INDEX

Abel, Niels Henrik, 234 algebraic manipulation, of integrand, finding by numerical methods, 367 abscissa, Web-H1 412 and improper integrals, 473 absolute convergence, 556, 557 alternating current, 324 as a line integral, 981 ratio test for, 558, 559 alternating harmonic series, 554 parametric curves, 369, 610 absolute error, 46 alternating series, 553–556 polar curve, 632 Euler’s Method, 501 alternating series test, 553, 560, 585 from vector viewpoint, 760, 761 absolute extrema amplitude arc length parametrization, 761, 762 Extreme-Value Theorem, 205 alternating current, 324 finding, 763, 764 finding on closed and bounded sets, simple harmonic motion, 124, properties, 765, 766 876–879 Web-P5 arccosine, 57 finding on finite closed intervals, 205, sin x and cos x, A21, Web-D7 Archimedean spiral, 626, 629 206 analytic geometry, 213, Web-F3 Archimedes, 253, 255 finding on infinite intervals, 206, 207 Anderson, Paul, 376 palimpsest, 255 functions with one relative extrema, angle(s), A1–A2, Web-A1–Web-A2 arcsecant 57 208, 209 finding from trigonometric functions, arcsine, 57 on open intervals, 207, 208 Web-A10 arctangent, 57 absolute extremum, 204, 872 of inclination, A7, Web-A10 area absolute maximum, 204, 872 between planes, 719, 720 antiderivative method, 256, 257 absolute minimum, 204, 872 polar, 618 calculated as double integral, 906, 907 absolute minimum values, 872 rectangular coordinate system, computing exact value of, 279 absolute value, Web-G1 A3–A4, Web-A3–Web-A4 definition, 277 and square roots, Web-G1, Web-B2 standard position, A3, Web-A3 finding with Green’s Theorem, 1008 absolute value function, A10, Web-B4 between vectors, 692–693 function of two variables, 805 continuity, 44 angular coordinate, 618 left endpoint approximation, 280 derivative of, 93 angular frequency, simple harmonic as a line integral, 981, 982, 984, properties, A10, Web-B5 motion, 124 1007–1008 absolute zero, 883 annuity, 238 midpoint approximation, 280 acceleration, 227 antiderivative(s), 258, 259 net signed, 282, 283, 287, 307 centripetal, 802 of vector-valued functions, 756 polar coordinates, 634, 635 constant, 308, 309 antidifferentiation, 256, 259 with polar double integrals, 915 due to gravity, 56, 310 aphelion, 666, 798 rectangle method, 257 instantaneous, 227 apogee, 666, 799 rectangular method for finding, 254 motion along curves, 781, 782 artificial Earth satellite, 151, 798 right endpoint approximation, 280 normal and tangential components, COPYRIGHTEDapproximation(s) MATERIALsurface of revolution, 370, 372 784–787 area under a curve, 255 total, 297, 307 rectilinear motion, 227, 228 left, right, and midpoint, 455 between two curves, 342, 344, 345 sign of, 228 local linear, 839 under a curve, 278 tangential scalar/vector component, local linearity, 153–154 area problem, 254 785 local quadratic, 563 argument, A8, Web-B2 vector, 782 Maclaurin and Taylor polynomial, arithmetic mean (or average), 315, 664 acceleration function, 227 564, 566, 567, 569–571 aspect ratio distortion, Web-K6 addition midpoint, 455, 457, 458 Astronomia Nova, 665 formulas for sine and cosine, A6, Riemann sum, 287 astronomy, planetary motions, 665, 666, Web-A8 of roots using Intermediate-Value 794–798 of functions, A11, Web-C1 Theorem, 46–47 asymptote(s), 30, 193 Agnesi, Maria, 744, 750 of roots using Newton’s Method, a curve as an, 30 air resistance, 508 234–237 curvilinear, 196 Alexeev, Vasili, 376 Simpson’s rule, 458, 459, 461 horizontal, 22, A21, Web-D6 algebraic functions, A21, Web-D6 trapezoidal, 455 oblique, 196 integration formulas, 413 arc length, 364, 365, 367, A2, Web-A2 slant, 196

I-1 I-2 Index

vertical, 9, A20, Web-D6 butterfly curve, 628 osculating, 78 asymptotes of a hyperbola, 641 circular cone, 726–729 finding, 647 cancellation equations, A24, Web-E2 circular helix, 745 asymptotic curves, 30 carbon dating, 493 circular paraboloid, 726–729 autonomous cardioid, 623 circulation, 1044 differential equation, 503 area, 634 circulation density, 1044 auxiliary equation, Web-P2 families of, 624 closed ball, 817 average rate of change, 84 intersection with circle, 635 closed disk, 817 average value, 314, 316, 901, 910 carrying capacity, 30, 483 closed form, sigma notation, 277 and average velocity, 317 Cartesian coordinate system, 674, closed interval(s), Web-F4 average velocity, 82, 314, 317 Web-H1 absolute extrema on finite, 205 geometric interpretation, 83 Cassini, Giovanni Domenico, 629 continuity on, 42 axis (axes) Cassini ovals, 629 closed parametric curves, 998 coordinate, Web-H1 catenary, 400 closed sets, 817 of an ellipse, 641 Cauchy, Augustin, 564, 566 finding absolute extrema, 876–879 of a hyperbola, 641 Cauchy–Riemann equations, 836 closed surfaces, 1030 of a parabola, 641 polar form, 854 clothoid, 672 polar, 618 Cavalieri, Bonaventura, 358 Cobb–Douglas production model, 892 of revolution, 352 Cavalieri’s principle, 358 coefficient of friction, 99, 124 rotation of, 657, 658 center coefficients, Web-J1 circles, Web-I3 correlation, Web-N2 Badel, Hannskarl, 408 ellipse, 640 leading, Web-J1 ball hyperbola, 641 polynomials, A19, Web-D5 close, 817 center of curvature, 777 cofunction, 125 open, 817 center of force, 794 Commentaries on the Motions of Mars, base b exponential function, 330 center of gravity, 383, 960 665 Bell, Alexander Graham, 71 lamina, 384–387, 960, 962, 963 common logarithms, 67 Bernhard Riemann, Bernhard, 287 solids, 963–965 comparison test, 547, 548, 560 Bernoulli equation, 513 centimeter-gram-second (CGS), 375 complete elliptic integral of the first kind, 479, 598 Bernoulli, Daniel, 612 central angle, cylindrical wedges, 939 complete linear factorization, Web-J2 Bernoulli, Jakob, 674 central force field, 794 Completeness Axiom, 528 Bernoulli, Jakob I, 611, 612 central forces, 794 completing the square, Web-I4 Bernoulli, Johann (John), 66, 162, 674, centripetal acceleration, 802 complex numbers, Web-F1–Web-F2 Web-B2 centripetal force, 802 compliance, Web-H11 Bernoulli, Johann (John) I, 611, 612 centroid(s), 389 component functions, 746 Bernoulli, Nikolaus, 612 geometric property, 387 components of r(t), 746 Bessel, Friedrich Wilhelm, 580 of a region, 387–389 components, of vectors, 682 Bessel equation of order one, 602 lamina, 963 composition of functions, A11, Web-C2 Bessel equation of order zero, 601 of the region R, 963 continuity, 44 Bessel functions, 580 solids, 963–965 derivatives, 118 derivative of, 593 chain rule, 118 u-substitutions for, 268, 269, 271 Beverton–Holt model, 530 alternative version, 119 compounding interest bifolium, 638 derivatives, 845–847, 850 annually, 238, 495 binomial coefficient, 590 functions of n variables, 849 continuously, 76, 495, 496 binomial series, 588, 589 generalized derivative formulas, 120 n times each year, 76 binormal vectors, 771 implicit differentiation, 850 quarterly, 496 formula summary, 778 implicit partial differentiation, semiannually, 496 bipartite cubics, Web-K12 851–852 compressible fluids, 1023 bisecond-degree equation, 656 partial derivatives, 848–849 computer algebra system(s) (CAS), 451, Bolzano, Bernhard, 95 proof, Web-L3 Web-K1 Bopp, Thomas, 669 two-variable, Web-L10 differentiation using, 122 boundary, 818 change of parameter, 762, 763 integration using, 272, 412, 450, 451 boundary point, 817 change of variables linear systems, 442 bounded functions, 291 double integrals, 952–954 Maple,Web-K1 bounded sets, 872 single integral, 948 Mathematica,Web-K1 finding absolute extrema, triple integrals, 955, 956 polynomial roots, Web-J5 876 chaos concave down, 174, 175 Bowditch curves, 617 and Newton’s Method, 237 concave up, 174, 175 Bowditch, Nathaniel, 617 circle(s), 639, Web-I2–Web-I4 concavity, 174, 193 Boyle’s law, Web-D3, Web-D10 of curvature, 777 conchoid of Nicomedes, 670 brachistochrone problem, 611 degenerate cases, Web-I4–Web-I5 concurrent forces, 687 Brahe, Tycho, 666 and discriminant, Web-O1 conditionally convergent, 558 branches, of hyperbola, 641 families, 624 cone, 726, 728, 733, 739 breakpoints, Web-B5 involute, 617 conic sections, 639–641 Index I-3

