Proofs to Accompany Chapter 30, Series 1131 lim n→∞ S n ≤ lim n→∞ a 1 +  n 1 f(x)dx =a 1 +A The sequence of partial sums is bounded and increasing. Therefore, by the Bounded Increasing Partial Sums Theorem,  ∞ k=1 a k converges. Suppose  ∞ 1 f(x)dx =∞.Refer to Equation (H.1) to obtain  n−1 1 f(x)dx ≤a 1 +a 2 +···+a n =S n Taking the limit as n →∞gives ∞≤ lim n→∞ S n . So S n grows without bound and  ∞ k=1 a k diverges. Now we show that the behavior of the integral can be determined by that of the series. Because f(x)>0,decreasing, and continuous on [1, ∞), lim b→∞  b 1 f(x)dx is either finite or grows without bound. Therefore, if we can find an upper bound, the integral converges. If it has no upper bound, it diverges. Suppose  ∞ k=1 a k converges. Denote its sum by S. From Equation (H.1) we know  n 1 f(x)dx ≤a 1 +a 2 +···+a n−1 lim n→∞  n 1 f(x)dx ≤ lim n→∞ n−1  k=1 a k = S. If lim n→∞  n 1 f(x)dx is bounded, so too is lim b→∞  b 1 f(x)dx (given the hypothe- ses). Suppose  ∞ k=1 a k diverges. Because the terms are all positive, we know lim n→∞  n k=1 a k =∞.From Equation (H.1) we know a 2 +···+a n ≤  n 1 f(x)dx lim n→∞ n  k=2 a k ≤ lim n→∞  n 1 f(x)dx. Weconclude that lim b→∞  b 1 f(x)dx =∞;the improper integral diverges. Index Abel, Niels, 382, 1078n Absolute convergence conditional, 953 explanation of, 952–953 implies convergence, 1128–1129 Absolute maximum point, 347 Absolute maximum value, 252, 347 Absolute minimum point, 347 Absolute minimum value, 347, 352 Absolute value function explanation of, 61–62 maximum and minimum and, 350 Absolute values analytic principle for working with, 66 elements of, 65 functions and, 129 geometric principle for working with, 66–68 Acceleration due to force of gravity, 150n, 406–407 explanation of, 76 Accruement, 746 Accumulation, 746 Addition of functions, 101–103 principles of, 1059–1061 Addition formulas, 668–669, 709 Additive integrand property, 738 Algebra equations and, 1053 explanation of, 1051–1052 exponential, 247, 309–312 exponents and, 1053–1054 expressions and, 1052–1053, 1056–1070 order of operations and, 1054 solving equations using, 1071–1083 (See also Equations) square roots and, 1054–1055 Alternating series error estimate, 954–955 explanation of, 953 Alternating series test, 953–956, 977 Amount added, 746 Amplitude definition of, 603 modifications to, 604, 606 Ancient Egyptians, 95, 97n, 627, 854 Angles complementary, 630 of depression, 631 of elevation, 631 initial side of, 619 measurement of, 619–622 right, 620 terminal side of, 619–620 trigonometric functions of, 622–623 vertex of, 619 Antiderivatives definition of, 761, 762 for integrand, 805–806 list of basic, 783–786 table of, 789–790 use of, 763, 764, 766, 770, 771 Antidifferentiate, 805 Approximations constant, 920 of definite integrals, 805–816, 820–825 Euler’s method and, 1022 examples of, 828–829 higher degree, 923–924 linear, 159, 163 local linearity and, 279–282 net change, 715–718 Newton’s method, 1121–1125 polynomial, 693–694, 919–931 second degree, 921–922 successive, 170, 176, 208, 715–718 tangent line, 282–284, 920–921, 937 Taylor polynomial, 924–931 third degree, 922–923 Arbitrarily close, 249 Arbitrarily small, 249, 250 Arccosine, 646, 647 Archimedes, 97n, 245, 1079n Arc length definite integrals and, 865–867 definition of, 594, 708 explanation of, 621–622 Arcsine, 646, 647 Arctangent, 646 Area of circle, 17 of oblique angles, 662–663 of oblique triangles, 662–664 slicing to find, 843–845 Area function amount added, accumulation, accruement and, 745 characteristics of, 747–755 definition of, 745 explanation of, 743–744 general principles of, 744–745 Astronomy, 1107 Asymptotes horizontal, 64–65, 407–409, 411 overview of, 64–65 vertical, 64, 407, 408, 617 Autonomous differential equations explanation of, 997 qualitative analysis of solutions to, 1002–1014 Average rate of change calculation of, 170 definition of, 75 explanation of, 73–76, 176 Average value definition of, 777 of functions, 775–780 Average velocity to approximate instantaneous velocity, 208 calculation of, 171–176 explanation of, 171 over time interval, 76, 77 Avogadro’s number, 150n 1133 1134 Index Babylonians, 95, 97n, 620 Bacterial growth, 306, 324–325 Balance line, 603, 604 Balance value change in, 606 definition