Index

Note: Boldface indicates a definition, italics indicates a figure, and 1t indicates a table.

Absolute convergence, 585–586, 588, 598t

Absolute extreme values, 263–264

finding, 268–272

for functions of two variables, 879–880, 883–886

Absolute value functions, 17

Absolute values

in inequalities, A-7–A-8

Acceleration, 179–181, 785–786

Coriolis, 795

due to gravity, 181, 335

tangential and normal components of, 788–792, 789

Acute angles, A-28

Addition. See Sums

Additive identities, 701, 706

Additive inverse properties, 701, 706

Adiabatic expansion, 452

Air resistance, 1079–1081

Aldrin, Buzz, 144

Algebra, used in calculus, A-1–A-11

Algebraic functions, 20, 221

Alternating harmonic series, 583, 598t

Alternating sequences, 539

Alternating series, 582–583

error for, 584–586

harmonic, 583, 598t

Alternating series test, 582–584, 598t

Amplitudes, 52, 54

Analytic geometry

and optimization, 323

used in calculus, A-15–A-25

Anderson, Janet, 608

Angles

acute, A-28

central, A-26

direction, 718

between planes, 742

positive vs. negative, A-25

quadrantal, A-26

reference, A-30

right, A-12

standard positions of, A-25

between vectors, 716–718

Angular momentum, 794

Angular speed, A-27

Angular velocity, 729–730

and curl, 1051

Antiderivatives, 328–329. See also Integrals; Integration

finding, 329–331

and integrals, 362, 364

and integrating by parts, 472

Apothems, 552

Approximations

of alternating convergent series, 584–586

of areas under graphs, 344–348

and calculator function, 537, 629, 634–635

for definite integrals, 627–628

for e, 625–626

of errors, 233–234

flat Earth, 798

linear, 232–232

for logarithms, 626–627

for π, 439, 609, 629

using Chebyshev polynomials, 634–635

using differentials, 844

using Newton’s Method, 234–237

using Riemann sums, 353–359

using Taylor/Maclaurin expansions, 623–628

using the Intermediate Value Theorem, 101–102

using the Trapezoidal Rule, 508–514

Arc lengths, 438–442, 440, A-26

and line integrals, 978

for parametric equations, 651–653

for parametric vector functions, 775–776

for polar equations, 675–676

for vector functions, 770–772

Archimedes, 412, 438, 439, 439

Area problem, 69, 343

and Method of Exhaustion, 439

Areas, 344–350. See also Surface areas

enclosed by polar equations, 678–680

errors in approximating, 347–348

between graphs, 405–411, 409

under graphs, 345, 348–350

maximizing, 319, 320–321, 322–323

between polar equations, 681–682

using double integrals, 918

using double integrals in polar coordinates, 927

using Green’s Theorem, 1010–1011

using integrals, 365

Arguments, 3

Armstrong, Neil, 144

Associative properties

of vector addition, 701, 706

Asymptotes

horizontal, 125

vertical, 119–120

Atom, Bohr model of, 795

Attractive forces, 976

Average rate of change

of functions, 11–12

limits of, 89

Average values, 373–374, 482

Average velocity, 145

Axes. See also x-axes; y-axes

of cones, 684

of conics, 685

coordinate, 695

major, 685, A-23

minor, 685, A-23

polar, 661

of symmetry, 18

transverse, 685, A-24

UVW, 792

Balls of radius δ, 819

Bernoulli, Daniel, 1085, 1085

Bernoulli, Jakob, 225, 590, 1082, 1082, 1085

Bernoulli, Johann, 225, 225, 300, 643

Bernoulli equations, 1082–1083

Bernoulli’s error, 590

Bernoulli’s inequality, 297

Bessel, Friedrich, 1038

Bessel functions, 611

Binomial coefficients, 620, A-43

Binomial series, 620

Maclaurin expansions for, 619–621

Binomial Theorem, 619, A-42–A-43

Binormal vectors, 784

Bisection method, 101

Body mass index (BMI), 239

Bohr model of the atom, 795

Boundary conditions, 331

Boundary points, 825

Bounded sequences, 546–547

Bounded sets, 883

Boundedness Theorem, 547

Boyle’s Law, 220

Brachistochrones, 643

Brahe, Tycho, 801–802

Calculators. See also Computer algebra systems (CAS)

errors for, 635

function of, 537, 629, 634–635

graphing, 64–67

Calculus

graphing functions using, 308–317

operational, 84

of variations, 643

Carbon-14 dating, 201

Cardioid microphones, 693

Cardioids, 220, 671, 675t

areas enclosed by, 679, 927

Carrying capacity, 1084

CAS. See Computer algebra systems (CAS)

