Smith R., Minton R. Calculus

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Subject Index I-7

area of regions bounded by, of y, 320

average cost, 211

average value of, 279–281

Bessel, 602–603

combination of, 36

component, deﬁned, 750

composition of, 36

continuity of, 830

ﬁnding, 37

identifying, 37

continuity of

deﬁned, 68, 828

of composite functions, 72

removable, 70

continuous, absolute extrema of,

187

cost, 211

cumulative distribution, 478

decreasing

deﬁned, 195

deﬁned, 13

density plots and, matching of, 816

derivatives and, rewriting for, 131

differentiable, deﬁned, 118

discontinuous, 68

domain of, 13

error, 598

exponential

base of, 391–392

deﬁned, 391

differentiation of, 395

integral of, 394

gamma, 478

generating, 586

hyperbolic, 411–417

derivative of, 412

integral involving, 413

inverse, 414–416

formula for, 415

increasing

deﬁned, 195

integrable, 273, 906

inverse

deﬁned, 384–385

ﬁnding, 386

graphing of, 387

tangent line to, 389

unknown, 387

iterations of, 42

log integral, 384

maximum rate of change of, 868

minimum rate of change of, 868

of several variables, 809–818

extrema of, 874–883

of two variables, 809–810

graph of, 811

limit of, 828

omega, 478

one-to-one, 385

periodic

deﬁned, 28, 607

fundamental period of, 28

piecewise-continuous, 278

piecewise-deﬁned, limit of, 65–66

polynomial, deﬁned, 13

potential, 292, 984

of vector ﬁeld, 1010

power, arc length of, 346

probability density, 481, 482

standard deviation for, 486

range of, 13

rational

continuity of, 69, 71

critical number of, 190

deﬁned, 15

graph of, 23–24, 208, 213–214

integration of, 442–448

limit of, 61

with no vertical asymptotes, 25

reliability, 478

Riemann sum for, with positive and

negative values, 274

Riemann-zeta, 563

root of, 14

scalar, 1026

sigmoid, 398

sine, 29

special, 602

square root, 15

chain rule with, 144

derivative of, 119–120

square-wave, 610

squeeze theorem for, 64

sum of values of, 263

table of data, deﬁned by, 810

transformations of, 36–41

trigonometric, 26–34

derivatives of, 147–152

in powers, 294

inverse, 399–403

calculus of, 405–409

derivative of, 406

integrals involving, 407–409

simpliﬁcation of, 402

limit of, 63, 91

loss of signiﬁcance involving, 101

polynomial and, sum of, 220–221

vector-valued

antiderivative of, 765

calculus of, 758–767

continuous, 759–760

deﬁned, 750

deﬁnite integral of, 766

derivative of, 761

differentiation of, 761

ellipse, 751

elliptical helix, 751–752

graphing of, 750–751

increment of, 760

indeﬁnite integral of, 766

limit of, 759

line, 752

matching to graph, 752

wave, 1098

well-deﬁned, 290

with no inverse, 385

with second derivative test as

inconclusive, 207

with vertical tangent line at inﬂection

point, 209

with zero derivative at local

maximum, 187

zeros of

approximate, 25

approximating

by method of bisection, 74–75

with Newton’s method, 179

Bessel functions, 603

by factoring, 16

by quadratic formula, 16

cubic polynomial, 17

determining number of, 164

ﬁnding, by method of bisections,

74–75

Fundamental period, 28

Fundamental Theorem for Line

Integrals, 1007

Fundamental Theorem of Algebra, 261

Fundamental Theorem of Calculus,

284–290

Future value, 501

Gabriel’s horn, 351, 477

Galileo, 1097

Gamma function, 478

Gauss, Karl Friedrich, 261, 269

Gauss’ Law for electricity, 1050

Gauss’ Theorem. See Divergence

Theorem

Gaussian quadrature, 310

General solution, for differential

equations, 492, 1075

General term, 532

Generating function, 586

Geometric series, 547

convergent, 547–548

divergence of, 548–549

Geometry, differential, 788

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I-8 Subject Index

Geosynchronous orbit, 779

Gibbs phenomenon, 618

Gibbs, Josiah Willard, 717

Gini index, 272

Global behavior

deﬁned, 23, 198

Gradient

applications, 871

deﬁned, 865, 869

scalar functions with, 1026

vector ﬁelds with, 1026

Gradient derivatives, 864–871

Gradient ﬁeld(s)

