P1: PBU/OVY P2: PBU/OVY QC: PBU/OVY T1: PBU JWDD023-IND2 JWDD023-Salas-v13 October 19, 2006 22:8

■ INDEX

A Asymptotic graphs, 564 Characteristic equation, 459–460 Abel, Niels Henrik, 628 Average, arithmetic, 286, 290 Circle, 9, 21 Abel’s theorem, 628 Average revenue per unit, 229 parametrized, 498 Abscissa, 17 Average value of a , 285 unit, 35, 37 Absolute convergence, 597–598 Average-value point of view, 307 Cissoid, 152 Absolute extreme values, 175–178 Average velocity, 210 Closed , 6 Absolute maximum, 175–176 Axis (axes): Collapsing sums, 241 Absolute minimum, 175–176 of ellipse, 470 Collisions, 500 Absolute value, 5–6, 13–16 of parabola, 470 Complex roots, 8 Absolute value function, 25 rotation of, A:1 Composition (of functions), 43–45 Acceleration, 209–213 Compound interest, 374–376 Addition formulas, 411 B Concavity, 190–193 Algebra, 7–8 Bacterial growth, 372–373 Conditional convergence, 598 Algebraic combinations of functions, Base p, 366–368 Cone(s): 41–42 Basic comparison theorem, 589–590 right circular, 10 Algebraic functions, 333 Bernoulli brothers, 526, 554n Constant: Alternating , 598 Bifolium, 496 decay, 373–374 basic theorem on, 598–599 , 633–634 gravitational, 213–214 estimating the sum of, 599–600 Birds, flight paths of, 190–191 growth, 371 Ambient temperature, 447 Bisection method, 98, 102 of integration, 268 Angle(s), 35 Bounded , 579 spring, 320 of incidence, 387 Bounded functions, 98–99 Continuity, 82–88 of refraction, 387 Boundedness, 7, 98–100 and differentiability, 110–112 Angular velocity, 217 Bounded , 533–534, on intervals, 88 , 254–257 540–541 of inverse functions, 337–340, A:10 Arbitrary powers, 364–366 Boyle’s law, 222 one-sided, 86 Arc cosecant, 384–385 Brachystochrone, 526 at a point, 82–87 Arc cosine, 383 Break-even points, 228 Continuous compounding, 374 Arc cotangent, 383 Continuous functions, 75, 77 Archimedean spiral, 516 C integrability of, A:11–14 Archimedes, 301 of variations, 526 Convergence: Archimedes, spiral of, 474 Cantor middle third set, 585 absolute, 597–598 , 509–513 Carbon dating, 377 conditional, 598 Arc secant, 384–385 Cardioids, 487 interval of, 618–622 Arc sine, 378–381 Cartesian coordinate system, see Rectangular of , 617–622 Arc speed, 514–515 coordinates radius of, 618 Arc (s), 381–383, 628–629 Catenary, 389–390, 517 speed of, 585 Area(s), 234–236 Cauchy, Augustin Louis, 553n Convergent sequences, 540, 544 calculating, by integration, 260–264 Cauchy mean-value theorem, 160, Convergent series, 580–584 of circular region, 281 557–558 , basic theorem on, by integration with respect to y, 292–295COPYRIGHTEDCauchy , 553 MATERIAL 598–599 in polar coordinates, 492–494 Center: basic comparison theorem for, of representative rectangles, 292–293 of ellipse, 470 589–590 of surface of revolution, 517–520 of hyperbola, 471 test for, 585–588 Argument (of a function), 25 Centroid(s): kth term of, 583 Aristotle, 47 of curve, 520–523 comparison theorem for, 590–591 Arithmetic average, 286, 290 in polar coordinates, 495 properties of, 582–584 Arithmetic-geometric mean, 553 of region, 312–315 for, 595–596 Arithmetic mean, 17 of solid of revolution, 317 for, 593–594 Astroid, 151 of surface of revolution, 525 Coordinates, 5, 17. See also Polar coordinates; Asymptotes, 34–35, 471 of triangle, 24 Rectangular coordinates horizontal, 197–198 (s), 133–137 Coordinate axes, 17 of hyperbola, 471 and trigonometric functions, Coordinate line, 5 oblique, 201 144–145 Cosecant(s), 36 vertical, 195–198 Change-of-variables formula, 278 arc, 384–385 I-1 P1: PBU/OVY P2: PBU/OVY QC: PBU/OVY T1: PBU JWDD023-IND2 JWDD023-Salas-v13 October 19, 2006 22:8