Catenaries, 244, 246, 444

curvature of, 783

Catenoids, 660

Cauchy sequences, 553

Cauchy–Schwarz inequality, 723

Cavendish, Henry, 450, 450

Ceiling functions, 17

Centers. See also Centroids

of circles, A-21

of curvature, 780

of ellipses, A-23

of hyperbolas, A-24

of mass, 458–460, 459, 931–932, 945–946, 1028

of spheres, 697

Central angles, A-26

Central fields of force, 795

Centripetal forces, 786

Centroids, 460–464. See also Centers

of surfaces, 1029–1030

Chain Rule

for composite functions, 198–201

and method of substitution, 388

for multiple composite functions, 204–205

proof of, B-4–B-5

for vectors, 763–764

Chain Rule I [one independent variable], 849–851

Chain Rule II [two independent variables], 851–853

Change. See also Average rate of change

in a variable, 841

Change-of-base formula, 46, A-11

Change of variables methods. See also Substitutions

and differential equations, 1064–1066

and Jacobians, 962–966

Chapter Projects

calculators, 537, 634–635

cooling towers, 404, 469–470

economy, 254, 342

exoplanets, 809, 861

Hall effect, 694, 755–756

ice thickness, 862, 901–902, 1058, 1096–1097

lunar module, 144, 195–196

microphones, 636, 692–693

oil spills, 68, 143

population growth, 197, 253

population limits, 471, 535

river flow, 343, 403

road design, 757, 807–808

star masses, 903, 971–792

tornados, 973, 1056–1057

Chebyshev polynomials, 634–635

Circles, A-15, A-21

of curvature, 780

equations for, A-21–A-22

motion along, 786–787

osculating, 780–782

polar equations for, 668t

unit, A-21

Circular cones, 745

Circular functions, 243, A-29. See also Trigonometric functions

Circular motion, 729–730

Circulation, 1050–1051

Cissoids, 783

Clairaut, Alexis Claude, 835, 835

Classes of functions, 14–15

Closed curves, 992–993, 996

Closed intervals, A-5

continuity on, 96–97

Closed sets, 825

Closed surfaces, 1031–1032

Cobb, Charles Wiggins, 833–834

Cobb–Douglas productivity model, 833–834, 838, 857

Coefficients

binomial, 620, A-43

of friction, 328

leading, 18

Collins, Michael, 144

Common factors, A-1

Common logarithms, 46, A-11. See also Logarithms; Natural logarithms

Commutative properties

of vector addition, 701, 706

of vector dot products, 716

Comparison Tests, 529, 575–577, 597t

Completing the square, A-2–A-3

and integration, 496

Complex functions, 3

Components of vectors, 703, 712

Composite functions, 25–27

continuity of, 99

derivatives of, 198–201

Compressions, 29

Computer algebra systems (CAS), 64–67, 235

finding arc lengths using, 772

graphing quadric surfaces using, 750–751

graphing vector fields using, 975

integration using, 522

Concavity, 287–289, 288

Conditional convergence, 587–588

Conditional equations, A-32

Conditions

boundary, 331

initial, 333

Cones, A-15

circular, 745

elliptic, 745

finding volumes of, 434

parametrization of, 1020–1021

right circular, 684

Congruent triangles, A-13

Conic sections (Conics), 685

polar equations for, 686–688, 804

Connected regions, 993

Conservative vector fields, 989–991, 990

and Green’s Theorem, 1013–1014

potential functions for, 995–996

and Stokes’ Theorem, 1049–1050

tests for, 996–1000

Constant forces, 1002

Constant functions, 10–11, 15

derivatives of, 163–164

integrals of, 377

limits of, 81, 821

Constant Multiple Rule, 165, 331

Constant vector forces, 1002–1003

Constants. See also e; π

gravitational, 450, 802, 977

of integration, 379

Lipschitz, 139

permittivity of free space, 1035

spring, 446

Stefan–Boltzmann, 171, 238, 844

Continuity, 93–102

of composite functions, 99

and derivatives, 158–159, B-7–B-8

and differentiability, 845–846

on domains, 97–98, 826

of exponential functions, 113–114

of functions of several variables, 824–827, 825, 826, 845–846

on intervals, 96–98

of inverse functions, 99–100, B-3–B-4

of logarithmic functions, 113–114

of polynomial functions, 97

of power series, 606

properties of, 98–100

of trigonometric functions, 111–113

of vector functions, 761–762

Continuously compounded interest, 228, 229, 382

Contour lines, 812–813. See also Level curves; Traces

and gradients, 869

Convergence, 524

absolute, 585–586, 588

of alternating series, 582–588

comparison tests for, 575–577

conditional, 587–588

of improper integrals, 524–525, 527, 528

and integral test, 569–570

intervals of, 602–604

of monotonic sequences, B-11–B-12

of power series, 600–602

radii of, 602

ratio test for, 591–593

root test for, 593–595

of sequences, 540–543, 541

of series, 554–556, 555, 558, 566–569, 571

of Taylor/Maclaurin series, 614–616

tests for, 596–597, 597t–598t

Coordinate axes, 695

Coordinate curves, 1017

Coordinate planes, 696

Coordinates

conversion of, between rectangular and cylindrical, 950–951

conversion of, between rectangular and polar, 664–666

conversion of, between rectangular and spherical, 956–958

cylindrical, 950

polar, 661

rectangular, 695–697, A-16

spherical, 956

Copernicus, Nicolaus, 801

Coriolis acceleration, 795

Corollaries, 83

Cosecant functions, A-28

graphs of, 56

hyperbolic, 244

inverse, 60, 218

Cosine functions, 50t, 51–52, A-28

continuity of, 111–113

derivatives of, 187

graphs of, 52–53

hyperbolic, 243

inverse, 60, 218

limits of, 108

Maclaurin expansion for, 617

sum and difference formulas for, A-34

Taylor expansions for, 618–619

Cosines, direction, 719

Cotangent functions, 55, A-28

graphs of, 56

hyperbolic, 244

inverse, 60, 218

Coulomb’s Law, 452, 1043–1044

Critical numbers, 267–268

Critical points, 880–882

Cross products, 724–730, 725

algebraic properties of, 726–727