deﬁnition of, 984

ﬁnding, 984–985

potential function of, 985–986

Granville, Evelyn, 772

Graph(s)

comparing, 39

ﬁrst derivatives in, 212

global behavior in, 23, 198

hidden behavior in, 198

horizontal asymptotes, 212

horizontal translation of, 38–39

intercepts in, 212

limit determination with, 54

local behavior in, 23, 198

of cardioid, 654–655

of contour plots, 815

of cubic polynomial, 23

of cylindrical surface, 734

of ellipsoid, 735

of elliptic cone, 738

of ﬂow lines, 982

of hyperboloid, 738–739

of inverse function, 387

of lima¸cons, 653–654, 655

of line, 10

of local extrema, 197, 875–876

of paraboloid, 737

of parametric surface, 799–800

of plane curve, 626

of polar coordinates, 652

of polynomial, 212–213

of polynomial and trigonometric

function sum, 220–221

of rational function, 23–24, 208,

213–214

of Taylor polynomials, 588

of three variable functions, 811

of two variable functions, 811

of vector-valued function, 750–751,

752

removing hole in, 69–70

vector ﬁelds, 978–979

vertical asymptotes, 24, 212

vertical translation of, 37–38

with difﬁcult-to-see features,

218–219

with no inﬂection points, 206

with no tangent lines, 115

with two vertical asymptotes,

215–216

Graphing calculators

automatic graphing window, 21

ﬁxed graphing window, 21

generating graph, 21

intersections on, 25

pixels in, 21

solving equations on, 26

Graphing window, 21

Gravitation, 680, 794

Green’s Theorem, 1014–1021

Green, George, 1014

Gross domestic product (GDP), 272

Growth and decay problems, 491–496

Growth constant, 492

Growth, logistic, 505–507

Guess, initial, 880

Half-life, 494

Half-open interval of convergence, 581

Halmos, Paul, 90

Hamilton, William Rowan, 697

Hardy-Weinberg law, 886

Harmonic content, 619

Harmonic motion, simple, 1079, 1098

Harmonic series, 550

alternating, 565

Hau, Lene, 708

Heat conductivity, 1041

Heat equation, 862, 1062, 1063

Heat index, 818

Heat, speciﬁc, 1063

Helix

curvature of, 783

elliptical

deﬁned, 752

vector-valued function deﬁning,

751–752

Hermite polynomial, 1107

Hermite’s equation, 1107

Higher order derivatives, 131–132

Higher order differential equations,

1081

Higher-order partial derivatives,

837

Homeomorphism, 411

Homogeneous differential equations,

1082

Hooke’s Law, 362

Horizontal asymptotes

ﬁnding, 81

in graphing, 212

Horizontal component, of vector, 637

Horizontal line test, 385

Horizontal tangent, 190, 661

Horizontal translations, 38–39

Hydrostatic force, 367

Hyperbola

center of, 674

deﬁned, 673

equation of, 674

foci of, 673

Hyperbolic cosecant function, 412

Hyperbolic cosine function, 411

inverse, 414

Hyperbolic cotangent function, 412

Hyperbolic functions, 411–417

derivative of, 412

integral involving, 413

inverse, 414–416

formula for, 415

Hyperbolic paraboloid, 740–741, 801

Hyperbolic secant function, 412

Hyperbolic sine function, 411

inverse, 414

Hyperbolic tangent function, 412

inverse, 414

Hyperboloid

equation for, 739

of one sheet, 738–739

of two sheets, 739, 740

Hypersurface, 928

Hypocycloid, 640

Identities, trigonometric functions and,

Image, of transformation, 963

Implicit differentiation, 155–160, 860

Implicit plot, 736

Implicit solution, 504

Improper integrals, 467–476

comparison test for, 474

convergence of, 469, 471

deﬁned, 467

divergence of, 468, 469, 471

with discontinuous integrand,

467–470

with inﬁnite limit of integration,

470–473

Improved Euler’s method, 520

Impulse, 364

Impulse-momentum equation, 283, 364

Incompressible vector ﬁeld, 1026

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Subject Index I-9

Increasing function

deﬁned, 195

Increasing sequence, 538

Increment

computation of, 849

deﬁned, 848

of vector-valued functions, 760

Indeﬁnite integrals

deﬁned, 253

evaluating, 254

of difference, 256

of sum, 256

of vector-valued function, 766