I-2 ■ INDEX

Cosecant(s) (continued ) and differentiation, 106–108 Dipole, 490 calculating involving powers and equality of, 164–165 Directrix (of parabola), 470, 491 products of, 415 of exponential functions, 358 Dirichlet function, 73, 83, 161 of, 143 of higher order, 127–128 Discontinuities, 82–83, 87 hyperbolic, 392 of linear combination, 116 Discontinuous functions: Cosine(s), 36 of product of two functions, 117–119 integrability of, 267–268 arc, 383 of quotients, 121–122 integration of, 245 calculating integrals involving powers and as rate of change, 130–132, 217, 218–219 Disk(s), 300 products of, 411–414 acceleration, 209–213 Disk method, 300–302 derivative of, 142 in economics, 228–229 Distance: hyperbolic, 388–391 free fall, 213–215 between point and line, 472 Cotangent(s), 36 speed, 210 in rectangular coordinates, 18 arc, 383 velocity, 209–213 and speed, 236–237 calculating integrals involving powers and of rational powers, 149–150 Divergence, 579 products of, 415 and reciprocal rule, 119–121 Divergence, test for, 583 derivative of, 143 of scalar multiples, 115–116 Divergent sequences, 540–541 hyperbolic, 392 of sums, 115–116 Divergent series, 580–584 Critical points, 168–169 and tangent to curve, 105–106 basic comparison theorem for, 589–590 Cross sections, volume by parallel, of trigonometric functions, 142–145 limit comparison theorem for, 590–591 296–304 and chain rule, 144–145 ratio test for, 595–596 Cube, 9 cosecant, 143 root test for, 593–594 Cubics, 33–34 cosine, 142 test for divergence, 583 Cubing function, 161 cotangent, 143 Domain, 25, 28–29 Curve(s): secant, 143 Double-angle formulas, 411 centroid of, 520–523 sine, 142 Double zero, 139 figure-eight, 152 tangent, 143 Doubling time, 372, 376 integral (solution), 451 Descartes, Ren´e, 3 Duhamel’s principle, 519 least-time, 526 Determinants, A:3–6 Dummy variable, 240 orthogonal, 151 Difference quotient, 106 parallel, 165 Differentiability, 106 E parametrized, 496–500 and continuity, 110–112 e, rational bounds for, 356–362, 364 petal, 488 of inverse functions, 339, A:10 Eccentricity, 491 same-time, 526 Differentiability theorem, 625 of ellipse, 477 sketching, 201–208 Differentials, 223–226 of hyperbola, 477 speed along plane, 514–515 , 106 Economic applications, 228–229 tangent to, 105–106 Differential equations, 443–449 Elements (of set), 3 Cusps, vertical, 199 applications of, 447–449 Ellipse, 22 Cycloid, 525–526 definition, 443 geometry of, 469, 470 inverted, 525 first-order, 443–446 parametrized, 498 Cylinder(s): homogeneous equations: in polar coordinates, 491 right, 296–297 y + ay + by = 0, 459–465 Elliptical reflectors, 474–475 right circular, 10 order of, 443 Endpoint extreme values, 175 separable, 452–456 Endpoint maximum, 174–175 D applications of, 454–456 Endpoint minimum, 174–175 Darboux method, 243 definition, 452 Energy, kinetic, 217 d/dx notation, 124–127 solutions of, 443 Engineering applications, see Scientific and Decay constant, 373–374 Differentiation, 106–108. See also engineering applications Decimal representation, 4–5 Derivative(s) Entries (of matrix), A:3 Decreasing functions, 160–165 chain rule for, 133–137 Equations of second degree, A:1–2 Decreasing sequences, 533–536 formulas for, 115–122 Equilateral triangle, 8, 24 Definite integral, see under Integral(s) differences, 116 Equipotential lines, 491 Degree (of polynomials), 32–34 , 117–119 Error estimates, 437–440 Degree measure (of angles), 35 , 121–122 Essential discontinuities, 82 Density: reciprocal rule, 119–121 Euclid, 47 of real numbers, 5 scalar multiples, 115–116 Euler equation, 466 weight, 328 sums, 115–116 Even functions, 27–28, 201 Dependent variable, 25 trigonometric functions, 142–145 Existence and uniqueness theorem, 461 Derivative(s). See also Differentiation implicit, 147–150 Exponents, laws of, 7 d/dx notation for, 124–127 of logarithm functions, 347–349 Exponential density functions, 573 definition of, 106 logarithmic, 352–354 (s), 356–362 of differences, 116 of power series, 623–626 with base p, 366–367 P1: PBU/OVY P2: PBU/OVY QC: PBU/OVY T1: PBU JWDD023-IND2 JWDD023-Salas-v13 October 19, 2006 22:8

