INDEX

A114 |||| INDEX

cable (hanging), 255

division of, A55, A58

convergent series, 688

calculator, graphing, 46, 315, 624, 646

equality of, A55

properties of, 693

calculus, 9

imaginary part of, A55

coordinate(s), A2

invention of, 399

modulus of, A56

Cartesian, A11

cancellation equations, 62

multiplication of, A55, A58

polar, 639

for inverse trigonometric functions, 67

polar form, A57

rectangular, A11

for logarithms, 64

powers of, A59

coordinate axes, A11

cans, manufacturing, 333

principal square root of, A56

Cornu’s spiral, 637

Cantor, Georg, 696

real part of, A46

cosine function, A26

Cantor set, 696

roots of, A60

derivative, 192

capital formation, 554

composition of functions, 41, 197

graph, 33, A31

cardiac output, 552

continuity of, 125

power series, 740

cardioid, 213, 643

derivative of, 199

cost function, 228, 327

carrying capacity, 233, 568

compound interest, 238, 306, 618

critical number, 274

Cartesian coordinate system, A11

compressibility, 225

cross-section, 422

Cartesian plane, A11

computer algebra system, 91, 624

cubic function, 28

Cassini, Giovanni, 649

for graphing sequences, 680

current, 201

catenary, 255

for integration, 491

curvature, 638

Cauchy, Augustin-Louis, 113, A45

computer, graphing with, 46, 315, 646

curve(s)