applications, 651 trigonometric functions, 51 cotangent, A3, Web-A3 degenerate, 639, Web-O2 vector-valued functions, 750 continuity, 51 focus–direction characterization, 661 of vector-valued functions, 751 derivative, 115 polar equation, 663 continuity equation, for incompressible hyperbolic, 399 quadratics, 656–660, Web-O2 fluids, 1036 Coulomb’s law, 973, Web-D10 reflection properties, 651 continuous compounding, 76, 496 CRC Standard Mathematical Tables and sketching in polar coordinates, continuous functions Formulae, 412, 445 663–665 average value, 315, 316 Crelle, Leopold, 234 translated, 648–650 properties, 43 critical point, 184, 874 conjugate axis, 641 contour maps, 807, 809 cross product, 701, 702 connected domain, 998 contour plot, 807 algebraic properties, 702–703 conservation of energy principle, 1003, using technology, 809 coordinate independent, 706 1048 convergence, 517, 527, 528, 531, 532 derivatives, 754 conservative fields, 974 absolute, 556, 557 geometric properties, 703–705 conservative vector fields, 974–997 and algebraic properties of series, 541 cross-product term, 658, 659 path independence and, 997–998 conditional, 558 cross section, 345, 349 test for, 999–1002 and eventual behavior of sequences, cubic polynomial, A19, Web-J1 Constant Difference Theorem, 243, 244 527 curl, 974–976 constant function, 14, 100, 172, A16, improper integral, 469–471 viewed as circulation, 1043 Web-D1 infinite series, 532 curl field, 1045 constant of integration, 259 monotone sequences, 527, 528 curvature, 773, 775–777 constant of proportionality, A18, radius/interval of, 576, 578 2-space interpretation, 777 Web-D3, Web-H11 sequences, 517 curve(s) inverse proportionality, A18, Web-D3 convergence set, 576 as asymptote, 30 constant polynomial, Web-J1 convergence tests integral, 263, 482 constant surfaces, 735–736 alternating series, 553–556 length of, 364–366, 610, 760, 761 constant term, polynomials, Web-J1 comparison test, 547, 548 level, 807–809 constant u-curves, 920, 924, 949 integral test, 542–544 limits along, 815, 816 constant v-curves, 920, 924, 949 limit comparison test, 548, 549 orthogonal, 134 constant(s) p-series, 543 parametric, 606, Web-M2 derivative, 100 ratio test, 549, 550 piecewise smooth, 992 factor, in a derivative, 102 root test, 551 polar, 619 factor, in a limit, 15 summarized, 560 position versus time, 226 and graph reflections, A13, Web-C7 coordinate axes, Web-H1 smooth, 364 velocity versus time, 307 and graph translations, A13, Web-C5 coordinate line, Web-F3 curve fitting, 881 integration formulas, 413 coordinate planes, 674 curve sketching, 192 second-order linear homogeneous coordinates, Web-H1–Web-H2 curved lamina, 1013 equations, Web-P2 Cartesian, 674, Web-H1 curvilinear asymptotes, 400 constrained absolute maximum converting between, 736–737 cusp, 197 (minimum), 885–887 cylindrical, 735 cycle, 324 constrained relative maximum polar, 618 cycloid, 610 (minimum), 885–887 spherical, 735, 739 apple of discord, 612 constrained-extremum principle Copernicus, 666 cycloids three variables and one constraint, corner points, 93 prolate, 615 887–888 Cornu, Marie Alfred, 672 role in mathematics history, 612 two variables and one constraint, Cornu spiral, 672, 803 cylinder, 349 885–887 correlation, Web-N1 cylindrical coordinates, 735 continuity, 40–44, 46, 47 correlation coefficient, Web-N2 converting, 736 applications, 41 cosecant, A3, Web-A3 equations of surfaces in, 738–739 compositions, 44 continuity, 51 cylindrical elements of volume, 939 and differentiability, 94, 839, 840 derivative, 115 cylindrical shells, 358, 360–362 exponential function, 329 hyperbolic, 399 cylindrical surfaces, 676–678 functions of three variables, 822 cosine, A3, Web-A3 cylindrical wedges, 938–939 functions of two variables, 819–820, continuity, 51 822 derivative of, 114, 117, 118 da Pisa, Leonardo (Fibonacci), 524 Intermediate-Value Theorem, 45–47 family, A21, Web-D6 damping constant, Web-P8 on an interval, 42 formulas, Web-A8 decay constant, 490 inverse functions, 45 hyperbolic, 399 interpreting, 491 from the left/right, 42 integrating powers of, 423, 424, 428 decibels, 71 logarithm function, 327, 331 integrating products with sine, 425, decimal representations, Web-F2 polynomials and rational functions, 426 decreases without bound, 8 43 rational functions of, 448 decreasing functions, 172 proof of basic property, trigonometric identities, A6, Web-A6 invertibility, A26, Web-E5 Web-L2–Web-L3 cost function, 219 decreasing sequence, 524 I-4 Index definite integrals functions of two or three variables, from velocity versus time curve, 307 defined, 286 837–839 displacement vector, 680 and Fundamental Theorem of geometric interpretation, 93 distance, 3-space, 675 Calculus, 294, 296, 297, 300, lack of, 93 distance formula, 675, Web-I1 301 vector-valued functions, 751 distance traveled integration by parts, 419 of vector-valued functions, 752 finding by integration, 306 properties of, 288, 290 differentiable motion along curves, 784 substitution, 319–321 on (a, b), 92 rectilinear motion, 307 degenerate conic sections, 639, Web-O2 everywhere, 92, 839 from velocity versus time curve, 307 degree, polynomials, A19, Web-J1 on an interval, 95 divergence, 974–976, 1031 degree measure, A1, Web-A1 on R, 839 absolute, 556 del ∇ operator, 976 at x0,92 as flux density, 1035 density differential equation(s), 481–482 improper integral, 469–471 circulation, 1044 autonomous, 503 of infinite series, 532 flux, 1035 Bernoulli, 513 of sequences, 517 lamina, 383, 1013 homogeneous, 513 divergence test, 539, 540, 560 Divergence Theorem, 1031–1033 of lamina, 959 initial-value problems, 264 finding flux, 1033–1035 mass, 394 and integration, 263 inverse-square fields, 1036, 1037 weight, 394 order of, 481 division density function, 959 separable, 487–490 / of functions, A11, Web-C1 dependent variable, A8, Web-B2 slope direction field for, 264, 498 solutions, 481 by zero, Web-F2 derivative function, 89–91 domain, A10, Web-B6 defined, 89, 90 differentials, 155, 157, 840, 841 error propagation, 155–156 effect of algebraic operations on, domain, 90 Web-B7 “prime” notation, 90 formulas, 157, 158 differentiation, 96 functions of two or more variables, slope producing, 90 806 derivative(s) computer algebra systems, 122 implicit, 131, 132, 848, 850 inverse functions, A24, Web-E2 compositions, 118 as inverse property of integration, 301 inverse trigonometric functions, 57 constant function, 100 logarithmic, 137–138, 142 natural, A10, Web-B6 constant multiple rule, 102 of power series, 591, 592 physical considerations in defined, 90, 96 and related rates, 146–150 applications, Web-B9 difference quotient, 90 rules, 112 restricted, A27, Web-E6 of dot and cross products, 754 techniques of, 100, 101, 103–105, restricting to make functions equation of the tangent line, 90 108–110, 112, 118 invertible, A27, Web-E6 exponential functions, 141 Diophantine analysis, 240 vector-valued functions, 746 inverse function, 141 dot products, 691 Diophantus, 213 algebraic properties, 691 inverse trigonometric functions, 143 direct proportionality, Web-H11 coordinate independent, 706 left-hand, 95 direction angles, 693–694 derivatives, 754 linear function, 91 direction cosines, 693 sign of, 693 logarithmic functions, 136 direction field, 264, 498 double integrals, 896 notation, 90, 96 direction of increasing parameter, 606, area calculated as, 906, 907 nth order, 105 Web-M2 change of variables, 952, 954, 955 one-sided, 95 in 3-space, 744 converting from rectangular to polar of position function, 92 directional derivatives, 855–858 coordinates, 915 power rule, 101, 138 directrix, 640, 661 evaluating, 896–899 of a product, 108, 109 discontinuities, 41, A18, Web-D3, of a quotient, 110 limits of integration, 903–906, 913 Web-H4 over nonrectangular regions, 902–904 of a reciprocal, 111 infinite, 41 right-hand, 95 in polar coordinates, 911, 912 and integrability, 291 properties of, 899 trigonometric functions, 113, jump, 41 115–117 in rectangular coordinates, 912 limits at, 49, 821 reversing order of integration, 906 of vector-valued functions, 752 physical significance, 41 double-angle formulas, A6, Web-A9 velocity as, 92 removable, 41, 49 doubling time, 491–492 Descartes René, 130, 213, 393, 611, = at x c,41 drag force, 508 Web-F3 discriminant, 659, Web-O1–Web-O2 dummy variables, 298 determinants, 700, 701 disease, first-order differential equations dx, 155 deterministic phenomena, Web-N1 modeling spread of, 484 dy, 155 difference quotient, 90 disk dy/dx, 96, 155 differentiability, 92–96 closed, 817 dyne-centimeters (erg), 376 and continuity, 94, 839 open, 817 dynes, 375 continuity of first-order partial displacement, 82 derivatives, 840 finding by integration, 302, 306 e, 136, 329, 330 functions defined implicitly, 133 motion along curves, 784 Earth Index I-5