INDEX ■ I-3

definition, 356 increasing, 160–165 Horizontal translations, 44 of, 358 integrability of continuous, A:11–14 Horsepower, 326 integral of, 360–362 integrability of discontinuous, 267–268 Hyperbola, 22 as inverse of logarithm function, 356 linear combinations of, 41–42 geometry of, 469–471 Exponential growth/decay, 370–376 logistic, 454 parametrized, 498–499 bacterial growth, 372–373 lower sum of, 236, 238 in polar coordinates, 491 compound interest, 374–376 odd, 27–28, 201 Hyperbolic functions, 388–395 population growth, 371–373 periodic, 37n, 201 cosecant, hyperbolic, 392 radioactive decay, 373–374 period of a, 37n cosine, hyperbolic, 388–391 Extreme values, 174–180 piecewise defined, 25 cotangent, hyperbolic, 392 absolute, 175–178 polynomial, 32–34 inverses of, 394–395 endpoint, 175 range of, 25 secant, hyperbolic, 392 finding, 179–180 rational, 34–35 sine, hyperbolic, 388–391 local, 167–172 scalar multiples of, 41–42 tangent, hyperbolic, 392–394 critical points, 168–169 sketching graphs of, 201–208 Hyperbolic reflectors, 476 derivative, 168 symmetric, 27–28 Hypocycloid, 524 and first-, 170–172 trigonometric, 35–39 and second-derivative test, 171–172 upper sum of, 236, 238 I in max-min problems, 182–187, 190–191 vertical line test for, 26 Identities, trigonometric, 37–38 Extreme-value theorem, 99–100 Fundamental Theorem of Integral Calculus, iff statements, 48 proof, A:9 254–258 Image (of function), 25 derivation of, 246–252 Implicit differentiation, 147–150 F proof, 255 Improper integrals, 565–571 , 7 Future value, 377 over unbounded intervals, 565–569 Factoring formulas, 8 of unbounded functions, 569–571 Fibonacci sequence, 554 G Improper rational functions, 422 Figure-eight curve, 152 Gabriel’s horn, 567 Inclination, 18 First-derivative test, 170–172 Galileo Galilei, 213 Increasing functions, 160–165 First mean-value theorem for integrals, General solution, 445 Increasing sequences, 533–536 285–286 Geometric mean, 564 Indefinite integrals, 268–273 First-order equations, 443–449 Geometric progression, 579 u-substitution with, 274–277 Fixed-point property, 101 , 579–584 Independent variable, 25 Fluid force, 327–329 Geometry, 8–10 (s), 560–563 Focus (foci), 491 Gompertz equation, 451 of type 00, 562 of ellipse, 470 Graphs: of type 0/0, 554 of hyperbola, 471 of functions, 26–27 of type 0 ∗∞, 561 of parabola, 470 of hyperbolic sine/cosine, 388–389 of type 1∞, 562 Folium of Descartes, 517 of inverse functions, 337 of type ∞0, 562–563 Foot-pounds, 321 of logarithm functions, 345–349 of type ∞−∞, 561–562 Force(s): in polar coordinates, 484–490 of type ∞/∞, 560–561 fluid, 327–329 sketching, 201–208 Index of refraction, 387 of gravity, 322–325 of trigonometric functions, 38–39 Induction, mathematical, 49–51 lines of, 490–491 Graphing utilities, 27 Inequalities, 5, 11–16 Free fall, 213–215 Gravitational constant, 213–214 and absolute value, 13–16 Fresnel function, 559 Gravity, counteracting force of, 322–325 triangle inequality, 16 Function(s), 24–30. See also Continuity; Greatest lower bound, 530–531, 533–534 Infinite discontinuities, 83 Transcendental functions Gregory, James, 629 Infinite limits, 58 absolute value, 25 Growth constant, 371 Infinite series, 575–600, 606–634 algebraic, 333 absolute convergence of, 597–598 algebraic combinations of, 41–42 H alternating series, basic theorem on, applications of, 29–30 Half-angle formulas, 411 598–599 behavior of, as x −±∞, 177–178 Half-life, 373 basic comparison theorem for, bounded, 98–99 Half-open (half-closed) interval, 6 589–590 composition of, 43–45 Harmonic motion, simple, 216 binomial series, 633–634 continuous, 75, 77 Harmonic series, 587 conditional convergence of, 598 critical points of, 168–169 Heaviside function, 90 convergence of, 579–580 decreasing, 160–165 Higher order derivatives, 127–128 convergent:: domain of, 25, 28–29 Hooke, Robert, 320 absolute convergence, 597–598 even, 27–28, 201 Hooke’s law, 320–321 alternating series, basic theorem on, graphs of, 26–27 Horizontal asymptotes, 34–35, 197–198 598–599 image of, 25 Horizontal line test, 334 basic comparison theorem, 589–590 P1: PBU/OVY P2: PBU/OVY QC: PBU/OVY T1: PBU JWDD023-IND2 JWDD023-Salas-v13 October 19, 2006 22:8