of, 603 explanation of, 604 Base, 1053 Benchmark, 53 Bernoulli, Johann, 17n Bessel, Freidrich, 961 Bessel function, 961, 976 Bhaskara, 95 Binomial series explanation of, 946–947 use of, 948–949 Bottle calibration, 4, 20–21, 29n, 32, 209–210 Bounded Increasing Partial Sums Theorem, 965, 966, 1131 Bounded Monotonic Convergence Theorem, 965 Boyle, Robert, 546n Boyle’sLaw,546n Braces, 18n Brahe, Tycho, 1107 Briggs, Henry, 447 Calculators. See also Graphing calculators irrational numbers and, 96, 97 limits and, 267–269 logarithms using, 440–443, 450 order of operations on, 1054 roots on, 1058 trigonometric functions on, 605n, 608, 647, 919 use of graphing, 87–89 Calculus differential, 111n, 806 Fundamental Theorem of, 757–758, 761–771 (See also Fundamental Theorem of Calculus) historical background of, 211, 245 integral, 806 Calibration, bottle, 4, 20–21, 29n, 32, 210–211 Cardano, Girolamo, 95, 382 Carrying capacity, 150n, 1008–1010 Cellular biology, 984 Chain Rule application of, 517–519, 523–527, 539–543, 551, 785, 805, 1028 definition of, 517, 709 derivatives and, 521–522, 692, 706 interpretation of, 517 in reverse, 787–796 substitution to reverse, 792–794 Charles, Jacques, 546n Charles’sLaw,546n Circle area of, 17 explanation of, 1100, 1102 illustration of, 1099 symmetry of unit, 623–624 Circumference formula, 17n Closed form of sum, 564 Closed intervals, 18 Coefficients differential equations with constant, 1045–1049 explanation of, 1053 leading, 379, 1063 of the xk term, 380 Cofunction identity, 669 Common log of x, 442 Comparison Tests, 969–972, 977 Comparison Theorem, 912 Complementary angles, 630 Completeness Axiom, 965n Completing the square, 231–232, 1075 Composite functions Chain Rule to differentiate, 788 derivatives of, 513–517, 521–527 Composition, of functions, 108–113 Concave down, 52, 195 Concave up, 51, 52, 195 Concavity cubic and, 376 second derivative and, 356–358 Conditional convergence, 953 Conic sections astronomy and, 1107 degenerate, 1099 from geometric viewpoint, 1100–1101 overview of, 1099–1100 reflective properties of, 1106–1107 relating geometric and algebraic representations of, 1102–1105 Conjecture, 335 Constant factor property, 738 Constant function, 8 Constant Multiple Rule, 291–292 Constant of proportionality, 306n Constant rate of change calculating net change in case of, 712–714 linear functions and, 143–144, 169 Constant solutions. See Equilibrium solutions Continuity definition of, 270, 273 Extreme Value Theorem and, 271 Intermediate Value Theorem and, 271 Sandwich Theorem and, 272–273 Continuous functions average value of, 775–776 of closed interval, 352 explanation of, 54–55 invertible, 426 Convergence absolute, 952–953, 1128–1129 alternating series and, 954–956 conditional, 953 explanation of, 566, 568, 569, 904 improper integrals and, 904–907 in infinite series, 574 of Maclaurin series, 944 of power series, 945–946, 956, 1129 radius of, 945, 956–958 of Taylor series, 944n, 947 Convergence tests basic principles of, 964–965 Comparison Test and, 969–971 Integral Test and, 965–969, 972 Limit Comparison Test and, 971–972 Ratio and Root tests and, 972–976 summary of, 977 transition to, 962 use of, 975 Convergence Theorem for Power Series, 1129 Converse, 30n Cosecant, 629 Cosine functions. See also Trigonometric functions definition of, 594 domain and range of, 598 graphs of, 597–598, 603–609 symmetry properties of, 598–599 Cosines, Law of, 658–662, 709 Cotangent, 629 Coterminal angles, 620 Critical points concavity and, 356–358 of cubics, 375–376 extreme values and, 349–353 of polynomials, 386–387 Cubic formula, 382 Cubics characteristics of, 375–377 explanation of, 373 function as, 373–377 roots of, 381–382 Simpson’s Rule applied to, 823 Curves calculating slope of, 169–177 slicing to find area between, 843–845 solution, 501–503 Decomposition of functions, 119–121 into partial fractions, 898–901 Definite integrals arc length and, 865–867 area between curves and, 843–850 average value of function and, 775–780 definition of, 728 Index 1135 evaluation of, 763 explanation of, 725–727 finding exact area and, 727–729 finding mass when density varies and, 827–838 fluid pressure and, 871–874 Fundamental Theorem of Calculus used to compute, 770 integration by parts and, 882–883 