Blank B.E., Krantz S.G. Calculus: Single Variable

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solutions, 589

universal applicability of, 614615

differential notation, integration by

parts, 470471

differentials

derivatives, 180181

theory, 250251

differential triangle, 626

differentiation. See also

antidifferentiation

of basic functions, 200207

constants for, 190191

deﬁned, 178179

derived functions, 181

Fundamental Theorem of

Calculus, 410

implicit, 237243, 282283

linear differential equations, 606 609

logarithmic, 225227, 426

Newton’s method of, 277

numeric, 194196

periodic functions, 332

polynomial functions, 203

Power Rule of, 424

power series, 682683

powers of x, 200203

products and quotients, 191194

rules for, 189197, 359

summary of formulas, 186

trigonometric functions, 203204

discontinuity, point of, 106

discount rate, improper integrals and,

523524

discrete data

average value of function, 564

probability theory, average values

in, 567

Simpson’s rule with, 454455

trapezoidal approximation, 450451

discrete triangle inequality, order

properties of integrals, 404405

disks, method of, 619

volume slicing, 538539

x-axis rotation, 540541

y-axis rotation, 541542

distance

deﬁnition of, 4

velocity and acceleration

antiderivatives, 361

363

work

calculations, 580585

distinct linear factors, partial fractions

for, 499500

distributive law, summation, 377

divergence

alternating series test, 661665

comparison test for, 655657

improper integrals, 515, 521523

inﬁnite sequences, 631633

integral test of, 646648, 649650

interior singularity of integrands, 516

Limit Comparison Test, 657658

limits of sequences and, 128129

Newton-Raphson method, 352

series, 637638

harmonic series, 636637

inﬁnite series, 634635

nonnegative terms in, 645

p-series, 648649

Ratio Test, 667669

Root Test, 670671

tail end of, 645646

test of, 643646, 662665, 674675

division, skew-asymptotes, 333

domain

arithmetic operations, 49

continuity, 105106

even/odd functions, 58

of functions, 3439, 51, 77

graphing functions, 327328

of inverse functions, 52

pairing functions, 59

parameterization, 59

population density function, 568569

skew-asymptotes, 333

trigonometric functions, 68

Doppler effect, related rates rule, 282

Double Angle formulas, trigonometry,

doubling time, exponential growth, 150

e (number)

derivatives, 204206

exponential function and, 145147

natural logarithms and exponential

functions, 422423

economics

increments in, 248

Newton-Raphson applications,

353354

trapezoidal approximation, 450452

effective yield, bond valuation, 353354

Einstein, Albert, 438

electric circuits, linear differential

equations, 610611

ellipse(s), 1617

elliptical trajectory, numerical

integration techniques, 446449

empty set, 45

endpoints

arbitrary base points, power series, 681

area approximation, 380381

curve sketching, 330

extreme values, 308

of intervals, 5

closed intervals, 308311

interval of convergence and,

677678

maxima and minima, 307315

numerical integration, 447449

parametric curves, 554556

precise deﬁnition of area, 381384

Riemann integral, 393394

solutions at, 311312

surface area calculations, 557559

unbounded integrands, 517

Epitome Astronomiae Copernicanae

(Kepler), 467

equilibrium position, spring constant,

583584

Eratosthenes, 466

erg units, 580

error(s)

absolute, 7, 29

approximation, 7

bounds of inequalities, approximation

techniques, 454

error term estimation, 701702

function, in power series, 691692

roundoff, 196

truncation, 7

Eudoxus, 465

Euler, Leonhard, 161, 628

inﬁnite series and, 719720

even function, 58

exact

area calculations, 383384

existence theorem, 114

exp(x), 146, 421

expectation of X, 567

exponential decay, 150151

exponential functions, 138151, 461

antidifferentiation, 360361

calculus approach to, 417426

Chain Rule and, 215216

compound interest and, 147149

derivatives of, 204206, 215216, 422

e (number) and, 145 147, 422423

integrals of, 422

irrational exponents, 140142

logarithmic functions, 142145

Mean Value Theorem applications

and, 294295

montone convergence property,

138140

natural exponential function, 145147

power of arbitrary bases and, 423424

power series expansion, 705706

properties of, 421422

exponential growth, 150

exponents, law of, 424425

Index I-5

expressions, derivatives of, 186

Extreme Value Theorem, 114115

endpoint solutions, 311

integrals, 405406

periodic functions, 331332

Riemann sums, 390392

Rolle’s theorem and, 293294

extremum (extreme variables)