I-4 ■ INDEX

Infinite series (continued ) over unbounded intervals, 565–569 J conditional convergence, 598 of unbounded functions, 569–571 Joules, 321 integral test, 585–588 indefinite, 268–273 Jump discontinuities, 83 kth term of, 583 u-substitution with, 274–277 limit comparison theorem for, 590–591 as limit of Riemann sums, A:14 K power series, 617–622 linearity of, 257–258 Kinetic energy, 217 properties, 582–584 mean-value theorems for, 253 ratio test, 595–596 first, 285–286 L root test for, 593–594 second, 287–289 Lagrange, Joseph-Louis, 106n, 197, 605 definition, 578 Riemann, 244 Lagrange formula for the remainder, divergence of, 580–584 Integral curves, 451 605 test for, 583 Integral sign, 240 Laplace transforms, 572–573 divergence test for, 583 Integral tables, 400 Latus rectum (of parabola), 476 divergent:: Integral test, 585–588 Least-time curve, 526 basic comparison theorem for, 589–590 Integrand, 240 Least upper bound, 528–530 limit comparison theorem for, Integrating factors, 445 of sequence, 533–534 590–591 Integration, 237 Least upper bound axiom, 529–530 ratio test for, 595–596 area problems using, 260–264 Left-hand limit(s), 70–71 root test for, 593–594 calculating areas by, with respect to y, Leibniz, Gottfried Wilhelm, 3 geometric series, 580–584 292–295 Leibniz notation, 124–125, 340 integral test for, 585–588 constant of, 268 Lemniscate, 151, 482, 488 limit comparison theorem for, 590–591 of discontinuous functions, 245 L’HÙpital,G. F. A., 554n partial sums of, 578–579 formulas for, 398–399 L’ Hopital’s ˆ rule, 555–558, 563 power series, 616–622 of logarithm functions, 349–352 Lima¸con, 488 convergence/divergence of, 617–622 numerical, 433–440 Limit(s), 54–82 definition, 617 with partial fractions, 423–429 definition of, 64–71 differentiation of, 623–626 by parts, 402–408 existence of, 77–78 integration of, 627–631 of power series, 627–631 idea of, 54–56 vs., 631–632 with rationalizing substitutions, infinite, 58 ratio test for, 595–596 430–432 of integration, 240 rearrangements of, 600 symmetry in, 283–284 left-hand, 70–71 root test for, 593–594 term-by-term, 627–631 lower, 240 sigma notation for, 575–577 of trigonometric functions, 350–352 nonexistent, 60 sum of, as limit, 578–579 by trigonometric substitution, 417–421 one-sided, 56–57, 70–71, 79 Taylor series:: Intercepts (of line), 18, 19 right-hand, 70–71 power series vs., 631–632 Interest, compound, 374–376 of sequences, 536–538 in x, 606–611 Interior points (on an interval), 6 definition, 538 in x-a, 613–616 Intermediate-value theorem(s), 97–99, A:8 theorems, 536–539 Initial position, 213 proof, A:9 stability of, 545 Initial-value problems, 446 Intersections, 500 theorems involving, 73–79, 81–82 Initial velocity, 213 Interval(s), 6 trigonometric, 92–96 Integers, 4 continuity on, 88 uniqueness of, 73 Integrability, 266 of convergence, 618–622 upper, 240 of continuous functions, A:11–14 Inverse functions, 334–340 Limit comparison theorem, 590–591 of discontinuous functions, 267–268 continuity of, 337–340, A:10 Line(s), 18–21 Integral(s): definition, 334 distance between point and, 472 and , 254–257 differentiability of, 339, A:10 equations for, 19 definite, 237–245 graphs of, 337 equipotential, 491 definition, 237 hyperbolic functions, 394–395 of force, 490–491 distance, calculation of, 236–237 notation for, 335, 394 intercepts of, 18, 19 formulas for, 239, 242 one-to-one functions, 334–337 normal, 113 general properties of, 281–284 trigonometric functions, 378–385 parallel, 19 notation, 239 arc cosecant, 384–385 parametrized, 497–498 Riemann sums, 243–245 arc cosine, 383 perpendicular, 19 u-substitution with, 277–279 arc cotangent, 383 perpendicular bisector of, 24 of exponential function, 360–362 arc secant, 384–385 of, 18 fundamental theorem of integral calculus for arc sine, 378–381 straight, 497–498 evaluation of, 254–258 arc tangent, 381–383, 628–629 tangent, 106, 109–110 255, 285–286 Inverted cycloid, 525 Linear combination, 116 derivation, 246–252 Irrational numbers, 4–5 Linearity of integral, 257–258 improper, 565–571 Isosceles triangles, 24 Linear polynomials, 32–34 P1: PBU/OVY P2: PBU/OVY QC: PBU/OVY T1: PBU JWDD023-IND2 JWDD023-Salas-v13 October 19, 2006 22:8