interpretations of, 729, 743 numerical methods of approximating, 805–816, 820–825 properties of, 738–741 qualitative analysis and signed area and, 731–736 substitution in, 794–796 work and, 867–871 Definitions, 3 Degrees of angles, 620 radians vs., 685 Demand function, 104 Density, 827–838 Dependent variables, 5 Derivatives of bx, 473–475 calculation of, 190–192 of composite functions, 513–517, 521–527 Constant Multiple Rule and, 290–293 of cubic, 374–375 definition of, 187 explanation of, 188–190, 481, 523 of exponential functions, 334–338 historical background of, 211 instantaneous rate of change and, 209–211, 711, 983 of inverse trigonometric functions, 703–706 limit definition of, 190, 191 local linearity and, 279–284 of logarithmic functions, 467–471 meaning and notation and, 208–209 modeling using, 288–289 Product Rule and, 292–295 properties of, 273, 291–292 of quadratics, 219–221 qualitative interpretation of, 194–201 Quotient Rule and, 297–298 second, 221, 288–289, 356–358 of sums, 290–292 theoretical basis of applications of, 1087–1093 as tool for adjustment, 282 of trigonometric functions, 683–685, 708 working with limits and, 272 of xn for n, 295–297 Derivative tests first, 351, 358 second, 357, 358 Descartes, Rene, 61 Differential calculus, 111n, 806 Differential equations applications for, 984–988 autonomous, 997 with constant coefficients, 1045–1049 definition of, 498 explanation of, 498–500 first order, 988, 1018–1022 homogeneous, 1045–1049 modeling with, 503–506, 983–990 nonseparable, 1022 obtaining information from, 988–990 population growth and, 503–506, 984 power series and, 959–961 qualitative analysis of solutions to autonomous, 1002–1014 radioactive decay and, 503, 504 second order, 988, 1045–1049 separable, 1018–1022, 1030 solutions to, 498–503, 988, 991–997, 1002–1014, 1018–1022 systems of, 1024–1040 Differentiation of composite functions, 513–517 examples of, 535–538 flawed approaches to, 536 implicit, 541–554 (See also Implicit differentiation) logarithmic, 538–541 of logarithmic functions and exponential functions, 476–481 polynomials and, 385, 386 of power series, 956–959 Product Rule and, 292–294 of trigonometric functions, 683–698, 703–706 Differentiation formulas, 523–524 Direct Comparison Test, 969 Directrix, 1100, 1101 Dirichlet, Lejeune, 17n Discontinuities points of, 407–410 removable and nonremovable, 407 Discontinuous functions, 54–55 Distributive Law, 1056, 1065 Divergence explanation of, 566, 568, 904 of harmonic series, 953 improper integrals and, 904–907 in infinite series, 574 nth Term Test for, 574, 964, 977 Division of fractions, 1061–1063 of functions, 103–105 of polynomials, 382–383 by zero, 1053 Domain of function, 5 natural, 17 of trigonometric functions, 598 Dominance property, 738 Double-angle formula, 669, 709 Double root, 380 Dummy variables, 784 Einstein’s theory of special relativity, 938 Ellipses explanation of, 1100–1103 illustration of, 1099 reflecting properties of, 1106 Endpoint reversal property, 738, 744 Epidemic models, 1024–1025, 1034–1038 Epidemiology, 986, 1024–1025, 1034–1038 Equal Derivatives Theorem, 1088, 1092–1093 Equations. See also Differential equations explanation of, 1053 exponentiation to solve, 449–455 expressions vs., 537–538 fitted to sinusoidal graph, 606–607 fundamental principle for working with, 1053 growth, 306 higher degree, 1078–1080 linear, 159, 1071–1073 of lines, 144–145, 148–150 logarithms to solve, 449–457 quadratic, 1073–1078 with radicals, 1081–1083 solutions to, 1071–1083 trigonometric, 651–655 Equilibrium, stable, 106–108 Equilibrium solutions definition of, 1004 example of, 1004–1005 explanation of, 989 stability and, 1006–1008 Erd ¨ os, Paul, 688n Error bounds, 823–825 Euler, Leonhard, 17n, 492n Euler’s Formula, 1048 Euler’s method, 1022 Even functions, 65 Existence and Uniqueness Theorem, 994 Exponential algebra application of, 247, 309 basics of, 309–312 Exponential functions derivative of, 334–338 explanation of, 312–314, 497 food decay and, 303, 326–327 generalized power functions and, 524 graphs of, 462–463 growth of money in bank account and, 303, 320–322, 325, 503 1136 Index Exponential functions (continued) laws of logarithmic and, 444–447 manipulation of, 314–316 modeling with, 559 radioactive decay