derivatives of, 763–764

geometric properties of, 727–730

triple, 732, 803

Cross sections. See Sections

Cube root functions, 16

Cube roots, A-9

Cubes, flux across, 1034

Cumulative distributive functions, 378

Curl, 1046

and angular velocity, 1051

interpretations of, 1050–1051

Curvature, 776–778, 777

of catenaries, 783

of cissoids, 783

and osculating circles, 780–782

of plane curves, 779–780

radii of, 780

of space curves, 778–779

Curves

closed, 992–993, 996

coordinate, 1017

learning, 220–221, 1070

level, 812–814, 813

logistic growth, 306

orthogonal, 818

plane, 637, 758

simple, 996

smooth, 647–648, 768

space, 759, 778

Curvilinear motion, 640–642

Cycles, 53

Cycloids, 642

arc lengths of, 652

curvature of, 783

tangent lines to, 649

Cylinders, 748–750, A-15

elliptic, 749

flux across, 1034

hyperbolic, 749

lateral surface area of, 980–981

parabolic, 749

tangent planes of, 1022–1023

Cylindrical coordinates, 950

and changing variables, 962–963

conversion of, to rectangular coordinates, 950–951

and parametrization, 1020–1021

triple integrals using, 951–954

Cylindrical shells, 424

Decreasing functions, 10–11

tests for, 279–282

Decreasing sequences, 548

Dedekind, Richard, 1038

Definite integrals, 355. See also Indefinite integrals; Integrals

approximations for, 627–628

finding, using limits of Riemann sums, 356–359

method of substitution for, 391–393

properties of, 369–374

trigonometric substitution for, 492–493

Degenerate cases, 685

Degrees

of differential equations, 1059

of polynomials, 18

Degrees, vs. radians, A-26–A-27

δ-neighborhoods, 819

Density. See also Mass density

mass vs. weight, 449

of stars, 971–972

Dependent variables, 3, 810

Derivatives, 150–151. See also Differentiation

and continuity, 158–159, B-7–B-8

directional, 863–866, 871

as functions, 154–160

graphs of, 155–157

higher-order, 177–179, 178

and local extreme values, 284–286

partial, 829–837, 830

of vector functions, 762–764, 767–768

Determinants, 185, 725–726

Difference formulas, trigonometric, A-34

Difference quotients, 3. See also Derivatives

limits of, 89

and partial derivatives, 830

Differences

continuity of, 98–99

derivatives of, 166–167

limits of, 82, 821

of series, 568

of vectors, 701

Differentiability, 154, 762

and continuity, 845–846

of functions of several variables, 841–842

Differential equations, 331–333

degrees of, 1059

exact, 1072–1073, 1074–1075

first-order linear, 1077–1082

and integrating factors, 1075–1076

linear, 1059

logistic, 1083–1084

order of, 332, 1059

ordinary, 1059–1060

separable, 399, 1062–1063

solving, 382–384, 394–395, 1062–1063

solving, using change of variables methods, 1064–1066

solving, using power series methods, 1089–1093

vector, 796–797

Differential forms, 1062

Differentials, 230–232, 231

approximations using, 844

and error analysis, 844–845

of functions of three variables, 846–847

partial, 843

surface area, 1023–1024

total, 843

Differentiation. See also Derivatives

Chain Rule for, 198–201, 204–205, 388, B-4–B-5

Chain Rule I for, 849–851

Chain Rule II for, 851–853

of composite functions, 198–201

of constant functions, 163–164

of differences of functions, 166–167

of exponential functions, 168–169, 202

of hyperbolic functions, 245–246

implicit, 209–218

implicit, for functions of several variables, 853–855

of integrals, 380–381

of inverse functions, 214–215, B-4

of inverse hyperbolic functions, 247–248

of inverse trigonometric functions, 216–218

logarithmic, 225–227

of logarithmic functions, 222–225

of polynomial functions, 167

of power functions, 164–166

of power series, 606–607

of products of functions, 173–175

of quotients of functions, 175–177

of reciprocal functions, 176–177

of sums of functions, 166–167

of trigonometric functions, 185–190

Dini’s surface, 1026

Direction, 699, 759

Direction angles, 718

Direction cosines, 719

Directional derivatives, 863–866

for functions of three variables, 871

Directrix, 685

Dirichlet functions, 361

Discontinuities

in functions of several variables, 826

jump, 95

removable, 95

Discriminants, 882, A-3

Disk method for finding volumes, 414–417

Disks of radius δ, 819

Distance formula, A-16

and arc lengths, 439

between points and planes, 739–740

in space, 696

Distributive properties

of vector addition and scalar multiplication, 706

of vector dot products, 716

Divergence [of series], 524

of alternating series, 582–588

comparison tests for, 575–577

of improper integrals, 524–525, 527, 528

to infinity, 545–546

of power series, 600–602

ratio test for, 591–593

root test for, 593–595

of sequences, 541, 545–547

of series, 554–556, 555, 558, 566–569, 571

tests for, 596–597, 597t–598t

Divergence [operator], 1038

interpretations of, 1042–1044

Divergence Theorem, 1037–1038, 1052

applications of, 1038–1041

Division. See Quotients

Domains, 2, A-4

continuity on, 97–98, 826

of functions, 3, 4–5, 19–20

of functions of two or three variables, 810, 811–812

of logarithmic functions, 44

parameter, 1016

of trigonometric functions, A-31–A-32

of vector functions, 758

Dot products, 715–721

derivatives of, 763

directional derivatives as, 866

and divergence operator, 1038

Double-angle formulas, A-35

Double integrals, 906, 912

and centers of mass, 929–934

and Jacobians, 964–965

over non-rectangular regions, 911–919

over rectangular regions, 906–909

and surface areas, 936–939

and surface integrals, 1027–1030

using polar coordinates, 923–927

Douglas, Paul H., 833–834

Downward-pointing normal vectors, 1030–1031

