www.elsolucionario.net About the Cover The maglev (magnetic levitation) train uses electromagnetic force to levitate, guide, and propel it Compared to the more conventional steel-wheel and track trains, the maglev has the potential to reach very high speeds—perhaps 600 mph In this series, author Soo T Tan uses the maglev as a means to introduce the concept of the limit of a function and continues to weave this common thread through the topic of integration Tan’s innovative introduction of the limit concept in the context of finding the velocity of the maglev captures student interest from the very beginning This intuitive approach demonstrates the relevance of calculus in the real world and is applied throughout the text to introduce and explain some of the fundamental theorems in calculus, such as the Intermediate Value Theorem and the Mean Value Theorem www.elsolucionario.net Openmirrors.com Now that you’ve bought the textbook GET THE BEST GRADE BY SAVING TIME AND MONEY! 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Join the thousands of students who have benefited from www.iChapters.com Simply search by author, title, ISBN, then filter the results by “Study Tools” and select the format best suited for you Source Code: 10PMA0002 www.iChapters.com — Your First Study Break www.elsolucionario.net www.elsolucionario.net Openmirrors.com RULES OF DIFFERENTIATION Basic Formulas d (c) ϭ dx d du (cu) ϭ cu dx dx d u a bϭ dx √ d du d√ (u Ϯ √) ϭ Ϯ dx dx dx d d√ du (u√) ϭ u ϩ√ dx dx dx √ du d√ Ϫu dx du √ d n du (u ) ϭ nu nϪ1 dx dx d f(t(x)) ϭ f ¿(t(x))t¿(x) dx Exponential and Logarithmic Functions d u du (e ) ϭ eu dx dx d u du (a ) ϭ (ln a)a u dx dx 10 d du ln Ϳ u Ϳ ϭ u dx dx 11 d du (log a u) ϭ dx u ln a dx Trigonometric Functions 12 d du (sin u) ϭ cos u dx dx 14 d du (tan u) ϭ sec u dx dx 16 d du (sec u) ϭ sec u tan u dx dx 13 d du (cos u) ϭ Ϫsin u dx dx 15 d du (csc u) ϭ Ϫcsc u cot u dx dx 17 d du (cot u) ϭ Ϫcsc u dx dx Inverse Trigonometric Functions 18 d du (sinϪ1 u) ϭ dx dx 21 Ϫ u 21 d du (cscϪ1 u) ϭ Ϫ dx Ϳ u Ϳ2u Ϫ dx 19 d du (cosϪ1 u) ϭ Ϫ dx dx 21 Ϫ u 22 d du (secϪ1 u) ϭ dx Ϳ u Ϳ2u Ϫ dx 20 d du (tanϪ1 u) ϭ dx ϩ u dx 23 d du (cot Ϫ1 u) ϭ Ϫ dx ϩ u dx Hyperbolic Functions 24 d du (sinh u) ϭ cosh u dx dx 27 d du (csch u) ϭ Ϫcsch u coth u dx dx 25 d du (cosh u) ϭ sinh u dx dx 28 d du (sech u) ϭ Ϫsech u u dx dx 26 d du (tanh u) ϭ sech2 u dx dx 29 d du (coth u) ϭ Ϫcsch2 u dx dx Inverse Hyperbolic Functions 30 d du (sinhϪ1 u) ϭ dx 21 ϩ u dx 33 d du (cschϪ1 u) ϭ Ϫ dx Ϳ u Ϳ2u ϩ dx 31 d du (coshϪ1 u) ϭ dx 2u Ϫ dx 34 d du (sechϪ1 u) ϭ Ϫ dx dx u21 Ϫ u 32 du d (tanhϪ1 u) ϭ dx Ϫ u dx 35 du d (cothϪ1 u) ϭ dx Ϫ u dx www.elsolucionario.net TABLE OF INTEGRALS Basic Forms 10 Ύ cot u du ϭ ln Ϳ sin u Ϳ ϩ C 11 Ύ sec Ύ sin u du ϭ Ϫcos u ϩ C 12 Ύ csc Ύ cos u du ϭ sin u ϩ C 13 Ύ sec u tan u du ϭ sec u ϩ C Ύ tan u du ϭ ln Ϳ sec u Ϳ ϩ C 14 Ύ csc u cot u du ϭ Ϫcsc u ϩ C Ύe 15 Ύ 2a 16 Ύ u2u Ύ Ύu u n du ϭ du u u nϩ1 ϩ C, n nϩ1 Ϫ1 ϭ ln Ϳ u Ϳ ϩ C du ϭ eu ϩ C u u du ϭ tan u ϩ C u du ϭ Ϫcot u ϩ C du Ϫu 2 ϭ sinϪ1 u ϩC a Ύa Ύ sec u du ϭ ln Ϳ sec u ϩ tan u Ϳ ϩ C 17 Ύa Ύ csc u du ϭ ln Ϳ csc u Ϫ cot u Ϳ ϩ C 18 Ύa 23 Ύ u(a ϩ bu) ϭ a ln ` a ϩ bu ` ϩ C 24 Ύ u (a ϩ bu) ϭ Ϫau ϩ a 25 Ύ u(a ϩ bu) 26 Ύ u (a ϩ bu) 31 Ύ du 1a ϩ bu du ϭ 21a ϩ bu ϩ a u u1a ϩ bu 32 Ύ b 1a ϩ bu 1a ϩ bu ϩ du ϭ Ϫ u u2 33 Ύ u 1a ϩ bu du u du ϭ a ϩC ln a du 2 u secϪ1 ϩ C a a ϭ Ϫ a2 du u ϭ tanϪ1 ϩ C a a ϩ u2 du uϩa ϭ ln ` `ϩC uϪa 2a Ϫu Forms Involving a ؉ bu 19 Ύ a ϩ bu ϭ b 20 Ύ a ϩ bu u du a ϩ bu Ϫ a ln Ϳ a ϩ bu Ϳ ϩ C u du ϭ C (a ϩ bu)2 Ϫ 4a(a ϩ bu) ϩ 2a ln Ϳ a ϩ bu Ϳ D ϩ C 2b 21 Ύ (a ϩ bu) 22 Ύ (a ϩ bu) u du ϭ a ϩ ln Ϳ a ϩ bu Ϳ ϩ C b (a ϩ bu) b ϭ u du b aa ϩ bu Ϫ du du u du ϭ du 2 b ln ` a ϩ bu `ϩC u 1 a ϩ bu Ϫ ln ` `ϩC u a(a ϩ bu) a ϭϪ 2b a ϩ 2bu u ϩ `d ϩ C c ln ` a a ϩ bu a u(a ϩ bu) a2 Ϫ 2a ln Ϳ a ϩ bu Ϳ b ϩ C a ϩ bu Forms Involving 1a ؉ bu 27 Ύ u1a ϩ bu du ϭ 15b 28 Ύ 1a ϩ bu ϭ 3b 29 Ύ 1a ϩ bu ϭ 15b 30 u du u du Ύ 2 (3bu Ϫ 2a)(a ϩ bu) 3>2 ϩ C (bu Ϫ 2a) 1a ϩ bu ϩ C (8a ϩ 3b 2u Ϫ 4abu) 1a ϩ bu ϩ C 1a ϩ bu Ϫ 1a ` ϩ C if a Ͼ ln ` du 1a ϩ bu ϩ 1a 1a ϭ μ a ϩ bu u1a ϩ bu tanϪ1 ϩC if a Ͻ Ϫa B 1Ϫa Ύ u ndu Ύ cu n(a ϩ bu)3>2 Ϫ na u nϪ1 1a ϩ bu dud b(2n ϩ 3) Ύ 1a ϩ bu ϭ www.elsolucionario.net Openmirrors.com du n ϭ 34 Ύ u 1a ϩ bu 2u n 1a ϩ bu 2na Ϫ b(2n ϩ 1) b(2n ϩ 1) u nϪ1 du Ύ 1a ϩ bu 1a ϩ bu 35 Ύ u 1a ϩ bu ϭ Ϫa(n Ϫ 1)u 36 Ύ du n nϪ1 Ϫ b(2n Ϫ 3) 2a(n Ϫ 1) Ύu nϪ1 du 1a ϩ bu (a ϩ bu)3>2 (2n Ϫ 5)b Ϫ1 1a ϩ bu du ϭ c ϩ n u a(n Ϫ 1) u nϪ1 Ύ 1a ϩ bu dud , n u nϪ1 Forms Involving 2a2 ؉ u2, a > 37 Ύ 2a 38 Ύ ϩ u du ϭ u a2 2a ϩ u ϩ ln u ϩ 2a ϩ u 2 ϩ C 2 u du ϭ ln u ϩ 2a ϩ u 2 ϩ C u a2 2a ϩ u Ϫ ln u ϩ 2a ϩ u 2 ϩ C 2 Ύ u2a 2a ϩ u ϩ a ϭ Ϫ ln ` `ϩC a u ϩ u2 44 Ύu du 2a ϩ u ϩ ln u ϩ 2a ϩ u 2 ϩ C u 45 Ύ (a 51 Ύ u2a 52 Ύu 53 Ύ (a Ύ a ϩ 2a ϩ u 2a ϩ u `ϩC du ϭ 2a ϩ u Ϫ a ln ` u u 40 Ύ 2a ϩ u u ϩ u2 43 a4 ln u ϩ 2a ϩ u 2 ϩ C 39 du ϭ Ϫ du Ύ 2a Ϫ Ύ 2a 42 u (a ϩ 2u 2) 2a ϩ u u 2a ϩ u du ϭ 41 ϩ u2 ϭ du 2a ϩ u du ϩ u 2) 3>2 ϭϪ ϭ 2a ϩ u a 2u u a 2a ϩ u ϩC ϩC Forms Involving 2a2 ؊ u2, a > 46 47 Ύ 2a Ϫ u du ϭ Ύu u a2 u 2a Ϫ u ϩ sinϪ1 ϩ C a 2 2a Ϫ u du ϭ u a4 u (2u Ϫ a 2)2a Ϫ u ϩ sinϪ1 ϩ C a 8 Ύ a ϩ 2a Ϫ u 2a Ϫ u `ϩC du ϭ 2a Ϫ u Ϫ a ln ` u u 49 Ύ u 2a Ϫ u du ϭ Ϫ 2a Ϫ u Ϫ sinϪ1 ϩ C u a u2 50 Ύ 2a 48 u du u a2 u ϭ Ϫ 2a Ϫ u ϩ sinϪ1 ϩ C 2 a 2 Ϫu a ϩ 2a Ϫ u ϭ Ϫ ln ` `ϩC a u Ϫu du 2 du 2a Ϫ u 2 ϭϪ 2a Ϫ u ϩ C a 2u Ϫ u 2)3>2 du u 3a u ϭ Ϫ (2u Ϫ 5a 2)2a Ϫ u ϩ sinϪ1 ϩ C a 8 54 Ύ (a 59 Ύ 2u du Ϫ u 2) 3>2 ϭ u a 2a Ϫ u ϩC Forms Involving 2u2 ؊ a2, a > 55 Ύ 2u 56 Ύ 57 58 Ϫ a du ϭ u a2 2u Ϫ a Ϫ ln ͉ u ϩ 2u Ϫ a ͉ ϩ C 2 u 2u Ϫ a du u a4 ϭ (2u Ϫ a 2)2u Ϫ a Ϫ ln ͉ u ϩ 2u Ϫ a ͉ ϩ C 8 Ύ 2u Ϫ a a du ϭ 2u Ϫ a Ϫ a cosϪ1 ϩC ͿuͿ u Ύ 2u Ϫ a u 2 2u Ϫ a ϩ ln ͉ u ϩ 2u Ϫ a ͉ ϩ C u du ϭ Ϫ du Ϫ a2 u du 60 Ύ 2u 61 Ύ u 2u 62 Ύ (u Ϫ a2 ϭ ln ͉ u ϩ 2u Ϫ a ͉ ϩ C ϭ du 2 Ϫ a2 du Ϫa ) www.elsolucionario.net 3>2 u a2 2u Ϫ a ϩ ln ͉ u ϩ 2u Ϫ a ͉ ϩ C 2 ϭ 2u Ϫ a ϭϪ a 2u ϩC u a 2u Ϫ a 2 ϩC Forms Involving sin u, cos u, tan u 63 Ύ sin 64 Ύ cos 65 Ύ tan 66 Ύ sin 67 Ύ cos 68 Ύ tan 69 Ύ sin sin(a Ϫ b)u sin(a ϩ b)u ϩ ϩC 2(a Ϫ b) 2(a ϩ b) u du ϭ 1 u Ϫ sin 2u ϩ C 73 Ύ cos au cos bu du ϭ u du ϭ 1 u ϩ sin 2u ϩ C 74 Ύ sin au cos bu du ϭ Ϫ u du ϭ tan u Ϫ u ϩ C 75 Ύ u sin u du ϭ sin u Ϫ u cos u ϩ C u du ϭ Ϫ (2 ϩ sin2 u) cos u ϩ C 76 Ύ u cos u du ϭ cos u ϩ u sin u ϩ C 3 u du ϭ (2 ϩ cos2 u) sin u ϩ C 77 Ύu u du ϭ tan2 u ϩ ln Ϳ cos u Ϳ ϩ C 78 Ύu n nϪ1 u du ϭ Ϫ sinnϪ1 u cos u ϩ n n 79 Ύ sin 70 Ύ nϪ1 cos u du ϭ cosnϪ1 u sin u ϩ n n 71 Ύ tan u du ϭ tannϪ1 u Ϫ nϪ1 72 Ύ sin au sin bu du ϭ n