and, 322–324 writing and rewriting, 324–326 Exponential growth examples of, 306–307 explanation of, 303–306 Exponentiation, 449–455 Exponent Laws, 1058 explanation of, 309, 310 list of, 481 Exponents explanation of, 1053–1054 multiplication and working with, 1057–1059 Expressions addition and subtraction, 1059–1061 division and complex fractions, 1061–1063 equations vs., 537–538 explanation of, 1052–1053 factoring, 1064–1070 multiplying and factoring, 1056–1057 multiplying and working with exponents, 1057–1059 terminology and, 1063–1064 working with, 1056–1070 Extraneous roots, 1082 Extrema analysis of, 346–348 definition of, 347 global, 351–354 local, 351, 353, 391–392 Extreme, absolute, 353 Extreme Value Theorem continuity and, 271 explanation of, 349, 352, 1087, 1089, 1090 Factorial notation, 694 Factoring difference of perfect squares, 1068–1069 explanation of, 1056–1057 out common factor, 1065–1066 quadratics, 1066–1068 use of, 1064–1065, 1069–1070 Finite geometric sum, 561, 563 First derivative, 288–289 First derivative test, 351, 358 First order differential equations explanation of, 988 solutions to separable, 1018–1022, 1030 Fixed costs, 102, 103 Flipping, 126–129 Fluid pressure, 871–874 Focus, 1100 fog, 108 Folium of Descartes, 543, 544 Food decay, 303, 326–327 Foot-pound, 868 Fractions adding and subtracting, 1059–1061 complex, 1061–1063 division of, 1061–1063 historical background of, 95 integration using partial, 898–901 multiplication of, 1057 simplification of, 1057 Free fall, 237–240 Functional notation examples of, 9–10, 17 explanation of, 8–9 Functions. See also specific types of functions absolute value and, 66–68, 350 addition and subtraction of, 101–103 altered, 126–131 area, 743–755 asymptotes and, 64–65 average value of, 775–780 Bessel, 961, 976 composite, 513–517, 519–525 composition of, 108–113 constant, 8 continuous/discontinuous, 54–57, 352, 426 decomposition of, 119–121 definition of, 5 derivatives as, 187–211 (See also Derivatives) domain and range of, 5, 17–21 equality of, 17 even, 65 examples of, 61–64 explanation of, 2–3, 5–9 exponential, 303, 312–328 to fit parabolic graph, 233–235 graphs of (See Graphs) increasing/decreasing features of, 51–54 inverse, 32, 110, 421–426, 429–432, 434 invertible, 423–426, 429–434, 440 linear, 143–151 locally linear, 142 logarithmic, 439–457, 462–466 (See also Logarithmic functions) as machine, 6–7 as mapping, 6–8 of more than one variable, 33 multiplication and division of, 103–105 names of, 5 odd, 65 overview of, 1–2 portable strategies for problem solving and, 21–28 positive/negative features of, 49–51 quadratic, 217–240 rational, 407–417 reciprocal, 62, 130–131 representations of, 15–17 sinusoidal, 603–604 solution to differential equation as, 988 Fundamental Theorem of Calculus definite integrals and, 761–763 explanation of, 757–758, 770–771 power of, 762, 767–770 use of, 763–767, 1097 Galois, ´ Evariste, 1078n Galois theory, 1078n Generalized power functions, 524 Generalized power rule, 524 Generalized Ratio Test, 975–976 General solutions, 500–502, 996 Geometric formulas, 17n, 1086 Geometric series applications of, 579–586 to compute finite geometric sums, 570 convergence and, 977 definition of, 566 explanation of, 566–570, 962, 964 Geometric sums applications of, 579–586 examples of, 563–564 finite, 561, 563 geometric series to compute finite, 570 introductory example of, 559–563 Global extrema, 351–354 Global maximum point, 347 Global maximum value, 347, 352 Global minimum point, 347 Global minimum value, 347, 352 Graphing calculators. See also Calculators Taylor polynomials on, 937 trigonometric functions on, 605n, 608 use of, 87–89, 382 Graphs of derivative functions, 196–201 of functions, 28–33, 62, 63, 65, 84, 87–89 of invertable functions, 425–426 of linear functions, 144–147 of logarithmic functions, 462–466 of periodic functions, 596 of polynomials, 384–385, 391–399 position, 84–85 of quadratics, 219, 221, 223, 231–235 rate, 85–87 of rational functions, 410–417 of reciprocal functions, 130–131 of squaring functions, 63 tips for reading, 85, 87 of trigonometric functions, 597–598, 603–609, 616–618, 647, 708 Gravitational attraction, 406 Gregorian calendar, 53n Index 1137 Guy-Lussac, Joseph, 546n Guy-Lussac’sLaw,546n Happiness