power rule for, 254

Indeterminate forms

deﬁned, 82, 457

limit of, 459

other, 461–464

simpliﬁcation of, 461

Index of summation, 260

Induction assumption, 263

Induction, mathematical, 263–264

Inequality

alternate method of solving, 5

Cauchy-Schwartz, 707, 712

linear, 3

quadratic, 4

real number system and, 2–5

triangle, 5, 708, 712

two-sided, 3–4

with absolute value, 5

with fraction, 4

with sum inside absolute value, 5

Inequality constraint, 890–891

Inertia ellipsoids, 745

Infant mortality phase, 421

Inﬁnite products, 553

Inﬁnite series, 266

convergent, 546

deﬁned, 545

divergence of, 546

sums of, 545

Inﬁnity

limits at, 81–84, 93–96

Inﬂection points, 205, 206

Information theory, 478

Information, qualitative, 510

Initial condition, 493, 503, 1077

Initial guess, 880

Initial point, of vector, 688

Initial value problem (IVP), 503, 1077

Inner partition, 908, 926, 929

Instantaneous rate of change, 114,

132–133

Instantaneous velocity, 113

Integers, 2, 261, 1034–1035

Integrable function, deﬁned, 273, 906

Integral Mean Value Theorem, 280

Integral test, 554–562

Integral(s)

Boltzmann, 478

completing the square with, 423

deﬁnite

approximation of, with Midpoint

Rule, 274

computing exactly, 285

deﬁned, 903

integration by parts for, 430–431

of vector-valued functions, 766

signed area and, 275–276

substitution in, 295–296

Taylor series for approximating,

601

with variable upper limit, 286

double, 901–914

area with, 918

change of variables in, 967

deﬁned, 906, 909

evaluation of, 910, 912

in polar coordinates, 926–931

irregular partitions and, 904

order in, 913

over general regions, 908–913

over rectangle, 903–907

volume with, 904–905, 918

improper, 467–476

convergence of, 469, 471

deﬁned, 467

divergent, 468, 471

with discontinuous integrand,

467–470

with inﬁnite limit, 470–473

indeﬁnite

deﬁned, 253, 273

evaluating, 254

of difference, 256

of sum, 256

of vector-valued functions, 766

power rule for, 254

line, 990–1001

deﬁned, 990

determining sign of, graphically,

1000–1001

evaluation of

over piecewise-smooth curve,

993–995

with Green’s Theorem,

1017–1018

with respect to arc length, 992

with Stokes’ Theorem, 1055

fundamental theorem for, 1007

Green’s Theorem and, 1017

independence of path, 1003–1011

with respect to x, 997

with respect to y, 997

work and, 999–1000

of exponential functions, 394

of inverse trigonometric functions,

407–409

Riemann-Stieltjes, 426

substitution in evaluation of, 293

surface, 1032–1041

deﬁnition of, 1032, 1040

evaluation of, 1034–1035

using complement of surface,

1062

with polar coordinates, 1035

with spherical coordinates, 1038

with Stokes’ Theorem, 1056

surface area with, 1038–1039

tangent line for, 289

Taylor series for approximation of,

602

triple, 928–946

center of mass and, 944–946

change of variables in, 970

deﬁned, 928

in cylindrical coordinates,

950–951

in spherical coordinates, 957–960

inner partition of, 929

order of integration in, 942–943

over rectangular box, 929

over tetrahedron, 940–941

volume with, 943–944, 952–953

with ﬁrst integration with respect

to x, 941–942

with polar coordinates, 948

with discontinuous integrand, 278

with logarithms, 378

with powers of trigonometric

functions, 433–437

with variable upper and lower limits,

288

Integrand, 253, 278, 295

discontinuous, 467–470

expansion of, 423

with even power of cosine, 435

with even power of secant, 436

with even power of sine, 435

with odd power of cosine, 434

with odd power of sine, 434

with odd power of tangent, 436

with single term, 428

Integrating factor, 989

Integration

by parts, 426–431

for deﬁnite integral, 430–431

repeated, 428

by substitution, 292–296, 422, 447

constant of, 253

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I-10 Subject Index

Integration—Cont.