closed intervals, continuous functions,

308311

Fermat’s investigation of, 292, 371

of functions, 299303, 308

mean value theorem, 290291

Rolle’s theorem, 293

face value, bond valuation, 353354

factorials

ratio of, 678

Ratio Test and, 669

factoring, area between two curves,

442443

falling bodies, velocity and acceleration

and, 362363

Fermat, Pierre de, 81

ﬁrst derivative test for local extrema,

301303

integral calculus and, 466467

life of, 371

maxima/minima investigations,

371372

mean value theorem, 291293

points of inﬂection investigations,

371372

principle of least time, 372373, 627

tangent problem solutions, 275277

Theory of Indivisibles and, 466

Fermat’s theorem

absolute extrema, 291293

critical points for local extrema,

301303

development of, 371

extreme values and, 293294

local extrema, 324325

maxima/mimima location, 291293

Fibonacci sequence, 40

ﬁnance, improper integrals in, 523524

ﬁnite limits

comparison test for convergence

and, 655

l’Ho

pital’s Rule for, 342343, 345

First Derivative Test

closed intervals, 309

curve sketching, 327328

local extrema, 324325

maxima and minima, 301303

ﬁrst order differential equations

linear equations, 606615, 621

separable equations, 588598, 621

ﬁxed costs, 313314

ﬁxed positive constant, Taylor series

expansion, transcendental

function, 704706

ﬂoating point decimal representation, 8

ﬂuents, 277

ﬂuid mechanics, 584

585

ﬂuxions,

277

Folium of Descartes, 276277

foot-pound, 580

force

Newton’s Law, 581583

work calculations, 580585

forward difference quotient, 195

Fourier, Jean Baptiste Joseph, 467468

fractions

l’Ho

pital’s Rule for common

denominator, 344345

partial fractions, 499503

Fresnel functions, 593

frustrum, 556559

fulcrum, 572

functional composition, 51

functions

absolute value function, 3637

antiderivatives of, 357358

antidifferentiation of, 357363

average value of, 561569, 620

combining (See combining functions)

continuous functions, 105106,

111114

from data, 4044

deﬁned, 3435

derivatives for, 299300

derived functions, 181

differential equations, 589

differentials and approximation

of, 246251

differentiation

basic functions, 200207

rule for, 210212

examples of, 35

exponential functions, 138151

power series expansion, 705706

graphing functions, 327336

graphs of, 3739

concavity, 321322

integrable functions, 399402

integral test and, 647648

logistic growth, 597

598

maxima

and minima of, 299303

multivariable functions, 45

periodic, 331332

piecewise-deﬁned, 3637

power series representation, 686695

expansions of, 686687

of real variables, 3536

sequences, 3940

sign (signum) function, 36

sums of, 170171

transcendental, 253266

Taylor series expansions, 704706

trigonometric, 6771

Fundamental Theorem of Algebra, 50

partial fractions, 506512, 534

Fundamental Theorem of Calculus,

396397, 460

area function, 408

average value of function, 563564

basic equations, 408409

Chain Rule and, 411412

continuous functions, 396397,

408409

deﬁnite and indeﬁnite integrals, 411

differential equations, 592593

examples, 410411

Half Angle formulas, 411

improper integrals with inﬁnite

integrands, 514515

integral evaluations, 399402

integration reversal, 402403

inverse operations, 410

Kepler and, 467

Mean Value Theorem, 409

Newton and, 467

power series error function, 691692

Galileo, 82, 467

Gateway Arch (Saint Louis), 627

Gauss, Carl Friedrich, 534535

Gauss, Eugen, 535

Gauss, Wilhelm, 535

Geometria Indivisibilius

Continuorum, 373

geometric constructions

area calculations, 376385

integral test and, 647648

limits, 133135

Newton-Raphson method, 348349

power series as, 686689

sums, 378379

geometric series, 638639

divergence test, 644

eom

etrie (Descartes), 81, 371,

467, 625

eostatique, 372

Gini coefﬁcient, 451

global maximum, mean value

theorem, 290

I-6 Index

graphs and graphing

area between two curves, 441443

calculators and software, 334336

closed intervals, 310311

concavity, 320325

curve sketching, 327331

of differentiability, 183186

distance formula and circles, 1213

equation of circle, 13

of functions, 3739, 327336

exponential functions, 421422

inverse function, 56

periodic functions, 331332

of limits, 89

method of completing the square,

1315

natural logarithmic function, 420

Newton-Raphson method, 348354

parametric curves, 61

planar coordinates, 1119

precise deﬁnition of area, 381384

of refracted rays, 373

sketching guidelines, 327328

skew-asymptotes, 332333

gravity and gravitational force

differential equations, 595596

linear velocity law, 614615

Newton’s Law of, 581583

velocity and acceleration and,

362363

Gregory, James, 533534, 719720

growth constant, exponential growth, 150

Half Angle Formula

Fundamental Theorem of Calculus, 411

squares of trigonometric functions,

480481

trigonometry, 70

half-life, exponential decay, 151

half open/half closed intervals, 5

Halley, Edmond, 161

Hamilton, William Rowan, 278

harmonic series, 636637

alternating series test, 662665

convergence

conditional convergence, 665

interval of, 676678

divergence tests of, 644

comparison test for divergence,

656657

Limit Comparison Test, 658

p-series as, 649

Ratio Test for, 669

Harriot, Thomas, 533534

Heaviside, Oliver, 501

Heaviside’s method, partial fractions

for linear factors, 501502

Heiberg, John Ludvig, 466

higher derivatives, 230234, 271.