Mercator projections, 429 eventually holding properties, 527 mixing problems, 506–507 spherical coordinates, 739 evolute, 780 modeling with, 490–493, 506–509 Earth-observing satellite, 117 existence theorem, 205 separable, 488–490 earthquake intensity, 71, 75 exp, 66 slope fields, 498 eccentricity, 661 exponential decay, 490, 492 first-order initial-value problem, 482 as measure of flatness, 662 exponential functions, 64 first-order model, pendulum period, 599 ecliptic, 795 analyzing growth with L’Hôpital’s first-order partial derivatives, 831 economics, optimization problems, 219, rule, 164, 165 flow fields, 1021 220 approximating with power series, 586 flow line, 979 electric intensity vector, 865 with base b,64 fluid, 392 electrostatic field, 973–974, 1021 derivatives, 141–142, 171 dynamics, 392 elementary functions, 331 equation solving, 68–70 force, vertical surface, 395–396 elements, of sets, Web-F4 graph, 65 homogeneous, 394 ellipses, 639, 640 integration formulas, 413 pressure, 394 area, 654, 1008 natural, 66 statics, 392 and discriminant, Web-O1 radioactive decay, 171 fluid flow, 971 eccentricity as measure of flatness, exponential growth, 71, 490, 492 steady state, 1023 662 extrema, 872 flux, 1023, 1024 focus–directrix property, 662 Extreme-Value Theorem, 205, 212, 873 density, 1035 polar equation, 663 extremum (extrema), 183, 205 of F across σ, 1024 reflection properties, 651 absolute, 204 finding with Divergence Theorem, sketching, 644, 645 relative, 183, 193 1033–1035 sketching in polar coordinates, extremum problem integrals, 1024 663–665 applied, 212, 214–220 outward, 1033–1035 standard equations, 643, 644 one-variable, 205 focal axis, 641 translated, 649 three-variable with one constraint, focus, 640, 661 ellipsoids, 726, 728 883, 887–889 ellipse, 640 identifying, 732 two variable with one constraint, hyperbola, 641 elliptic cones, 726, 728, 729 883–886 parabola, 640 identifying, 732 extrusion, 677 focus–directrix property, of conics, 661 elliptic paraboloids, 726, 728, 729 Folium of Descartes, 130, 132, 638 identifying, 732 Factor Theorem, Web-J4 Folsom points, 495 elliptical orbits, 666, 794, 797 factorial (n!), 520, 565 foot-pounds, 376 empty set, Web-F4 factors, polynomials, Web-J4 force(s) end behavior, 22 false gaps, graphing utilities, Web-K7 constant, 375–376 and limits, 23, 26 false line segments, graphing utilities, fluid, 396 Endpaper Integral Table, 445–449 Web-K8 line of action, 710 energy, 375 families pressure, 392–393 conservation of, 1003 of cardioids, 624, 625 resultant of two concurrent forces, kinetic, 375, 380 of circles, 624 687 potential, 1003 of curves, A16, Web-D1 variable, 376–378 relationship to work, 380 of functions, A16, Web-D1 force field equal vectors, 681 of limaçons, 624, 625 work performed by, 990 equiangular spirals, 638 of rose curves, 624 force vector, 680 equilateral hyperbola, 648, A17, of spirals, 625 four-cusped hypocycloid, 134, Web-C11 Web-D2 families of functions, A16, Web-D1 four-petal rose, 621, 624 equilibrium, 385 cosine, A21–A22, Web-D6–Web-D8 fractals, and Newton’s Method, 237 equipotential curves, 815 exponential, 64 free-fall model, 310 erg, 376 sine, A21–A22, Web-D6–Web-D8 integration, 310, 311 error Fermat’s last theorem, 213 with resistance, 508–509 absolute, 46, 501 Fermat’s principle, 225 free motion, Web-P8 in local linear approximations, 154 Fermat, Pierre de, 212, 611 Frenet, Jean Frédéric, 772 measurement, 156 Fibonacci sequence, 524 Frenet frame, 772 percentage, 157, 501 finite intervals, Web-F4 Frenet–Serret formulas, 780 propagated, 156 finite solid, 930 frequency relative, 157 first-degree equation alternating current, 324 error function [erf(x)], 332, 451 x and y,Web-H10 simple harmonic motion, 124, error propagation, 156 first derivative test, 185 Web-P5 escape speed, 797 first moment, of lamina, 962 sin x and cos x, A21, Web-D7 Euler, Leonhard, 66, 538, 884, Web-B1 first octant, 675 Fresnel cosine function, 332, 333 Euler’s equidimensional equation, first-order differential equations, 481, Fresnel sine function, 332, 333 Web-P7 482, 489, 504–509 Fresnel, Augustin, 332 Euler’s Method, 499–501 functions of two variables, 498 Fubini, Guido, 898 even function, A15, Web-C9 linear, 504 function(s), A8, Web-B1 I-6 Index

approximating with power series, 584, relationship between definite and properties of interest, 193 585, 587, 589 indefinite integrals, 297 quadratic equation, Web-I5–Web-I7 arithmetic operations on, A11, vector-form, 756 quadric surfaces, 726, 728–730 Web-C1 Fundamental Theorem of Work rational functions, 193, 199 bounded, 291 Integrals, 997 reflections, A13, Web-C6 compositions, A11, Web-C2–Web-C3 roots, A9, Web-B4 constant, A16, Web-D1 g, acceleration due to gravity, 56 sequence, 516 decreasing, 172, 173, A26, Web-E5 Gabriel’s horn, 475, 922 stretches and compressions, A13, defined by integrals, 331, 333 Galileo, 611 Web-C7 defined by power series, 579, 580 Galileo, 669 symmetry, A14, Web-C7 defined explicitly and implicitly, Galois, Evariste, Web-J5 in 3-space, 675 129–130 Gamma function, 479 translations, A13, Web-C5 described by tables, 806, 807 gases, 392 vector-valued function of two differentiable, 92 Gateway Arch, 408 variables, 923 domain, A10, Web-B6 Gauss, Carl Friedrich, 287, 1032 vector-valued functions, 747 elementary, 331 Gaussian curve, 1032 vertical tangents and cusps, 197 x-intercepts, A9, Web-B4 even, A17, Web-D2 Gauss’s law zeros, A9, Web-B4 even and odd, A15, Web-C9 for electric fields, 1037 gravitational field, 971 expressing as composition, A13, for inverse-square fields, 1036 great circle, 743 Web-C4 Gauss’s Theorem, 1031–1033 great circle distance, 743 graphing utilities, Web-K1 general equation circle, Web-I5 Great Pyramid at Giza, 171 graphs of, A8, Web-B3 greatest integer function, Web-C12 homogeneous of degree n, 893 lines, Web-H10 general form, equation of plane, 718 greatest lower bound, 529 increase without bound, 8, 71 Green, George, 1007 increasing, 172, 173, A26, Web-E5 general logarithms, 331 general solution, 482 Green’s Theorem, 1005–1007 interpolating, Web-N1 general term, 514, 540 finding area using, 1008 inverse, A23, Web-E1 geometric curves, Web-H2 finding work using, 1007 invertible, A25, Web-E4 geometric mean, 664, 665 multiply connected regions, logarithmic, 67 geometric series, 533, 535 1009–1010 natural domain, A10, Web-B6 geosynchronous orbit, 799 and Stokes’ Theorem, 1043 odd, A15, Web-C9 gradient fields, 973 Gregory, James, 587 one-to-one, A25, Web-E4 gradients, 858–861 Gregory of St. Vincent, 325 partial derivatives, 825 to find tangent lines to intersections of grid lines, Web-K2 piecewise-defined, Web-B5 surfaces, 869 growth constant, 490 range, A10, Web-B6 graphing using calculus and technology interpreting, 491 relative maxima and minima, 183 together, 200 growth rate, population, 126 restriction, A27, Web-E6 graphing utilities Hale, Alan, 669 smooth, 364, 759 aspect ratio distortion, Web-K6 of three variables, 806 Hale–Bopp comet, 669 contour plots, 809 half-angle formulas, A6, Web-A9 of two variables, 498, 806 errors of omission, Web-K8 vertical line test, A9, Web-B4 half-closed intervals, Web-F4 false gaps and line segments, Web-K7 half-life, 492 ways of describing, Web-B1 functions of two variables, 810, 812 functions of n real variables, 806 half-open intervals, Web-F4 inverse function graphing, Web-K9 Halley’s comet, 667 functions of three variables nonelementary functions, 336 hanging cables, 400, 671 continuity, 822 parametric curve generation, 745, harmonic motion, simple, 124, 487 gradient, 858 Web-K9, Web-M3 harmonic series, 536 level surfaces, 810 polar curves, 626 heat equation, 836 multiple integrals, 894, 930–933 rational functions, 199 height partial derivatives, 830 resolution, Web-K7 cylindrical wedges, 939 functions of two variables sampling error, Web-K6 right cylinder, 349 continuity, 819–820, 822 and true shape of graph, Web-K8 helicoid, 929 contour maps, 808, 809 graphs, Web-H2 Hippasus of Metapontum, Web-F1 gradient, 858 conic sections in polar coordinates, homogeneous differential equation, 513 graphing utilities, 810, 812 663–665 homogeneous functions, 893 graphs, 807 exponential functions, 65 homogeneous lamina, 383, 959 maxima and minima, 872–879 functions, A8, Web-B3 Hooke, Robert, 377, Web-H11 multiple integrals, 894–896, 898–900, functions of two variables, 807 Hooke’s law, 377, Web-H11 902–904 hyperbolic functions, 399, 400 horizontal asymptotes, 22, A21, Web-D6 partial derivatives, 825, 827, 828, 830 inverse functions, 45, A26, Web-E6 polar curves, 630 Fundamental Theorem of Calculus, 294, logarithmic functions, 67 horizontal line test, A26, Web-E4 296, 300, 301 parametric curves in 3-space, 744 horizontal range, 793 dummy variables, 298 polar coordinates, 619, 621 Hubble, Edwin P., Web-N8 and integrating rates of change, 301 power functions, A17, Web-D2 Hubble’s constant, Web-N8 Index I-7