deﬁned, 253

generalizing rule of, 423

in engineering, 361–369

in physics, 361–369

lower limit of, 273

numerical, 298–309

error bounds for, 305–308

of power series, 583–584

of rational functions, 442–448

partial, 906

reduction formula and, 430, 451

tables, 301, 450–456

trigonometric techniques of, 433–440

with computer algebra systems,

450–456

with partial fractions, 442–448

Intercepts, 212

Interest

compound, 496–498

continuous compound, 496

Interior point, 828

Intermediate Value Theorem, 74

Interpolation, linear, 177

Interval

closed, 2

absolute extremum on, 191

continuity on, 72

open, 3

derivative on, 119

Interval of convergence, 581

half-closed, 581

Inverse cosine function

deﬁned, 400

evaluation of, 400

Inverse function

deﬁned, 384–385

ﬁnding, 386

graphing of, 387

tangent line to, 389

unknown, 387

Inverse hyperbolic cosine function, 414

Inverse hyperbolic functions, 414–416

formula for, 415

Inverse hyperbolic sine function, 414

Inverse hyperbolic tangent function,

414

Inverse problem, 375

Inverse relationship, 384

Inverse secant function

deﬁned, 401

evaluation of, 402

Inverse sine function, 399

evaluation of, 400

hyperbolic, 414

Inverse square ﬁeld, ﬂux of,

1049–1050

Inverse square law, 987

Inverse tangent function

deﬁned, 401

evaluation of, 401

hyperbolic, 414

integral related to, 408

simpliﬁcation of expression with,

402–403

Inverse trigonometric functions,

399–403

calculus of, 405–409

derivatives of, 406

integrals involving, 407–409

simpliﬁcation of, 402

Irrational numbers, 2

Irregular partitions, 272, 904

Irrotational ﬂow, 1058

Irrotational vector ﬁelds, 1024,

1027–1028

Iterates, 42

Iterations, of functions, 42

IVP. See Initial value problem (IVP)