Cauchy’s Mean Value Theorem, A45

concavity, 291

asymptotic, 315

Cavalieri’s Principle, 432

Concavity Test, 291, A44

Bézier, 624, 639

center of gravity, 542

concentration, 224

bullet-nose, 51, 204

center of mass, 542

conchoid, 626, 648

Devil’s, 213

centroid of a plane region, 543

conditionally convergent series, 715

length of, 525

Chain Rule, 197

cone, 431

orthogonal, 214

change of base, 66

conic section, 654, 662

parametric, 621

change of variables in

directrix, 662

polar, 641

integration, 401

eccentricity, 662

smooth, 525

charge, 224

focus, 662

swallowtail catastrophe, 629

chemical reaction, 224

polar equation, 663

curve fitting, 25

circle, A16

shifted, 659, A21

curve-sketching procedure, 308

area of, 469

conjugates, properties of, A56

cusp, 626

circular cylinder, 422

constant function, 173

cycloid, 624

cissoid, 629, 648

Constant Multiple Law of limits, 100

cylinder, 422

closed interval, A3

Constant Multiple Rule, 176

cylindrical shell, 433

Closed Interval Method, 275

consumer surplus, 550

cochleoid, 670

continued fraction expansion, 686

decay, law of natural, 236

coefficient

continuity

decay, radioactive, 236

of friction, 196, 278

of a function, 119

decreasing function, 20

of inequality, 399

on an interval, 121

decreasing sequence, 681

of a polynomial, 28

from the left, 121

definite integral, 366

of a power series, 723

from the right, 121

properties of, 373

combinations of functions, 41

continuous compounding of

Substitution Rule for, 404

comets, orbits of, 668

interest, 306, 618

definite integration

comparison properties of the

continuous random variable, 555

by parts, 456

integral, 375

convergence

by substitution, 404

Comparison Test, 514, 706

absolute, 714

degree of a polynomial, 28

comparison tests for series, 705

conditional, 715

delta ( ) notation, 148

Comparison Theorem for

of an improper integral, 509, 512

demand curve, 327, 550

integrals, 514

interval of, 725

demand function, 327, 550

Completeness Axiom, 682

radius of, 725

De Moivre, Abraham, A59

complex conjugate, A55

of a sequence, 677

De Moivre’s Theorem, A59

complex exponentials, A57

of a series, 688

density

complex number(s), A55

convergent improper integral, 509, 512

linear, 223

argument of, A57

convergent sequence, 677

liquid, 540

INDEX |||| A115

dependent variable, 12

logistic differential, 568, 600

derivative(s), 143,154

Lotka-Volterra, 609

of a composite function, 199

nth-degree, 210

of a constant function, 173

of a parabola, A18

domain of, 154

parametric, 631

of exponential functions, 180, 201,

point-slope, 19, A12

A51, A53

polar, 641

as a function, 154

predator-prey, 609

higher, 160

second-degree, A16

of hyperbolic functions, 254

slope-intercept, A13

of an integral, 380

two-intercept form, A16

of an inverse function, 212

equilateral hyperbola, A21

of inverse trigonometric func-

equilibrium point, 610

tions, 211, 213

equilibrium position, 196

left-hand, 165

equilibrium solution, 568, 609

of logarithmic functions, 215, A49, A54

error

notation, 157

in approximate integration, 521

of a power function, 174

percentage, 251

of a power series, 729

relative, 251

of a product, 183, 184

in Taylor approximation, 749

of a quotient, 185

error bounds, 499, 503

as a rate of change, 148

error estimate

right-hand, 165

for alternating series, 712

second, 160

for the Midpoint Rule, 521

as the slope of a tangent, 146

for Simpson’s Rule, 525

third, 161

for the Trapezoidal Rule, 521

of trigonometric functions, 189, 193

escape velocity, 539

Descartes, René, A11

estimate of the sum of a series, 700, 708,

descent of aircraft, determining

712, 717

start of, 206

Eudoxus, 3

Devil’s curve, 213

Euler’s constant, 704

Difference Law of limits, 100

Euler’s formula, A61

Difference Rule, 177

Euler’s Method, 575

differentiable function, 157

even function, 19, 308

differential, 250

exponential decay, 591

differential equation, 234, 342, 566, 569

exponential function(s), 33, 92, 180

autonomous, 575

with base a, A53

Bernoulli, 607

derivatives of, 180, 201, A51, A53

first-order, 580

graphs of, 53, 180

general solution of, 569

integration of, 370, 385, 400

linear, 602

limits of, 137, A51

logistic, 592, 687

power series for, 736

order of, 569

properties of, A51

second-order, 569

exponential graph, 53

separable, 580

exponential growth, 591

solution of, 569

exponents, laws of, 54, A51, A53

differentiation, 157

extrapolation, 27

formulas for, 187, RP5

extreme value, 271

implicit, 207, 208

Extreme Value Theorem, 272

logarithmic, 217

of a power series, 729

family of functions, 50, 320

term by term, 729

family of hypocycloids, 629

differentiation operator, 157

family of solutions, 568

Direct Substitution Property, 102

fat circles, 211, 531

direction field, 572, 573, 600

Fermat, Pierre, 4, 153, 273

A116 |||| INDEX

Fermat’s Principle, 331

Gompertz, 600

geometric series, 688

Fermat’s Theorem, 273

graph of, 12

Gompertz function, 600

Fibonacci, 686

greatest integer, 105

Gourdon, Xavier, 739

Fibonacci sequence, 676

Heaviside, 45, 92

graph(s)