index, 51 Height, 77 Herodotus, 95n Higher degree equations, 1078–1080 Hindus, 95n Hipparchus, 627 Hooke’s Law, 1045 Horizontal asymptotes explanation of, 64–65 rational functions and, 407–409, 411 Horizontal line test, 33, 426 Horizontal shift, 607–608, 669 Hypatia, 95n Hyperbolas explanation of, 1103 illustration of, 1099 reflective properties of, 1107 Hypotenuse, 628 Identity function, 61 Implicit differentiation explanation of, 541–545 particular variables and, 546–548 process of, 545–546 related rates of change and, 550–554 Improper integrals explanation of, 903 infinite interval of integration and, 903–907 method of comparison and, 912–914 methods to approach, 908–912 unbounded and discontinuous integrands and, 907–908 use of, 903 Increasing and Bounded Partial Sums Test, 574 Increasing/Decreasing Theorem, 1088 Indefinite integrals definition of, 783 using integration by parts, 878–882 Independent variables, 5 Indeterminate forms, 490n Induction, 296, 1095–1097 Induction hypothesis, 296 Inequalities, Intermediate Value Theorem and, 55–56 Infinite series definition of, 566 general discussion of, 572–574 geometric, 566–570 Inflation, 487 Inflection point, 357, 376 Initial condition, 502 Initial conditions, 501–503 Instantaneous rate of change calculation of, 170–178, 208 derivative and, 209–211, 711, 983 explanation of, 170, 176, 523 velocity and, 170, 173 Instantaneous velocity average velocity to approximate, 208 explanation of, 170, 173–174 limit and, 245–246 Integers, 1063 Integral calculus, 806 Integrals definite (See Definite integrals) improper, 903–914 indefinite, 783–785, 878–882 substitution to alter form of, 798–802 trigonometric, 886–890 Integral Test for convergence of series, 1129–1131 explanation of, 965–968, 977 proof of, 968 use of, 968–969, 972 Integrands antiderivative of, 805–806 definition of, 783 explanation of, 729, 770 expressed as sum, 798 unbounded and discontinuous, 907–908 Integration infinite interval of, 903–907 of power series, 956–959 solving differential equations by, 995 Integration by parts. See also Product Rule definite integrals and, 882–883 explanation of, 877–878 formula for, 878, 879, 882 indefinite integrals and, 878–879 repeated use of, 879–882 use of, 805, 846n Interest rates, 487–489, 494–495, 579–586 Intermediate Value Theorem continuity and, 271 explanation of, 55 inequalities and, 55–56 Interval notation, 18–19 Interval of convergence explanation of, 945–946 Taylor series and, 944n, 947 Intervals closed, 18 increase and decrease on, 51–54 open, 18, 270 Inverse functions arriving at expression for, 429–432 definition of, 423 explanation of, 32, 110, 421–426, 440 graphs and, 425 horizontal line test and, 426 interpreting meaning of, 434 slicing and, 846–850 Inverse sine, 646 Inverse trigonometric functions derivatives of, 703–706 explanation of, 645–649, 708 Irrational numbers, 1063–1064 definition of, 87 historical background of, 95, 96 nature of, 88–89, 96–99 proof of, 88n working with, 247 Isosceles triangles, 635n Joule, 868 Julian calendar, 53n Kepler, Johannes, 447, 1107 Kepler’s Laws, 961 al-Khowarizmi, 95 Kinetic energy, 938 Lagrange, Joseph, 211 Lambert, Johann, 97 Law of Cooling (Newton), 326, 985–986, 1002–1004, 1025–1028 Law of Cosines explanation of, 658–659, 709 proof of, 660–661 use of, 661–662 Law of Sines explanation of, 659–660, 709 proof of, 664 use of, 664–665 Leading coefficients, 379 Least common denominator (LCD), 1059, 1060 Left- and right-hand sums explanation of, 727, 728, 820 limits and, 258–262, 270, 728 net change and, 718–722 use of, 806–808, 812–815 Legs, 628 Leibniz, Gottfried Wilhelm, 209, 211, 245, 292n, 953 Leibniz’s notation, 209, 211, 221 Lemmas, 688n L’H ˆ opital’s Rule definition of, 1112 proof of, 1114–1116 use of, 490n, 1112–1114, 1117, 1118 Limit Comparison Test, 971–972, 977 Limit principle, 490, 492n, 493n Limits application of, 245–246 approaches to, 265–269 computation of, 251–255, 489–491 definition of, 250–255, 524 explanation of, 246–250 function of, 491–495 left- and right-handed, 258–262, 270 one-sided, 259–260 1138 Index Limits (continued) principles for working with, 272–273 trigonometric functions and, 688–691 two-sided, 262 Linear approximations, 159, 163–164 Linear equations definition of, 1071 