Jacobi, Carl Gustav, 966

Jacobian of transformation, 967

Jones, Vaughan, 464

Julia set, 712

Just-in-time inventory, 251

Kepler’s laws of planetary motion, 680,

794–797

Kepler, Johannes, 194, 794

Kinetic energy, 361, 1013

Klein bottle, 803

kth term test, 549–550

L’H ˆopital’s Rule, 459, 535–536

L’H ˆopital, Guillaume de, 459

Lagrange multiplier(s), 887–894

deﬁned, 889

method of, 888

Lagrange points, 184, 799

Lagrange, Joseph-Louis, 888

Lambert shading, 874

Lamina, 920

center of mass of, 920, 921–922

moments of inertia of, 923

Laplace equation, 989

Laplace transform, 478

Laplace, Pierr´e-Simon, 470

Laplacian, 858, 862, 873, 1026

Least squares method, 878

Legendre polynomials, 607

Leibniz notation, 122

Leibniz, Gottfried Wilhelm,

122, 140

Level curves, 814

directional derivatives and, 867

Level surfaces, 817–818

Lima¸cons, 653–654, 655, 664

Limit

approaching, 53

approaching value of, 56

at inﬁnity, 81–84, 93–96

by factoring, 61

by rationalizing, 62–63

comparison test for, 560

computation of, 59–66

concept of, 52–57

continuity and, 822–831

describing velocity, 66

evaluating, 53

formal deﬁnition of, 87–96, 823

graphic determination of, 54,

91–93

inﬁnite, 94

of improper integral, 470–473

L’H ˆopital’s rule and, 459

loss of signiﬁcance errors, 98–103

nonexistent, 54, 55, 92, 825

of indeterminate forms, 459

of integration, 273

of natural log, 381

of nth root of polynomial, 62

of piecewise deﬁned functions,

65–66

of polynomial, 61, 824

of product that is not product of

limits, 63–64

of quotient that is not quotient of

limits, 82

of rational function, 61

of sequence, 534

of trigonometric functions, 63,

80, 91

of two variable function, 828

one-sided, 53, 56, 79

precise deﬁnition of, 89

proving as correct, 89

proving existence of, 827

simple, 88

Taylor series for conjecture of value

of, 600–601

variable, integral with, 288

verifying, 64–65, 88

where two factors cancel, 55

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Subject Index I-11

Limit cycle, 668

Line integrals, 990–1001

deﬁned, 990

determining sign of, graphically,

1000–1001

evaluating

over piecewise-smooth curve,

993–995

with Green’s Theorem, 1017–1018

with respect to arc length, 992

with Stokes’ Theorem, 1055

fundamental theorem for, 1007

Green’s theorem and, 1017

in space, 997–998

independence of path, 1003–1011

with respect to x, 997

with respect to y, 997

work and, 999–1000

Line segments, parametric equations of,

628

Line(s)

curvature of, 782

distance from point to, 720

equations of, 9–18

ﬂow

deﬁnition of, 981

differential equations with, 983

Euler’s method in approximation

of, 983–984

graphing of, 982

graphing of, 10

in space, 726–731

nonintersecting but not parallel, 727

normal, 846, 870

orthogonal, 727

parabola and, intersection of, 17–18

parallel, 11, 12, 727

perpendicular, 11, 12

point-slope form, 11

secant, 108

slope of, 9, 10

slope-intercept form of, 11

symmetric equations of, 726

tangent, 47–51

equation of, 110, 137, 152

ﬁnding with implicit

differentiation, 156

for function deﬁned as integral,

289

graphical approximation of, 111

horizontal, 190

numerical representation of, 111

to inverse function, 389

to parametric curve, 635

velocity and, 107–115

vertical, 190, 209

vector-valued function deﬁning, 752

Linear approximation

deﬁned, 175, 847, 850

ﬁnding, 175–176, 847–848

for linear interpolation, 177

of cube roots, 176

of sine function, 176

tangent planes and, 844–852

Linear convergence, 184

Linear density, 243

Linear equations, in three dimensions,

729

Linear inequality, 3

Linear interpolation, 177

Linear momentum, 775, 778

conservation of, 778

Linear ordinary differential equations,

989

Linear polynomial, 14

Linear regression, 878, 879–880

Linear transformations, 27

direct, 972

Local behavior, in graphing, 23, 198

Local extremum

approximation of, 200

deﬁned, 22

discriminant in ﬁnding, 877–878

graph of, 197, 875–876

in functions of several variables, 874

of polynomial, 189

using ﬁrst derivative test, 200

Local maximum

deﬁned, 22, 187

function with zero derivative at, 187

in functions of several variables, 874

Local minimum

deﬁned, 22, 187

function with undeﬁned derivative at,

188

in functions of several variables,

874

Log integral function, 384

Logarithm(s)