First Derivative Test, 288

hyperbolic, 254

of an equation, A10, A16

for Absolute Extreme Values, 324

implicit, 207

of exponential functions, 53

first-order linear differential equation, 602

increasing, 20

of a function, 12

fixed point of a function, 171, 286

inverse, 61

of logarithmic functions, 66

flash bulb, current to, 84

inverse hyperbolic, 254

of a parametric curve, 632

flux, 551, 552

inverse trigonometric, 67, 68

polar, 641, 646

FM synthesis, 318

limit of, 88, 110

of power functions, 30, RP3

focus

linear, 24

of trignometric functions, A30, RP2

of a conic section, 662

logarithmic, 34, 63, A48, A53

graphing calculator, 46, 315, 624, 646

of an ellipse, 656, 662

machine diagram of, 12

gravitational acceleration, 438

of a hyperbola, 658

marginal cost, 229, 327

gravitation law, 442

of a parabola, 655

marginal profit, 327

greatest integer function, 105

folium of Descartes, 208, 672

marginal revenue, 327

Gregory, James, 732

force, 438

natural logarithmic, 64

Gregory’s series, 732

exerted by fluid, 540

nondifferentiable, 60

growth, law of natural, 234, 591

Fourier, Joseph, 230

odd, 19, 308

growth rate, 226

Fourier sequence, 467

one-to-one, 60

relative, 234

four-leaved rose, 643

periodic, 308

fractions (partial), 473, 474

piecewise defined, 17

half-angle formulas, A29

Fresnel, Augustin, 383

polynomial, 28

half-life, 236

Fresnel function, 383

position, 145

hare-lynx system, 612

frustum

power, 29, 173

harmonic series, 791

of a cone, 431

profit, 327

Heaviside, Oliver, 92

of a pyramid, 432

quadratic, 28

Heaviside function, 45, 92

function(s), 12

ramp, 45

Hecht, Eugene, 250, 754

absolute value, 17

range of, 11

height of a rocket 459

algebraic, 31

rational, 31, 473

higher derivatives, 160

arc length, 528

reciprocal, 31

Hooke’s Law, 439

arrow diagram of, 12

reflected, 38

horizontal asymptote, 132

average cost, 330

representation as a power series, 728

horizontal line, A13

average value of, 433, 557

representations of, 11, 13

Horizontal Line Test, 60

Bessel, 674, 724

revenue, 327

Hubble Space Telescope, 276

combinations of, 41

root, 30

Huygens, 625

composite, 41, 197

shifted, 37

hydrostatic pressure and force, 539

constant, 173

sine integral, 389

hyperbola, 213, 658, 662, A20

continuous, 119

step, 18

asymptotes, 629, A20

cost, 228, 327

stretched, 38

branches, 629, A20

cubic, 28

transcendental, 34

directrix, 662

decreasing, 20

transformation of, 37, 38

eccentricity, 662

demand, 327

translation of, 38

equation, 658, A20

derivative of, 146

trigonometric, 32, A26

equilateral, A21

differentiable, 157

value of, 12

foci, 658, 662

discontinuous, 119

Fundamental Theorem of Calculus, 381,

polar equation, 663

domain of, 11

384, 387

reflection property, 662

elementary, 487

vertices, 658

even, 19, 308

G (gravitational constant), 231, 442

hyperbolic function(s), 254

exponential, 33, 52, 180

Gabriel’s horn, 537

derivatives, 254

extreme values of, 271

Galileo, 625, 633

inverse, 257, 258

family of, 50, 320

Galois, Evariste, 210

hyperbolic identities, 255

fixed point of, 171, 286

Gause, G. F., 596

hyperbolic substitution, 471

Fresnel, 397

Gauss, Karl Friedrich, A35

hypocycloid, 629

INDEX |||| A117

i (imaginary number), A46

of exponential functions, 370, 385, 400

slope of, A12

I/D Test, 287

formulas, 452, 484, RP6–10

tangent, 4, 83, 144

ideal gas law, 233

indefinite, 391

liquid force, 539, 540

implicit differentiation, 207, 208

limits of, 366

Lissajous figure, 673

implicit function, 207

numerical, 496

lithotripsy, 658

improper integral, 508

by partial fractions, 473

local maximum and minimum, 271

impulse of a force, 601

left-hand derivative, 165

logarithm(s), 34, 63

Increasing/Decreasing Test, 287

left-hand limit, 192, 113

laws of, 64, A49

increasing function, 20

Leibniz, Gottfried Wilhelm, 4, 157, 399,

natural, 64, A48

increasing sequence, 681

580, 748

notation for, 64

increment, 147

Leibniz notation, 157

logarithmic differentiation, 217

indefinite integrals, 392

lemniscate, 213

logarithmic function(s), 34, 63

table of, 392

length

with base a, A53

independent variable, 11

of a curve, 325

derivatives of, 215, A49, A53

indeterminate difference, 298

of a line segment, A7, A12

graphs of, 64, 66

indeterminate forms of limits, 298

of a parametric curve, 635

integration of, 218

indeterminate product, 302

of a polar curve, 652

limits of, 101, A50

indeterminate power, 303

l’Hospital, Marquis de, 299, 307

properties of, 68, A49

index of summation, A34

l’Hospital’s Rule, 299, 307

logistic difference equation, 687

inequalities, rules for, A4

origins of, 307

logistic differential equation, 568, 592

infinite discontinuity, 120

libration point, 340

logistic model, 600

infinite interval, 530, 531

limaçon, 647

logistic sequence, 687

infinite limit, 94, 116, 136

Limit Comparison Test, 707

LORAN system, 661

infinite sequence (see Sequence)

Limit Laws, 99, A39

Lotka-Volterra equations, 609

infinite series (see Series)

Limit Laws for sequences, 778

inflection point, 219

limit(s), 3

machine diagram of a function, 12

initial condition, 570

calculating, 99

Maclaurin, Colin, 736

initial point of a parametric curve, 662

of exponential functions, 136, 137

Maclaurin series, 734, 736

initial-value problem, 570

of a function, 88, 110

table of, 741

instantaneous rate of change, 85, 148,

infinite, 94, 116, 136

major axis of ellipse, 657

148, 221

at infinity, 130, 131, 137

marginal cost function, 228, 327

instantaneous rate of growth, 221

of integration, 366

marginal profit function, 327

instantaneous rate of reaction, 225

left-hand, 93, 113

marginal propensity to consume

instantaneous velocity, 85, 145, 221

one-sided, 93, 113

or save, 695

integer, A2

precise definitions, 109, 113, 116, 138,

marginal revenue function, 327

integral(s)