Hubble’s law, Web-N8 and Fundamental Theorem of inflection points, 175, 177, 178, 193 Humason, Milton L., Web-N8 Calculus, 296, 297 applications, 178 hyperbolas, 639, 641 properties, 261, 262 inhibited population growth, 483–484 and discriminant, Web-O1 independence of path, 997 initial conditions, 264, 482, Web-P4 equilateral, 648, A17, Web-D2 independent variable, A8, Web-B2 initial point, vectors, 680 polar equation, 663 indeterminate form, 18, 161, 163–166 initial side, of angles, A1, Web-A1 reflection properties, 651 of type ∞ / ∞, 163 initial-value problems, 264 sketching, 647, 648 of type ∞ − ∞, 166 Euler’s Method for, 499–501 sketching in polar coordinates, of type 0/0, 18, 161 first-order differential equations, 482 ∞ 663–665 of type 00, ∞0,1 , 166 second-order linear homogeneous standard equation, 646, 647 of type 0 · ∞, 165 differential equations, Web-P4 translated, 649 index instantaneous acceleration, 227 hyperbolic cosecant, 399 for sequence, 515 motion along curves, 782 hyperbolic cosine, 399 of summation, 275 instantaneous rate of change, 1, 84 hyperbolic cotangent, 399 inequalities instantaneous speed hyperbolic functions, 398, 399 with absolute values, Web-G3 motion along curves, 782 applications arising in, 400 algebraic properties, Web-F5 instantaneous velocity derivative formulas, 401 solving, Web-F6 definition, 83 graphs, 399, 400 triangle, Web-G4 geometric interpretation, 83 integration formulas, 402, 413 infinite discontinuities, improper motion along curves, 782 inverse, 403–405 integrands, 471 rectilinear motion, 92 and unit hyperbola, 401 infinite discontinuity, 41 instantaneous velocity function, 92 hyperbolic identities, 400, 401 infinite intervals, Web-F4 integer powers, 63 hyperbolic navigation systems, 652 absolute extrema on, 206, 207 integers, Web-F1 hyperbolic paraboloids, 726, 728, 729 integrals over, 468–470 integrability and discontinuities, 290 identifying, 732 infinite limits, 7, 35, 36 integrable functions, 286 hyperbolic secant, 399 informal view, 8 integrable on [a, b], 286 hyperbolic sine, 399 infinite sequence, 514 integral(s) hyperbolic spiral, 625, 630 infinite series, 514, 531–533, 535, 536 complete elliptic, of the first kind, hyperbolic substitution, 436 absolute convergence, 556, 557 479, 598 hyperbolic tangent, 399 algebraic properties of, 541, 542 curves, 263, 482 hyperboloid, 739 alternating harmonic series, 554 double, 896, 899 hyperboloid of one sheet, 726, 728 alternating series, 553–556 functions defined by, 331, 332 identifying, 732 binomial series, 588, 589 iterated, 897, 902, 903 hyperboloid of two sheets, 726, 728 comparison test for, 547, 548, 560 Mean-Value Theorem for, 299, 315 identifying, 732 conditional convergence of, 558 notation, 259 hyperharmonic series, 543 convergence/divergence of, 532 partial definite, with respect to x and convergence tests for, 539–544, y, 896 ideal gas law, 824, 883 547–549, 551, 553, 555–558, relationship between definite and ill posed problems, 218 560 indefinite, 296, 297 image of S under T, 949 divergence test for, 540, 560 Riemann, 287 image of x under f , A8, Web-B2 general term of, 540 sign ( ), 259, 260 implicit differentiation, 130, 132, 133, geometric series, 533–535 single, 899 850, 851 harmonic series, 536 surface, 1013 chain rule, 850 hyperharmonic series, 543 tables, 413, 445–448, 450 implicit functions, 129 integral test for, 542–544 triple, 930 implicit partial differentiation, 828, 830 limit comparison test for, 548, 549, of vector-valued functions, 755, 757 chain rule, 851–852 560, Web-L6 integral test, 542–544, 560 improper integrals, 467, 469–473 Maclaurin series, 574–577, 583, 584, integrand, 259 arc length and surface area, 473 586–589, 594, 596–599 integrating factor, 505 integrands with infinite nth partial sum, 532 integration, 259 discontinuities, 471 power series, 575–580 constant of, 259 over infinite intervals, 468–470 p-series, 543 and differential equations, 263 improper rational functions, integrating, ratio test for, 549, 550 and direction fields, 265 442, 443 root test for, 551 hyperbolic substitution, 436 incompressible fluids, 1023 roundoff error with, 585 as inverse property of differentiation, continuity equation, 1036 sums of, 531–533 301 increases without bound, 8, 71 Taylor series, 573–577, 583, 584, 594, iterated (or repeated), 897, 902, 903 increasing functions, 172 595, 597, 598 limits of, 286 invertibility, A26, Web-E5 terms of, 531 notation for, 259, 260 increasing sequence, 524 truncation error with, 585 partial, 896 increment, 96, 838 infinite slope, 608, Web-H5 of power series, 593, 594 Euler’s method, 500 infinity, limits at, 22–27 of rates of change, 302 indefinite integrals, 259, 260 inflection point at x0, 175 reversing the order of, 906 I-8 Index

and slope fields, 264 irrotational vector field, 1044 least upper bound, 529 u-substitution, 268, 270, 271 isobar, 814 left-endpoint approximation, 455 integration by parts, 415, 417 isothermal curves, 815 left-hand derivatives, 95 definite integrals, 419 iterated integrals, 897, 902, 903 left-handed coordinate systems, 674 LIATE, 416, 417 with nonconstant limits of integration, Leibniz, Gottfried Wilhelm, xxi, 1, 162, reduction formulas, 420 902 256, 611, 612 repeated, 417 reversing order of integration, 906 derivative notation, 96 tabular, 418 integral notation, 96, 259 integration formulas, 260, 413 Jacobi, Carl Gustav Jacob, 949 lemniscate, 134, 623 inverse trigonometric functions, 272, Jacobian, 948 length 413 three-variable, 955 arc, 365, 366, A2, Web-A2 vector-value functions, 756 in two variables, 950, 951 of a vector, 684 integration methods, 412, 413 joule, 376 level curves, 807, 809 intensity, of sound, 70 jump discontinuity, 41 gradients normal to, 860–861 intercepts, Web-H4 level surfaces, 810 interior, 818 Kepler, Johannes, 666 normal, 867 interior point, 817 Kepler’s equation, 250 normal line to, 867 Intermediate-Value Theorem, 46 Kepler’s laws, 250, 666, 794 tangent plane, 866–867 International Space Station, 378 first law, 666, 794, 797 lever arm, 385 International System of Units (SI), 375 second law, 666, 794, 797 LIATE stategy for integration by parts, interpolating function, Web-N1 third law, 666, 794, 798 416 intersection, intervals, Web-F5 kinetic energy, 375, 380 Libby, W. F., 492 interval of convergence, 576 Kramer, Gerhard (Mercator), 429 limaçons, 624, 625 finding, 576, 577 limit comparison test, 548, 549, 560 power series in x − x0, 577, 578 L’Hôpital, Guillaume Francois Antoine proof, Web-L6 intervals, Web-F4 De, 162, 611, 612 limit(s), 1, 3, 5–9, 11, 13–19, 22–24, intrinsic error, 461 L’Hôpital’s rule, 161–167 26–29, 31–36, 74, 750, inverse cosine function, 57 Lagrange, Joseph Louis, 805, 884 817–819, 822 inverse function(s), 67, A23, Web-E1 Lagrange multiplier, 885 area defined as, 277, 278 cancellation equations, A24, Web-E2 Lagrange error bound, 570 computing, 13, 18, 23, 26 derivative, 141 Lagrange multipliers method, 884–886 along curves, 815–816 continuity, 45 three variables and one constraint, decimals and, 3 domain, A24, Web-E2 887–889 at discontinuities, 49, 821 domain and range, A24, Web-E2 Lagrange’s identity, 709 does not exist, 6, 7 existence, A25, Web-E4 Lamé’s special quartic, 134 end behavior, 22 finding, A24, Web-E3 lamina, 383 epsilon-delta definition, 32 graphing utilities, Web-M4 center of gravity of a, 384–387, 962, fail to exist, 5–7, 27, 53 graphs, A26, Web-E6 963 functions of three variables, 822 horizontal line test, A26, Web-E4 centroid of, 387–389, 963, 965 functions of two variables, 818–819 notation, A23, Web-E1 curved, 1013 horizontal asymptotes, 22 parametric equations, 607 density of, 383, 959, 1013 indeterminate forms, 161–164, 166 range, A24, Web-E2 first moment of a, 386, 962 indeterminate forms of type 0/0, 18 inverse hyperbolic functions, 403–404 homogeneous/inhomogeneous, 383, infinite, 7, 35, 36 derivatives and integrals, 405–406 959 at infinity, 22–23 logarithmic forms, 404, 405 mass, 383, 960, 1013–1014, 1018 informal view of, 4 inverse proportionality, A18, Web-D3 moment(s) of, 385–387, 962 intuitive approach, 1–9 inverse secant function, 57 Laplace, Pierre-Simon de, 212, 894, 977 nth root, 15 inverse sine function, 57 Laplace transform, 476 one-sided, 5, 6, 32 inverse tangent function, 57 Laplace’s equation, 836, 977 piecewise-defined functions, 19 inverse transformation, 949 polar form, 854 polynomials as x→±∞,24 inverse trigonometric functions, 56, 143 Laplacian ∇2 operator, 977 polynomials as x→a, 15, 17 derivatives, 143 latitude, 921 product, 15 evaluating, 58, 59 Law of Areas, 666, 794, 797 proof of basic theorems, Web-L2 identities, 59, 60 Law of Cosines, 61, Web-A8 quotient, 15 inverse-square fields, 973 for a tetrahedron, 743 radicals, 18 Gauss’s law, 1036 Law of Orbits, 666, 794, 797 rational functions as x→±∞,24 invertible functions, A25, Web-E4 Law of Periods, 666, 794, 798 rational functions as x→a, 15, 17 involute of a circle, 617 Law of Sines, Web-A13 of Riemann sums, 287, 896, 931 inward orientation, 1030 Law of Universal Gravitation, 795, 971, rigorous discussion, 31–33, 35, 36 inward unit normal, 771 972, Web-D1–Web-D10 sampling pitfalls, 5 irrational exponents, 63, 330 leading coefficient (term), Web-J1 sequences, 516, 517, 519 derivatives, 138 least squares line of best fit, 882, simple functions, 13 irrational numbers, Web-F1 Web-N2 Squeezing Theorem, 52 irreducible quadratic factors, Web-J2 least squares principle, 225 sum, 15 Index I-9