derivative of

of absolute value, 377

differentiation of, 377, 380–381,

381–382

integrals involving, 378

natural

deﬁned, 376

inverse of, 391–393

limiting behavior of, 381

rewriting expressions with, 380

Logistic equation, 244, 245, 505

Logistic growth, 505–507

Lorentz curve, 272

Loss-of-signiﬁcance errors, limits and,

98–103

Loss-of-signiﬁcant-digits error, 100

Lower limit of integration, 273

M-test, 586

Maclaurin series, 588, 598

Maclaurin, Colin, 555

Magnetic ﬁeld, ﬂux of, 1051, 1059

Magnitude, of vector, 688, 700

Magnus force, 356, 722, 723

Major axes, of ellipse, 671

Mandelbrot set, 712

Mandelbrot, Benoit, 286

Map, topographical, 821

Marginal cost, 239

Marginal proﬁt, deﬁned, 239

Mass

center of, 365

of solid, 945–946

triple integrals and, 944–946

point, 365

Mass density, 243

Mathematical analysis, 87

Mathematical induction, 263–264

Matrix

determinant of, 715

Maximum

absolute, 185, 882

local

deﬁned, 22, 187

function with zero derivative at,

187

in functions of several variables,

874

Maximum increase, direction of,

869–870

Maximum rate of change of function,

868

Maxwell’s equations, 1065–1066

McDuff, Dusa, 159

McNulty, Kay, 515

Mean

arithmetic, 478, 896

Mean Value Theorem, 162–167, 280

Median, 484

Method of bisections, ﬁnding zeros by,

74–75

Method of cylindrical shells, 338–343

Method of disks, 328–330

Method of Lagrange multipliers,

888

Method of undetermined coefﬁcients,

1083

Method of washers, 330–335, 340

Midpoint Rule, 273, 299–300, 304

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I-12 Subject Index

Minimum

absolute, 185, 882

distance, 888

local

deﬁned, 22, 187

function with undeﬁned derivative

at, 188

in functions of several variables,

874

Minimum rate of change of function,

868

Minor axes, of ellipse, 671

Mixed second-order partial derivatives,

837

M¨obius, August Ferdinand, 1039

Modeling, with differential equations,

491–498

Moment of inertia

about the x-axis, 922

about the y-axis, 922

of lamina, 923

Moment(s)

ﬁrst, 365, 944

second, 922

with respect to x-axis, 921

with respect to y-axis, 921

Momentum

angular, 775, 778

conservation of, 776, 925

impulse and, 364

linear, 775

conservation of, 778

Monotonic sequence, 538, 540

Monte Carlo method, 916

Morawetz, Cathleen, 1057

Mori, Shigefumi, 1082

Motion

equations of, 773–776

in space, 769–777

Kepler’s laws of, 794–797

Newton’s second law of, 252, 770

planetary, 794–797

projectile

analyzing, 771–772

in three dimensions, 776

in two dimensions, 355

initial velocity to reach given

height, 354

Newton’s second law and, 352

parametric equations and, 626–627

terminal velocity in, 361

velocity at impact, 352–353

vertical motion in, 353

with air resistance, 356

simple harmonic, 1079, 1098

Moving trihedral, 788

Multiplicity, of zeros, 68, 184, 466

Multiplier effect, 553

Multipliers, Lagrange, 887–894

deﬁned, 889

method of, 888

Music synthesizers, 617–618, 619

Napier, John, 378

Natural frequency, 1087

Natural logarithms

deﬁned, 376

inverse of, 391–393

limiting behavior of, 381

Negative exponent, 254

Negative orientation, 1014

Newton’s Law of Cooling, 494

Newton’s method, 178–181

Newton’s second law of motion, 252,

352, 770

Newton, Isaac, 96, 178

Newton-Raphson method, 179

Nonexistent limit, 54, 55, 92, 825

Nonhomogeneous differential

equations, 1082–1088

Nontrivial solution, 1098

Normal component of acceleration,

790–794

Normal distribution, 482

Normal line, 846, 870

Normal plane, 789

Normal vector, 713, 786–797

Normalization, of vector, 693

Notation

derivatives, 121–123

Leibniz, 122

sigma, 259–264

summation, 260

vector, 688

Number(s)