Dummy variables, 362

e, 41–42

approximations for, 625–626

expressed as a limit, 227–228

Eccentricity, 685

of orbits, 688, 689

in polar equations, 686–687

Einstein, Albert, 353, 629

Electric flux, 1035–1036

Electric force fields, 976

and the Divergence Theorem, 1043–1044

Elementary functions, 508

Ellipses, 538, A-23–A-24

as conic sections, 685

motion along, 786, 790

osculating circles for, 781–782

polar equations for, 686–687

Ellipsis, A-38

Ellipsoids, 744–746

of revolution, 745

Elliptic cones, 745

Elliptic cylinders, 749

Elliptic paraboloids, 745–746

Elliptical integrals of the second kind, 443

Endpoints, left and right, A-5

Energy

kinetic, 794

Law of Conservation of, 1006

potential, 1005–1006

rotational kinetic, 934

and work, 1004–1006

Enhanced Fujita (EF) scale, 1057

Enthalpy, 376

Epicycloids, 647

e-δ definition of limits, 130–132

uses of, 132–137

Equations. See also Differential equations; Formulas; Functions; Identities

Bernoulli, 1082–1083

conditional, A-32

of continuity, 1042

and derivatives, 168

differential, 331–333, 1059–1060

exponential, 45–48

Gompertz, 1088

Hagen–Poiseuille, 171

and inverse functions, 35–37

Kepler’s, 239

Laplace’s, 839, 840

for lines, A-18–A-20

for lines in space, 733–737

logarithmic, 45–48, 46

logistic, 1084–1086

parametric, 637

for planes in space, 737–739

Poiseuille, 171

polar, 666–668

quadratic, A-3–A-4

rectangular, 638–640

Schrödinger, 1061

for spheres, 697

symmetric, for lines, 734–736, 735

for tangent lines, 200, 211–212, 226, 831–832

for tangent planes, 876

trigonometric, 61–63

in two variables, A-16

van der Waals, 220

vector, 733–734

wave, one-dimensional, 857

Equilateral triangles, A-12

Equilibrium, 446

static, 710

Errors

for an alternating series, 584–586

approximating, 233–234

in approximating areas, 347–348

Bernoulli’s, 590

for calculators, 635

and differentials, 844–845

Euler’s, 623

and the Mean Value Theorem, 512

percentage vs. relative, 233–234

in using Simpson’s Rule, 516–517

in using the Trapezoidal Rule, 511–512

Euler, Leonhard, 42, 42, 565, 1085

Euler’s error, 623

Euler’s number, 565

Even functions, 9–10

integrals of, 393–394

Even/Odd identities, A-33

Event horizons, 262

Evolutes, 784

Exact differential equations, 1072–1076, 1073

Existence

of definite integrals, 356

of functions, 5

of limits, 822–824

Exoplanets, 688, 804, 809, 861

Expectation values, 532

Explicit forms of functions, 4, 811

Exponential density model, 972

Exponential equations, 45–48

Exponential functions, 38–41, 39, 42

continuity of, 113–114

derivatives of, 168–169, 202

properties of, 40

vs. power functions, 39

Exponential laws, 382, 1068–1070

Exponents, A-8–A-9

laws of, 40, A-8

Extreme Value Theorem, 265–266

Extreme values, 263–267, 264

finding, 268–272

for functions of two variables, 879–880, 883–886

Factorial series, 598t

Factorial symbols, 178, A-40

Factoring, A-1

Factors, common, A-1

Fermat, Pierre de, 266, 266, 324

Fibonacci numbers, A-39

Fibonacci sequences, 552, A-39

First-derivative test, 284–286

First moments, 930

First-order linear differential equations, 1077–1082

First-order partial derivatives, 830–834

Flat Earth approximation, 798

Floor functions, 17

continuity of, 95

Flow rates in mixtures, 1081–1082

Flux, 1033–1034

electric, 1035–1036

using the Divergence Theorem, 1040

Foci

of conics, 685

of ellipses, A-23

of hyperbolas, A-24

of parabolas, A-22

Folium of Descartes, 221, 1015

Forces

attractive, 976

central fields of, 795

centripetal, 786

constant, 1002

constant vector, 1002–1003

gravitational, 450

hydrostatic, 453–456, 454

Lorentz, 732

repulsive, 976

restoring, 446

resultant, 700, 710

spring, 446–447

variable, 1002

variable vector, 1003

work done by, 444–446, 720–721

Formulas. See also Equations; Functions; Identities

change-of-base, A-11

difference, trigonometric, A-34

distance, A-16

double-angle, A-35

general forms of, A-21

for geometric figures and solids, A-15

for integration by parts, 472–478

quadratic, A-3–A-4

recursive, 235, 1091, A-39

reduction, 477–478

for sequences sums, A-41

standard forms of, A-3, A-21

sum, trigonometric, A-34

Wallis’s, 479

Fourier’s Law of Conductivity, 1096

Fourth derivatives, 177–179, 178. See also Snap

of vector functions, 763

Free fall, 334–336, 335

with air resistance, 1079–1081

Frustums, 656

Fubini, Guido, 907

Fubini’s Theorem, 907–908

for triple integrals, 942

for x-simple regions, 913–914

for y-simple regions, 915–916

Functions, 3–4. See also Equations; Formulas; Identities; specific function

absolute value, 17

algebraic, 20, 221

approximations of, linear, 232–232

approximations of, using Taylor/Maclaurin expansions, 623–625

average rate of change of, 11–12

average values of, 373–374

Bessel, 611

ceiling, 17

circular, A-29

classes of, 14–15

complex, 3

composite, 25–27

concavity of, 287–289, 288

constant, 10–11, 15

continuity of, 93–102

cosecant, 56, A-28

cosine, 51–52, A-28

cotangent, 55, A-28

cube root, 16

cumulative distributive, 378

derivatives as, 154–160

derivatives of, 163–169

directional derivatives of, 864–865

Dirichlet, 361

domains of, 3, 4–5, 19–20

elementary, 508

evaluating, 3–4

existence of, 5

expectation values of, 532

exponential, 38–41, 39, 42

expressed as Taylor/Maclaurin series, 613–614

finding limits of, 81–89

floor, 17

gamma, 565

graphs of, 5–6, 7–9, 312–313, 316–317

greatest integer, 17

Gudermannian, 387

harmonic, 839

Heaviside, 84, 159

homogeneous, 1063–1064

identity, 15

improper, 499

integrals of, 369–372, 380–381