See also second derivatives

calculation of, 241242

Newton’s notation, 232

notation, 230231

second derivative approximation,

232233

velocity and acceleration, 231232

Hilbert, David, 12, 162

Hipparchus, 70

Hobbes, Thomas, 625 626

Hooke, Robert, 82

Hooke’s Law, 583584

horizontal asymptotes, 121124

curve sketching, 327329, 331

e number and, 145147

periodic functions, 332

horizontal line

to curves, 173

inverse trigonometric function, 253

horizontal translation, 57

Hubble, Edwin, 4042

Hubble’s constant, 42

Hudde, Johann, 719

Huygens, Christiaan, 82

hyperbolas, deﬁned, 17

hyperbolic functions, 260262

derivatives, 262263

inverse, 263265

trigonometric, 260262, 272

hypoteneuse

irrational number, 2

Pythagorean Theorem, 1213

trigonometric substitution, 491

implicit differentiation, 237243, 271

higher derivatives calculation,

241242

method of, 282283

parametric curves, 242243

related rates and, 282284

improper integrals

ﬁnancial applications, 523254

inﬁnite integrands and, 514515,

530531

integral test of, 646648, 649650

over entire real line, 524525

unbounded integrands, 514519

unbounded intervals, 521526

income data, trapezoidal approximation,

451452

income stream, improper integrals and,

523524

increasing functions

concavity, 321322

curve sketching, 327328

derivatives, 299300

identiﬁcation, 298299

natural logarithms, 420

transcendental functions, 314315

increasing sequences, 138

incremental costs, 313314

increments

differentials theory, 250251

in economics, 248

indeﬁnite integral

antiderivatives, 357

calculation of, 359360

Fundamental Theorem of Calculus,

411

integration by parts, 474

power series, 682683

Riemann integrals, 393396

table of, 433435

indeﬁnite integration, by substitution,

429430

independent variable, 41, 56

indeterminate forms

l’Ho

pital’s Rule, 338

345

Taylor

polynomial evaluation of,

708709

index of refraction, Newton-Raphson

method, 352353

index of summation, 377

convergence test, 655

geometric series, 638

inﬁnite series, 633

integral test, 647, 649

power series

differential equation and, 692693

differentiation and antidifferentia-

tion, 682683

function expansion, 687

of functions, 687, 689

index variable, inﬁnite sequences,

632633

indirect substitution, trigonometric

functions, 493494

Indivisibles, Theory of, 466467

inductive sequences, 40

inequalities

alternating series test, 662 663

coefﬁcient of, 451

Newton-Raphson accuracy, 350351

order properties of integrals, 404405

real numbers, 3 5

Riemann integrals, 394

symbols for, 533

triangle, 7

inﬁnite differentiability, 682683

Index I-7

infinite integrands, improper integrals

with, 514515

inﬁnite limits

asymptotes and, 118124

l’Ho

pital’s Rule for, 342343, 345

inﬁnitely long solid of revolution, 627

inﬁnite sequence, 3940

inﬁnite series vs., 635

limits of, 631633

partial sums, 633634

inﬁnite series

applications, 629630

comparison tests

for convergence, 653655

for divergence, 655657

convergence of, 634635

deﬁned, 633634

divergence test, 643646, 655657

history of, 718719

integral test of, 646648

Limit Comparison Test, 657658

nonnegative terms, 644645

p-number in, 648649

power series, 674683

Ratio Test, 667669

Root Test, 670671

tail end of, 645646

inﬁnitesimals, 161

differentials theory, 250251

Rolle’s Theorem and, 373

inﬁnite sums

applications, 629630

basic properties, 631637

geometric series, 135

inﬁnite-valued limits, 118119

average value of function, 562569

inﬁnity, limits at, 121124

natural logarithms, 420

inﬂection points

curve sketching, 327

331

deﬁned,

322

Fermat’s investigation of, 371372

periodic functions, 332

strategic determination of, 323324

initial condition, differential equation,

590593

initial value problems

differential equation, 590591

power series and, 692693

linear differential equations, 607609

logistic growth function, 597598

Taylor series expansions, power series,

705706

inside function, chain rule, 211

instantaneous rate of change, 168170

acceleration and velocity, 361363

instantaneous velocity, 166168

integers

basic properties, 2

consecutive sequences, 39

natural logarithms, 419420

positive, 2, 39

integral calculus, 396

Fermat and, 466467

integrals

applications, 537

arc length and surface area, 552559

area calculations, 440444

average value of function, 561569,

564569

center of mass, 572577

deﬁnite integral, 393396, 430432

Fundamental Theorem of Calculus,

412

improper, 514519, 521526,

530531

indeﬁnite, 357, 359360, 393396

inﬁnite intervals, 521523

integration

numerical integration, 446449

by parts, 471476

reversal, 402

403

rules

for, 399406

linear ﬁrst order differential equations,

606615

Mean Value Theorem for, 405406

order properties, 403405

over entire real line, 524525

power series, 682683

of powers of x, 358360

precise deﬁnition of area, 381384

properties of, 375, 460 461

Riemann integral, 385396

separable ﬁrst order differential

equations, 588598

sine and cosine, 483485

substitution for, 428437

summation notation, 377

tables, 433435

volumes, 538548

work calculations, 580585

integral sign, Riemann integral, 393

Integral test, Root Test and, 671

integral test

extension of, 649650

inﬁnite series, 646648

p-series, 648649

integrands

antiderivatives, 357

endpoints, unboundedness at, 517

inﬁnite, 514515

interior singularities, 515516

Riemann integrals, 396

unbounded, 514519

integration

area

approximation, 379381

problem, 376385

constant of, 362363