n Ύ tan nϪ2 Ύ Ύ sin cos nϪ2 nϪ2 u du cos(a ϩ b)u cos(a Ϫ b)u Ϫ ϩC 2(a Ϫ b) 2(a ϩ b) Ύ n sin u du ϭ Ϫu n cos u ϩ n u nϪ1 cos u du n cos u du ϭ u n sin u Ϫ n u nϪ1 sin u du Ύ n u cosm u du ϭϪ u du ϭ u du nϪ1 sinnϪ1 u cosmϩ1 u ϩ nϩm nϩm mϪ1 sinnϩ1 u cosmϪ1 u ϩ nϩm nϩm Ύ sin Ύ sin n nϪ2 u cosm u du u cosmϪ2 u du sin(a ϩ b)u sin(a Ϫ b)u Ϫ ϩC 2(a Ϫ b) 2(a ϩ b) Forms Involving cot u, sec u, csc u 80 Ύ cot 81 Ύ cot 82 Ύ sec 83 Ύ csc 3 u du ϭ Ϫcot u Ϫ u ϩ C 84 Ύ cot u du ϭ Ϫ cot 2u Ϫ ln Ϳ sin u Ϳ ϩ C 85 Ύ sec 86 Ύ csc 92 Ύ 93 Ύu u du ϭ 1 sec u tan u ϩ ln Ϳ sec u ϩ tan u Ϳ ϩ C 2 u du ϭ Ϫ1 cot nϪ1 u Ϫ nϪ1 n u du ϭ nϪ2 tan u sec nϪ2 u ϩ nϪ1 nϪ1 Ύ sec n u du ϭ Ϫ1 nϪ2 cot u csc nϪ2 u ϩ nϪ1 nϪ1 Ύ csc n Ύ cot nϪ2 u du nϪ2 u du nϪ2 u du 1 u du ϭ Ϫ csc u cot u ϩ ln Ϳ csc u Ϫ cot u Ϳ ϩ C 2 Forms Involving Inverse Trigonometric Functions 87 Ύ 88 Ύ cos 89 Ύ tan 90 91 Ύ sinϪ1 u du ϭ u sinϪ1 u ϩ 21 Ϫ u ϩ C Ϫ1 Ϫ1 u du ϭ u cosϪ1 u Ϫ 21 Ϫ u ϩ C u du ϭ u tanϪ1 u Ϫ Ύ u cos Ϫ1 u du ϭ n u2 ϩ u tanϪ1 u Ϫ ϩ C 2 sinϪ1 u du ϭ cu nϩ1 sinϪ1 u Ϫ nϩ1 u nϩ1 du Ύ 21 Ϫ u d , n ln(1 ϩ u 2) ϩ C 2u Ϫ u21 Ϫ u u sinϪ1 u du ϭ sinϪ1 u ϩ ϩC 4 2u Ϫ u21 Ϫ u cosϪ1 u Ϫ ϩC 4 u tanϪ1 u du ϭ 94 Ύu n cosϪ1 u du ϭ cu nϩ1 cosϪ1 u ϩ nϩ1 nϩ1 Ύ 21 Ϫ u d , u du n 95 Ύu n tanϪ1 u du ϭ cu nϩ1 tanϪ1 u Ϫ nϩ1 nϩ1 Openmirrors.com Ϫ1 Ύ 21 ϩ u d , u du n www.elsolucionario.net Ϫ1 Ϫ1 Forms Involving Exponential and Logarithmic Functions 96 Ύ ue 97 Ύu 98 Ύ 99 Ύe au n du ϭ eau du ϭ Ύ ϩ be 101 Ύ ln u du ϭ u ln u Ϫ u ϩ C eau (a sin bu Ϫ b cos bu) ϩ C a2 ϩ b 102 Ύ eau (a cos bu ϩ b sin bu) ϩ C a ϩ b2 103 Ύ u ln u du ϭ ln Ϳ ln u Ϳ ϩ C n au n u e Ϫ a a eau sin bu du ϭ au 100 (au Ϫ 1)eau ϩ C a2 cos bu du ϭ Ύu nϪ1 au e du du au ϭuϪ ln(1 ϩ beau) ϩ C a u nϩ1 [(n ϩ 1)ln u Ϫ 1] ϩ C (n ϩ 1)2 u n ln u du ϭ Forms Involving Hyperbolic Functions 104 Ύ sinh u du ϭ cosh u ϩ C 109 Ύ csch u du ϭ ln ͉ 105 Ύ cosh u du ϭ sinh u ϩ C 110 Ύ sech 106 Ύ u du ϭ ln cosh u ϩ C 111 Ύ csch 107 Ύ coth u du ϭ ln Ϳ sinh u Ϳ ϩ C 112 Ύ sech u u du ϭ Ϫsech u ϩ C 108 Ύ sech u du ϭ tan 113 Ύ csch u coth u du ϭ Ϫcsch u ϩ C 118 Ύ 22au Ϫ u 119 Ύ 22au Ϫ u 120 Ύ 22au Ϫ u Ϫ1Ϳ sinh u Ϳ ϩ C u͉ ϩ C u du ϭ u ϩ C u du ϭ Ϫcoth u ϩ C Forms Involving 22au2 ؊ u2, a > 114 Ύ 22au Ϫ u du ϭ uϪa 22au Ϫ u 2 ϩ 115 Ύ u22au Ϫ u du ϭ a2 aϪu cosϪ1 a bϩC a 2u Ϫ au Ϫ 3a 22au Ϫ u ϩ Ύ 22au Ϫ u aϪu du ϭ 22au Ϫ u ϩ a cosϪ1 a bϩC u a 117 Ύ 22au Ϫ u u du ϭ Ϫ ϭ cosϪ1 a ϭ Ϫ22au Ϫ u ϩ a cosϪ1 a ϭϪ u du u 2du a3 aϪu cosϪ1 a bϩC a 116 ϩ 121 Ύ u22au Ϫ u du 222au Ϫ u aϪu Ϫ cosϪ1 a bϩC u a www.elsolucionario.net aϪu bϩC a du aϪu bϩC a (u ϩ 3a) 22au Ϫ u 2 3a aϪu cosϪ1 a bϩC a ϭϪ 22au Ϫ u ϩC au I 10 Index Newton’s Law of Inertia, 1014 Newton’s method, 144 for approximating the zeros of a function, 389–390 iteration, 390 Newton algorithm, 390 Newton’s Second Law of Motion, 62, 194, 557, 565, 681, 708, 1010, 1019, 1250 Nonrigid matter, work moving, 551–553 Norm of a partition, 457 Normal component of acceleration, 1016 Normal line, 1108 to a curve, 183, 194 Normal vector, 945 Notation derivative, 166 function, 17 indefinite integral, 406 infinite limits at infinity, 339 inverse function, 75 for a line integral, 1265 logarithm, 90 for partial derivatives, 1056 second-order, 1062–1063 sigma, 431 nth partial sum, 743 nth term of a convergent series, 748–749 nth term test for Divergence, 748–749 nth-degree Maclaurin polynomial, 805 nth-degree Taylor polynomial, 805 Number, critical, 285 Number e, 87, 182, A22 limit involving, 249, 338, A25 Numerical integration, 480–491 Simpson’s Rule, 487 Trapezoidal Rule, 482 Octants, 910 Odd function, 22, 36, 230 integration of, 471 One-sided limits, 104, 105 One-to-one correspondence, A1 One-to-one function, 77 One-to-one transformation, 1205 Open intervals, 138, A1 Open region, 1048, 1255 Optimization problems, guidelines for solving, 362 See also Maximum; Minimum Orbits of the planets, 889, 1021 Order of a differential equation, 664 Ordered triple, 909 Orientable surface, 1291 Orientation of a curve, 845, 1243 Oriented surface, 1291 Origin of a polar coordinate system, 863 on the real number line, A1 Orthogonal curves, 243 Orthogonal planes, 947 Orthogonal surfaces, 1112 Orthogonal trajectory, 674–675 Orthogonal vectors, 922 Osculating circle, 1003 Osculating plane, 1022 Pappus, Theorem of, 573 Parabola, 61, 824 axis of, 825, 826 directrix of, 824, 826 eccentricity of, 884, 886, 888 focus of, 824, 826 polar equation of, 886–887 reflective property of, 828 shifted, 836 standard equation of, 825 vertex of, 61–62, 824, 826 Parabolic cylinder, 955, 956 Parabolic spiral, 883 Paraboloid, 954, 960–961, 964, 965 Parallel lines, 10 in space, 943 Parallel planes, 947 Parallel vectors, 898, 935 Parallelogram Law, 899, 901 Parameter, 844, 846 arc length, 997 Parametric equations, 844, 1275 graph of, 845 of a line, 941 for a surface, 1275 of a trajectory, 845 Parametric surface(s), 1275 area of, 1283 evaluating surface integrals for, 1290, 1295 representation of, 1277–1280 smooth, 1280 surface area of, 1283 tangent plane to, 1280 Partial derivative(s) of a function of more than two variables, 1061 of a function of two variables, 1056 notations for, 1056, 1062–1063 second-order, 1062 Partial differential equation, 1063 Partial fractions, 622 Partial integration, 1154 Partial sums, sequence of, 743 www.elsolucionario.net Openmirrors.com Particular solution of a differential equation, 410, 665 Partition, 456 norm, 457 polar, 1165 regular, 437, 457, 1145, 1184, 1186, 1199 Pascal, Blaise (1623–1662), 252, 559 Pascal’s Principle, 559 Patch, 1178 Path, 1254 Percentage error, 266, 267 Perihelion, 889–890, 891 Period, 35 Periodic function, 33, 175 Permeability of free space, 820 Permittivity of free space, 556, 820, 1296 Perpendicular lines, 11 Perpendicular vectors, 922 Perpetuity, 751 Phase curves, 715 Phase plane, 715 Phase portrait, 715 Phase trajectories, 715 Piecewise defined function, 21–22, 103, 401, 469 Piecewise-smooth curve, 1240 Piriform, 510 Pitiscus, Bartholomeo (1561–1613), 35 Planar lamina, 569 center of mass of, 569 moment of, 569 Plane center of mass of a system in a, 567 coordinate, 900 distance between a point and, 949 osculating, 1022 phase, 715 tangent, 1108, 1110 to approximate a surface, 1110 to a parametric surface, 1280 Plane curve, 845, 980 Plane(s) in space angle between two, 948 equation of, 945, 947 orthogonal, 947 parallel, 947 Point of diminishing returns, 321 fixed, 162, 304 inflection, 317, 318, 347 of intersection, 55, 880–881 