critical

deﬁned, 188

of rational function, 190

irrational, 2

rational, 2

real, 2–7

computer representation of, 99

sequences of, 532–542

Numerical differentiation, 122–123

Numerical integration, 298–309

error bounds for, 305–308

Oblique asymptote, 83

Octants, 698

Octave, 8

Omega function, 478

One-sided limits, 53, 56, 79

One-to-one functions, 385

One-to-one transformation, 963

Open disk, 828

Open interval, 3

differential on, 119

Opposite vector, 690

Optimization, 223–230

constrained, 887–894

with inequality constraint, 890–891

with two constraints, 893–894

Orbit, geosynchronous, 779

Ordered pair, 6

Orientable surface, 1039

Orientation of curve

deﬁnition of, 990

negative, 1014

positive, 1014

vector-valued curve, 751

Orthogonal lines, 727

Orthogonal planes, 730

Orthogonal projection, 712

Orthogonal vectors, 706, 792

Orthogonality condition, 607

Osculating circle, 789

Osculating plane, 789

p-series, 556–557

Pappus’ Theorem, 337

Parabola

closest point on, 225–226

curvature of, 784

deﬁned, 668

directrix of, 668

equation for, 669, 670

focus of, 668

line and, intersection of, 17–18

minimum distance on, 226–227

opening left, 670

reﬂective property of, 670

Paraboloid(s)

applications of, 741–742

circular, 737

graph of, 737

hyperbolic, 740–741, 801

volume between two, 930

Parallel lines, 11, 12, 727

Parallel planes, 730

Parallel vectors, 691

Parallelepiped, volume of, 721

Parallelogram, area of, 720

Parameter, in parametric equations,

626

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Subject Index I-13

Parameterizations, 628, 780

Parametric equations, 457

calculus and, 634–639

circles deﬁned by, 627–628

conic sections in, 668–675

deﬁned, 625

ellipses deﬁned by, 627–628

for intersecting surfaces, 755

for line segment, 628

for x-y equations, 629

of hyperbolic paraboloid, 801

parameter of, 626

plane curves and, 625–630

projectile motion and, 626–627

sine and, 627

slope of, 635

surface area with, 641–647

Parametric plot, 736

Parametric surfaces, 799–802

graphing of, 799–800

Partial derivatives, 833–840

applications, 836–837, 839

computing of, 835–836

deﬁned, 835

from table of data, 839–840

higher-order, 837

mixed second-order, 837

of three variable functions,

838–839

Partial differential operator, 835

Partial fraction decomposition,

442–448

Partial integration, 906

Partial sum, 546–547

error estimation in, 557

Partition(s)

inner, 908, 926, 929

irregular, 272, 904

regular, 266

Parts, integration by, 426–431

repeated, 428

Pascal’s Principle, 367

Pascal, Blaise, 481

Path

deﬁnition of, 1004

independence of, in line integrals,

1003–1011

of steepest ascent, 868

pdf. See Probability density function

(pdf)

Pendulum, undamped, 1094

Perelman, Grigori, 735

Period, of periodic function, 30–31,

607

Periodic function

deﬁned, 28, 607

fundamental period of, 28

Perpendicular lines, 11, 12

Perpendicular vectors, 706

Phase portrait, 487, 522

Piecewise-continuous function, 278

Piecewise-deﬁned functions, limit of,

65–66

Piecewise-smooth curve, line integrals

over, 993–995

Piecewise-smooth surface, 1037

Pixels, 21

Planck’s law, 398, 606

Plane curves

arc length of, 643

deﬁned, 626

graph of, 626

unusual, 629

Plane, Cartesian, 6–7

Plane(s)

equation of, 729

in R

, 728–731

in space, 726–731

intersection of, 730

normal, 789

orthogonal, 730

osculating, 789

parallel, 730

tangent, 844–852

equation of, 846–847

gradient and, 870

normal line, 870

vectors in, 688–695

Planetary motion, 680, 794–797

Plot

Bode, 1096

contour, 814

density, 814, 816

Point masses, 365

Point of diminishing returns,

233

Point-slope form, of line, 11

Point(s)

boundary, 828

colinear, 9, 10

critical, 875, 878

derivative at, 118–119

distance from, 720

evaluation, 269, 270

ﬁxed, 42

in three dimensions, 698

inﬂection, 205, 206

initial, 688

interior, 828

Lagrange, 184, 799

on parabola, closest,

225–226

saddle, 741, 876

source, 1026

Polar coordinates

arc length in, 666

area in, 664, 927

calculus and, 660–666

conic sections in, 677–681

conversion to/from rectangular

coordinates, 649–650

deﬁned, 649

double integrals in, 926–931

graphing of, 652

horizontal tangent lines and, 661

intersections in, 665

plotting points in, 649

surface area in, 935

surface integral evaluation with, 1035

transformation involving, 965

triple integrals with, 948

volume in, 928

Polar form of vector, 694

Polynomial(s)