direction reversal, 402403

Fundamental Theorem of Calculus,

410

history of, 533535

inﬁnite, 429

430

logarithms,

426

lower and upper limit, 393

Newton on substitution of, 467

numerical techniques of, 446455, 462

by parts, 470476, 529, 648

point rule, 446449

power series of functions, 687688

reduction formulas, 475476

Riemann integral, 392394, 396

rules for, 399406

by substitution, 428437

techniques of, 469

trapezoidal rule and, 449452

trigonometric functions, 436437

two solutions to, 433

intercept form of line, 27

interest

bond valuation, 353354

compound interest, 147149

improper integrals and, 523254

interior singularities, integrands with,

515516

Intermediate Value Theorem, 112114

Bolzano and, 161162

exponential and logarithmic functions,

142145

integrals, 405406

natural logarithms, 420

Newton-Raphson accuracy, 351

interval of convergence

arbitrary base point, 679681

radius of convergence and, 676678

intervals

antiderivatives of, 357358

bounded, 46

closed, 5

endpoints, 5

inﬁnite, 521523

integrable functions, 399402

open, 5

precise deﬁnition of area, 381384

sets of real numbers and, 46

unbounded, 56, 521526

Inverse Function Derivative Rule,

222224

inverse

hyperbolic functions, 265

inverse trigonometric functions, 259

I-8 Index

natural logarithm and, 417418

inverse functions

combining functions, 5256

continuity and differentiability,

220223

derivatives of, 220227

exponential functions, 421

graph of, 56

hyperbolic functions, 263265

sine and cosine, 254256

tangent function, 257

trigonometric functions, 253254,

258259, 271

inverse operations, Fundamental

Theorem of Calculus, 410

inverse sine and cosine, transcendental

functions, 254256

inverse substitution, method of, 489495

inverse tangent function, 257

inverse trigonometric functions,

253254, 258259

trigonometric substitution, 489495

irrational exponents, 140142

power of arbitrary bases and, 423424

irrational numbers

basic properties, 3

decimal expansion, 7

successive approximation and, 80

irreducible coefﬁcients, polynomial

functions, 50

irreducible quadratic polynomials

irreducibility veriﬁcation, 508509

partial fractions, 506512

iterative calculations, Newton-Raphson

method, 350, 351

joule, 580

Kepler, Johannes, 82, 467, 533

solids of revolution and, 625

Kepler’s equation, Newton-Raphson

accuracy, 351

Kirchhoff’s voltage law, 611

La Dioptrique, 372

Lagrange, Joseph Louis, 373

Lambert, Johann Heinrich, 80

law of exponents, 424425

Law of Refraction, 372373

law of the contrapositive, 643644

leading coefﬁcient, polynomial

functions, 50

least positive integer solution, 80

least squares lines, 2831

least squares method, 535

lines, 2831

least upper bound property, 111, 162

Legendre, Adrien-Marie, 80

Leibniz, Wilhelm von, 180181, 278,

467468, 533

catenary and, 626

differential triangle and, 626

inﬁnite series and, 718719

Leibniz notation

derivatives, 180181

differential approximation, 250251

Fundamental Theorem of Calculus,

410

history of, 467468

Product Rule, 191194

Leibniz’s rule

implicit differentiation, 238239

second derivatives, 233234

Letters of Descartes, 372

l’Ho

pital’s Rule, 338345, 467

common denominator, 344345

comparison test for divergence

and, 656657

conjugation and, 344345

indeterminate forms, 338343

inﬁnite sequences, 632633

integral test and, 648648

logarithm applications, 343344

present value, 523524

Limit Comparison Test

applications of, 657658

comparison test for convergence

and, 654

limit of integration

Fundamental Theorem of Calculus

and, 410, 412

reversed direction, 402

Riemann integral, 393

trigonometric substitution, 490491

limit(s)

algebraic

manipulations, 87

applications, 8990

basic properties, 8390

continuous function calculations,

135136

convergence and divergence, 128129

deﬁnition, 85, 94

degrees of accuracy, 8788

differentials theory, 250251

exponential functions, 138151

graphical methods, 89

inﬁnite limits, 118124

inﬁnite sequences, 631633

integrals, 375

l’Ho

pital’s Rule, 338345

linearization and, 250

non-existent limits, 9899

one-sided limits, 8687, 9596

precise deﬁnition of area, 383384

Ratio Test and, 669

of sequences, 127136

trigonometric, 100102

limit theorems, 94102

open intervals, 97

Pinching Theorem, 99100

sequences and, 131133

Lindemann, Carl Louis Ferdinand

von, 81

linear drag law, 613615

linear equations

differential equations, 606615

constant coefﬁcients, 611612

electric circuits, 610611

linear drag law, 613615

mixing problems, 609610

Newton’s laws for temperature

change, 612613

solutions, 606609

general equation, 26

normal equation, 27

linear

factors

distinct linear factors, 499500

Heaviside’s method, 501502

linear factors, 498503

partial fractions, 498503

repeated linear factors, 498, 502503

simple linear building blocks, 498

linear functions, derivatives for, 190191

linearization

differential approximation, 251

functional approximation, 248250

linear polynomials, 50

linear velocity law, 613615

line(s), 2131

general equation, 26

integrals over, 524525

intercept form of, 27

least squares (regression), 2831

normal equation, 2728

normal lines to curves, 173174

parallel, 23

perpendicular, 22, 173174

point slope form of, 23

secant lines, 171173

slope-intercept form of, 2526

tangent lines, 171173

two-point form of, 24

Liouville, Joseph, 80

local extrema

curve sketching, 327328

Fermat’s theorem, 292

ﬁrst derivative test for, 301303

Index I-9

local extrema (continued )