Point(s) in space coordinates of, 910 distance between, 911 Point-slope equation of a line, Index Poiseuille, Jean, 271 Poiseuille’s Law, 292, 377, 1042 Polar axis, 863 symmetry with respect to, 867 Polar coordinate system, 863–864 origin of, 863 polar axis of, 863 pole, 863 Polar coordinates, 863 area in, 874–877 area of a surface of revolution in, 879 conic sections in, 884 converting rectangular coordinates to, 865 points of intersection of graphs in, 880–881 transformation of a double integral to, 1165, 1167 Polar curve arc length of, 878 area bounded by, 875, 877 symmetry in, 867 tangent line to, 870 Polar equations, 865 of conics, 886 graph of, 865–867 tangent lines at the pole, 871 tests for symmetry, 867 Polar partition, 1165 Polar rectangle, 1164 Polar region, area of, 874–877 Pole (or origin), 863 symmetry with respect to, 867 Polygonal curve, 485 Polynomial Maclaurin, 805 Taylor, 805 Polynomial approximation, 805 centered at c, 805 Taylor’s Theorem, 807 Polynomial function, 26, 59 constant term, 59 continuity of, 139 degree, 59 limit of, 114 of two variables, 1050 Population density, 1163 Position function, 100, 197, 404 for a projectile, 1011 Position vector, 900, 913, 980 Positive orientation, 1291 for a simple closed curve, 1264 of a surface, 1292 Potential energy, 1262 Potential function, 1223, 1253, 1258–1261 Power function, 65, 218 Power Rule, 177, 178, 191, 248 Power series, 781 centered at c, 781 coefficients of, 781 convergence of, 782, 790, A16 for cosine, 795 differentiation of, 786 for an exponential function, 795 integration of, 786 interval of convergence of, 783, 786 properties of functions defined by, 782–783, 785–786 radius of convergence of, 783, 786 representation of functions as, 795–796 for sine, 795 Predator-prey equations, 713 critical points of, 713 equilibrium points of, 713 phase curves of, 715 solution of, 713 Predator-prey model, 228, 250, 478, 712–717 Pressure exerted by a fluid, 557–564 Principal unit normal vector, 1014 Product Law for limits, 113, A7–A8 Product Rule, 186 Profit function, 202 Projectile, motion of, 1010 Prolate cycloid, 853 Properties of absolute value, A4 of a continuous function of two variables, 1050 of a convergent series, 749 of the cross product, 936 of the dot product, 920 of double integrals, 1150–1151 of the gradient, 1097 of inverse trigonometric functions, 81 relating the natural logarithmic function and the natural exponential function, 91 of a sequence, 737 of vector operations, 902 p-series, 755, 776 Pure resonance, 214, 388 Pursuit curve, 547, 576, 650–651 Pythagorean identities, 37 Quadratic factors, irreducible distinct, 626 repeated, 627 Quadratic function, 61 Quadric surface, 952, 957 aids for sketching, 963–965 www.elsolucionario.net I 11 cone, 960, 964 cylinder, 955 ellipsoid, 957–958, 963 hyperboloid, 958, 959, 963 paraboloid, 954, 961, 964, 965 Quotient Law of limits, 113, A8–A9 Quotient Rule, 188 Radial coordinate, 863 Radian measure of angles, 28 Radical, limit of a function involving a, 113, A9–A10 Radius of convergence, 783, 786 of curvature, 1003 of gyration, 1176 Range of a function, 17, 1030 Rate of change average, 153 derivative as, 164–165 instantaneous, 154, 164 Ratio Test, 773, 777 Rational function, 65 continuity of, 139 finding limits at infinity of, 336 limit of, 115 of two variables, 1032, 1050 Reaction rate, 677 Real number line, A1 Real number system, Completeness Axiom for, 738 Real-valued function of a real variable x, 17 Rearrangement of series, 777 Rectangle, polar, 1164 Rectangular coordinates converting to cylindrical coordinates, 968 converting to polar coordinates, 865 converting to spherical coordinates, 971 Rectilinear motion, 257, 426 Recursively defined sequence, 728 Reduction formulas, 600–601 Reference angle, 32 Reflecting a function, 45–48 Reflection, 45–48 about the line y ϭ x, 76 about the x-axis, 45 about the y-axis, 45 Reflective property of an ellipse, 831–832 of inverse functions, 76 of a parabola, 828 Refraction, 377, 1136 I 12 Index Region area of, 428, 438, 1155 centroid of, 572 between two curves, 499 closed, 1048 connected, 1255 multiply-connected, 1269 open, 1048, 1255 simple or elementary, 1265 simply-connected, 1257 Region of integration r-simple region, 1201 r-simple, 1167 u-simple, 1167, 1170 x-simple, 1157, 1158 y-simple, 1157, 1158 z-simple, 1186–1187, 1196 Regression line, 56, 377, 1122 Regular partition, 437, 457, 1145, 1184, 1186, 1199 Related rates problems, 251–252 guidelines for solving, 252 Relative error, 266, 267 Relative extrema finding, 309, 356–357 First Derivative Test for, 308 of a function, 284 of two variables, 1112, 1116 Second Derivative Test for, 321 Relative growth rate, 690 Relative maximum, 284, 347, 1112, 1116 First Derivative Test for, 308 Second Derivative Test for, 321 Relative maximum value, 1112, 1116 Relative minimum, 284, 347 First Derivative Test for, 308 Second Derivative Test for, 321 Relative minimum value, 284, 1112, 1116 Remainder estimates, 808 alternating series, 768 Removable discontinuity, 136 Representation of functions as power series, 795–796 Resonance curve, 376 Resonance, pure, 214, 388 Rest energy, 800 Restitution, coefficient of, 751 Revenue function, 202 Revolution axis of, 511 solid of, 511 surface of, 542 Rhodenea, 883 rho-simple region, 1201 Riccati equation, 723 Riemann, Bernhard (1826–1866), 445 Riemann sum, 445, 1148, 1149, 1185, 1186 Riemann zeta function, 757 Right circular cylinder, 365 Right strophoid, 510 Right-hand limit of a function, 104 Right-handed coordinate systems, 910 Rivlin’s equation, 402 Roberval, Gilles Persone de (1602–1675), 500 Rolle, Michel (1652–1719), 296 Rolle’s Theorem, 296 Root Law for limits, 113, 114, A9–A10 Root of an equation, 389 Newton’s method of approximating, 389–396 Root Test, 775, 777 Rose curve, 869, 872, 873 r-simple region of integration, 1167 Rule(s) of differentiation for vector functions, 991 of integration, 408 Simpson’s, 487 of summation, 433 Trapezoidal, 482 for vector addition and scalar multiplication, 902 Rulings of a cylinder, 955 Saddle point, 1115, 1116 Salk, Jonas, 313 Scalar, 898 Scalar components of a vector, 900 Scalar multiple, 898 Scalar multiplication, 901, 902, 913 Scalar product of two vectors, 920 Scalar projection, 925 Scalar triple product, 937–938 Scatter diagram, 56, 1122 Scatter plot, 60 Schwarz’s inequality, 496, 1218 Secant function, 34 derivative of, 210, 220 graph