inverse, 32–37

inverse trigonometric, 58–63

library of, 15–17

linear, 15

logarithmic, 42–45

logistic, 1083–1084

of n variables, 811

natural logarithmic, 42

one-to-one, 32–37

operations on, 24–25

orthogonal, 221

partial derivatives of, 829–837

periodic, 50, 162

piecewise-defined, 4

polynomial, 17–19

potential, 990–991

power, 15

Power Rule for, 202–204

power series, 604–606

probability density, 378, 532

production, 220, 833–834

proper, 499

properties of, 9–11

quadratic, 18

ranges of, 3

rational, 19–20, 499

real, 3

reciprocal, 16, 56

reconstruction of, from gradients, 995–996

related, 543–544

Riemann’s zeta, 565, 575

root, 16

roots of, 9

secant, 56, A-28

of several variables, 811

sine, 50–51, A-28

sinusoidal, 54

square root, 16

standard normal density, 318, 622

step, 17

stream, 1056

tangent, 55, A-28

of three variables, 810–811

transcendental, 20

transformations of, 27–30

trigonometric, 49–56, A-27–A-37

of two variables, 810

vector, 3, 758–762

Fundamental periods, 50

Fundamental Theorem of Calculus, 362–363, 364, 1051

proof of, B-9–B-10

using, 363–366

Fundamental Theorem of Line Integrals, 990–993, 1051

converse of, 993–995

Gabriel’s Horn, 526, 660

Galilei, Galileo, 180, 336, 804

Gamma function, 565

Gauss, Karl Friedrich, 353, 1038, 1038

Gauss’ Theorem. See Divergence Theorem

General forms

of the circle formula, A-21

of equations for planes, 738

of quadric surface formulas, 744

General terms of series, 554

Generators of cones, 684

Geocentric Theory of Planetary Motion, 801

Geometric series, 557–560, 579t, 598t

Geometry

in calculus, A-11–A-15

formulas for, A-15

Geometry, analytic

in calculus, A-15–A-25

and optimization, 323, 891–892

Gompertz equation, 1088

Gradient vector fields, 976

Gradients, 866–867

for functions of three variables, 871

and potential functions, 990

properties of, 868–870

reconstructing functions from, 995–996

Gram–Schmidt orthogonalization, 724

Graphing technology, 64–67

using, 22, 48

Graphs, A-16. See also Plots

approximating areas under, 344–348

arc lengths of, 439–442

areas between, 405–408

areas under, 345, 348–350

of circles, A-21–A-22

of derivatives, 155–157

of ellipses, A-23–A-24

of equations in two variables, A-16–A-17

of exponential functions, 40–41

of functions, 5–6, 7–9, 312–313, 316–317

of functions of two variables, 812

holes in, 126

of hyperbolas, A-24–A-25

of limits, 73–76

of lines, A-18–A-20

of logarithmic functions, 43–44

of parabolas, A-23

of parametric surfaces, 1017–1018

of plane curves, 637

in polar coordinates, 661–663

of polar equations, 666–668

of polynomials, 17–19, 308–309

of quadric surfaces, 744–751

of rational functions, 309–312

of resultant vectors, 700

scatter plots, 21–22

of sequences, 538

sinusoidal, 53–54

symmetry of, A-17–A-18

and transformations of functions, 27–30

of trigonometric functions, 50–56, 314–315

using calculus, 308–317

of vector fields, 975

of vector functions, 758–760

Gravitation, 450

Gravitational constant, 450, 802, 977

Gravitational fields, 975

Gravitational potential, 977

Gravity, acceleration due to, 181, 335

Greatest integer functions, 17

Green, George, 1008

Green’s Theorem, 1008–1009, 1051

and areas, 1010–1011

and line integrals, 1009–1010, 1012–1013

and multiply connected regions, 1011–1014

Gregory, James, 608, 608

Gregory’s series, 608, 629

Growth terms, 1084

Gudermann, Christoph, 387

Gudermannian function, 387

Hagen–Poiseuille equation, 171

Half-closed intervals, A-5

continuity on, 96

Half-lines, A-25

Half-lives, 384, 1094

Half-open intervals, A-5

Hall Edwin H., 694, 755

Hall effect, 694, 755

Hall probes, 694

Hall voltages, 756

Halley’s comet, 835

Harmonic functions, 839

Harmonic motion

damped, 129

simple, 189–190

Harmonic series, 561–562, 579t, 598t

Heaviside, Oliver, 84, 84

Heaviside function, 84, 159

Heaviside layer, 84

Helices, circular, 760, 770, 771

Heliocentric Theory of Planetary Motion, 801

Hessians. See Discriminants

Higher-order derivatives, 177–179, 178

using implicit differentiation, 212–213

of vector functions, 763

Holes, 126

Homogeneous functions, 1063–1064

and differential equations, 1064–1066

Homogeneous laminae, 460

centroids of, 461–464

Hooke’s Law, 446

Horizontal asymptotes, 125

Horizontal-line tests, 33, 408

Horizontal tangent lines, 187–188, 648

Huygens, Christian, 643

Hydrostatic forces, 453–456, 454

Hydrostatic pressure, 453–456

Hyperbolas, A-24–A-25

as conic sections, 685

polar equations for, 686–687

Hyperbolic cylinders, 749

Hyperbolic functions, 243–244

derivatives of, 245–246

identities for, 244–245

inverse, 246–248

Maclaurin expansion for, 617

Hyperbolic paraboloids, 746–747

Hyperboloids, 746–748

of one sheet, 747

of revolution, 747

of two sheets, 747–748

Hypercardioid microphones, 693

Hypocycloids, 220, 647

Hypotenuse, A-12

Ideal Gas Law, 183, 838, 857

Identities, A-32. See also Equations; Formulas; Functions

hyperbolic, 244–245

Lagrange’s, 732

polarization, 724

trigonometric, A-32–A-36

Identity functions, 15

limits of, 81

Images. See Values

Implicit differentiation, 209–218

Implicit forms of functions, 4, 811

Improper integrals, 523–524

comparison test for, 529

convergence of, 524–525, 527, 528

finding, 524–525

geometrical interpretation of, 525–527

of the second kind, 527–528

Improper rational functions, 499

Incompressibility, 1056

Increasing functions, 10–11

tests for, 279–282

Increasing sequences, 548

Indefinite integrals, 379. See also Antiderivatives; Definite integrals; Integrals