simultaneous, 159 solutions to, 1071–1073 Linear functions characteristics of, 143–144 definition of, 144 explanation of, 373 graphs of, 144–147 piecewise, 159, 161 zero of, 381 Linearity differential calculus and local, 111n intuitive approach to local, 142–143 Linear models, 159–164 Lines equations of, 144–145, 148–150 graphs of, 147 parallel, 148 perpendicular, 148 point-slope form of, 149 secant, 75 slope-intercept form of, 149 slope of, 145–150 vertical, 147–148 Lithotripsy, 1106–1107 Liu Hui, 95 Local extrema, 351, 353, 391–392 Local Extremum Theorem, 1087–1090 Local linearity derivatives and, 279–284 differential calculus and, 111n explanation of, 142–143, 1114 use of, 143 Local maximum point, 347 Local maximum value, 347, 354 Local minimum point, 347 Local minimum value, 347, 354 Logarithmic differentiation definition of, 538 to find y 1 , 538–539 use of, 539–540 Logarithmic functions definition of, 440, 442 derivative of, 467–475 explanation of, 440–443 graphs of, 462–466 historical background of, 447 introductory example of, 439–440 laws of exponential and, 444–447 Logarithmic laws, 445, 481 Logarithms calculator use and, 440–443, 450 converted from one base to another, 450–454 definition of, 441–442 differentiation and, 476–480 properties of, 444–448 solving equations using, 449–457 summary of, 480–481 uses for, 447–448 Logistic growth model, 898, 1009, 1019–1020, 1024 Long division, 382–383 Lower bound, 716, 717 MacLaurin, Colin, 920 Maclaurin series alternation series and, 953 binomial series and, 946, 947 convergence and, 944 explanation of, 941, 948 procedure for finding, 941–942, 957, 961 Marginal cost, 210 Mass, 827–838 Mathematical induction, 296, 1095–1097 Mathematical models, 2 Mayans, 95n Mean Value Theorem, 346, 866n, 1087, 1090–1093 Midpoint sum, 807–811, 813–815 Modeling with derivatives, 288–289 with differential equations, 503–506, 983–990 of population interactions, 1038–1040 Money, growth of, 303, 320–322, 325 Multiplication with exponents, 1057–1059 with expressions, 1056–1057 with fractions, 1057 with functions, 103–105 Napier, John, 447 Natural domain, 17 Natural log of x, 442 Negative numbers, 95 Net change with constant rate of change, 712–714 difference between left- and right-hand sums and, 718–722 explanation of, 712 with nonconstant rate of change, 715–718 overview of, 711–712 Newton, 868 Newton, Isaac, 211, 245, 1107 Newtonian physics, 938–939 Newton-meter, 868 Newton’s Law of Cooling, 326, 985–986, 1002–1004, 1025–1028 Newton’s method applications for, 1124–1125 for approximating root of f, 1123–1124 explanation of, 1121–1122 use of, 1122–1123 Newton’s Second Law, 868, 1034 Nonremovable discontinuities, 407 Nonseparable first order differential equations, 1022 Notation factorial, 694 functional, 8–10, 17 interval, 18–19 summation, 575–577, 769 nth partial sum, 566 nth Term Test for Divergence, 574, 964, 977 Nullclines, 1030–1032, 1035 Number line, 18 Numbers irrational, 87, 88, 96–99 rational, 87, 95, 96 real, 87, 88 Oblique triangles area of, 663–664 explanation of, 657 Law of Cosines and Law of Sines and, 658 Obtuse angles, 662–663 Odd functions, 65 Open interval, 18, 270 Operator notation, 209 Optimization analysis of extrema and, 341–348 application of, 361–364 concavity and second derivative and, 356–358 extrema of f and, 348–354 overview of, 341 Order, of differential equations, 988 Order of operations, 1054 Parabolas definition of, 219 with derivatives, 220–221 explanation of, 1100, 1101 graphs of, 231–235, 342–343 illustration of, 1099 reflecting properties of, 1106 turning point of, 221 vertex of, 221, 223–225 Parabolic arc, 1107 Parabolic graphs, 233–235 Parallel lines, 148 Parameter, 126n Partial fraction decomposition, 898–901 Partial sums, 964–966 Particular solutions, 498, 501, 502 Index 1139 Perfect squares equations with, 1075 factoring difference of, 1068–1069 Period, 603, 604 Periodic functions, 596 Periodicity-based identity, 669 Periodicity-reducing identity, 669 Perpendicular lines, 148 Piecewise linear function, 161 Points, 347 Point-slope form of lines, 149 Polynomial approximations explanation of, 919–920 of sin x around x = 0, 920–924 Taylor, 924–931 of trigonometric functions, 693–694 Polynomials characteristics of and differentiation of, 