140

mass, center of, 542

approximations to, 372

properties of, 99

mathematical induction, 695

change of variables in, 400

right-hand, 93, 163

principle of, 77

comparison properties of, 375

of a sequence, 6, 359, 677

mathematical model, 14, 24

definite, 366

of a trigonometric function, 190

maximum and minimum values, 271

derivative of, 401

linear approximation, 246

mean life of an atom, 517

evaluating, 369

linear density, 223

mean of a probability density

improper, 508

linear differential equation, 602

function, 557

indefinite, 391

linear equation, A14

Mean Value Theorem, 282

patterns in, 494

linear function, 24

for integrals, 443

properties of, 373

linearization, 48

mean waiting time, 557

of symmetric functions, 405

linear model, 24

median of a probability density

table of, 484, RP6–10

linear regression, 27

function, 558

units for, 396

line(s) in the plane, A12

method of cylindrical shells, 433

Integral Test, 697

equations of, A12, A13, A14

method of exhaustion, 3, 102

integrand, 366

horizontal, A13

method of least squares, 27

discontinuous, 511

normal, 181

midpoint formula, A16

integration, 366

parallel, A14

Midpoint Rule, 372, 496

approximate, 495

perpendicular, A14

error in using, 497

by computer algebra system, 491

secant, 4, 83

mixing problems, 584

A118 |||| INDEX

modeling

nondifferentiable function, 160

percentage error, 251

with differential equations, 567

normal distribution, 559

perihelion, 667

motion of a spring, 568

normal line, 181

perilune, 661

population growth, 55, 567, 591, 600,

nth-degree equation, finding

period, 308

597, 616

roots of, 210

periodic function, 308

vibration of membrane, 724

nth-degree Taylor polynomial, 254, 737

perpendicular lines, A14

model(s), mathematical, 24

number

phase plane, 610

comparison of natural growth vs. logistic,

complex, A55

phase portrait, 610

596

integer, A2

phase trajectory, 610

empirical, 25

irrational, A2

piecewise defined function, 17

exponential, 33

rational, A2

Planck’s Law, 757

Gompertz function, 600

real, A2

point of inflection, 291

linear, 24

numerical integration, 496

point-slope equation of a line, 18, A12

logarithmic, 34

Poiseuille, Jean-Louis-Marie, 227

polynomial, 28

odd function, 19, 308

Poiseuille’s Laws, 253, 332, 552

power function, 29

one-sided limits, 92, 113

polar axis, 639

predator-prey, 233, 609

one-to-one function, 60

polar coordinates, 639

rational function, 31

open interval, A3

polar equation(s), 639

seasonal growth, 600

optics

of conics, 663

trigonometric, 32, 33

first-order, 754

graph of, 641

von Bertalanffy, 616

Gaussian, 7554

polar form of a complex number, A57

modulus, A56

third-order, 755

pole, 639

moment

optimization problems, 271, 322

polynomial, 28

about an axis, 543

order of a differential equation, 569

population growth, 591

of a lamina, 543

ordered pair, A10

of bacteria, 227, 591, 598

of a mass, 543

Oresme, Nicole, 692

of fish, 599

of a system of particles, 543

origin, A2, A10

of insects, 483

momentum of an object, 601

orthogonal curves, 214

models, 567

monotonic sequence, 681

orthogonal trajectory, 214, 583

world, 55, 235

Monotonic Sequence Theorem, 683

ovals of Cassini, 649

position function, 145

movie theater seating, 446

positive angle, A25

multiplication of power series, 745

Pappus, Theorem of, 546

potential, 520

multiplier effect, 695

Pappus of Alexandria, 546

pound, 438

parabola, 655, 663, A18

power consumption, approximation of, 396

natural exponential function, 56, A50

axis, 655

power function, 29, 173

derivatives of, 180, A51

directrix, 655

Power Law of limits, 101

graph of, 180

equation, 656

Power Rule, 174, 218

properties of, A51

focus, 655, 662

power series, 723

natural growth law, 234, 591

polar equation, 663

coefficient of, 723

natural logarithm function, 64, A48

reflection property, 268, 269

differentiation of, 729

derivatives of, 215, A49

vertex, 655

division of, 745

limits of, 424

paradoxes of Zeno, 6, 7

integration of, 729

properties of, A49

parallelepiped, 422

interval of convergence, 725

negative angle, A25

parallel lines, A14

multiplication of, 745

net area, 367

parameter, 621

radius of convergence, 725

Net Change Theorem, 394

parametric curve, 621

representations of functions as, 728

net investment flow, 554

parametric equations, 621

predator, 608

Newton, Sir Isaac, 4, 9, 102, 153, 157,

paraxial rays, 249

predator-prey model, 233, 609

380, 399, 748

partial fractions, 473, 474

pressure exerted by a fluid, 540

newton (unit of force), 438

partial integration, 453, 454

prey, 609

Newton’s Law of Cooling, 237

partial sum of a series, 688

prime notation, 146, 177

Newton’s Law of Gravitation, 231, 442

parts, integration by, 453, 454

principal square root of a complex

Newton’s method, 334, 335

patterns in integrals, 494

number, A56

Newton’s Second Law, 438

pendulum, approximating the period

principle of mathematical induction, 77,

Nicomedes, 626

of, 249, 253

80, A36