of, 34 integral of, 421 inverse of, 79, 80, 140 limit of, 118 Secant line, 154 slope of, 152, 154 Second derivative, 191, 323–324 of a vector function, 990 Second Derivative Test, 321, 1116 Second Fundamental Theorem of Calculus, 467 Second moment, 1175 Second-order partial derivatives, 1062 www.elsolucionario.net Openmirrors.com Separable differential equation, 666 Sequence, 726 bounded, 735 convergent, 729 decreasing, 734 divergent, 728 Fibonacci, 741 greatest lower bound, 738 increasing, 734 least upper bound, 738 limit laws for, 730 limit of, 728, 729 monotone convergence theorem for, 735, 738 monotonic, 734 nth term of, 726 of partial sums, 743 recursively defined, 728 Squeeze Theorem for, 732 terms of, 726 Series, 742 absolutely convergent, 771 alternating, 765, 777 error estimate in approximating, 768 binomial, 794 Comparison Test, 758 conditionally convergent, 772, 778 convergent, 743 properties of convergent series, 749 divergence of, 743 nth term test for, 748–749 geometric, 745, 776 convergence of, 745 harmonic, 747, 771 alternating, 765, 766 infinite, 742 properties of, 749 Integral Test, 753 Maclaurin, 791, 792 nth partial sum, 743 nth term of, 742 nth term of convergent, 748–749 power, 781 convergence of, 782 p-series, 755 Ratio Test, 773 Root Test, 775, 777 sum of, 743 summary of tests for, 776–777 Taylor, 790 telescoping, 744, 776 terms of, 742 Shell method, 528, 531 Shifted conics, 835–838 Sigma notation, 431 Significance, loss of, 107 Index Simple curve, 1257 Simple region, 1265 Simple solid region, 1300 Simply-connected region, 1257 Simpson, Thomas (1710–1760), 487 Simpson’s Rule, 487 error bounds for, 487 Sine function, 34 derivative of, 208, 210, 220 graph of, 34 inverse of, 78, 79, 80, 140 limit of, 118, 120 series for, 795 Sink, 1306 Skew lines, 944 Slant asymptotes, 354–355 Slope(s), field, 679 of a horizontal line, 11 of a line, of parallel lines, 10 of perpendicular lines, 11 of a secant line, 152, 154 of a tangent line, 152 of a vertical line, Slope-intercept, Slope-intercept equation of a line, 7, Slug, 565 Smooth curve, 536, 856, 997 length of, 536–540, 857 Solenoidal, 1230 Solid of revolution, 511 volume of, 512 Solution(s) curves, 665 of a differential equation, 410, 664 Euler’s Method, 682, 684 first-order linear, 702 general, 410, 665 particular, 410, 665 equilibrium, 694 of an inequality, A3 of predator-prey equations, 713 Solution set, A3, A4 Somerville, Mary Fairfax (1780–1872), 600 Source, 1306 Space curve, 980 arc length of, 996 smooth, 1243 Special theory of relativity, 66, 124, 332, 800 Speed, 198, 1006, 1007 Sphere, 912 equation of, 912 surface area of, 858–859 volume of, 514 Spherical coordinate system, 970 converting to rectangular coordinates, 971 triple integral in, 1200 Spherical wedge, 1199 Spherical zone, 547 Spiral, 873 cornu, 861, 1005 epi-spiral, 874, 894 logarithmic, 873 parabolic, 883 of Poinsot, 894 toroidal, 984 Spring constant, 59, 61, 550 Square root function, 17, 18 Square-wave function, 112 Squeeze Theorem, 119, A10 for sequences, 732 Standard basis vectors, 903, 914–915 Standard equation of an ellipse, 830 of a hyperbola, 834 of a parabola, 825 of a sphere, 912 Static friction, coefficient of, 295 Steady-state current, 707 Steepest descent, path of, 244 Stefan’s Law, 678 Step function, 485 Stiffness (spring constant), 59, 61, 550 Stokes, George G (1819–1903), 1310 Stokes’ Theorem, 1308, 1313 Streamlines, 1220 Stress-strain curve, 505 Substitution integration by, 417 method of, 161 trigonometric, 614, 615, 637 Subtraction formulas for sine and cosine, 36, 37 Sum of an alternating series, approximating, 767–769 continuity of, 138 of an infinite series, 743 integral of, 454 kth term of, 431 Riemann, 445, 1148, 1149, 1185, 1186 Rule, 179 Sum Law for limits, 113 Sum of vectors, 899 Summary of integration techniques, 636–637 of tests for series, 776–777 Summation formulas, 434 www.elsolucionario.net I 13 index of, 431, 432 rules of, 433 Supply equation, 712 Surface closed, 1300 level, 1038, 1108 orientable, 1291 parametric, 1275 parametric representations of, 1277–1280 quadric, 952, 957 smooth, 858, 997 Surface area, 542–545, 1178–1182 of a parametric surface, 1283 of a sphere, 858–859 Surface integral(s), 1288 evaluating for graphs, 1293 for parametric surfaces, 1290, 1295 summary of, 1312, 1313 of a vector field, 1292 Surface of revolution, 542 area of, 544, 858 parametric representation of, 858 polar representation of, 879 Swallowtail catastrophe curve, 853, 1273 Symmetric equations of a line in space, 942 Symmetry with respect to the polar axis, 867 with respect to the pole, 867 with respect to the vertical line, 867 with respect to the x-axis, 22 with respect to the y-axis, 22 tests for, 867 Table of differentiation formulas, 407 Table of Integrals, 407 use of, 632–635 Tangent function derivative of, 210, 220 graph of, 34 integral of, 421 inverse of, 79, 80, 140 limit of, 118 Tangent line(s), 152 approximation, 267 to a curve, 149, 853–854 horizontal, 180, 855 to a parametric curve, 853 to a polar curve, 870 at the pole, 871 slope of, 152 vertical, 169, 171, 855 Tangent plane, 1108 to approximate a surface, 1110 to a parametric surface, 1280 to a surface, 1104 I 14 Index Tangent vector, 989 unit, 989, 1014 Tangential component of acceleration, 1016 Tautochrone problem, 850, 851 Taylor, Brook (1685–1731), 790 Taylor polynomial, 805 error in approximating, 808 Taylor remainder of f at c, 808 Taylor series, 790 convergence of, 791–792 guidelines for finding, 795–796 Taylor’s Theorem, 807 Techniques of integration, summary, 636–637 Telescoping series, 744, 776 Term of a sequence, 726 Term of a series, 742 Terminal point of a curve, 845 of a vector, 898 Terminal side of an angle, 27 Terminal velocity, 345, 677 Test(s) for concavity, 316 conservative vector field, 1258, 1260 for convergence Alternating Series Test, 765 Comparison Test, 758 geometric series, 745 Integral Test, 753 Limit Comparison Test, 761 p-series, 755 Ratio Test, 773, 777 Root Test, 775, 777 summary of, 776–777 for coplanar vectors, 938 for divergence, 748, 776 for even and odd functions, 22 for increasing and decreasing function, 307 for parallel vectors, 935 for symmetry, 867 Theorem of Pappus, 573 u-simple region of integration, 1167, 1170 Third derivative, 191 Three-dimensional rectangular coordinate system, 910 Three-leaved rose, 873 Thurstone Learning Curve, 97 TNB frame, 1023 Topographic map, 