inﬂection points and, 323

Second Derivative Test of, 323324

local extreme value, Fermat’s theorem,

292

local maximum value, 290291

Second Derivative Test for extrema

and, 323324

local minimum value, 290291

Second Derivative Test for extrema

and, 323324

locus, Cartesian planes, 12, 76

logarithms, 142145, 461

arbitrary bases, 424425

calculus approach to, 417426

comparison test for divergence

and, 656657

derivatives, 223225

differentiation, 225226, 426

exponential functions and, 421422

integration of, 426

l’Ho

pital’s Rule for indeterminate

forms and, 343345

logistic growth equation, 597598

natural logarithm (See natural

logarithms)

powers with arbitrary bases, 423424

trigonometric substitution, 494495

logistic growth, differential equations

and, 596598

Lorenz curve, 451

Lorenz function, 451

loss of signiﬁcance, 8

graphing calculators, 334336

quotients and, 195196

Love’s Labour’s Lost (Shakespeare), 533

lower limit of integration, 393

Fundamental Theorem of Calculus,

411412

reversal, 402403

lower Riemann sum, 390392

Maclaurin, Colin, 720

Maclaurin series

binomial series, 707

history of, 719720

polynomials, 693695

transcendental function expansion,

704706

Madhavan, S., 718

Maple computer algebra system

Newton-Raphson method, 352353

tables integrals and, 435

marginal cost

incremental economic analysis, 248

instantaneous rate of change and,

169170

proﬁt maximization, 313314

marginal product of labor, incremental

economic analysis, 248

marginal proﬁt, incremental economic

analysis, 248

marginal revenue, incremental economic

analysis, 248

Marlowe, Christopher, 533

mass

center of, 572577, 620 621

dissolved substances, 609

Newton’s Law, 581583

mathematical model, data functions, 40

maturity, bond valuation, 353354

maxima and minima

applied problems, 307315

closed intervals, 308311

critical points, 301303

curve sketching, 327328

endpoint solutions, 311312

Fermat’s investigation of, 371372

First Derivative Test, 301303

of functions, 299303

location of, 291292

mean value theorem, 290293

periodic functions, 331332

proﬁt maximization, 313314

transcendental functions, 314315

mean, calculation of, 567

Mean Value Theorem, 289295

applications, 294295

average value of function, 564565

function increase/decrease, 299

300

increasing

and decreasing functions,

299300

integrals, 405406

maxima and minima, 290 293

Riemann integrals, 395396

Rolle’s theorem and, 293294, 373

Taylor’s theorem, 698699

Mengoli, 718

Mercator, Gerardus, 533

Mercator’s projection, 533

Mersenne, 371372

Method Concerning Mechanical

Theorems, dedicated to

Eratosthenes, 466

“Method of Cascades” (Rolle), 373

Method of Completing the Square,

1315

Method of Cylindrical Shells

solid of revolution and, 620

volume calculations, 545548

Method of disks

solid of revolution, 619

volume slicing, 538539

x-axis rotation, 540541

y-axis rotation, 541542

Method of Exhaustion, 465

Method of Implicit Differentiation,

282283

Method of Increments, The (Taylor), 720

Method of Inverse Substitution,

trigonometric substitution, 489495

Method of least squares. See least

squares method

Method of Partial Fractions, 507508

Method of Separation of Variables,

separable differential equations,

591598

Method of Slicing, volume calculations,

538539, 545

Method of Washers, solid of revolution,

542543, 620

Midpoint Rule, 447449, 451452, 462

parametric curves, 555556

midpoint(s)