of summations, 279 local quadratic approximations, 563 mass density, 394 trigonometric functions, 51 locally linear, 153 mass density function, 960 two-sided, 5–6, 31–32 logarithmic differentiation, 137, 142 mass of a wire, 980 of vector-valued functions, 750 logarithmic function(s), 67, 68 as a line integral, 984 vector-valued functions, 750 base b,67 Mathematica,Web-K1 x n as x→±∞,24 derivatives, 135–136 mathematical models, 881, Web-N1 x→±∞, 34, 35 equation solving, 68–70 linear functions as, Web-N1–Web-N4 limits of integration, 286 integral, 416 quadratic and trigonometric functions line integrals, 980, 982, 983, 990, 992 properties, 68 as, Web-N4–Web-N6 area as, 981, 982, 984, 1007 properties of exponential functions maximum along closed paths, 998 compared, 68 absolute, 204, 872 evaluating, 982–985, 987–988, 997 logarithmic growth, 71 relative, 183, 872 of F along C, 988–989 logarithmic scales, 70, 71 mean value, 316, 901, 910 Fundamental Theorem of, 997 logarithmic spiral, 626, 638 Mean-Value Theorem, 241–243 geometric interpretation, 989, 990 logarithms, 67 consequences, 243 Green’s Theorem, 1005 approximating with power series, 587 proof, 241 independence of path, 997 change of base formula, 70 velocity interpretation, 242 mass of a wire, 980, 984 properties, 68 Mean-Value Theorem for Integrals, 299, orientation of C, 983, 986 logistic 315 along piecewise smooth curves, 992 curve, 178, 496 measurement error, 156 with respect to arc length, 980–981 differential equation, 484 mechanic’s rule, 238 with respect to x, y,andz, 985, 988 growth, 178, 179, 183, 484 members, of sets, Web-F4 around simple closed curves, 1007 longitude, 921 Mercator (Gerhard Kramer), 429 in 3-space, 984 lower bound Mercator projection, 429 work as, 990, 991 greatest, 528 mesh lines, 725 line of action, 710 monotone sequence, 528 mesh size, of partition, 286 line segments, 713 sets, 528 method vector form, 748 lower limit of integration, 286 of cylindrical shells, 359–362 linear algebra, 442, 701 lower limit of summation, 275 of disks, 352 linear combination, 690 of integrating factors, 505–506 linear depreciation schedule, Web-D9 m, slope of line, Web-H5 of least squares, 882 linear differential equation, 504, Web-P1 Machin’s formula, 591 Newton’s, 234, 236 linear equation, Web-H10 Maclaurin, Colin, 564 of washers, 353 linear factor rule, 438 Maclaurin polynomials, 564–566 midpoint approximation, 455, 461, 464 linear factors, 438–440 sigma notation, 567, 569 error estimate, 461–463 linear functions, A19, Web-D5 Maclaurin series, 574–577, 583, 584, errors, 456 mathematical models, 586–589, 594, 596–599 midpoint formula, Web-I2 Web-N1–Web-N4 binomial series, 588, 589 Miller, Norman, 224 practical problems, Web-H10 differentiation, 591, 592 millibars, 814 linear interpolation, 807 exponential functions, 586 minimax point, 730 linear polynomial, A19, Web-D5 integrating, 593, 594 minimum linearly dependent functions, Web-P1 logarithm approximation, 587 absolute, 204, 872 linearly independent functions, Web-P1 practical ways to find, 596–598 relative, 183, 872 lines, Web-H5–Web-H10 trigonometric function minor axis, 641 angle of inclination, A7, Web-A10 approximation, 584, 586 minutes (angle), A1, Web-A1 degenerate conics, 639, Web-O2 various functions, 589 mixed second-order partial derivatives, determined by point and slope, Magellan spacecraft, Web-N3 831 Web-H8 magnitude, A10, Web-B4, Web-G1 equality, 831 determined by point and vector, of vector, 684 mixing problems, first-order differential 710–711 major axis, 640 equations, 506–507 determined by slope and y-intercept, Mantle, Mickey, 671 Möbius, August, 1022 Web-H8–Web-H9 manufacturing cost, 219 Möbius strip, 1020, 1022 families of, 624 Maple,Web-K1 modeling with differential equations, parallel to the coordinate axes, mapping, transformations, 949 481, 483–485, 490–493, Web-H7 marginal analysis, 220 506–509 vector equations of, 713 marginal cost, 220 differential equations, 481 liquids, 392 marginal productivity of capital, 892 doubling time, 491, 492 Lissajous curve, 615 marginal productivity of labor, 892 exponential growth/decay, 490–492 Lissajous, Jules Antoine, 615 marginal profit, 220 half-life, 492 Lituus spiral, 625 marginal revenue, 220 Newton’s Law of Cooling, 485 local linear approximation(s), 153, 564, Mars, 669 pharmacology, 484 842, 843 mass population growth, 483–484 local linearity, 153, 841, 843 of curved lamina, 1013–1014 spread of disease, 484 differentials, 155, 157 of a lamina, 383, 960 vibration of springs, 485 I-10 Index moment Newton’s Law of Universal Gravitation, optimization problems, 172 lamina, 385–387 795, 971, 972, Web-B1, absolute maxima and minima, about a line, 385 Web-D10 205–208 about a point, 384 Newton’s Method, 235, 236 applied maximum and minimum moment of inertia, 967 difficulties with, 236 problems, 212–221 moment, of lamina, 962 nonnegative number, Web-F5 categories, 212 monkey saddle, 725 nonorientable surfaces, 1022 economics applied, 219, 220 monotone sequences, 525 nonparametric surfaces, orientation of, five-step procedure for solving, 214 convergence, 527, 528 1026–1027 ill posed, 218 monotonicity testing, 525, 526 norm, of vector, 684 involving finite closed intervals, properties that hold eventually, 527 normal 212–217 Moon to level surface, 867 involving intervals not both finite and orbit eccentricity, 800 normal line closed, 217–219 motion to level surface, 867 Lagrange multipliers, 884, 885, along a curve, 781–789 normal plane, 772 887–889 constant acceleration, 308 normal scalar component of maxima and minima of functions of free-fall, 310, 311, 508, 509 acceleration, 785 two variables, 872–876, 878, 879 rectilinear, 226–229, 306, 308, 309 normal vector, 867 order rotational in 3-space, 707 normal vector component of of the derivative, 105 simple harmonic, 124, 487 acceleration, 785 differential equations, 481–482 m ,80 normalizing vectors, 685 tan ordinate, Web-H1 multiplication, of functions, A11, nth Maclaurin polynomial, 565, 570, 573 nth order derivative, 105 Oresme, Nicole, 539 Web-C1 orientation, 1022 multiplicity, 187, Web-J2 nth partial sum, 532 nth remainder, 570, 582 of a curve, 744 geometric implications of, 187 nonparametric surfaces, 1026–1027 multiply connected domain, 999 estimating, 583, 584 nth Taylor polynomial, 568, 570, 574 piecewise smooth closed surfaces, null set, Web-F4 1030 natural domain, A10, Web-B6 / numbers positive negative, 1009, 1023 functions of two or more variables, complex, Web-F1 relative, 1039 806 integers, Web-F1 smooth parametric surface, 1022 vector-valued function, 746 natural, Web-F1 in 3-space, 744 natural exponential function, 66 rational, Web-F1 orientation, of a curve, 606, Web-M2 formal definition, 329 oriented surfaces, 1022, 1026–1027 real, Web-F1 natural logarithm, 67 origin, 618, 674, Web-F3, Web-H1 numerical analysis, 461, 586 approximating with Maclaurin series, symmetry about, 621, 622 numerical integration, 454–463 orthogonal 587 absolute error, 456, 459 curves, 134 derivative, 326 error, 456, 459, 461 trajectories, 134 integral, 416 midpoint approximation, 455 orthogonal components, 694–695 properties, 327 Riemann sum approximation, 454 orthogonal projections, 696 natural