243, 1035–1037 Toroidal spiral, 984 Torque, 934–935 Torricelli, Evangelista (1608–1647), 517 Torricelli’s law, 294 Torricelli’s Trumpet, 654, 1280 Torus, 534 surface area of, 1286 volume of, 573 Total differential, 1071 Trace of a surface, 953, 958, 959, 960 Tractrix, 661 Trajectory(ies), 674, 845 phase, 715 of a projectile, 1011 Transcendental functions exponential functions, 84–89, A22 hyperbolic functions, 576 inverse functions, 75 inverse trigonometric functions, 79 logarithmic functions, 89–92, 245–249, A19 Transformation, 17, 1205 inverse, 1205 Jacobian, 1207 one-to-one, 1205 Transformation of a graph of a function, 43–48 basic types, 44, 45 horizontal shift, 43–44 horizontal stretching and compressing, 44–45, 47 reflection about the x-axis, 45 reflection about the y-axis, 45 vertical shift, 44 vertical stretching and compressing, 44, 45, 47 Transverse axis, of a hyperbola, 832, 833 Trapezoidal Rule, 482 error in, 483 Triangle area of a, 935–936 inequality, 132, A5 Trigonometric functions, 31 continuity of, 139 derivatives of, 210, 220 graphs of, 34 integrals of, 421 inverse, 79 derivatives of, 237 integrals involving, 422 properties of, 81 limits of, 118 Trigonometric identities, 37, 610 Trigonometric integrals, 604–613 guidelines for evaluating, 606–607, 608 Trigonometric substitution, 614, 615 Triple integral, 1185 change of variables in, 1212 in cylindrical coordinates, 1197 over a general bounded region, 1186 www.elsolucionario.net Openmirrors.com over a rectangular box, 1184 in spherical coordinates, 1200 Triple product, 938 Trochoid, 852 Tschirnhausen’s cubic, 242, 524 Twisted cubic, 1001 Two-sided limit, 104, 105 Unit circle, 28, 31 Unit inner normal, 1291 Unit normal vector, 1014 Unit outer normal, 1291 Unit tangent vector, 989, 1014 Unit vector, 902 angular form of, 904 Unrestricted growth, 690 Upper bound least, 738 of a sequence, 738 van der Waals equation, 195, 557 Variable dependent, 18 force, 550 independent, 18 Vector(s), 898 acceleration, 1006–1007 addition, 899, 901–902, 913 angle between, 921 basis, 903, 914 components, 900, 913 coplanar, 938 cross product of, 930 difference of, 899, 901 direction angles of, 923 direction cosines of, 923 displacement, 927 dot product of, 920, 922 equal, 898, 913 force, 903 gradient, 1095 horizontal component of, 903 initial point, 898 inner product of, 920 length of, 901, 913 magnitude of, 898, 901 normal, 945 operations, 902 orthogonal, 922 parallel, 898, 935 position, 900, 913, 980 principal unit normal, 1014 projection of, 925 scalar multiplication, 901, 913 scalar product of, 920 scalar projection, 925 Index scalar triple product, 937–938 standard basis, 903, 914–915 sum, 899 tangent, 989 terminal point, 898 unit, 902 angular form of, 904 unit tangent, 989, 1014 velocity, 1006, 1007 vertical component of, 903 wind velocity, 904 zero, 898 Vector field, 1220, 1221 conservative, 1223, 1253, 1255–1258, 1260 curl of, 1232 divergence of, 1228, 1234 flux of a, 1292 gradient, 1222 gravitational, 1252 incompressible, 1230, 1306 irrotational, 1234 line integral of, 1246 potential function for, 1223, 1258–1261 solenoidal, 1230 surface integral of, 1292, 1293, 1295 Vector function(s), 980, 981 continuity of, 985 definite integral of, 992 differentiation of, 988, 989 rules of, 991 integration of, 992 limit of, 984 parametric equations of, 980 Vector-valued function See Vector function(s) Velocity, 1007 average, 100, 463 escape, 346, 424, 655 field, 1221 instantaneous, 101 of a projectile, 1010 terminal, 345, 677 vector, 1006, 1007 Velocity constant, 388 Velocity curve, 165 Velocity function, 197 Vertex, 27 of an ellipse, 829, 830 of a hyperbola, 832, 837 of a parabola, 61–62, 824, 826 of the paraboloid, 961 Vertical asymptote, 331 Vertical line(s), 2, 5, symmetry with respect to, 867 Vertical Line Test, 21 Vertical shift of a graph of a function, 43, 44 Vertical stretching and compressing, 44, 45, 47 Vertical tangent line, 169, 171, 318, 855 Vertical vector components, 903 Volterra, Vito (1860–1940), 713 Volume by double integrals, 1157, 1158 under the graph of z ϭ f(x, y), 1146 by triple integrals, 1190 Volume of a solid, 512 by cross sections, 519 disk method, 513, 515 shell method, 528, 531 Theorem of Pappus and, 573 washer method, 516 www.elsolucionario.net I 15 von Bertalanffy function, 94, 328 von Bertalanffy growth model, 678 von Bertalanffy, Ludwig (1901–1972), 666 Wallis, John (1616–1703), 106, 538 Washer method, 516 Weierstrass, Karl Theodor Wilhelm (1815–1897), 645 Wessel, Caspar (1745–1818), 902 Wilson lot-size formula, 1043 Witch of Agnesi, 852 Work, 548, 549, 550, 553, 927, 1246 x-axis moment about, 567 reflection about, 45 x-intercept, x-simple region, 1157, 1188 xy-plane, 910 xz-plane, 910 y-axis moment about, 567 reflection about, 45 y-intercept, 7, y-simple region, 1157, 1188 yz-plane, 910 Zero of a function, 54 approximating bisection method, 144 with Newton’s method, 390–391 of a continuous function, existence of, 144 determining number of, 302 Zero vector, 898 z-simple region, 1186–1187, 1196 www.elsolucionario.net Openmirrors.com Inequalities and Absolute Value ALGEBRA If a Ͻ b and b Ͻ c, then a Ͻ c Arithmetic Operations If a Ͻ b, then a ϩ c Ͻ b ϩ c aϩb a b ϭ ϩ c c c If a Ͻ b and c Ͼ 0, then ca Ͻ cb a c ad ϩ bc ϩ ϭ b d bd If a Ͼ 0, then If a Ͻ b and c Ͻ 0, then ca Ͼ cb a a b b a d ad ϭ a ba b ϭ b c bc c a b d Ϳ x Ϳ ϭ a if and only if x ϭ a or Ϫa a ͿxͿ Ͻ a if and only if ( Ϫa Exponents and Radicals x mx n ϭ x mϩn x Ϫn xm ϭ x mϪn xn (x m) n ϭ x mn n ϭ n x n n n n n 1xy ϭ 1x 1y Ϫa Ͻ x Ͻ a ) a x x Ͼ a or ) Ϫa ( a n x Ͻ Ϫa x GEOMETRY n m x n>m ϭ 2x n x Ϳ x Ϳ Ͼ a if and only if x x a b ϭ n y y (xy) ϭ x y n x ϭ Ϫa x 1x ϭ n By 1y n Geometric Formulas Factoring Formulas for area A, circumference C, and volume V: x Ϫ y ϭ (x Ϫ y)(x ϩ y) 2 Triangle Circle Sector of Circle x Ϫ y ϭ (x Ϫ y)(x ϩ xy ϩ y 2) x ϩ y ϭ (x ϩ y)(x Ϫ xy ϩ y 2) a h r r b Binomial Theorem A ϭ 12 bh ϭ 12 ab sin u (x ϩ y)2 ϭ x ϩ 2xy ϩ y (x Ϫ y)2 ϭ x Ϫ 2xy ϩ y (x ϩ y)3 ϭ x ϩ 3x 2y ϩ 3xy ϩ y 2r u C ϭ 2pr s ϭ ru a h b b A ϭ bh Sphere n(n Ϫ 1) p (n Ϫ k ϩ 1) n where a b ϭ k 1ؒ2ؒ3ؒpؒk Aϭ Cylinder Ϫb Ϯ 2b Ϫ 4ac 2a (a ϩ b)h Cone r r h Quadratic Formula If ax ϩ bx ϩ c ϭ 0, then x ϭ (u in radians) Trapezoid h n(n Ϫ 1) nϪ2 x y ϩp n ϩ a bx nϪky k ϩ p ϩ nxy nϪ1 ϩ y n k A ϭ pr Parallelogram (x Ϫ y)3 ϭ x Ϫ 3x 2y ϩ 3xy Ϫ y (x ϩ y)n ϭ x n ϩ nx nϪ1y ϩ s V ϭ 43 pr A ϭ 4pr www.elsolucionario.net V ϭ pr 2h h r V