of intervals, 5

numerical integration, 447449

minimization, law of refraction and,

372

373

minus

sign

alternating series test, 662665

errors in using, 397

mixing problems, 609

mixing problems, linear differential

equations, 609610

mks measurement system, 580

modeling, trigonometric functions, 7172

moments

center of mass and, 572574

of ﬂuents, 277

two point systems, 572

Monotone Convergence Property

e number and, 146

intermediate value theorem and, 162

irrational exponents, 140142

Riemann integrals, 394

sequences, 138140

monotone sequences

alternating series test, 661665

error term estimation, 702

multicase function, 36 37

multiplication

constants for, 190191

scalar multiplication, power series,

681682

multivariable functions, 45

Napier, John, 144

natural exponential function, deﬁned, 421

I-10 Index

natural logarithm

calculus and, 417418

differentiation and, 149150

e (number) and, 422423

exponential functions and,

424425, 461

graphing, 420

properties of, 418420

natural numbers, 2

navigation, projections in, 533

negative area calculations, 442443

negative integers, natural logarithms,

419420

negative values, differentiability and,

182183

negative velocity, 167168

nested interval property, 111

intermediate value theorem and, 162

Newton, Isaac, 82

differentiation method of, 277

Fundamental Theorem of Calculus

and, 467

inﬁnite series and, 718719

Leibniz and, 278, 533534

Newton-Raphson algorithm and,

373374

power series and, 719720

newton-meter, 580

Newton notation

derivatives, 181

higher derivatives, 232

Newton-Raphson method

accuracy, 349352

bond valuation application, 353354

calculation of, 349350

geometry, 348349

history of, 373374

pitfalls, 352

quadratic formula, 348354

Newton’s binomial series, 707

Newton’s Law of Cooling, 612613

Newton’s Law of Gravitation,

581583

differential equations, 595596

Newton’s Law of Heating, 612613

Niele, William, 626

nondecreasing sequences, 138

nondenumerable numbers, 80

nonexistent limits, 9899

nonhorizontal line, 221

nonincreasing sequences, 138

nonnegative functions

alternating series test, 661665

center

of mass, 574577

comparison test for, 653655, 657

divergence and, 657

integration reversal, 402403

order properties of integrals, 405

in series, 644645

surface area calculations, 558559

nonnegative square root, 34

non-quotients, l’Ho

pital’s Rule for

indeterminate forms, 342

nonradical quadratic expressions,

trigonometric substitution,

493495

nonterminating, nonrepeating

decimals, 3

nonterminating decimals, 23

Northumberland, Earl of, 533

Notation, history of, 467469

number lines, real numbers on, 3

numbers

approximation, 78

basic principles, 211

computer algebra systems, 8

intervals, 46

real number sets, 34

summation notation, 377

Triangle Inequality, 7

numerator

inﬁnite sequences, 632633

l’Ho

pital’s Rule, 339345

numerical integration techniques,

446455, 462

point rule, 446  449

Simpson’s rule, 452454

trapezoidal rule, 449452

numeric differentiation, 194196

oblique-asymptote, 332333

odd function, 58

sine/cosine integrals, 483485

trigonometric powers, 482483

one-sided continuity, 109110

one-to-one property, 254, 257259, 263

open intervals, 5

integrals, 405406

limit theorem, 97

Mean Value theorem, 293294

Second Derivative Test of extrema

and, 323324

optics principles, Fermat’s research on,

372373

orbital calculations, Newton-Raphson

accuracy, 351

ordered pair(s), 1112

order properties of integrals, 403405

Oresme, Nicole, 81, 718

outside function, chain rule, 211

overshoot, Newton-Raphson method,

352

pairing functions, 5960

palimpsest, 466

Pappus, 371

parabolas

area between two curves, 442443

axes role reversal, 443444

Cartesian equation of, 1617

semicubical, 626

Simpson’s Rule, 452454

Parallelipiped, approximation of,

545548

parallel lines, slope of, 23

parallelogram rotation, noncoordinate

axis, 544545

parametric curves

arc length calculations, 554556

graphs of functions, 61

implicit differentiation, 242243

pairing functions, 5960

partial fractions

decomposition, coefﬁcients of,

509512

distinct linear factors, 499500

Heaviside’s method, 501502

history of, 534

irreducible quadratic factors, 506512

linear factors, 498503

logistic growth equation, 596598

Method of, 507508, 530

repeated linear factors, 498, 502503

simple linear building blocks, 498

partial sums

alternating series test, 661 665

comparison test for convergence,

653655

divergence test, 643644

inﬁnite series, 633635

integral test and, 650

nonnegative terms in series, 645

power series, 677678, 691692

telescoping series, 635636

partitions

average value of function, 562563

cone volume approximation, 538539

Simpson’s Rule, 454455

solid of revolution, 540

542

summation,

377

parts, integration by, 470476, 529, 648

Pascal, Blaise, 626

Pearl, Raymond, 289

Pell equation, 80

periodic functions, 331332

perpendicular lines

to curves, 173174

slope of, 22

perpetuity, present value, 523524

Index I-11

physical sciences, differential equations

in, 593596

pi (π)

deﬁned, 2

as irrational number, 8081

Piazzi, Giuseppe, 534535

piecewise-deﬁned function, 3637

Pinching Theorem, 99100

sequences, 132133

planar coordinates

basic properties, 1119

distance formula and circles, 1213

equation of circle, 13

method of completing the square,

1315

parabolas, ellipses, and hyperbolas,

1617

points and loci, 76

regions in the plane, 1819

planar distance formula, arc length

calculations, 552553

point rule, numerical integration,

446449

point(s)