383–387 critical points of, 386–387 cubics and, 373–377 degree of, 380, 1063 of even and odd degrees, 385 explanation of, 373, 379–380, 385–386, 1063 graphs of, 384–385, 391–399 long division of, 383 overview of, 373 zeros of, 380–383, 391 Population growth differential equations and, 503–506, 984 growth equations and, 306 instantaneous rate of change and, 210 logistic, 1008–1011, 1019–1020, 1024 Population interactions, 1038–1040 Position versus time, 210 Positive integers, 295–297, 1063 Power-reducing formula, 669 Power series convergence and, 945–946, 956, 1129 definition of, 945 differential equations and, 959–961 differentiation and integration of, 956–959 manipulating, 956 Uniqueness Theorem and, 945 Predictions, 139–143 Present value, 582–585 Probability density function, 903 Production cost versus amount produced, 210 Product Rule. See also Integration by parts differentiating f(x)· g(x), 292–294 explanation of, 294 proof of, 295 use of, 297, 513, 543, 548, 692, 805 Proof by induction, 1095–1097 Proportionality constant of, 306n direct, 16 explanation of, 306 Ptolemy, 95n, 627 Pythagoras, 95, 97 Pythagorean identities explanation of, 599, 669, 709 trigonometric identities and, 886, 887 use of, 668 Pythagoreans, 89, 95–97 Pythagorean Theorem historical background of, 95 Law of Cosines and, 658 use of, 554, 599, 631, 635, 637, 672 Quadratic equations explanation of, 1073–1077 factoring, 1077–1078 Quadratic formula, 1074, 1076–1077, 1080 Quadratics calculus perspective to, 217–221 definition of, 219 disguised, 1080–1081 examples of, 217–219 explanation of, 373 factoring, 1066–1068 free fall and, 237–240 graphs of, 219, 221, 223, 231–235 noncalculus perspective to, 223–225 overview of, 217 use of quadratic formula to solve, 1076–1077 zero of, 381 Qualitative analysis, 1002–1014 Quotient Rule application of, 337, 522, 692, 1115 explanation of, 297–298 Radians converted to degrees, 621 converting degrees to, 621 definition of, 620 degrees vs., 685 Radicals, 1081–1083 Radioactive decay differential equations and, 503, 504 rate of, 322–324 Radio waves, 605n Radius of convergence, 945 Range of functions, 5, 17–21 of trigonometric functions, 598 Rate of change average, 73–76, 170, 176 constant, 143–144, 169, 712–714 decomposition to find, 121 implicit differentiation and, 550–554 instantaneous, 170–178, 209–211, 523, 711, 983 interpreting slope as, 153 nonconstant, 715–718 predictions and, 140–142 of quadratics, 219 Rational functions asymptotes and, 407–409 decomposition of, 898–900 discontinuity and, 407–410 explanation of, 406–407 graphs of, 410–417 Rational numbers, 1063 definition of, 87 historical background of, 95, 96 Ratio Test, 973–974, 977 Real numbers, 87, 1063 Real number system historical background of, 95–96 irrationality and, 96–99 Reciprocal functions explanation of, 62 graphs of, 130–131 Reduction formula derivation of, 880–882, 888 use of, 888–890 Relation, 988n Relative maximum point, 347 Relative minimum point, 347 Removable discontinuities, 407 Revolution, 856–862 Richter scale for earthquakes, 448 Riemann, Bernhard, 727, 955 Riemann sums explanation of, 727–729 limit of, 728n, 831, 1097 to obtain approximations of definite integrals, 806 use of, 828, 830, 832, 866 Right angles, 620 Right triangles applications for, 627–628 definitions of, 628–630, 707 45 ◦ ,45 ◦ ,635–637 30 ◦ ,60 ◦ ,637–643 trigonometry of, 631–633 Rolle, Michel, 1089n Rolle’s Theorem, 1087, 1089–1091, 1127, 1128 Roots of multiplicity, 380 Root Test, 975, 977 Sandwich Theorem, 272–273, 757, 944 Secant, 629 Secant line, 75, 208 Second derivative, 221 Second derivative test, 357, 358 Second order differential equations with constant coefficients, 1045–1049 explanation of, 988 Separable differential equations, 1018– 1022 1140 Index Sequences, 964, 965, 1129–1131 Series absolute and conditional convergence and, 952–953 alternating, 953–956 binomial, 946–949 convergence tests and, 962, 964–977, 1129–1131 geometric, 566–570, 579–586, 962, 964 Maclaurin, 941–942, 944, 946–948, 953, 957, 961 power, 945–946, 956–961 Taylor, 941–944, 947–949, 962 Shifting, 126–129 Shrinking, 126–129 Signed area, 727, 732 Similar triangles, 628 Simple root, 380 Simpson’s Rule explanation of, 820–821 requirements for, 822–823 use of, 821–823 Sine