INDEX ■ I-5

Local extreme values, 167–172 Mixing problems, 448–449 Partial fractions, 423–429 critical points, 168–169 Monotonic sequences, 534 Partial sums, 578–579 derivative, 168 Particular solution, 445 and first-derivative test, 170–172 N Partitions, 237 and second-derivative test, 171–172 , 368 regular partition, 245 Local maximum, 167–168 Natural logarithm functions, 344 Period, 37n Local minimum, 167–168 Natural numbers, 4n Periodic functions, 37n, 201 Logarithm function, 342–354 Newton, Sir Isaac, 3 Periodicity of trigonometric functions, 37 definition, 344 Newtons (unit), 321 Perpendicular bisector, 24 differentiation of, 347–349, 354 Newton-meters, 321 Perpendicular lines, 19 graphing, 345–349 Newton-Raphson method, 229–231 Petal curves, 488 integration of, 349–352 and sequences, 547–548 Piecewise functions, 25 as inverse of exponential function, 356 Newton’s Law of Cooling, 447–448 Pinching theorem, 91–93 natural, 344 Newton’s Law of Gravitational Attraction, for sequences, 543–544 range of, 343 139 Point(s): and trigonometric functions, 350–352 Nondecreasing sequences, 534 continuity at, 82–87 Logarithmic differentiation, 352–354 Nonexistent limits, 60 distance between line and, 472 Logarithm of x to the base p, 367–368 Nonincreasing sequences, 534 of inflection, 191–193 Logistic equation, 454 Normal lines, 113 interior, 6 Logistic functions, 454 Norman window, 32 Polar coordinates, 478–483 Lower bound, 7, 530–531 Notation: area in, 492–494 greatest, 530–531, 533–534 for inverse functions, 335 centroids in, 495 Lower limit, 240 Leibniz, 124–125, 340 ellipse in, 491 Lower sum, 236–238 sigma, 575–577 graphing in, 484–490 Number line, 5 hyperbola in, 491 M Numerical integration, 433–440 intersection of polar curves, 489–490 m × n matrix, A:4 parabola in, 491 Maclaurin, Colin, 607 O rectangular coordinates, relation to, Maclaurin series, 607 Oblique asymptotes, 201 480–482 Major axis (of ellipse), 470 Odd functions, 27–28, 201 symmetry in, 482–483 Mapping, 25 One-sided continuity, 86 Polygonal path, 509–510 Marginal cost, 228 One-sided limits, 56–57, 69–71, 79 Polynomial functions (polynomials), Marginal profit, 228 One-to-one functions, 333–337 32–34 Marginal revenue, 228 continuity of, 337–340 derivative of, 118–119 Mass moment, 289 definition, 334 Population growth, 371–373 Mathematical induction, 49–51 graphs of, 337 Positive integers, 4 Mathematical proof, 47–48 horizontal line test for, 334 Power series, 616–622 Matrices, A:3–6 inverses of, 334–337 convergence/divergence of, 617–622 Maximum: One-to-oneness, test for, 336 definition, 617 absolute, 175–176 Open interval, 6 differentiation of, 623–626 endpoint, 174–175 Order (of a ), 443 integration of, 627–631 local, 167–168 Ordered pair, 17 Taylor series vs., 631–632 Max-min problems, 182–187, 190–191 Ordinate, 17 Present value, 377 Mean: Origin, 5, 17 Pressure, fluid, 327–329 arithmetic, 17 Orthogonal curves, 151 Prime number, 532 of probability density function, 573 Orthogonal trajectories, 151, 458 Principle of Least Time, 186n Mean-value theorem(s), 154–158 Probability density functions, 573 Cauchy, 557–558 P Product rule, 117–119 for integrals, 253 π, estimating, 634 Profit function, 228 first mean-value theorem, 285–286 Pappus’s theorem on surface area, 523 Proofs, 47–48 second mean-value theorem, 287–289 Pappus’s theorem on volumes, 315–317 Proper rational functions, 422–423 Median (of triangle), 24 Parabola, 22, 469–470 p-series, 587 Members (of set), 3 in polar coordinates, 491 Pumping problems, 323–325 Method of fluxions, 3 Parabolic mirrors, 472–474 Pyramid, volume of, 298 Midpoint formula, 18 Parabolic trajectories, 503 Pythagorean theorem, 18 Minimum: Parallel curves, 165 absolute, 175–176 Parallel lines, 19 Q endpoint, 174–175 Parametrized curves, 496–500 Quadrants, 17 local, 167–168 definition, 496 Quadratic formula, 8 Minor axis (of ellipse), 470 intersections/collisions, 500 Quadratic functions, 33 Mirrors, parabolic, 472–474 to, 503–508 Quotient rule, 121–122 P1: PBU/OVY P2: PBU/OVY QC: PBU/OVY T1: PBU JWDD023-IND2 JWDD023-Salas-v13 October 19, 2006 22:8