numbers, Web-F1 Simpson’s rule, 458–461 orthogonal surfaces, 871 n-dimensional, 806 tangent line approximation, 457 osculating circle, 78, 777 negative, of a vector, 681 trapezoidal approximation, 455 negative angles, A1, Web-A1 osculating plane, 772 outer boundary, simple polar regions, negative changes of parameter, 763 oblate spheroid, 734 911 negative direction, 1023, 1039, Web-F3 oblique asymptote, 196 output, of function, A8, Web-B2 arc length parametrization, 761 octants, 675 outside function, A12, Web-C4 negative orientation odd function, A15, Web-C9 nonparametric surfaces, 1026 outward flux, 1033–1035 Ohm’s law, 39 outward flux density, 1035 parametric surface, 1023 On a Method for the Evaluation of net signed area, 282, 283 outward orientation, 1030 Maxima and Minima, 212 overhead, 219 net signed volume, 896 one-sided derivatives, 95 Newton’s Law of Cooling, 99, 316, 485 one-sided limits, 5, 32 Pappus, Theorem of, 389–390 Newton’s Second Law of Motion, 380 one-third rule, 461 Pappus of Alexandria, 390 Newton, Isaac, xx, 1, 66, 256, 611, 612, one-to-one functions, 140–141, A25, parabolas, 639, Web-I5 666, Web-B2 Web-E4 defined, 640 derivative notation, 96 one-dimensional wave equation, 832 and discriminant, Web-O1 Maclaurin disciple of, 564 1-space, 674 focus–directrix property, 662 solution of the brachistochrone one-to-one transformations, 949 Kepler’s method for constructing, 672 problem in his own handwriting, open ball, 817 polar equation, 663 612 open disk, 817 reflection properties, 651 Newtonian kinetic energy, 604 open form, sigma notation, 277 semicubical, 609 newton-meters, 376 open interval, Web-F4 sketching, 642, 643 newtons, 375 open sets, 817 sketching in polar coordinates, 663 Index I-11

standard equations, 641, 642 linear factors, 438–440 points translated, 648 quadratic factors, 440–442 distance between two in plane, Web-I1 parabolic antenna, 651 partial integration, 896 point-slope form, Web-H8 parabolic mirror, 651 partial sums, 532 polar angle, 618 parabolic spiral, 625, 629 partition polar axis, 618 paraboloid, 726, 728, 729, 739 of the interval [a, b], 286 polar coordinates, 618 parallel lines, Web-H6 regular, 287 area in, 630, 634, 635 parallel vectors, 681 Pascal, Blaise, 393 graphs, 619 parameter(s), 605, A16, Web-D1 pascal (Pa), 393 relationship to rectangular, 618 arc length as, 761 Pascal’s Principle, 398 sketching conic sections, 663–665 change of, 762, 763 path independence, of work integral, symmetry tests, 622, 623 parametric curves, 606, 744, Web-M1 996, 997 polar curves arc length, 369, 610 path of integration, 996 arc length, 632 change of parameter, 762 path of steepest ascent, 864 area bounded by, 630, 634, 635 closed, 998 peak voltage, 324 conic sections, 663 generating with graphing utilities, pendulum, Taylor series modeling, 598 generating with graphing utilities, 626 Web-K9 percentage error, 157 intersections, 636 limits along, 815, 816 Euler’s Method, 501 tangent lines, 630–631 line integrals along, 982–983 perigee, 666, 799 tangent lines at origin, 631 orientation, 606, 744, Web-M2 artificial Earth satellite, 151 polar double integrals, 911, 912 piecewise-defined, 614, Web-M6 perihelion, 666, 799 evaluating, 912–914 scaling, Web-K10, Web-M5 period, 160 finding areas using, 915 simple, 999 alternating current, 324 polar form tangent lines, 608–610 first-order model of pendulum, 599 of Cauchy–Riemann equations, 854 3-space, 744, 745 pendulum, 160 of Laplace’s equations, 854 translation, Web-K10, Web-M4 simple harmonic motion, 124 polar rectangle, 911 parametric equations, 605, 606, Web-M1 simple pendulum, 479 polar Riemann sums, 912 expressing inverse functions sin x and cos x, A21, Web-D7 pole, 618 parametrically, Web-K9 periodicity, 193 polygonal path, 365 expressing ordinary functions permittivity constant, 973 polynomial in x, A19, Web-D4 parametrically, 607, Web-M3 perpendicular lines, Web-H6 polynomial of degree n,Web-J1 graphing utilities, 626, 745 pH scale, 71, 74 polynomials, Web-J1 intersections of surfaces, 745 phenomena coefficients, Web-J1 of lines, 711, 713 deterministic, Web-N1 continuity, 43 orientation, 606, 744, Web-M2 probabilistic, Web-N1 degree, Web-D5, Web-J1 projectile motion, 788 physical laws, Taylor series modeling, Factor Theorem, Web-J4 of a tangent line, 753 598 geometric implication of multiplicity parametric surfaces, 920 π of a root, 187 orientation, 1022 approximating, 254, 255, 587 graphing, 187–189 of revolution, 922 famous approximations, Web-F9 limits as x→±∞,24 surface area, 920, 925–927 Piazzi, Giuseppi, 1032 limits as x→a, 15, 17 tangent planes, 923, 925 piecewise-defined functions, Web-B5 Maclaurin, 564–566 partial definite integrals, 896 limits, 19 method for finding roots, Web-J5 partial derivative sign, 826 piecewise smooth closed surfaces, 1030 properties, 188, 193 partial derivatives piecewise smooth functions, line quick review, A19, Web-D5 chain rule, 848–849 integrals along, 992 Remainder Theorem, Web-J3 and continuity, 829 pixels, Web-K6 roots, 234 estimating from tabular data, 827–828 planes Taylor, 567, 568 functions, 825 angle between, 719, 720 population growth, 483–484 functions of two variables, 824, 825, determined by a point and a normal carrying capacity, 30, 483 827, 828, 830 vector, 717–718 first-order differential equations, 483 functions with more than two distance between two points in, inhibited, 483–484 variables, 830 Web-I1 rate, 126 higher-order, 830–831 distance problem, 720–721 the logistic model, 484 mixed, 831 parallel to the coordinate planes, 717 uninhibited, 483 notation, 826, 829 perpendicular, 693 position as rates of change and slopes, transformation, 948–950 finding by integration, 306 826–827, 829 planetary orbits, 250, 617, 665, 666, position function, 82, 226, 781 of vector-valued functions, 923 794–798 derivative of, 92 partial differential equation, 832 plot, Web-H2 position vector, 747 partial differentiation, implicit, 828–830 Pluto, 669 position versus time curve, 226 partial fraction(s), 437, 438 point-normal form analyzing, 228 decomposition, 438 of a line, 724 positive angles, A1, Web-A1 improper rational functions, 442, 443 of a plane, 717 positive changes of parameter, 763 I-12 Index positive direction, 1023, 1039, Web-F3 quadric surfaces, 726 right-handed, 674 arc length parametrization, 761 graphing, 726, 728–730 rectangular coordinates, 674–675 positive number, Web-F5 identifying, 732 converting cylindrical and spherical, positive orientation reflections in 3-space, 731 736 multiply connected regions, 1009 translations, 730 relationship to polar, 618 nonparametric surfaces, 1026–1027 quartic polynomials, A19, Web-D5 rectifying plane, 772 parametric surface, 1023 quintic polynomials, A19, Web-D5 rectilinear motion, 82, 226–229, potential energy, 1003 quotient rule, 110 306–310 potential function, 974 acceleration, 227 pounds, 375 r, polar coordinate, 618, 619 average velocity, 82 power functions, A16, Web-D1 radial coordinate, 618 constant acceleration, 308, 309 fractional and irrational exponents, radial unit vector, 803 distance traveled, 307 63, 330 radian measure, A1, Web-A1 free-fall, 311 noninteger exponents, A18, Web-D4 radians, A1, Web-A1 instantaneous speed, 226 power rule, 101, 138 radicals, limits involving, 18 instantaneous velocity, 92 power series, 575, 578 radioactive decay, 171, 492 position function, 82 convergence of, 576, 578 radius, A2, Web-A2 speed, 226, 227 differentiation, 591, 592 circles, Web-I3 velocity, 92, 226, 227 exponential function approximation, radius of convergence, 576, 578 recursion formulas, 521 586 radius of curvature, 777 recursively defined sequences, 521 functions defined by, 579, 580, 589 radius vector, 747, 923 reduction formulas, 420 integrating, 593, 594 Radon-222, 494 integral tables, matches requiring, 447 interval of convergence, 576, 577 Ramanujan, Srinivasa, 591 integrating powers and products of logarithm approximation, 587 Ramanujan’s formula, 591 trigonometric functions, π approximation, 587 range, A10, Web-B6 423–425 radius of curvature, 777 horizontal, 793 reference point, arc length and Taylor series, 594 inverse functions, A24, Web-E2 parametrization, 761 trigonometric function physical considerations in reflections, A13, Web-C6 approximation, 584, 585 applications, Web-B8 of surfaces in 3-space, 731 pressure, 392–393 rate(s) of change, 83, 86 region, 389 principal unit normal vector, 769, 924 applications, 86 regression probabilistic phenomena, Web-N1 average, 84 