of, 568

Riemann, Georg Bernhard, 301

Riemann sum(s), 301

for multiple integrals, 990, 1026

right circular cylinder, 354

right-hand derivative, 134

right-hand limit, 71, 91

right-hand rule, 801, 824

RMS voltage, 502

Roberval, Gilles de, 319, 669

rocket science, 977

Rolle, Michel, 214

roller coaster, design of, 148

roller derby, 1048

Rolle’s Theorem, 214

root function, 30

root law for limits, 79

Root Test, 754

roots of a complex number, A51

roots of an nth-degree equation, 167

rubber membrane, vibration of, 760

ruled surface, 848

ruling of a surface, 840

rumors, rate of spread of, 179

saddle point, 960

sample point, 294, 988

satellite dish, parabolic, 846

scalar, 807

scalar equation of a plane, 834

scalar ﬁeld, 1064

scalar multiple of a vector, 807

A134

||||

INDEX

scalar product, 815

scalar projection, 818, 819

scalar triple product, 826

geometric characterization of, 827

scatter plot, 14

seasonal growth model, 636

secant function, A26

derivative, 152

graph, A31

secant line, 4, 62

secant vector, 860

second derivative, 129

of a vector function, 862

Second Derivative Test, 225

Second Derivatives Test, 960

second moment of inertia, 1019

second partial derivative, 920

second-order differential equation, 605

solutions of, 1147, 1152

sector of a circle, area of, 686

sensitivity, 182

separable differential equation, 616

sequence, 6, 711

bounded, 718

convergent, 713

decreasing, 717

divergent, 713

Fibonacci, 712

graph of, 716

increasing, 717

limit of, 6, 291, 713

limit laws for, 714

monotonic, 717

of partial sums, 724

terms of, 711

series, 7, 723

absolutely convergent, 750

alternating, 746

alternating harmonic, 747, 751

binomial, 778, 784

coefﬁcients of, 759

conditionally convergent, 751

convergent, 724

divergent, 724

geometric, 724

Gregory’s, 768

harmonic, 727

inﬁnite, 723

Maclaurin, 770, 772

p-, 735

partial sum of, 724

power, 759

rearrangement of, 755

strategy for testing, 756

sum of, 7, 724

Taylor, 770, 772

terms of, 723

trigonometric, 759

series solution of a differential

equation, 1169

set, A3

set, bounded or closed, 964

serpentine, 145

shell method, 365

shift of a function, 37

shifted conics, 695, A21

shock absorber, 1162

Sierpinski carpet, 732

sigma notation, 294, A34

simple curve, 1085

simple harmonic motion, 163, 1161

simple plane region, 1092

simple solid region, 1135

simply-connected region, 1085

Simpson, Thomas, 537, 538, 985

Simpson’s Rule, 536, 538

error bounds for, 539

sine function, A26

derivative of, 151

graph of, 32, A30

power series for, 776

sine integral function, 322

sink, 1139

skew lines, 833

slant asymptote, 247

slope, A12

of a curve, 113

slope ﬁeld, 609

slope-intercept equation of a line, A13

smooth curve, 561, 867

smooth function, 561

smooth parametrization, 867

smooth surface, 1111

Snell’s Law, 266

snowﬂake curve, 797

solid, 354

volume of, 359, 366, 1027, 1028

solid angle, 1145

solid region, 1135

solid of revolution, 359

rotated on a slant, 574

volume of, 359, 366, 574

solution curve, 608

solution of predator-prey equations, 645

source, 1139

space, three-dimensional, 801

space curve, 854

arc length of, 866, 867

speed of a particle, 118, 875

sphere

equation of, 804

ﬂux across, 1124

parametrization of, 1108

surface area of, 1112

spherical coordinate system, 1041

conversion equations for, 1042

triple integrals in, 1042

spherical wedge, 1043

spherical zone, 600

spring constant, 371, 604, 1161

Squeeze Theorem, 83, A42

for sequences, 715

standard basis vectors, 810, 811

standard deviation, 595

static friction, coefﬁcient of, 851

stationary points, 959

steady state solution, 1167

step function, 18

Stokes, Sir George, 1129, 1134

Stokes’ Theorem, 1128, 1129

strategy

for integration, 519, 520

for optimization problems, 256