finding, 379–380

method of substitution for, 387–391

trigonometric substitution for, 488–492

using, 380–381

Independence of paths, 991

Independent variables, 3, 810

Indeterminate forms, 298–299

limits of, 303–305

Indices

of roots, A-9

of summation, 554, A-40

Inequalities

absolute values in, A-7–A-8

Bernoulli’s, 297

Cauchy–Schwarz, 723

solving, 168, A-5–A-8

triangle, 547, A-7

Inertia

moments of, 933–934, 945, 946, 953–954

polar moments of, 933

Infinite series. See Series

Infinity

divergence to, 545–546

infinite limits at, 122–124, 123, 137

limits at, 117–119, 136–137

properties of limits at, 120–125

symbol for, 117, A-5

Inflection points, 289–290

Inhibition terms, 1084

Initial conditions, 333

Initial points, 699

Initial positions, 145

Initial sides, of rays, A-25

Inner normal unit vectors, 1032

Inputs, 2

Instantaneous rate of change, 149–150

Instantaneous velocity, 145–147, 146

Integrability, 906, 912

Integral notation, 355

Integral test, 569–570, 572–573, 597t

Integrals. See also Antiderivatives; Definite integrals; Indefinite integrals

approximations for, 627–628

definite, 355

double, 906–939

elliptical, of the second kind, 443

of functions, 369–372, 380–381

improper, 523–524

indefinite, 379

iterated, 907

line, 977–986

Mean Value Theorem for, 372–373

methods of substitution for, 387–393

properties of, 369–374, 380–381

Riemann, 355

surface, 1027–1030

tables of, 380t, 520–521

triple, 941–960

using limits of Riemann sums, 356–359

of vector functions, 796–797

Integrands, 355, 379

Integrating factors, 1075–1076

Integration, 379

constants of, 379

of even and odd functions, 393–394

of functions, 369–372, 380–381

of functions containing quadratic expressions, 496–498

of functions containing trigonometric expressions, 480–486

limits of, 355, 391–393

partial, 906–908

by parts, 472–478

of power series, 606

of rational functions, 499–505

by separation of variables, 382

of sums of functions, 369

and symmetry, 407

by trigonometric substitution, 488–493

using Computer Algebra Systems (CAS), 522

using numerical techniques, 508–517

using partial fractions, 499–505

using tables, 520–521

Intercepts, A-17

Interest, continuously compounded, 228, 229, 382

Interior points, 825

Intermediate Value Theorem, 100–102

Intersections

of lines, 736–737

of planes, 739

Interval notation, A-5

Intervals, continuity on, 96–98

Intervals of convergence, 602–604

Inverse functions, 32–37

continuity of, 99–100, B-3–B-4

derivatives of, 214–215, B-4

Inverse hyperbolic functions, 246–247. See also Hyperbolic functions

derivatives of, 247–248

Inverse trigonometric functions, 58–63. See also Trigonometric functions

derivatives of, 216–218

integrals of, 474

power series for, 608–609

Involutes, 784

Irreducible quadratic polynomials, 503–504

Isosceles triangles, A-12

Isotherms, 813, 817, 874

Iterated integrals, 907

Jacobi, Carl Gustav, 963, 963

Jacobians

in three variables, 965–966

in two variables, 963–965

Jerk, 183, 196

Joule, James Prescott, 445, 445

Jump discontinuities, 95

k-to-the-k series, 579t, 598t

Kelvin, William Thomson, Lord, 1046

Kennedy, John F., 144

Kepler, Johannes, 801

Kepler’s equation, 239

Kepler’s Laws of Planetary Motion, 801–804

Kinetic energy, 794

rotational, 934

and work, 1004–1005

Kirchhoff’s Laws, 1088

Koch snowflakes, 564

Lagrange, Joseph-Louis, 892, 892

Lagrange multipliers, 891–892

and optimization with one constraint, 892–895

and optimization with two constraints, 896–897

Lagrange’s identity, 732

Lagrange’s Theorem, 892

Laminae, homogeneous, 460, 929

centers of mass of, 931–932

centroids of, 461–464, 931

moments of, 930

Laminae, non-homogeneous, 1028–1029

Laplace transforms, 532

Laplace’s equation, 839, 840

Latitude, 960

Latus rectum, 432

Laws. See also Rules; Theorems

Boyle’s, 220

of Conservation of Energy, 1006

of Cosines, 716, A-37–A-38

Coulomb’s, 452, 1043–1044

of eventual diminishing marginal productivity, 834

exponential, 382, 1068–1070

of exponents, 40, A-8

Fourier’s, of Conductivity, 1096

Hooke’s, 446

Ideal Gas, 183, 838, 857

of inhibited growth or decay, 1068–1070

Kepler’s, of Planetary Motion, 801–804

Kirchhoff’s, 1088

Malus’s, 192

Newton’s, of Cooling, 128

Newton’s, of Heating, 1069

Newton’s, of Universal Gravitation, 450, 802, 975

Newton’s First, of Motion, 336–337

Newton’s Second, of Motion, 336, 786–788, 802

of Sines, A-36–A-37

Snell’s, 324–325

of uninhibited growth or decay, 253, 382, 1068

Leading coefficients, 18

Learning curves, 220–221, 1070

Learning theory, 1069–1070

Least-squares fits, 890

Left endpoints, A-5

Left-hand limits, 71

Left-hand rule, 695

Left-handed coordinate systems, 695

Legs [of triangles], A-12

Leibniz, Gottfried Wilhelm von, 69, 266, 582, 629

Leibniz notation, 163

Leibniz Test, 582

Lemniscates, 220, 675t, 677

Level curves, 812–814, 813. See also Contour lines; Traces

and gradients, 869–870

Level surfaces, 815

and gradients, 871

L’Hôpital, Guillaume François de, 300, 300

L’Hôpital’s Rule, 299–302

and convergence of sequences, 544–545

proof of, B-5–B-7

Libby, Willard F., 384

Limaçons, 671–673, 672, 675t

areas enclosed by, 680

Limit comparison tests, 577–579, 598t

Limiting velocity, 1080

Limits, 70–71, 76

e-δ definition of, 130–137, 132

existence of, 822–824

of functions, 81–89

of functions of several variables, 819–821, 820

graphs of, 73–76

of indeterminate forms, 303–305

at infinity, 117–119, 136–137

and L’Hôpital’s Rule, 300–303

properties of infinite, 120–125

of sequences, 541

types of, 71–73

uniqueness of, B-1

of vector functions, 761

Limits of integration, 355

changing, 391–393

infinite, 524–525

Line integrals, 977

along closed curves, 992–993

along piecewise-smooth curves, 984–985

along smooth curves, 979–981

converse of Fundamental Theorem of, 993–995

Fundamental Theorem of, 990–993

in the plane, 977–978

in space, 985–986

in two variables, 981–984

using Green’s Theorem, 1009–1010, 1012–1013

using Stokes’ Theorem, 1049

Line segments, 699

Linear approximations of functions, 232–232

Linear density model, 971–972