ϭ 13 pr 2h A ϭ pr2r ϩ h2 (lateral surface area) Distance and Midpoint Formulas Trigonometric Functions Distance between P1 ϭ (x 1,y1) and P2 ϭ (x 2, y2) : P(x, y) d ϭ 2(x Ϫ x 1) ϩ (y2 Ϫ y1)2 r Midpoint of P1P2: a x ϩ x y1 ϩ y2 , b 2 Lines Slope of the line through P1 ϭ (x 1, y1) and P2 ϭ (x 2, y2): mϭ y Ϫ y1 x2 Ϫ x1 y x sin u ϭ y r cos u ϭ x r tan u ϭ y x csc u ϭ r y sec u ϭ r x cot u ϭ x y y Slope-intercept equation of the line with slope m and y-intercept b: (Ϫ 21 , √23 ) (0, 1) ( 21 , √23 ) 90Њ √2 , √ (Ϫ √22, √22 ) 3 23 ( 2) 120Њ 60Њ 45Њ ( √3 , ) (Ϫ√23, 12 ) 56 135Њ 2 y ϭ mx ϩ b Point-slope equation of the line through P1 ϭ (x 1, y1) with slope m: y Ϫ y1 ϭ m(x Ϫ x 1) 30Њ 150Њ 0Њ 360Њ 2 (1, 0) x (Ϫ1, 0) 180Њ 330Њ 11 √3 7 210Њ ,Ϫ1 315Њ 225Њ Ϫ√ , Ϫ 6 2 2 5 240Њ 300Њ 7 5 √2 , Ϫ √2 , √2 4 √ Ϫ Ϫ 2 3 270Њ 3 2 , Ϫ √3 √ Ϫ ,Ϫ (0, Ϫ1) 2 2 Equation of a Circle ( Circle with center (h, k) and radius r: ( ) ( (x Ϫ h) ϩ (y Ϫ k)2 ϭ r ) ( ) ( ( ) ) ) TRIGONOMETRY Graphs of Trigonometric Functions Angle Measurement r r y s y=ß x π 2π x π Ϫ1 p radians ϭ 180° 1° ϭ p rad 180 rad ϭ 180° p y Right Triangle Definitions hyp y=ỗ x y y= x y= x y opp 2π x Ϫ1 π 2π x π Ϫ1 2π x adj sin u ϭ opp hyp cos u ϭ adj hyp tan u ϭ opp adj csc u ϭ hyp opp sec u ϭ hyp adj cot u ϭ adj opp y y=† x y=ˇ x 2π π www.elsolucionario.net Openmirrors.com y x π 2π x Fundamental Identities Addition and Subtraction Formulas cos u sin(x ϩ y) ϭ sin x cos y ϩ cos x sin y sin u tan u ϭ cos u cos u cot u ϭ sin u cos(x ϩ y) ϭ cos x cos y Ϫ sin x sin y sin u ϩ cos2 u ϭ 1 ϩ tan2 u ϭ sec u ϩ cot u ϭ csc u sin(Ϫu) ϭ Ϫsin u cos(Ϫu) ϭ cos u tan(Ϫu) ϭ Ϫtan u p sina Ϫ ub ϭ cos u p cosa Ϫ ub ϭ sin u csc u ϭ tana sin u sec u ϭ sin(x Ϫ y) ϭ sin x cos y Ϫ cos x sin y cos(x Ϫ y) ϭ cos x cos y ϩ sin x sin y tan x ϩ tan y Ϫ tan x tan y tan(x Ϫ y) ϭ tan x Ϫ tan y ϩ tan x tan y Double-Angle Formulas p Ϫ ub ϭ cot u sin 2x ϭ sin x cos x cos 2x ϭ cos2 x Ϫ sin2 x ϭ cos2 x Ϫ ϭ Ϫ sin2 x The Law of Sines B sin C sin A sin B ϭ ϭ a c b The Law of Cosines tan 2x ϭ C A tan x Ϫ tan2 x a c a ϭ b ϩ c2 Ϫ 2bc cos A tan(x ϩ y) ϭ b Half-Angle Formulas sin2 x ϭ Ϫ cos 2x www.elsolucionario.net cos2 x ϭ ϩ cos 2x www.elsolucionario.net Openmirrors.com INDEX OF APPLICATIONS ENGINEERING AND PHYSICAL SCIENCES Acceleration of a car, 207, 414, 477 Action of an impulse on an object, 147 Adiabatic processes, 259 Air pollution, 293, 312, 327, 359, 478, 490, 620 Air purification, 477 Air temperature and altitude, 173 Air traffic control, 916 Aircraft fleet damage rate (“bathtub curve”), 305 Airflow around an airfoil in a wind tunnel, 1220 Alternating current intensity, 612 Ampere’s Law, 1314 Angular velocity, 940 Angular velocity of a particle, 1237 Arch of a bridge, 842 Artificial satellites, 1023 Atmospheric pressure, 94, 203, 251 Autocatalytic chemical reaction, 293 Automobile headlight design, 828 Average acceleration of a car, 477 Average illumination, 621 Average mass of an electron, 640 Average power in AC circuits, 655 Average temperature in a room, 1196 Average tension on a cable, 584 Average velocity of a motorcycle, 157 Avoiding a collision, 373 Ballast dropped from a balloon, 413, 414, 423 Bond angle of a molecule of methane, 929 Cantilever beam, 312 Capacity of a fuel tank, 521, 525 Capacity of an artificial lake, 535 Charge in an electric current, 603 Chemical mixtures, 360 Child-Langmuir Law, 271 Clepsydra or water clock, 534 Coast Guard launch, 185 Coast Guard patrol search mission, 260 Coefficient of restitution, 751 Colliding billiard balls, 147 Collision of two particles, 414 Concentration of carbon monoxide in the air, 817 Constructing a cylindrical barrel, 400 Constructing a marina, 373 Constructing a new road, 195 Constructing a restaurant between two jetties, 376 Constructing a storage tank, 271 Constructing an AC transformer, 294 Contour maps, 1036, 1040, 1060, 1065, 1066, 1103, 1152 Coriolis acceleration, 1027 Coulomb’s Law, 556, 632, 919, 1296 Course taken by a yacht, 1091 Current in an electric circuit, 388 Cycloid motion, 1013 Cylinder pressure, 195 Damped harmonic motion, 585, 602, 603 Deflection of a beam, 206 Designing a conical figure, 372 Designing a grain silo, 372 Designing a loudspeaker, 370 Diffusion, 603 Discharging water from a tank, 678 Distance between two aircraft, 375 Distance between two cars, 498, 506 Distance covered by a cyclist, 441 Distance traveled by a bouncing ball, 742, 747, 751 Distance traveled by a tanker, 538 Diving, 205 Docking a boat, 258 Doppler effect, 1044, 1090 Earthquake’s effect on a structure, 272 Effect of stopping on average speed, 184 Elastic curve of a beam, 148, 415 Elastic deformation of a long beam, 654 Elastic hysteresis, 505 Electric charge, 1299 Electric charge on a disk, 1177 Electric charge over a region, 1173 Electric circuits, 259, 706, 707, 711, 712 Electric field, 1222 Electric field induced by a line charge, 820 Electric potential, 230, 1067, 1100 Electrical force of a conductor, 375 Electrostatic potential, 1058 Emptying a storage tank, 555 Engine efficiency, 1068 Equipotential curves, 1043 www.elsolucionario.net Error in calculating the capacity of a storage tank, 1072 Error in calculating the power of a battery, 1079 Error in computing the range of a projectile, 1072 Error in measuring an electric current, 272 Error in measuring the period of a pendulum, 1079 Error in measuring the resistance of a circuit, 1079 Escape velocity, 346, 655 Estimating the flow of water in a river, 509 Estimating the surface area of the moon, 266 Explosion of a gas main, 205 Fabricating corrugated metal sheets, 612 Falling raindrop, 677 Finding the true course and ground speed of an airplane, 904, 909 Flight of a model rocket, 312, 414 Flight of a projectile, 424, 1080 Flight of a rocket, 250 Flight path of a plane, 196, 361, 548, 852 Flight path of two airplanes, 944 Flow of fluid/liquid, 1299 Flow of water in a canal, 477 Fluid pressure, 558, 560–561, 562, 564 Fluid pressure on the gate of a dam, 561, 564 Flux of an electric field, 1299, 1304 Force acting on a body in an elliptical path, 295 Force exerted by a charge distribution, 803 Force exerted by an electric charge, 251 Force exerted on the viewing port of a submersible, 620 Force generated by a centrifuge, 1043 Force of attraction of a cylinder on a point, 803 Force on a proton moving through a magnetic field, 940 Forensic science, 228 Formaldehyde levels in homes, 194 Global warming, 70, 327, 478 