functions. See also Trigonometric functions definition of, 594 domain and range of, 598 graphs of, 597–598, 603–609 symmetry properties of, 598–599 Sines, Law of, 659–660, 664–665, 709 Singh, Jai, 627 Sinusoidal functions, 603–604 Sinusoidal graphs, 606–607 Slicing to find area between two curves, 843–845 to find mass when density varies, 827–838 to find volume, 853–856 to help with definite integrals, 846–850 Slope of f  , 194–196 of line, 145–150 modeling and interpreting, 153–155 of tangent line, 208, 684 Slope function explanation of, 161 of f(x),188, 189 Slope-intercept form of lines, 149 Solution curves examples of, 1005, 1006, 1010, 1027 explanation of, 501–503 Solutions to differential equations, 498–503, 988, 991–997, 1002–1014 equilibrium, 989, 1004–1008 to trigonometric equations, 651–655 Soper, H. E., 1037 Splitting interval property, 738 Square roots, 1054–1055 Squaring function explanation of, 61 graph of, 63 Squeeze Theorem, 272, 757 Stable equilibrium, 106–108 Standard position, 620 Stationary points, 350 Stifel, Michael, 96 Stretching, 126–129 Substitution to alter form of integral, 798–802 in definite integrals, 794–796 explanation of, 787 mechanics of, 790–791 Taylor series and, 947–949 trigonometric, 890–894 used to reverse Chain Rule, 792–794 Subtraction of functions, 101–103 principles of, 1059–1061 Subtraction formulas, 668–669 Successive approximations, 715–718 Summation notation examples of, 576–577 explanation of, 575–576 use of, 769 Sum Rule, 291, 292 Symmetry even and odd, 65, 386–387 polynomials and, 386–387 of trigonometric functions, 598–599, 623–624 Symmetry-based identity, 669 Symmetry property, 738 Tangent functions definition of, 615 geometric interpretation of, 615 graph of, 615–618 period of, 616 Tangent line to f at x = a, 176 slope of, 208, 684 Tangent line approximations, 282–284, 920–921, 937 Tartaglia, Niccolo Fontana, 95, 382 Taylor, Brook, 694, 920 Taylor expansions, 947 Taylor polynomials centered at x = 0, 925–928, 936, 948 centered at x = b, 928–931, 934, 941 definition of, 925 examples of, 935–939 Taylor remainders, 934–935, 942 Taylor series convergence and, 944n, 947, 949 explanation of, 941–944 obtaining new, 947–949 substitution and, 947–949 Taylor’s Inequality, 935 Taylor’s Theorem convergence and, 962 definition of, 935 proof of, 1127–1128 use of, 934–935, 937, 942 Temperature change, 73–74 Theon of Alexandria, 95n Theorem on Differentiation of Power Series, 956, 959 Total cost function, 102, 105, 406 Trajectories, 1027–1036 Trapezoidal sums explanation of, 808 use of, 812–815, 820 Triangles historical background of, 627 oblique, 657 perspective on, 627–628 similar, 628 solving, 631 trigonometry of general, 657–665 trigonometry of right, 627–633, 635–643 (See also Right triangles) Trigonometric equations explanation of, 651 solutions to, 651–655 Trigonometric functions. See also specific functions of angles, 622–723 on calculators, 605n, 608, 647 definitions of, 594 differentiation of, 683–698, 703–706 domain and range of, 598 f(x)=tan x, 615–618 graphs of, 597–598, 603–609, 616–618, 647, 708 inverse, 645–649, 703–706, 708 periodicity and, 596 polynomial approximations of, 693–694 properties of, 594–595 summary of, 707–709 symmetry properties of, 598–599, 623–624 Trigonometric identities addition formulas and, 668–669 explanation of, 599–600, 709 summary of, 669 trigonometric integrals and, 886–887 use of, 600–601, 667, 697 Trigonometric integrals explanation of, 886 miscellaneous, 890 sin x and cos x, 886–888 tan x and sec x, 888–890 Trigonometric substitution, 890–894 Trigonometry angles and arc lengths and, 619–624 applied to general triangle, 657–665 . change with constant rate of change, 712–714 difference between left- and right-hand sums and, 718–722 explanation of, 712 with nonconstant rate of change, 715–718 overview of, 711–712 Newton,. integrals explanation of, 903 infinite interval of integration and, 903–907 method of comparison and, 912–914 methods to approach, 908–912 unbounded and discontinuous integrands and, 907–908 use of, 903 Increasing. functions, 334–338 historical background of, 211 instantaneous rate of change and, 209–211, 711, 983 of inverse trigonometric functions, 703–706 limit definition of, 190, 191 local linearity and,