of discontinuity, 106

of inﬂection, 322323, 327, 332,

371372

Newton-Raphson method, 348349

Riemann integral, 385392, 394396

point-slope equation, Newton-Raphson

method, 349

point slope form of line, 23

polynomials

approximation, 5051

curve sketching, 329

differentiation and, 203

endpoint solutions, 310311

graphing calculators, 334336

limit theorem, 98

Rolle’s Theorem and, 373

Second Derivative Test for extrema,

324325

trigonometric substitution, 489495,

493494

functions

irreducible quadratic factors,

506512

partial fractions, 499

503

power

series representation,

686687

quadratic polynomials

partial fractions, 506512

rational functions, 506508

simple and repeated quadratic

terms, 506508

trigonometric substitution,

488495

Taylor series of, 693695, 698701

accuracy, 702704

error term estimation, 701702

indeterminate form evaluation,

708709

population density functions, 568 569

position, as velocity antiderivative,

361363

positive area calculations, 443444

positive integers, 2

area approximation, 379381

limit theorem, 98

natural logarithm, 419420

parametric curves, 554556

Riemann integral, 385392

Simpson’s Rule, 453454

positive numbers

l’Ho

pital’s Rule for indeterminate

forms indeterminate forms,

342343

Newton-Raphson accuracy, 351

periodic function, 331332

positive velocity, 167168

pound, 580

power of arbitrary bases, logarithms

with, 423424

Power Rule

Chain Rule and, 213

differentiation and, 202

implicit differentiation, 237239, 283

power of arbitrary bases and, 424

power series, 638639

addition and scalar multiplication,

681682

applications, 629630

arbitrary base point, 679

681

basic

properties, 674683

coefﬁcients and derivatives in,

689691

differential equations and, 692693

differentiation and antidifferentiation,

682683

exponential function expansion,

705706

function representation, 686695

history of, 719720

radius and interval of convergence,

674678

Ratio Test and, 669

sine function expansion, 706

Taylor series expansion, transcenden-

tal function, 705706

powers of x, 200203

antidifferentiation and, 357360

integrals, 358360

precise deﬁnition of area, 381384

predictability, continuity as, 182183

present value

improper integrals, 523

of perpetuity, 523524

principal, 147

principal associated angle, 6869

Principia Mathematica (Newton), 277

principle of least time (Fermat),

372373, 627

probability density function, random

variables, 565566

probability theory

average values in, 567

random variables, 565566

Product Rule

derivatives, 191194, 196

higher derivatives calculation, 242

second derivatives, 233

differentiation

implicit differentiation, 283

logarithmic differentiation, 227

history of, 278

inverse functions, 220

linear differential equations, 606609

proof of, 278

in reverse, 470471

trigonometric functions, 204

products, derivatives of, 191194

proﬁt maximization, 313314

projections, Mercator’s projection, 533

proportionality

constant, logistic growth

function, 597598

p-series, 648649

Euler on, 719

Ratio Test for, 669

Root Test for, 670671

pumping systems, ﬂuid mechanics,

584585

Pyramid, volume calculations, 545

Pythagoras, 80

Pythagorean Theorem

closed intervals, 310

distance formula, 1213

endpoint solutions, 311

trigonometric substitution, 491

quadrants, 11

Quadratic Formula

area between two curves, 442443

average value of function, 564565

curve sketching, 330

method of completing the square,

1415

Newton-Raphson method, 348354

points of inﬂection and, 322323

polynomial functions, 50 51

I-12 Index

quadratic polynomials

partial fractions, 506512

rational functions, 506508

simple and repeated quadratic terms,

506508

trigonometric substitution, 488495

Quotient Rule

derivatives, 193194, 196

hyperbolic function derivatives,

262263

exponential functions, 206

inverse functions, 220

local extrema

critical points, 302303

ﬁrst derivative test for, 302303

logarithmic differentiation, 227

Simpson’s Rule, 453454

trigonometric functions, 204

quotients

backward difference quotient, 195

center of mass calculations, 575577

central difference quotient, 195196

derivatives of, 191194

forward difference quotient, 195

l’Ho

pital’s Rule for indeterminate

forms, 339345

loss of signiﬁcance and, 195196

roundoff error, 196

radians

limit theorems, 101

measurement with, 65

radical quadratic expressions, trigono-

metric substitution, 492493

radioactive decay, 151

radius of convergence, 678683

interval of convergence and, 674678

Taylor series of polynomials, 693695

Raleigh, Walter, 533

Ramanujan, Srinivasa, 719

random variables

average value, 565566

population density function, 568569

range

of functions, 3435

population density function, 568569

Raphson, Joseph, 374

rate-of-change problems, Chain Rule

and, 214215

rational functions

horizontal asymptotes and, 124

limits of inﬁnity and, 122124

limit theorem, 98

partial fractions, 499503

quadratic terms in denominator,

506508

repeated linear factors, 502503

skew-asymptotes, 333

rational numbers, basic properties, 2

ratio(s), geometric series, 638639

Ratio Test, 667669

binomial series, 707

radius of convergence and, 677678

Root Test and, 671

Taylor series expansions, 705

real numbers

algebraic numbers, 80

in calculus, 2

completeness property of, 82

functions, 3435

limits of sequences, 127136

monotone convergence property,

138140

Riemann integrals and, 392394

sets of, 34

summation notation, 377

real-valued function, 35

real variables, functions of, 3435

recession velocity, 4041

Reciprocal Rule

derivatives and, 192  193, 196

powers of x

and, 201203

rates problem, 282

rectangle(s)