Gravitational field, 1221, 1252 Gravitational force, 161, 207, 803 Great circles, 1277 (continued) Index of Applications (continued) Halley’s Law, 313 Heat generated by an alternating current, 612 Heat-seeking missiles, 585 Heaviside function, 103, 112, 134, 137, 169 Hitting a moving target, 1026 Hooke’s Law, 59, 60, 61, 377, 551 Hot-air balloon, 444 Hydrostatic force on a window, 618 Ideal gas law, 1043, 1067 Isothermal curves, 1140 Jumping while on Mars, 414 Lambert’s Law of Absorption, 676 Land reclamation, 535 Launching a lunar landing module, 556 Leaking bucket, 555 Length of a power line, 582 Locating a radio station, 1121 Magnitude of a magnetic field, 1251 Mass and center of mass of a wire, 1241, 1249, 1250 Mass of a body moving near the speed of light, 229 Mass of a moving particle, 259, 345 Mass of a ramp, 1299 Maximizing drainage capacity, 372 Maximizing light intensity, 400 Maximum deflection of a beam, 289, 293, 295 Maximum error in calculating centrifugal force, 1077 Maximum power output, 373 Maximum stiffness of a beam, 372 Maximum strength of a beam, 371 Maxwell’s equations, 1237 Measuring the height of a derrick, 272 Monte Carlo integration, 1217 Motion of a projectile, 1011, 1013, 1014, 1017, 1023, 1026 Motion of a submersible, 478 Motorcyclist’s turn, 294 Newton’s Law of Attraction, 148 Newton’s Law of Gravitation, 65, 185, 918, 1043, 1221 Newton’s Law of Inertia, 1014 Newton’s Laws of Heating and Cooling, 477, 672, 677, 720 Newton’s Second Law of Motion, 194, 1250 Newton’s wine barrel, 524 Norman window, 72, 372 Nuclear fallout, 676 Oil spills, 42, 98, 257 Optics, 207 Optimal driving speed, 371 Optimal illumination, 374 Orbit of a satellite, 230 Orbit of Halley’s comet, 889 Orbit of Mercury, 891 Orbit of Pluto, 843, 891 Packaging, 72, 98, 365, 370, 371, 372, 400 Paddle wheel rotation, 1231 Parabolic reflectors, 841 Parachute jump, 681, 686, 689, 709, 711 Path of a baseball, 1013 Path of a boat, 225, 294, 860, 1027 Path of a heat-seeking object, 1098 Path of a water skier, 661 Path of an acrobatic plane, 400 Path of an ant, 894, 895 Path of steepest ascent, 1098, 1102 Pendulum, 208, 230, 271 Period of a communications satellite, 272 Period of a satellite, 185, 271 Period of the Earth’s orbit, 1023 Periods of planets, 71 Police cruiser flasher, 259 Potential energy, 229 Potential of a charged disk, 229 Pulling a sled, 908 Pure resonance, 214, 388 Pursuit curve, 547, 650 Quickest descent, 1098 Racetrack design, 372 Radioactive decay, 228, 670, 676, 723 Range of a projectile, 174, 374 Range of an artillery shell, 271 Rate of change in temperature, 1090 Rate of flow of heat, 1297, 1299 R-C series circuit, 689 Rectilinear motion, 257, 426 Relative error in measuring an electric current, 272 Resonance, 214, 376, 388, 604 Rings of Neptune, 265 Rivlin’s equation, 402 Rocket launch, 556, 603, 646 Room temperature, 1152 Rotating fluid, 205 Rutherford scattering, 835 Shot put, 1025 Simple harmonic motion, 212, 214, 228, 229, 427 Sinking boat, 711 Sliding chain, 95, 229 Snell’s Law of Refraction, 377, 1136 Special theory of relativity, 66, 83, 124, 332, 345, 360, 427, 800 Specific gravity, 1079 www.elsolucionario.net Openmirrors.com Speed of a motorboat when motor cut off, 721 Speed of a rocket during liftoff, 252 Speed of a wave, 803 Spherical coordinate system for the Earth, 973 Steady-state temperature, 1067, 1102, 1103 Stefan’s Law, 678 Stopping distance of a car, 205, 414 Storing radioactive waste, 375 Surface area of a satellite dish, 841 Surface area of the Jacqueline Kennedy Onassis reservoir, 509 Suspension bridge cables, 827, 841, 842, 1080 Televising a rocket launch, 253 Temperature at a point, 987 Temperature of a hot plate, 1177 Temperature of a thin metal plate, 1043 Terminal velocity, 345 Test dive of a submarine, 319 Test flights, 205, 303 Towing a cruise ship, 908, 929 Tracking a missile cruiser, 1082 Tracking a plane with radar, 260 Tracking planes in a holding pattern at an airport, 1026 Tracking the path of a submarine, 258, 396 Traffic flow, 155, 184, 276, 292, 312, 360 Training course for a yacht, 849 Trajectory of a human cannonball, 1013 Trajectory of a plane, 987 Trajectory of a projectile, 168, 310 Trajectory of a shell, 852, 1022, 1023 Trajectory of an electron, 861 Truss bridges, 15 Turbocharged engine versus standard engine, 508 Variable resistors, 260 Velocity of a particle, 1226 Velocity of a shell, 908 Velocity of a sports car, 490 Velocity of an attack submarine, 490 Velocity of dragsters, 508, 658 Velocity of exploding fireworks, 200 Velocity of flow, 1226 Videographing a moving boat, 238 Voltage in AC circuits, 477 Volume of a pontoon, 525 Volume of water in a pond, 1152 VTOL aircraft, 205, 303 Walking up a spiral staircase, 1250 Waste disposal, 629 (continued) Index of Applications (continued) Water delivered by a water sprinkler, 1171 Water level of a tank, 415 Water pollution, 294, 326, 345 Wind chill factor, 1068 Work done by a magnetic field, 621 Work done by a repulsive charge, 556, 620, 654 Work done by an electric field, 1250, 1251 Work done by an expanding gas, 557 Work done in pulling a wagon, 929 Work done in pulling luggage, 929 Work done on a particle by a gravitational field, 1252 Work required to empty a boiler, 556 Work required to stretch a spring, 551 LIFE SCIENCES Absorption of drugs, 360, 400, 479 Air inhaled during respiration, 399 Alzheimer’s patients, 26, 54, 185 Blood alcohol level, 94, 228 Body mass, 1042 Brain growth and IQs, 292 Carbon monoxide in the atmosphere, 180 Carbon-14 dating, 677 Cardiac output, 491 Cells of a honeycomb, 374 Clark’s Rule for pediatric drug dosages, 97 Cricket chirping and temperature, 70 Crop yield, 156 Death due to strokes, 94, 328 Doomsday equation, 678 Drug concentration in the bloodstream, 470, 658, 678, 751 Energy expended by a fish, 375 Epidemic growth, 94, 328 Epidemiology, 204 Extinction situation, 95 Flights of birds, 373 Flow of blood, 377, 477, 1079 Forestry, 155 Fruit fly population, 640, 699 Goldfish population, 700 Gompertz growth curves, 689, 700 Growth of a tumor, 94 Heights and weights of children, 272 Importance of time in treating heart attacks, 195 Lengths of fish, 94 Lengths of infants, 478 Life expectancy of females, 427 Logistic growth functions, 691, 695, 699, 723 Male life expectancy at 65, 1122 Measles deaths, 71, 328 Nicotine content of cigarettes, 71 Obese children in the United States, 69 Obesity in