coefﬁcients of, 13

constant, 14

cubic, 14, 17

graph of, 23

deﬁned, 13

degree of, 13

graph of, 212–213

Hermite, 1107

Legendre, 607

limit of, 61, 824

linear, 14

local extrema of, 189

quadratic, 14

quartic, 14

quintic, 14

Taylor, of degree n, 588

trigonometric function and, sum of,

220–221

Position

equilibrium, 1074

estimating overall change in, 277

Position vector, 689, 692, 750

Positive orientation, 1014, 1039

Potential energy, 361, 1013

Potential function, 292

of gradient ﬁeld

deﬁnition of, 984

ﬁnding, 985–986

of vector ﬁeld, 1010

Potential, electrical, 607

Power functions, arc length of, 346

Power functions, inside cosine, 294

Power rule

for derivatives, 127–133

for indeﬁnite integral, 254

with fractional exponent, 255

with negative exponent, 254

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CONFIRMING PAGES

I-14 Subject Index

Power series

convergence of, 580

deﬁned, 580

differentiation of, 582–583

integration of, 583–584

solutions of differential equations,

1098–1106

Power, deﬁnition of, 863

Predator-prey systems, 298, 521–522

Present value, 496, 553

Pressure, of gas, vs. volume and

temperature, 157

Price vector, 713

Price, relative change in, 241

Price-to-earnings ratio, 821

Principal unit normal vector, 786

Probability density function (pdf),

481, 482

standard deviation for, 486

Probability distributions

continuous, 481

discrete, 481

normal, 482

Probability, conditional, 478

Product (in chemistry), 246

Product rule(s), for derivatives,

135–140, 151

Proﬁt

deﬁned, 245

marginal, deﬁned, 239

Projectile motion, 265

analyzing, 771–772

in three dimensions, 776

in two dimensions, 355

initial velocity to reach given height,

354

Newton’s second law and, 352

parametric equations and, 626–627

terminal velocity in, 361

velocity at impact, 352–353

vertical motion in, 353

with air resistance, 356

Projection, of vector, 708–711

orthogonal, 712

Pythagorean comma, 8

Quadratic approximation, 211

Quadratic factor, partial fractions with,

445–446

Quadratic formula, 16

Quadratic inequality, 4

Quadratic polynomial, 14

Quadric surfaces, 735

Qualitative information, 510

Quartic polynomial, 14

Quaternions, 697

Quintic polynomial, 14

Quotient

difference, 110, 127

limit of, 82

Quotient rule, for derivatives, 135–140

Radians, 28

Radioactive decay, 494

Radius of convergence, 581, 602–603

Radius of curvature, 789

Radius of sphere, 702

Ramanujan, Srinivasa, 575

Range, of function, deﬁned, 13

Rate of change

in economics, 239–244

in sciences, 239–244

in volume, 157

instantaneous, 114, 132–133

interpreting, 114

maximum, 868

minimum, 868

Ratio test, 571–577

Rational function

continuity of, 69, 71

critical number of, 190

deﬁned, 15

graph of, 23–24, 208, 213–214

integration of, 442–448

limit of, 61

with no vertical asymptotes, 25

Rational numbers, 2

Rationalizing, ﬁnding limit by, 62–63

Rayleigh-Jeans law, 606–607

Reactants, 246

Real line, 2

Real number(s), 2–7

computer representation of, 99

sequences of, 532–542

Rectangle

approximating area under curve with,

267

double integrals over, 903–907

Rectangular coordinates

conversion to/from cylindrical

coordinates, 951–952

conversion to/from polar coordinates,

649–650

conversion to/from spherical

coordinates, 956

deﬁned, 649

Recurrence relation, 1100

Reduction formula, 430, 451

Reﬂective property, 670

Region

connected, 1004

continuity on, 829

double integrals over general,

908–913

elementary polar, 926

simply-connected, 1009, 1058

transformation of simple, 964

Regression, linear, 878, 879–880

Regular partition, 266