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

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surface area

Archimedes and, 625

calculations, 556559, 620

surface of revolution, 556559

surface of rotation, 558559

symmetry, axis of, 16

tables of integrals, 433435

tail(s)

comparison test for divergence

and, 657

end of series, 645646

integral test of, 649650

Root Test, 670671

of sequence, 129

taking limits, 2

tangent function

history of, 275276

integration of, 436437

inverse, 257

hyperbolic functions, 261262

tangent line(s), 171173

approximation, 248250, 251

Fermat’s theorem and, 371372

increasing and decreasing functions,

299300

Mean Value Theorem, 289

Newton-Raphson method, 348349,

352

nonvertical, 179

trigonometric functions, 203204

trigonometry, 68

vertical tangent lines, corners and,

174175

Taylor, Brook, 720

Taylor, Zachary, 535

Taylor series

accuracy in, 702704

applications, 630

approximations, 708

binomial series, 707

history of, 719720

polynomials and, 693695

indeterminate form evaluation,

708709

transcendental function expansions,

704706

Taylor’s theorem, 698701, 720

telescoping series, 635636

telescoping sum, 378

temperature change, Newton’s laws for,

612613

terminal radius, sine and cosine, 65

terminating decimals, 22

Theory of Indivisibles, 466467

Theory of Probability, random variables,

565566

Theory of Relativity, 438

time-series data, least squares lines,

2831

Torricelli,

Evangelista, 82, 466, 626

Torricelli’s Law, 593594

Trait

e d’Algebre, 373374

Trait

e de la Lumi

e, 372373

transcendental functions, 253266

hyperbolic functions, 260263

inverse hyperbolic functions, 263265

inverse sine and cosine, 254256

inverse tangent function, 257

inverse trigonometric functions,

253254, 258259

maxima and minima, 314315

Newton-Raphson method, 348

notation, 260

Taylor series expansions, 704706

transcendental number, 80

trapezoidal rule, numerical integration,

449452, 462

Treaties of Fluxions (Maclaurin), 720

triangle

area of, 383384

characteristic/differential, 626

inequality, 7

order properties of integrals, 404405

trigonometric functions, 6572

antidifferentiation, 360361

conversion to sine and cosine, 485

deﬁned, 6869

derivatives of, 203204

higher powers of, 481482

hyperbolic functions, 260262, 272

identities, 70

integrals involving sine and cosine,

483485

integration of, 436437

by parts, 474475

inverse, 253254, 258259, 271

limits in, 100

102

modeling

with, 7172

Newton-Raphson method, 352353

odd powers of, 482483

powers and products, 479485

radian measure, 65

sine/cosine functions, 6568

trigonometric substitution, 488495, 530

nonradical quadratic expression,

493495

radical quadratic expression, 492493

true value, approximation and, 7

truncation error, 7

two-function composition, chain rule

differentiation, 210212

two-point form of line, 2425

two-point systems, center of mass in, 572

two-sided limits, derivatives of exponen-

tial functions, 205206

unbounded integrands, 514519

Comparison Theorem for, 518519

endpoints, unboundedness at, 517

interior singularity, 516

unbounded intervals, 56

improper integrals, 521526

uncountable numbers, 80

uniform mass density

center of mass, 574577

moments, 573574

uniform partition

area approximation, 379381

average value of function, 562563

order properties of integrals, 403405

point rule, 446449

Riemann sums, 390392

sigma notation and, 377, 459

Simpson’s Rule, 453455

surface area calculations, 557559

unit circle, 13

trigonometry, 65

unit of work, deﬁnitions, 580585

upper limit of integration, 393

Fundamental Theorem of Calculus,

410, 412413

reversal, 402403

upper Riemann sum, 390392

u-substitution, 429

van der Waal’s equation, implicit

differentiation, 284

variables, 1

continuous, 1

Leibniz notation, 180181

parabolas, ellipses, and hyperbolas,

1617

planar coordinates, 1112

point-slope of line, 23

separation of, 591598

variational principle, catenary

formula, 627

variations, calculus of, 628

velocity

antidifferentiation, 361363

average velocity, 164165

higher derivatives, 231

instantaneous velocity, 166168

linear velocity law, 613615

positive and negative, 167168

Index I-15

vertexes, of parabola, 17

vertical asymptotes, 119121

curve sketching, 327328, 330331

graphing calculators, 335

natural logarithms, 420

periodic functions, 332

Vertical Line Test, 35

vertical translation, 57

Vie

te, Franc¸ois, 81, 371

viewing window/rectangle, 19

volume

Archimedes and, 625

cylindrical shells method, 545548

differential equations for,

593594

disk slicing method, 538539

formulas for, 465

integral applications, 538548

Kepler’s calculations for, 625

rotation, 544545

solids of revolution, 540542

washers methods, 542543

Wallis, John, 465467, 626

Washers, Method of, 542543, 620

Weierstrass, Karl, 82, 162

weight, variations in, work calculations,

581583

Wilkinson Microwave Anisotropy Probe

(WMAP), 43

work

deﬁnition of, 580585, 621

ﬂuid mechanics, 584585

integral calculations, 580581

spring constant, 583584

weight variations, 581583

worst-case estimates, approximation

techniques, 454

Wren, Christopher, 626

Wright, Edward, 533

x, powers of, 200203

antidifferentiation and, 357360

expectation of, 567

integrals, 358360

x-axis

center of mass calculations, 576577

Method of Cylindrical Shells rotation,

547548

role reversal, 443444

rotation, Method of Disks and,

540541

surface area calculations, 558559

x-coordinate, 11

Newton-Raphson method, 348349

x-intercept, natural logarithm, 420

y-axis

Method of Cylindrical Shells rotation,

546548

role reversal, 443444

rotation, Method of Disks and,

541542

surface area calculations, 558559

y-coordinate, 11

y-intercept

curve sketching, 328, 331

natural logarithm, 420

Zeno’s paradoxes, 81

geometric series, 135

inﬁnite series, 718

zero slope, mean value theorem, 289

zeroth derivative, 231

I-16 Index