I-6 ■ INDEX

R Round-off error, 437 disk method, 300–302 , 35–36 Rule of 72, 376 washer method, 302–303 measure, 35 Solution(s): Radioactive decay, 373–374 S of differential equations, 443 Radius of convergence, 618 Same-time curve, 526 existence and uniqueness of, 461 Range, 25 Scalar multiples, derivatives of, Solution curves, 451 of logarithm functions, 343 115–116 Speed, 210, 217 unbounded, 344 Scientific and engineering applications: along plane curve, 514–515 Rates of change per unit time, 217, energy: of convergence, 585 218–219 kinetic, 217 and distance, 236–237 Rational bounds for e, 364 half-life, 373 Sphere, 9 Rational functions, 34–35 Hooke’s law, 320–321 volume of, 301 Rationalizing substitutions, 430–432 temperature, 24, 447–448 Spiral, Archimedean, 516 Rational numbers, 4 work, 319–323 Spiral of Archimedes, 474 Rational powers, derivatives of, 149–150 Secant(s), 36 Spring constant, 320 Ratio test, 595–596, 619–620 arc, 384–385 Square, 9 Real line, 5 calculating integrals involving powers and Square matrices, A:4 Real numbers, 4–7 products of, 414–415 Squaring function, 161 absolute value of, 5–6 derivative of, 143 Standard deviation, 573 boundedness of, 7 hyperbolic, 392 Standard position, 34 classification of, 4 Second degree, equations of, A:1–2 of ellipse, 470 decimal representation of, 4–5 Second-derivative test, 171–172 of hyperbola, 470 density of, 5 Second mean-value theorem for integrals, of parabola, 469 intervals of, 6 287–289 Strophoid, 496, 506 on number line, 5 Second-order equations, 443 Substitution(s): sequences of, 532–536 Sectors, 9 rationalizing, 430–432 Rearrangements, 600 Segments, circular, 421 trigonometric, 417–421 Reciprocal rule, 119–121 Separable equations, 452–456 u-substitution, 274–279 Rectangle, 9 applications of, 454–456 Sums: Rectangular coordinates, 17–23 definition, 452 collapsing, 241 distance/midpoint formulas, 18 Sequence(s), 532–536 derivatives of, 115–116 lines, 18–21 bounded, 533–534, 540–541 Riemann, 243–245 polar coordinates, relation to, 480–482 convergent/divergent, 540–541 upper/lower, 236–238 Rectangular solid, 9 increasing/decreasing, 533–536 Surface(s): Recursively defined sequences, 537 limits of, 538–536 of revolution: Reflectors: definition, 538 area of, 517–520 elliptical, 474–475 theorems, 539–536 centroid of, 525 hyperbolic, 476 monotonic, 534 Surface area, Pappus’s theorem on, 523 Refraction, 387 and Newton-Raphson method, 547–548 Symmetric functions, 27–28 Removable discontinuities, 82 of partial sums, 578–579 Symmetry: Repeating decimals, 4 pinching theorem for, 543–544 in integration, 283–284 Representative rectangles, 292–293 recursively defined, 537 in polar coordinates, 482–483 Revenue per unit, average, 229 unbounded, 540–541 Riemann, Bernhard, 243, 244, 600 Series: T Riemann integral, 244 alternating, 598–600 Tangent(s), 36 Riemann sums, 243–245 with nonnegative terms, 585–591 arc, 381–383, 628–629 integral as limit of, A:14 Set(s), 3–4 calculating integrals involving powers and Right circular cone, 10 Shell method, 306–311, 316 products of, 414–415 Right circular cylinder, 10 Sigma notation, 575–577 to curve, 105–106 Right cylinder, 296–297 Simple harmonic motion, 216 derivative of, 143 Right cylinder with cross section, 296 Sine(s), 36 hyperbolic, 392–394 Right-hand limits, 70–71 arc, 378–381 lines, tangent, 106, 109–110 Right triangle, trigonometric functions in terms calculating integrals involving powers and to parametrized curves, 503–508 of, 38 products of, 411–414 vertical, 110, 198–199, 505 Rod: derivative of, 142 Tangent lines, 106 center of mass of, 287–288 hyperbolic, 388–391 Tautochrone, 526 mass of, 287 limit of, 92 Taylor polynomials in x, 602–606 Rolle, Michel, 156 Slope, 18, 106 Taylor series: Rolle’s theorem, 156–157 Snell’s Law of Refraction, 387 power series vs., 631–633 Root test, 593–594, 619 Solids of revolution: in x, 606–611 Rotation of axes, A:1 centroid of, 317 in x-a, 613–616 P1: PBU/OVY P2: PBU/OVY QC: PBU/OVY T1: PBU JWDD023-IND2 JWDD023-Salas-v13 October 19, 2006 22:8

