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Copyrighted Material 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,
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