Linear differential equations, 1059

first-order, 1077–1082

Linear functions, 15

Linear speed, A-27

Linearity properties

of vector addition, 706

of vector dot products, 716

Linearization, 233

Lines

contour, 812–813

equations for, A-18–A-20

equations for, in space, 733–737

half-, A-25

normal, 172, 737, 876–877

parallel, 736–737, A-20

parametric equations for, 734

perpendicular, A-20

polar equations for, 668t

secant, 70, 147

skew, 736–737

stream, 1056

symmetric equations for, 734–736, 735

tangent, 69–70, 147–149, 148, 200, 211–212, 226, 768, 831–832

Lipschitz constants, 139

Little, Jack, 66

Local extreme values, 263–264

conditions for, 266–267

and first-derivative test, 284–286

for functions of two variables, 879–880

and second-derivative test, 291–294

Logarithmic equations

solving, 45–48, 46

Logarithmic functions, 42–45. See also Natural logarithmic function

continuity of, 113–114

derivatives of, 222–228

properties of, 44–45

Logarithmic spirals, 674

arc lengths of, 676

Logarithms, A-10–A-11. See also Natural logarithms

common, 46

common vs. natural, A-11

properties of, A-10

Logistic differential equations, 1083–1084

Logistic equations, 1084–1086

Logistic functions, 1083–1084

Logistic growth, 306, 1083–1086

Longitude, 960

Lorentz forces, 732

Lower limits of integration, 355

Lower sums, 346–347

finding area using, 349–350

Lumpy spheres, 1026

Lunes, 495

Maclaurin, Colin, 613, 613

Maclaurin expansions, 613

Maclaurin series, 611–613

convergence of, 614–616

and differential equations, 1092–1093

finding, 616–619

functions expressed as, 613–614

Magnitude of vectors, 699, 707–708

Major axes, 685, A-23

Malthus, Thomas, 197

Malus’s Law, 192

Maps, topographic, 813

Marginal productivity, 833–834

Mass, 944, 980, 1028

Mass density, 449

and double integrals, 929–930

and line integrals, 980

Maximizing. See Optimization

Maximum values, 263–264

of directional derivatives, 867

for functions of two variables, 879–880, 883–886

Maxwell, James Clerk, 1046

Mean Value Theorem, 276–279

and error, 512

for integrals, 372–373

proof of, B-5–B-6

for triple integrals, 1042

Methods

disk, 414–417

Exhaustion, 439

Newton’s, 234–237

shell, 424–431, 425

slicing, 433–437

of substitution, 388, 391–393

washer, 417–422, 418, 426, 427–428

Microphones, 636, 692–693

Minimizing. See Optimization

Minimum values, 263–264

of directional derivatives, 867

for functions of two variables, 879–880, 883–886

Minor axes, 685, A-23

Mixed forms, 850

Mixed partials, 834–835

Möbius, August, 1032, 1038

Möbius strips, 1032

Models

mathematical, 20–22

of star density, 971–972

Moler, Cleve, 66

Moments, 458

about coordinate planes, 944, 1028

about the origin, 459

of inertia, 933–934, 945, 946, 953–954

Momentum, 794

angular, 794

Monomials, 17

Monotonic sequences, 548–549

convergence of, B-11–B-12

Motion

along a curve, 785–792

circular, 729–730

curvilinear, 640–642

harmonic, damped, 129

harmonic, simple, 189–190

planetary, 801–804

projectile, 112–113, 646, 650

rectilinear, 145

vertical, 179–181

Multiple zeros, 18

Multiplication. See Products

Multiplicity, zeros of, 18

Multipliers, Lagrange, 891–897

Multiply connected regions, 1011–1014

Mutations in a population, 562

Nappes, 685

Natural logarithmic function, 42. See also Logarithmic functions

derivatives of, 222–223

Natural logarithms, A-11. See also Logarithms

approximations for, 626–627

vs. common, A-11

Navigation, 960

Negative angles, A-25

Negative orientation, 1031

Newton, Isaac, 69, 235, 266, 450, 802

Newton’s First Law of Motion, 336–337

Newton’s Law of Cooling, 128, 395–396

Newton’s Law of Heating, 1069

Newton’s Law of Universal Gravitation, 450, 802, 975

Newton’s Method, 234–237

Newton’s Second Law of Motion, 336, 786–788, 802

Nondecreasing functions, 10–11

Nondecreasing sequences, 548

Nonincreasing functions, 10–11

Nonincreasing sequences, 548

Normal components of acceleration, 788–792, 789

Normal lines, 172

finding equations for, 1022–1023

to planes, 737

to tangent planes, 876–877

Normal vectors

and gradients, 870

outer unit vs. inner unit, 1031–1032, 1033

principle unit, 769–770

upward-pointing vs. downward-pointing, 1030–1031

Normalization, 708

Norms

of partitions, 355, 904, 941, 978

of vectors, 707

Notation. See also Symbols

integral, 355

for intervals, A-5

Leibniz, 163

operator, 163, 866

prime, 148

summation, A-40–A-41

for vectors, 699

nth roots, A-9

nth terms

finding, 539–540

of sequences, 538

of series, 554

Numbers, critical, 267–268

Oblique triangles, A-35–A-37

Odd functions, 9–10

integrals of, 393–394

Oetzi the Iceman, 386

Omnidirectional microphones, 692

One-sided limits, 71

One-to-one functions, 32–34

graphs of, 35–37

Open intervals, A-5

continuity on, 96

Open sets, 825

Operational calculus, 84

Operator notation, 163, 866

Optimization, 318–325

and Lagrange multipliers, 891–897

for two variables, 886–888

Orbits, 688, 689

motion along, 788, 791–792

Order of differential equations, 332, 1059

Ordered pairs, A-16

and inverse functions, 34–35

Ordinary differential equations, 1059–1060

Orientability, 1030

Orientation

in parametric equations, 637

positive vs. negative, 1031

of surfaces, 1030–1033

and vector functions, 759

Origins, 695

moments about, 459

symmetry with respect to, A-17

Orthogonal curves, 818

Orthogonal functions, 221

Orthogonal surfaces, 878

Orthogonal trajectories, 1066–1068, 1067

Orthogonal vectors, 717

and the cross product, 727

and the dot product, 718

Orthogonalization, Gram–Schmidt, 724

Osculating circles, 780–782

Oseen velocity fields, 1056–1057

Ostrogradsky, Mikhail, 1038

Outer normal unit vectors, 1031, 1033

Outputs, 2

p-series, 570–573, 579t, 598t

Pappus of Alexandria, 464

Pappus Theorem, 464

Parabolas, A-22–A-23

as conic sections, 685

graphs of, 18

polar equations for, 686–687

Parabolic cylinders, 749

Paraboloids

elliptic, 745–746

hyperbolic, 746–747

of revolution, 745

surface areas of, 938

Parallel lines, 736–737, A-20

Parallel planes, 738–739

Parallel vectors, 706–707, 729