area approximation, 379381

area calculations, 376385

center of mass calculations, 576577

integral test of, 646648

precise deﬁnition of area, 381384

Riemann sum and area of, 390392, 394

rectiﬁcation, arc length and, 625626

recursive sequences, 40

redemption, bond valuation, 353354

reduction formulas

higher powers of trigonometric

functions, 481482

integration by parts, 475476

sine/cosine integrals, 483485

refraction

Fermat’s research on, 372373

index of, Newton-Raphson method,

352353

regions

area approximation, 379381

area calculations, 376377

in the plane, 1819

precise deﬁnition of area, 381384

Riemann integral, 386392

regression line. See least squares line

related rates

Chain Rule and, 215, 282

derivatives applications, 282286

implicit differentiation, 282284

problem-solving guidelines, 284285

relative error, approximation and, 7

relative maximum, 290291

relative minimum, 290291

Relativity, Theory of, 438

remainder term, 698701

repeated linear factors, 498, 502503

Reverse Product Rule, 470471

revolution

solid of, 540542

surface of, 556

559

Riemann,

Georg Friedrich Bernhard,

468

Riemann integral, 384385, 460

arc length, 552553

area calculation, 440444

average value of function, 563

calculation of, 394396

deﬁned, 392394

history of, 468

inﬁnite integrands, 514515

integral applications, 523524, 537

Kepler and, 467

moments in center of mass, 574

navigation problems, 533

order properties of integrals, 404405

point rule, 447449

probability theory, average values

in, 567

solid of revolution, 540542

surface area calculations, 557559

volume approximation, 539

work calculations, 580585

Riemann sums, 385392

Riemann surfaces, 468

Riemann zeta function, 468

right endpoint approximation, 380381

Roberval, Gilles, 371372, 466467, 625

Rolle, Michel, 373

Rolle’s theorem

extreme values and, 293294

Mean Value Theorem and, 373

root approximation

bond valuation, 353354

Newton-Raphson method, 348349

Root Test, 670671

radius of convergence and, 678

rotation

axial, 540541

Method of Cylindrical Shells, 545548

noncoordinate axes, 544545

surface of, 558559

rounding, comparison test for

convergence, 654655

roundoff error, 8

quotients and, 196

Russell, Bertrand, 81

Index I-13

scalar multiplication, power series,

681682

scatter plots, 40

scientiﬁc revolution, 82

secant, 68

higher power of, 481482

integral calculation for, 533

integration of, 436437

inverse, 258259

lines, 171173

periodic function, 331332

square of, 480481

second central difference quotient

approximation, second

derivatives, 232233

second derivatives, approximation,

232233

Second Derivative Test

for concavity, 322325

curve sketching, 327328

for extrema, 323324

periodic functions, 332

second order process, Newton-Raphson

accuracy, 351

semicubical parabola, 626

separable differential equations,

591598

Separation of Variables, Method of,

separable differential equations,

591598

sequences

convergence and divergence, 128129

of functions, 3940

geometric series, 133135

limits of, 127136

limit theorems, 131133

monotone convergence property,

138140

special sequences, 129130

tail of, 129

series. See also inﬁnite series

alternating series, 661665

arbitrary base points, power series,

679681

basic properties of, 637638

binomial series, 707

convergence

absolute convergence, 664

comparison test for, 653655

conditional convergence, 665

interval

of, 676678

deﬁned, 631637

divergence

comparison test for, 655657

test of, 643646

geometric (series of powers),

638639

harmonic series, 636637

integral test of, 646648

Limit Comparison Test, 657658

nonnegative terms, 644645

Ratio Test, 667669

Root Test, 670671

tail end of, 645646

telescoping series, 635636

sets of real numbers

intervals, 46

notation for, 34

Shakespeare, William, 533

sigma notation

inﬁnite series, 633634

summation, 377

sigmoidal growth, 597598

signal processing, integration by parts,

474476

sign (signum) function, 36

Simpson’s Rule, 452454

discrete data, 454455

sine function

absolute convergence, 664

antiderivative of, 360361

conversion to, 485

derivatives of, 184186, 203204

higher power of, 481482, 529

hyperbolic functions, 260262

integrals involving, 483485

integration of, 436437

inverse sine, 254256

odd powers of, 482483, 529

periodic function, 331332

power series expansion, 706

Riemann integral, 396

square

of, 480481

trigonometry, 6568

singularity, integrands with interior

singularity, 515516

sketches

curves, 327331

graphing calculators vs., 335

skew-asymptotes, 332333

graphing calculators, 335

slant-asymptote, 332333

slant height, surface area, 556559

slicing methods, volume calculations,

538539, 545

slope ﬁelds, differential equations,

589590

slope(s), 2131

concavity, 320325

intercept form of line, 2526

parallel lines, 23

perpendicular lines, 22

point-slope form, 2324

smoothness, differentiability and,

182183

Snell’s law, 373

Snel van Riujen, Willebrord van, 82, 373

software, graphing software, 334336

solids of revolution

Kepler and, 625

Method of Cylindrical Shells,

545548, 620

Method of Disks and, 619

Method of Washers, 542543, 620

Torricelli’s Law, 626627

volume approximation, 540542

washers methods, 542543

solution curve, differential equations,

589

special sequences, 129130

spheres

Archimedes and, 625

surface area calculations, 558559

spring constant, 583584

square roots, 34, 13

squares, of trigonometric functions,

480481

Stade (Zeno), 81

subintervals

area approximation, 379

381

center

of mass calculations, 576577

precise deﬁnition of area, 382384

Riemann integral, 385392

substitution

Chain Rule method of, 432

deﬁned, 461

deﬁnite integrals, 430432

integral substitution, 431432

integration, 428437, 492

indeﬁnite integration, 429430

Newton on, 467

two solutions to integration, 433

trigonometric substitution, 488495,

493494

subtraction, constants for, 190 191

successive approximation, 81

irrational numbers and, 80

summation index value, geometric series,

638639

Sum rule, 196

inverse functions, 220

sums

derivatives for, 190191

of functions, 170171

geometric series, 378379

Riemann integral, 385392, 393,

394396

I-14 Index