line, 882, Web-N2 product, of functions, A11, Web-C1 differential equations, 481 quadratic, Web-N4 product rule, 109, 415, 417 instantaneous, 84 regular partition, 286 production model, 892 integrating, 302 related rates, 146–150 product-to-sum formulas, A7, Web-A9 partial derivatives, 826–827, 829 strategy for solving, 147 profit function, 219 related rate, 146–150 relative decay rate, 491 projectile motion ratio, geometric series, 533 relative error, 157 parametric equations of, 788, 790 ratio test, 549, 550, 560 relative extrema, 183, 193, 872 vector model, 787, 788 for absolute convergence, 558, 560 and critical points, 184 projective geometry, 393 proof, Web-L7 finding, 873–876 prolate cycloid, 615 rational functions, A20, Web-D5 first derivative test, 185 propagated error, 156 continuity, 43 second derivative test, 186 proper rational function, 438 graphing, 193, 199 second partials test, 875–876 p-series, 543 integrating by partial functions, relative growth rate, 491 pseudosphere, 929 437–443 relative maxima, 872 limits as x→±∞,24 relative maximum, 183 quadrants, Web-H2 limits as x→a, 15, 17 relative minima, 872 quadratic approximations, local, 563 proper, 438 relative minimum, 183 quadratic equation(s) properties of interest, 193 relativistic kinetic energy, 604 discriminant, Web-O1 of sin x and cos x, 448 relativity, theory of, 12, 29 eliminating cross-product terms, 658, rational numbers, Web-F1 Remainder Estimation Theorem, 570, 659 rays, 624 571, 583, 585 in x, Web-I5 real number line, Web-F3 proof, Web-L8 in x and y, 649, 656 real numbers, 528, Web-F1 Remainder Theorem, Web-J2 in y, Web-I7 Completeness Axiom, 528 removable discontinuity, 41, 49 quadratic factor rule, 440 decimal representation, Web-F2 repeated integration, 897 quadratic factors, 440–442 real-valued function of a real variable, repeated integration by parts, 417 quadratic formula, Web-F2 A8, Web-B3 repeating decimals, Web-F2 quadratic mathematical model, Web-N4 rectangle method for area, 254 represented by power series, 579 quadratic polynomial, A19, Web-D5, rectangular coordinate systems, 674, residuals, 882, Web-N2 Web-J1 675, Web-H1 resistance thermometer, Web-H14 quadratic regression, Web-N4 angles, A3, Web-A3 resistivity, Web-N7 quadratrix of Hippias, 169 left-handed, 674 resolution, in graphing utilities, Web-K7 Index I-13 restriction of a function, A27, Web-E6 continuity, 51 Taylor and Maclaurin polynomials, resultant, 687 derivative, 115 567, 569 revenue function, 219 hyperbolic, 399 simple harmonic model, 124 revolution integrating powers of, 426, 427 simple harmonic motion, 487 solids of, 351 integrating products with tangent, simple parametric curve, 999 surfaces of, 370, 738 427, 428 simple pendulum, 479 Rhind Papyrus, Web-F9 second derivative, 105 simple polar regions, 910 Richter scale, 71, 75 second derivative test, 186 simple root, 187, Web-J2 Riemann, Bernhard, 287 second partials test, for relative extrema, simple xy-solid, 932 Riemann integral, 287 875–876 simple xz-solid, 935 Riemann sum approximations, 454 second-degree equation, 656, 726 simple yz-solid, 935 Riemann sums, 287, 341, 454 second-order initial-value problem, simply connected domain, 999 double integral, 896 Web-P4 Simpson, Thomas, 460 triple integral, 931 second-order linear differential equation, Simpson’s rule, 458–461 error estimate, 461–463 Riemann zeta function, 582 Web-P1 error in, 459 right cylinder, 349 second-order linear homogeneous sine, A3, Web-A3 height, 349 differential equations, Web-P1 continuity, 51 volume, 349 complex roots, Web-P3 derivative of, 114, 117, 118 width, 349 distinct real roots, Web-P2 right endpoint approximation, 455 family, A21, Web-D6 equal real roots, Web-P2 formulas, A6, Web-A8 right-hand derivatives, 95 initial-value problems, Web-P4 right-hand rule, 704 hyperbolic, 399 second-order model, pendulum period, integrating powers of, 423, 424, 428 right-handed coordinate systems, 674 599 right triangle, trigonometric functions, integrating products with cosine, 425, second-order partial derivatives, 426 A3, Web-A3 830–831 rise, Web-H5 rational functions of, 448 seconds (angle), A1, Web-A1 trigonometric identities, A4, Web-A6 RL series electrical circuit, 510 sector, A2, Web-A2 single integrals, 899 Rolle, Michel, 239 segment, 238 singular points, 608 Rolle’s Theorem, 239 semiaxes, Web-M5 sinks, 1036 root test, 551, 560 semiconjugate axis, 646 skew lines, 712 root-mean-square, 324 semicubical parabola, 609 slope, Web-H5 roots, Web-J2 semifocal axis, 646 and angle of inclination, A7, approximating using semimajor axis, 643 Web-A10 Intermediate-Value Theorem, semiminor axis, 643 partial derivatives, 826–827, 829 46–47 separable differential equations, slope field, 264, 498 approximating using Newton’s 487–490 slope of a surface, 827, 856 Method, 234, 235 separation of variables, 487–488 slope-producing function, 90 of functions, A9, Web-B4 sequence of partial sums, 532 slope-intercept form, Web-H9 multiplicity of, 187 sequences, 514, 516–518, 520, 521 slowing down, 228 simple, 187 small vibrations model, 599 rose curves, 624 convergence, 517 defined recursively, 521 smooth change of parameter, 763 rotation equations, 657, 658 smooth curve, 364 rotational motion, 3-space, 707 general term, 514 graphs, 516 smooth function, 364, 759 roundoff error, 461 / smooth parametrizations, 759, 760 in power series approximation, 585 increasing decreasing, 524 limit, 516, 517, 519 smooth transition, 780 Rule of 70, 495 Snell, Willebrord van Roijen, 226 lower bound, 528 run, Web-H5 Snell’s law, 225, 226 monotone, 525–529 Ryan, Nolan, 310 solids of revolution, 351 of partial sums, 532 solution, Web-H2 properties that hold eventually, 527 Saarinan, Eero, 408 of differential equation, 481–482 saddle point, 730, 874 Squeezing Theorem, 520 / inequalities, Web-F6 sampling error, Web-K7 strictly increasing decreasing, 524 solution set, Web-H2 scalar components, 695 types of, 524 inequalities, Web-F6 scalar moment, 707–708 upper bound, 527 sound intensity (level), 70, 71, 75 scalar multiple, 681 sets, Web-F4 sources, 1036 scalar triple product, 705 bounded, 872 speed, 82, 226, 227 algebraic properties, 706 closed, 817 instantaneous, 226, 782 geometric properties, 705–706 open, 817 motion along curves, 781, 782 scale and unit issues, graphs, Web-B9 unbounded, 872 terminal, 509 scale factors, Web-K2 shells, cylindrical, 358, 360–362 speeding up, 228 scale marks, Web-K2 Shroud of Turin, 493 spheres, 675, 676, 739 scale variables, Web-K2 sigma notation ( ), 275 spherical cap, 357 scaling, parametric curves, Web-K10 changing limits of, 276 spherical coordinates, 735, 739 secant, A3, Web-A3 properties, 276 converting, 736 I-14 Index

equations of surfaces in, 738–739 Sun, planetary orbits, 666 tangential scalar component of spherical element of volume, 941 superconductors, 39 acceleration, 785 spherical wedge, 941 surface area tangential vector component of spirals, 625 and improper integrals, 473 acceleration, 785 equiangular, 638 parametric surfaces, 920, 925–927 tautochrone problem, 611 families of, 625 as surface integral, 1014 Taylor, Brook, 568 spring constant, 377 surface of revolution, 370–372 Taylor polynomials, 567 spring stiffness, 377 surfaces of the form z = f(x, y), sigma notation, 567, 569 springs, 485, Web-P5 918–920 Taylor series, 574, 575, 584 free motion, Web-P8 surface integrals, 1013 convergence, 582, 583 Sprinz, Joe, 314 evaluating, 1014–1015, 1017–1018 finding by multiplication and division, square roots, and absolute values, A10, mass of curved lamina, 1013–1014, 598 Web-B5, Web-G2 1018 modeling physical laws with, 598 / squaring the circle crescent, 638 surface area as, 1014 power series representations as, 594, Squeezing Theorem, 52 surfaces 595 practical ways to find, 595, 597 Squeezing Theorem for Sequences, 520 oriented, 1022 Taylor’s formula with remainder, 570, standard equations relative orientation, 1039 571 circle, Web-I3 traces of, 725, 726 telescoping sum, 536, Web-L6 ellipse, 644 surfaces of revolution, 370, 738–739 temperature scales, Web-H13 hyperbola, 646, 647 parametric representation, 922 parabola, 641, 642 terminal point, vectors, 680 symmetric equations, 716 terminal side, of angles, A1, Web-A1 sphere, 675 symmetry, 193, A14, Web-C7 standard positions terminal speed, 509 area in polar coordinates, 634, 635 terminal velocity, 77, 509 angles, A1, A3, Web-A3 about the origin, A14, Web-C7 ellipse, 644 terminating decimal(s), 538, Web-F2 about the x-axis, A14, Web-C7 