for problem solving, 54

for related rates, 185

for testing series, 757

for trigonometric integrals, 498, 499

streamlines, 1069

stretching a function, 38

strophoid, 689, 707

Substitution Rule, 333, 334, 336

subtraction formulas for sine

and cosine, A29

sum

of a geometric series, 725

of an inﬁnite series, 724

of partial fractions, 510

Riemann, 301

telescoping, 727

of vectors, 806, 809

Sum Law of limits, 78

Sum Rule, 137

summation notation, A34

supply function, 589

surface(s), 802

closed, 1122

graph of, 1119

level, 901

oriented, 1122

parametric, 1106

positive orientation of, 1122

quadric, 841. See also quadric

surface

smooth, 1111

surface area, 570, 574

of a parametric surface, 671, 1111

of a sphere, 1112

of a surface , 1113z 苷 f 共x, y兲

INDEX

||||

A135

A136

||||

INDEX

surface integral, 1117

over a parametric surface, 1117

of a vector ﬁeld, 1123

surface of revolution, 568

area of, 570, 574

parametric representation of, 1109

swallowtail catastrophe curve, 665

symmetric equations of a line, 831

symmetric functions, integrals of, 337

symmetry, 19, 244, 337

in polar graphs, 680

symmetry principle, 579

and transformations, 1049

table of differentiation

formulas, 144,

RP5

tables of integrals, 520,

RP6–10

use of, 525

tangent function, A26

derivative, 152

graph of, 33, A31

tangent line(s), 113

to a curve, 4, 61, 113

early methods of ﬁnding, 123

to a parametric curve, 666

to a polar curve, 680

to a space curve, 860

tangent line approximation, 189

tangent plane

to a level surface, 953

to a parametric surface, 1110

to a surface , 953

to a surface , 928

tangent plane approximation, 930

tangent problem, 4, 61, 113

tangent vector, 860

tangential component of

acceleration, 878

tautochrone problem, 661

Taylor, Brook, 772

Taylor polynomial, 195, 773, 969

applications of, 785

Taylor series, 770, 772

Taylor’s Inequality, 773

techniques of integration, summary, 520

telescoping sum, 727

temperature-humidity index, 902, 914

term of a sequence, 711

term of a series, 723

term-by-term differentiation and

integration, 765

terminal point

of a parametric curve, 658

of a vector, 806

terminal velocity, 623

Test for Divergence, 728

tests for convergence and divergence

of series

Alternating Series Test, 746

Comparison Test, 741

Integral Test, 733, 735

Limit Comparison Test, 743

Ratio Test, 752

Root Test, 754

summary of tests, 757

Test for Divergence, 728

tetrahedron, 830

third derivative, 130

Thomson, Sir William (Lord Kelvin),

1092, 1129, 1134

three-dimensional coordinate system, 802

TNB frame, 871

toroidal spiral, 856

torque, 827, 884

Torricelli, Evangelista, 669

Torricelli’s Law, 180

torsion of a space curve, 874

torus, 364, 509, 1117

total differential, 932

total electric charge, 1016, 1032

total fertility rate, 197

trace of a surface, 840

trajectories, orthogonal, 170

trajectory, parametric equations

for, 877

transcendental function, 34

transfer curve, 887

transformation, 1049

of a function, 37

inverse, 1049

Jacobian of, 1051, 1055

one-to-one, 1049

translation of a function, 37

Trapezoidal Rule, 533

error in, 533

tree diagram, 939

trefoil knot, 856

Triangle Inequality, 93, A8

for vectors, 822

Triangle Law, 807

trigonometric functions, 32, A26

derivatives of, 148, 152

graphs of, A30

integrals of, 496

inverse, 454, 455

limits involving, 150, 151

trigonometric identities, A28

trigonometric integrals, 496

strategy for evaluating, 498, 499

trigonometric series, 759

trigonometric substitution, 503

table of, 503

triple integral(s), 1026

applications of, 1031

in cylindrical coordinates, 1038

over a general bounded region, 1027

Midpoint Rule for, 1034

in spherical coordinates, 1043, 1044

triple product, 826

triple Riemann sum, 1026

trochoid, 664

Tschirnhausen cubic, 169, 353