America, 195 Oxygen content of a pond, 25, 194, 328, 345 People living with HIV, 699 Photosynthesis, 203 Poiseuille’s Law, 292, 377, 1042 Polio immunization, 313 Predator-prey population model, 228, 250, 478, 714, 718, 719 Rate of growth for a plant, 699 Rate of growth of a fish population, 697, 699 Reaction of a frog to a drug, 70 Reaction to a drug, 292 Residual concentration of a drug in the bloodstream, 751 Respiratory cycle, 427, 494 Risk of Down syndrome, 415 Serum cholesterol levels, 479 Serum cholesterol population study, 655 Shock wave lithotripsy, 832 Spread of an epidemic, 399, 699 Spread of disease, 678 Strain on vertebrae, 250 Surface area of a horse, 271 Surface area of a human, 276, 1042, 1067, 1079 Surgeries in physicians’ offices, 312 Trout population, 722 Unclogging arteries, 271 Velocity of airflow during a cough, 293 Velocity of blood, 276, 292, 477 von Bertalanffy growth model, 94, 328, 678 Weight of whales, 15 Yield of apple orchard, 72 BUSINESS AND ECONOMICS Accumulation years of baby boomers, 227 Acquisition of failing S&L’s, 147 Advertising, 173, 270, 320, 326, 415, 508 Annual return of an investment, 346 Annuities, 94, 721, 740 Assembly time of workers, 326 Average single-family property tax, 69 Bank deposits, 414 Black Monday, 347 Book design, 72 Book publishing, 371 Canadian oil-sands production, 478 www.elsolucionario.net Capital value, 655, 752 Charter flight revenue, 68 Cobb-Douglas production function, 1042, 1068, 1103, 1135 Compound interest, 89, 339, 346, 387, 740 Construction costs, 72, 98 Continuous compounded interest, 674, 740, 1043 Demand, 157, 258, 271, 399 Depreciation, 57, 70, 98, 277, 478 Dial-up Internet households, 96 Driving costs, 345 DVD sales, 190 Effect of advertising on profits, 270 Effect of inflation on salaries, 346, 399 Effect of smoking bans, 327 Establishing a trust fund, 346 Estimating changes in profit, 1139 Federal debt, 293 Financing a home, 272, 396 Forecasting commodity crops, 272 Forecasting sales, 98 Fuel costs of operating an oil tanker, 264 Growth of HMOs, 603 Growth of service industries, 814, 817 Health care costs, 26, 184 Hedge fund assets, 185 Hiring lobbyists, 71 Home mortgage payments, 261, 1030, 1043 Hotel occupancy rates, 17, 52, 303, 493 Housing costs, 98, 720 Income of American families, 95 Information security software sales, 1122 Instant messaging accounts, 69 Loan amortization, 396 Manhattan hotel occupancy rate, 17 Marginal cost, 207 Marginal profit, 207 Marginal revenue, 202, 207 Market equilibrium, 712 Maximizing profit, 292, 399, 1135 Maximizing revenue, 292, 371 Maximizing yield, 371 Minimizing costs of laying cable, 364 Minimizing costs of packaging, 371, 400 Minimizing cruise ship costs, 373 Mortgage payments, 261, 272 Multiplier effect of tax cuts, 751 Net investment flow, 493 Office rents, 293 Oil consumption in United States, 480, 488 Oil production shortfall, 507 Online ad sales, 415 (continued) Index of Applications (continued) Operations management consulting spending, 1123 Optimal inventory control, 374 Optimal subway fare, 371 Outsourcing of jobs, 70 Perpetuities, 751, 752 Postal regulations, 26 Production costs, 173, 359 Production functions, 1059, 1067, 1078 Production planning, 908 Profit of a vineyard, 72 Profit versus inventory and floor space, 1067 Projected profit, 277 Projected U.S gasoline use, 477 Quality control, 740 R & D expenditure, 174 Rate of return, 720 Real estate, 444 Revenue function, 70 Sales of functional food products, 313 Satellite TV subscribers, 96 Savings accounts, 676 Senior workforce, 399 Shortage of nurses, 228 Sickouts, 399 Social Security contributions, 15 Social Security trust fund assets, 63 Spending on fiber-optic links, 312 Spending on Medicare, 184 Stock prices, 228 Theme park attendance, 640 Trust fund, establishing, 346 U.S hotel rates, 26 U.S sales of organic milk, 415 U.S strategic petroleum reserves, 490 U.S Treasury two-year notes yield, 18 Use of diesel engines, 293 Wilson lot-size formula, 1043 Worker efficiency, 359 Worldwide PC shipments, 360 Yahoo! in Europe, 98 SOCIAL AND BEHAVIORAL SCIENCES Age of drivers in crash fatalities, 313 Aging drivers, 69 Aging population, 83, 227 Alcohol-related traffic accidents, 602 Americans over age 100, 94 Arson study, 1067 City planning, 345, 632 Crime fighting, 51 Crime rate, 359 Cyclical models, 700 Decline of Russian population, 185 Demographics, 204 Dissemination of information, 721 Effect of budget cuts on drug-related crimes, 327 Effect of immigration on population, 677 Energy production, 602 Fighting crime, 51 Foreign-born medical residents, 292 Gender gap, 15, 25 Global supply of plutonium, 51 Harbor cleanup, 26 Increase in juvenile offenders, 95 Lay teachers at Roman Catholic schools, 699 Learning curves, 677 Morning traffic rush, 312 Motorcycle deaths, 51 Nuclear plant utilization, 15 Online video viewers, 70 www.elsolucionario.net Openmirrors.com Overcrowding of prisons, 52 Percent of females in the labor force, 95, 229 Population density, 1163 Population density of a city, 1178 Population growth in Clark County, 71 Rising median age, 26 Rwandan genocide, 69 Surveillance cameras, 328 Testosterone use, 69 Thurstone learning curve, 97–98 Traffic flow, 155, 184, 276, 292, 312, 360, 493 TV on mobile phones, 493 TV-viewing patterns, 155 Voter registration, 640 GENERAL Baseball diamond, 258 Blowing a soap bubble, 257 Boston Marathon, 312 Breaking the speed limit, 303 Buffon’s needle problem, 591 Cavalieri’s Theorem, 525 Crossing the finish line, 414 Daylight hours in Chicago, 478 Describing the path of pop-up baseball, 411 Gabriel’s Horn, 654, 1280 International America’s cup class, 1043 Interval training, 612 Number of hours of daylight, 277 Postal regulations, 1117, 1122, 1129 Snowfall accumulation, 360 Spam messages, 51 Water level in a harbor, 312, 360 Women’s soccer, 328 ... to inverse functions Soo T Tan x Openmirrors.com www.elsolucionario.net Tan Calculus Series The Tan Calculus series includes the following textbooks: Calculus: Early Transcendentals © 2011 (ISBN... Openmirrors.com CALCULUS EARLY TRANSCENDENTALS SOO T TAN STONEHILL COLLEGE Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States www.elsolucionario.net Calculus: Early. .. Single Variable Calculus: Early Transcendentals © 2011 (ISBN 0-534-46570-6) Calculus © 2010 (ISBN 0-534-46579-X) Single Variable Calculus © 2010 (ISBN 0-534-46566-8) Multivariable Calculus © 2010