INDEX ■ I-7

Taylor’s theorem, 604–605 Triangle inequality, 16 V Temperature, 24 Trigonometric functions, 35–39 Value(s): ambient, 447 angles, 35 of f at x, 25 Term-by-term integration, 627–631 arc tangent, 381–383 of trigonometric functions, 37 Terminal velocity, 457 calculating integrals involving powers and Variables, 25 Terminating decimals, 4 products of: Velocity, 209–213 Theoretical error, 437–440 cotangent/cosecant, 415 angular, 217 Torus, 312, 422 sines/cosines, 411–414 average, 210 Trajectories: tangent/secant, 414–415 initial, 213 orthogonal, 458 derivatives of, 142–145 terminal, 457 parabolic, 503 and chain rule, 144–145 Verhulst, P. F., 454 Transcendental functions, 333, 342–354 cosecant, 143 Vertex (vertices): exponential functions, 356–362 cosine, 142 of ellipse, 470 with base p, 366–367 cotangent, 143 of hyperbola, 471 definition, 356 secant, 143 of parabola, 470 derivative of, 358 sine, 142 Vertical asymptotes, 34–35, 195–198 integral of, 360–362 tangent, 143 Vertical cusps, 199 as inverse of logarithm function, graphs of, 38–39 Vertical line test, 26 356 identities, trigonometric, 37–38 Vertical tangents, 110, 198–199, 505 hyperbolic functions, 388–395 integration of, 350–352 Vertical translation, 42–43 cosecant, hyperbolic, 392 inverse functions, 378–385 Volume(s): cosine, hyperbolic, 388–391 arc cosecant, 384–385 Pappus’s theorem on, 315–317 cotangent, hyperbolic, 392 arc cosine, 383 by parallel cross sections, 296–304 inverses of, 394–395 arc cotangent, 383 by shell method, 306–311 secant, hyperbolic, 392 arc secant, 384–385 sine, hyperbolic, 388–391 arc sine, 378–381 W tangent, hyperbolic, 392–394 arc tangent, 381–383, 628–629 Wallis sine formulas, 416 inverse trigonometric functions, periodicity of, 37 Washer method, 302–303, 315 378–385 in terms of arbitrary triangle, 38 Water, weight density of, 328 arc secant, 384–385 in terms of right triangle, 38 Watts, 326 arc sine, 378–381 values of, 37 Weight density, 328 arc tangent, 381–383 Trigonometric limits, 92–96 Weighted average, 287 logarithm function, 342–354 Trigonometric substitutions, 417–421 Well-defined sets, 4 definition, 344 Triple zero, 139 Whispering chambers, 475 differentiation of, 347–349 Work, 319–323 graphing, 345–349 U and Hooke’s law, 320–321 integration of, 349–352 Unbounded divergence, 579 per unit of time, 326 natural logarithm function, 344 Unbounded functions, 98 tank, pumping out, 323–325 range of, 343 integrals of, 569–571 Wronskians, 461–462 and trigonometric functions, Unbounded range, 344 350–352 Uniform circular motion, 217 X trigonometric functions, 350–352 Uniformly continuous functions, x-axis, 17 Translations, 471–472 A:11 x-coordinate, 17 horizontal, 44 Unit circle, 35, 37 xy-term, elimination of, A:2–3 vertical, 42–43 Unit circle relations, 411 Transverse axis (of hyperbola), 471 Unit pulse function, 90 Y Triangle(s), 8 Upper bound, 7, 528 y-axis, 17 centroid of, 24 least, 528–530, 533–534 y-coordinate, 17 equilateral, 24 Upper limit, 240 isosceles, 24 Upper sum, 236–238 Z median of, 24 u-substitution, 274–279 Zeno’s paradox, 575, 584 trigonometric functions in terms of, 38 with definite integrals, 277–279 Zeroes, 48 P1: PBU/OVY P2: PBU/OVY QC: PBU/OVY T1: PBU JWDD023-IND2 JWDD023-Salas-v13 October 19, 2006 22:8

8