terms hyperbola, 646 about the y-axis, A14, Web-C7 parabola, 641, 642 infinite sequences, 514 symmetry tests, A15, Web-C8 infinite series, 531 static equilibrium, 690 polar coordinates, 621–623 test(s) stationary point, 184 alternating series, 553–556, 560 steady-state flow, 1023 comparison, 547, 548, 560 step size, Euler’s method, 500 tabular integration by parts, 418 conservative vector field, 974, Stokes, George Gabriel, 1041 tangent, A3, Web-A3 999–1002 Stokes’ Theorem, 1040 continuity, 51 divergence, 539, 560 calculating work with, 1040, 1042 derivative, 115 integral, 542–544, 560 circulation of fluids, 1043 double-angle formulas, A6, Web-A9 hyperbolic, 399 limit comparison, 548, 549, 560, and Green’s Theorem, 1043 Web-L6 strictly decreasing sequence, 524 integrating powers of, 426–428 integrating products with secant, 427, ratio, 549, 550, 560, Web-L7 strictly increasing sequence, 524 root, 551, 560 strictly monotone sequence, 525 428 trigonometric identities, A4, Web-A6 symmetry, 620–623, A15, Web-C8 string vibration, 832 test points, Web-F7 substitution(s) tangent line(s), 80, 81 definition, 80 Theorem of Pappus, 389–390 definite integrals, 319–321 Theorem of Pythagoras equation, 80, 91 hyperbolic, 436 for a tetrahedron, 743 graph of vector-valued functions, 753 integral tables, 446–449 theory of relativity, 12, 29 intersection of surfaces, 869 trigonometric, 431–434 θ, polar coordinate, 618 as a limit of secant lines, 79 u-substitution, 268, 270, 271 thickness, cylindrical wedges, 939 substitution principle, 30 parametric curves, 608–610 thin lens equation, 153 subtraction, of functions, A11, Web-C1 parametric equations of, 753 third derivative, 105 sum polar curves, 630–631 3 × 3 determinant, 701 absolute value, Web-G4 polar curves at origin, 631 3-space, 674 / open closed form, 277 slope, 80 tick marks, Web-K2 partial sums, 532 vertical, 197, 608 TNB-frame, 772 telescoping, 285, Web-L6 tangent line approximation, 457 topographic maps, 807 of vectors, 681 tangent planes, 867 torque, 707 summation graph of the local linear torque vector, 707 formulas, 277, Web-L6 approximation, 868 Torricelli’s law, 497 index of, 275 to level surfaces, 866–867 torsion, 780 notation, 275 to surface z = F(x, y), 867–868 torus, 929 sums of infinite series, 531–533 to parametric surfaces, 923, 925 torus knot, 745 and convergence, 532 total differentials, 868 total differentials, 840, 868 sum-to-product formulas, A7, Web-A10 tangent vector, 753 trace of a surface, 725–726 Index I-15 tractrix, 409 uniform circular motion, 801 tangent, 753 Traité de Mécanique Céleste, 977 uninhibited growth model, 484 triple products, 709, 710 trajectory, 605, 781, Web-M1 uninhibited population growth, 483 unit, 685 transform, 476 union, intervals, Web-F5 velocity, 680 transformations, 412, 955 unit circle, A4, Web-A4, Web-I3 zero, 681 plane, 948–950 unit hyperbola, 401 vector components, 695 translated conics, 648–650 unit normal vectors, 769, 778 vector fields, 971 translation, A13, Web-C5 for arc length parametrized curves, circulation of, 1048 parametric curves, Web-K10, 771 conservative, 974 Web-M4 inward 2-space, 770, 771 divergence and curl, 974–976 quadric surfaces, 730–731 unit tangent vectors, 768, 778 flow fields, 1021 transverse unit vector, 803 for arc length parametrized curves, flow lines, 979 trapezoidal approximation, 455, 461, 771 gradient fields, 973 464 unit vectors, 685 graphical representation, 972 error estimate, 461–463 units, graphs, Web-B10 integrating along a curve, 988–989 error in, 456 universal gravitational constant, 795 inverse-square, 972–973 tree diagram, 846 Universal Law of Gravitation, Web-D10 vector moment, 707 triangle inequality, A10, Web-B5, universe, age of, Web-N8 vector triple products, 709, 710 Web-G4 upper bound vector(s) for vectors, 690 least, 529 normal, 693 trigonometric functions monotone sequence, 527 orthogonal, 693 approximating with Taylor series, sets, 528 vector-value functions 584, 585 upper limit of integration, 286 integration formulas, 756 continuity, 51 upper limit of summation, 275 vector-valued functions, 746 derivatives, 115 u-substitution, 268–272, 319–321, 412 antiderivatives of, 756 finding angles from, Web-A10 guidelines, 269 calculus of, 750, 752–754, 756 hyperbolic, 398–406 continuity of, 751 integration formulas, 413 value of f at x, A8, Web-B2 differentiability of, 752 inverse, 56–60, 272 Vanguard 1, 669, 800 domain, 746 limits, 51 variables graphs, 747 mathematical model, Web-N4 change of in double integrals, 952 integrals of, 755 right triangles, A3, Web-A3 change of in single integrals, 948 limits of, 750 trigonometric identities, A4, Web-A6 change of in triple integrals, 955, 956 natural domain, 746 trigonometric integrals, 423–429 dependent and independent, A8, tangent lines for graphs, 753, 754 trigonometric substitutions, 431–434 Web-B2 vector-valued functions of two variables, integrals involving ax 2 + bx + c, 435 dummy, 298 922 triple integrals, 930 separation of, 488 partial derivatives, 923 change of variables, 955, 956 vector(s), 680 velocity, 82, 226, 782 converting from rectangular to angle between, 692–693 average, 314, 317 cylindrical coordinates, 940 arithmetic, 682–683 finding by integration, 306 converting from rectangular to components, 682 function, 92, 226 spherical coordinates, 945 in coordinate systems, 682 instantaneous, 92, 226, 782 cylindrical coordinates, 938–940 decomposing into orthogonal motion along curves, 781, 782 evaluating, 931–933 components, 694–695 rectilinear motion, 92, 307 limits of integration, 933, 935, 940 determined by length and a vector in terminal, 77, 509 order of integration, 931, 935, 941 the same direction, 686 versus time curve, 307 spherical coordinates, 941, 942 determined by length and angle, 686 velocity field, 971 volume calculated, 933, 934 direction angles, 693 vertex (vertices) trisectrix, 134 displacement, 680 angles, A1, Web-A1 truncation error, 461, 477 equal, 681 ellipse, 641 power series approximation, 585 equation of a line, 713 hyperbola, 641 tube plots, 745 force, 680 parabola, 640, Web-I5 twisted cubic, 746 geometric view of, 680–681 vertical asymptotes, 9, A20, Web-D6 2 × 2 determinant, 700 initial point not at origin, 683 polar curves, 630 two-point vector form of a line, 748 magnitude, 684 vertical line test, A9, Web-B4 two-sided limits, 5–6 norm of, 684 vertical surface, fluid force on, 395–396 2-space, 674 normalizing, 685 vertical tangency, points of, 93 type I/type II region, 902 orthogonal projections, 696 vertical tangent line, 197, 608 position, 747 vibrations of springs, 485 unbounded sets, 872 principal unit normal, 769, 924 vibratory motion, springs, 485, Web-P5 undefined slope, Web-H5 radius, 747, 923 viewing rectangle, Web-K2 I-16 Index viewing window, 810, Web-K2 Weierstrass, Karl, 32, 449 Wren, Sir Christopher, 611 choosing, Web-K3 weight, 378 graph compression, Web-K5 weight density, 394 x-intercepts, 193, Web-H4 zooming, Web-K5 wet-bulb depression, 812 of functions, A9, Web-B4 viewpoint, 810 Whitney’s umbrella, 924 of polynomials, 189 vinst,83 width, right cylinder, 349 x-axis, 674, Web-H1 volume Wiles, Andrew, 213 x-coordinate, 674, Web-H1 by cylindrical shells method, 358, wind chill index (WCT), 806, 826–828, x-interval, for viewing window, Web-K2 360–362 Web-B6, Web-B13 xy-plane, 674, Web-H1 by disks and washers, 352–354 witch of Agnesi, 750 xz-plane, 674 by function of three variables, 805 work, 375, 376, 378–380 net signed, 896 calculating with Green’s Theorem, y-intercepts, 193, Web-H4 slicing, 349, 350 1007 y-axis, 674, Web-H1 solids of revolution, 351, 353, 354 calculating with Stokes’ Theorem, y-coordinate, 674, Web-H1 under a surface, 895 1040, 1042 y-interval, for viewing window, Web-K2 triple integral, 933, 934 done by constant force, 375–376 yz-plane, 674 volume problem, 894 done by variable force, 376–377 polar coordinates, 911 as line integral, 990–991 z-axis, 674 performed by force field, 990 z-coordinate, 674 Wallis cosine formulas, 431 vector formulation, 697 zero vector, 681 Wallis sine formulas, 431 work integrals, 995 zeros, Web-J2 washers, method of, 353 Fundamental Theorem of, 997 of functions, A9, Web-B4 wave equation, 832 path of integration, 996 zone, of sphere, 374 wedges work–energy relationship, 375, 379–380 zoom factors, Web-K5 area in polar coordinates, 633 World Geodetic System of 1984, 734 zooming, Web-K5 cylindrical, 938 world population, 491 in, Web-K5 spherical, 941 doubling time, 492 out, Web-K5