Parallelograms, 728–729

Parameter domains, 1016

Parameters, 637, 1016

Parametric equations, 637, 1016. See also Plane curves

arc lengths for, 651–653

conversion of, to rectangular equations, 638–640

for cycloids, 642

and line integrals, 977–979

for lines, 734

for normal lines, 877

for polar equations, 670–674, 675t

and surface areas, 658

time as parameter in, 640–641

Parametric surfaces, 1016–1017

finding equations for, 1018–1021

graphs of, 1017–1018

surface areas of, 1023–1024

Parametrization, 1016

smooth, 1021

Partial derivatives, 829–837, 830

first-order, 830–834

of functions of n variables, 836–837

second-order, 834–836

Partial differentials, 843

Partial fraction decomposition, 500

Partial fractions, 499

integration using, 499–505

Partial integration, 906–908

Partial sums, sequences of, 554

Partitions, 345–346

norms of, 355, 904, 941, 978

rectangular, 904–905

regular, 356

of Riemann sums, 355

of the x-axis, 405–408

of the y-axis, 408–411

Pascal, Blaise, 266, 643

Patches, 1023, 1027

Paths, independence of, 991

Pendulums, 191, 238, 643

Percentage error, 233–234

Periodic functions, 50, 162

Periods, 50, 54

Permittivity of free space, 1035

Perpendicular lines, A-20

π, approximations for, 439, 609, 629

Piecewise-defined functions, 4

continuity of, 94–95

Piecewise-smooth curves, 984

line integrals along, 984–985

Piecewise-smooth surfaces, 1028

Pinching Theorem. See Squeeze Theorem

Plane curves, 637, 758. See also Parametric equations

arc lengths for, 651–653

curvature of, 779–780

graphs of, 637

tangent lines to, 648–651

Planes

angles between, 742

coordinate, 696

equations for, 737–739

general equations for, 738

parallel, 738–739

tangent, 875–876

Planetary motion, 801–804

Plots, A-16. See also Graphs

scatter, 21–22

wireframe, 1017

Point-slope form of equation of a line, A-19

Points

boundary, 825

critical, 880–882

interior, 825

saddle, 881

test, 892

Poiseuille equation, 171

Polar axes, 661

Polar coordinates, 661

and changing variables, 962

conversion of, to rectangular coordinates, 664–666

double integrals using, 923–927

graphs using, 661–663

Polar equations, 666

arc lengths for, 675–676

areas between, 681–682

areas enclosed by, 678–680

of conics, 686–688, 804

graphs of, 666–668

parametric equations of, 670–674, 675t

Polar grids, 666

Polar moments of inertia, 933

Polarization identity, 724

Poles, 661

Polynomial functions, 17

Chebyshev, 634–635

continuity of, 97

derivatives of, 167

graphs of, 17–19, 308–309

irreducible quadratic, 503–504

limits of, 86

Taylor, 240–241

Position vectors, 703–705

Positive angles, A-25

Positive orientation, 1031

Potential, gravitational, 977

Potential energy, 1005

and orthogonal trajectories, 1067

and work, 1005–1006

Potential functions, 990–991

for conservative vector fields, 995–996

Power, 794

Power functions, 15

derivatives of, 164–166

limits of, 84–85

vs. exponential functions, 39

Power Rule, 177, 226–227

for functions, 202–204, 214

for rational exponents, 213–214

Power series, 600

continuity of, 606

convergence or divergence of, 600–602

and differential equations, 1089–1093

as functions, 604–606

integration of, 606

intervals of convergence for, 603–604

properties of, 606–609

Power series methods, 1089

Pressure, hydrostatic, 453–456

Prime notation, 148

Principal nth root of a real number, A-9

Principle unit normal vectors, 769–770

Probability density functions, 378, 532

Probability theory, 266

Product Rule, 174

and integration by parts, 472

Product-to-sum identities, A-35

Production functions, 220, 833–834

Productivity, marginal, 833–834

Products

continuity of, 98–99

cross (vector), 724–730, 725

derivatives of, 173–175, 763

dot (scalar), 715–721

limits of, 82–83, 821, B-2–B-3

Projectile motion, 112–113, 646, 650

vector equations for, 797–799

Proofs of theorems, B-1–B-13

by contradiction, 135

using chain rules, 855–856

Proper rational functions, 499

Pyramids, finding volumes of, 435–436

Pythagorean identities, A-33

Pythagorean Theorem, A-12

and distance between points in space, 696

and integration by trigonometric substitution, 488

Quadrantal angles, A-26

Quadrants, A-26

and trigonometric functions, A-30

Quadratic formula, A-3–A-4

Quadratic functions, 18. See also Polynomial functions

irreducible, 503–504

Quadric density model, 972

Quadric surfaces, 744–751

Quotient identities, A-33

Quotient Rule, 175

Quotients

continuity of, 98–99

derivatives of, 175–177

difference, 3

indeterminate forms of, 298–299

limits of, 87–88, 821

Radians, A-26–A-27

Radicals, A-9

Radicands, A-9

Radii, 697, A-21

of curvature, 780

Radii of convergence, 602

Ranges, 2

of functions, 3

of functions of two or three variables, 810

of projectiles, 798

of trigonometric functions, A-31–A-32

Raphson, Joseph, 235

Rates of change, 149. See also Average rate of change; Related rates

instantaneous, 149–150

partial derivatives as, 832–834

Rates of decay, 383–384

Ratio test for convergence, 591–593, 598t

and power series, 601

Rational functions, 19–20

asymptotes of, 126–127

continuity of, 97

graphs of, 309–312

integration of, 499–505

limits of, 87–88

proper vs. improper, 499

Rays, A-25

Real functions, 3

Reciprocal functions, 16, 56

derivatives of, 176–177

Reciprocal identities, A-33

Rectangles, A-15

Rectangular boxes, A-15

Rectangular coordinates, 695–697, A-16

conversion of, to cylindrical coordinates, 950–951

conversion of, to polar coordinates, 664–666

conversion of, to spherical coordinates, 956–958

and parametrization, 1020–1021

Rectangular equations, 638–640

conversion of, to parametric equations, 641–642

and surface areas, 659

Rectangular partitions, 904–905

Rectilinear motion, 145

and differential equations, 333–334

and first-derivative test, 286–287

Recursive formulas, 235, 1091, A-39

Reduction formulas, 477–478

Reference angles, A-30

Reflections, 30

Regions

connected, 993

multiply connected, 1011–1014

simply connected, 996–997

x-simple, 913–914

y-simple, 915–916

Regular partitions, 356

Related functions, 543–544