twisted cubic, 856

type I or type II plane region, 1002, 1003

type 1, 2, or 3 solid region, 1027, 1029

ultraviolet catastrophe, 793

underdamped vibration, 1163

undetermined coefﬁcients, method of,

1154, 1158

union of sets, A3

unit normal vector, 870

unit tangent vector, 860

unit vector, 811

value of a function, 11

variable

continuous random, 591

dependent, 11, 891, 939

independent, 11, 891, 939

independent random, 1022

intermediate, 939

variables, change of. See change

of variables

variation of parameters, method of,

1158, 1159

vascular branching, 267

vector(s), 806

acceleration as, 875

addition of, 806, 809

algebraic, 808

angle between, 815

basis, 810, 811

binormal, 870

components of, 808

coplanar, 827

cross product of, 822

difference of, 807

displacement, 806, 819

dot product, 815, 816

equivalent, 806

force, 1066

geometric representations of, 808

gradient, 949, 951

i, j, and k, 811

initial point of, 806

length of, 809

magnitude of, 809

z 苷 f 共x, y兲

F共x, y, z兲苷 k

⫺1

INDEX

||||

A137

multiplication of, 807, 809

-dimensional, 810

negative, 807

normal, 833, 870

orthogonal, 817

parallel, 807

perpendicular, 817

position, 809

principal unit normal, 870

projection of, 818, 819

properties of, 810

scalar multiple of, 807

standard basis, 811

subtraction of, 809

sum of, 806, 809

tangent, 860

terminal point of, 806

three-dimensional, 808

triple product, 826, 827

two-dimensional, 808

unit, 808

unit normal, 870

unit tangent, 860

velocity, 867

wind velocity, 800

zero, 806

vector equation

of a line, 830, 831

of a line segment, 833

of a plane, 834

of a plane curve, 854

vector ﬁeld, 1063, 1064

conservative, 1068

curl of, 1098

divergence of, 1101

electric ﬂux of, 1125

force, 1063, 1067

ﬂux of, 1123

gradient, 1067

gravitational, 1067

incompressible, 1102

irrotational, 1100

line integral of, 1077, 1078

surface integral of, 1123

velocity, 1063

vector function, 853

continuity of, 854

derivative of, 860

differentiation formulas for, 862

integration of, 863

limit of, 853

parametric equations of, 854

vector product, 822

properties of, 826

vector projection, 818, 819

vector triple product, 827

vector-valued function. See vector

function

velocity, 4, 63, 115, 171

average, 5, 63, 115, 171

instantaneous, 64, 115, 171

velocity ﬁeld, 1066

air ﬂow, 1063

ocean currents, 1063

wind patterns, 1063

velocity gradient, 177

velocity problem, 63, 114

velocity vector, 867

velocity vector ﬁeld, 1063

Verhulst, Pierre-François, 604

vertex of a parabola, 691

vertical asymptote, 73, 244

Vertical Line Test, 16

vertical plane, equation of, 802

vertical tangent line, 129

vertical translation of a graph, 37

vertices of an ellipse, 693

vertices of a hyperbola, 694

vibrating spring, 1161

vibration of a rubber membrane, 760

vibrations, 1161, 1162, 1164

viewing rectangle, 46

volume, 354

by cross-sections, 355

by cylindrical shells, 365, 366

by disks, 356, 359

by double integrals, 987

of a hypersphere, 1036

of a solid, 355, 360, 989

of a solid of revolution, 359, 574

of a solid on a slant, 574

by triple integrals, 1031

by washers, 358, 359

Volterra, Vito, 645

Von Bertalanffy model, 652

Wallis, John, 4

Wallis product, 495

washer method, 358

wave equation, 922

Weierstrass, Karl, 518

weight, 370

wind patterns in San Francisco Bay

area, 1063

wind-chill index, 892

witch of Maria Agnesi, 145, 664

work, 370, 371, 819

deﬁned as a line integral, 1077

Wren, Sir Christopher, 671

-axis, 837, A10

-coordinate, 837, A10

-intercept, A19

-mean, 1023

-plane, 802

-axis, 837, A10

-coordinate, 837, A10

-intercept, A19

-mean, 1023

-plane, 802

-axis, 837

-coordinate, 837

Zeno, 6

Zeno’s paradoxes, 6

zero vector, 806

zone, spherical, 600ˆ

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