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SCHAUM'S OUTLINE OF 3000 SOLVED PROBLEMS IN Calculus Elliot Mendelson, Ph.D. Professor of Mathematics Queens College City University of New York Schaum's Outline Series MC Graw Hill New York Chicago San Francisco Lisbon London Madrid Mexico City Milan New Delhi San Juan Seoul Singapore Sydney Toronto Copyright © 1988 by The McGraw-Hill Companies, Inc. All rights reserved. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the publisher. ISBN: 978-0-07-170261-4 MHID: 0-07-170261-X The material in this eBook also appears in the print version of this title: ISBN: 978-0-07-163534-9, MHID: 0-07-163534-3. All trademarks are trademarks of their respective owners. Rather than put a trademark symbol after every occurrence of a trademarked name, we use names in an editorial fashion only, and to the benefit of the trademark owner, with no intention of infringement of the trademark. Where such designa- tions appear in this book, they have been printed with initial caps. McGraw-Hill eBooks are available at special quantity discounts to use as premiums and sales promotions, or for use in corporate training programs. To contact a representative please e-mail us at [email protected] TERMS OF USE This is a copyrighted work and The McGraw-Hill Companies, Inc. (“McGrawHill”) and its licensors reserve all rights in and to the work. Use of this work is subject to these terms. Except as permitted under the Copyright Act of 1976 and the right to store and retrieve one copy of the work, you may not decompile, disassemble, reverse engineer, reproduce, modify, create derivative works based upon, transmit, distribute, disseminate, sell, publish or sublicense the work or any part of it without McGraw-Hill’s prior consent. You may use the work for your own noncommercial and personal use; any other use of the work is strictly prohibited. Your right to use the work may be terminated if you fail to comply with these terms. THE WORK IS PROVIDED “AS IS.” McGRAW-HILL AND ITS LICENSORS MAKE NO GUARANTEES OR WARRANTIES AS TO THE AC- CURACY, ADEQUACY OR COMPLETENESS OF OR RESULTS TO BE OBTAINED FROM USING THE WORK, INCLUDING ANY INFOR- MATION THAT CAN BE ACCESSED THROUGH THE WORK VIA HYPERLINK OR OTHERWISE, AND EXPRESSLY DISCLAIM ANY WAR- RANTY, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO IMPLIED WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. McGraw-Hill and its licensors do not warrant or guarantee that the functions contained in the work will meet your requirements or that its operation will be uninterrupted or error free. Neither McGraw-Hill nor its licensors shall be liable to you or anyone else for any inaccuracy, error or omission, regardless of cause, in the work or for any damages resulting therefrom. McGraw-Hill has no responsibility for the content of any information accessed through the work. Under no circumstances shall McGraw-Hill and/or its licensors be liable for any indirect, incidental, special, punitive, consequential or similar damages that result from the use of or inability to use the work, even if any of them has been advised of the possibility of such damages. This limitation of liability shall apply to any claim or cause whatsoever whether such claim or cause arises in contract, tort or otherwise. CONTENTS Chapter 1 INEQUALITIES 1 Chapter 2 ABSOLUTE VALUE 5 Chapter 3 LINES 9 Chapter 4 CIRCLES 19 Chapter 5 FUNCTIONS AND THEIR GRAPHS 23 Chapter 6 LIMITS 35 Chapter 7 CONTINUITY 43 Chapter 8 THE DERIVATIVE 49 Chapter 9 THE CHAIN RULE 56 Chapter 10 TRIGONOMETRIC FUNCTIONS AND THEIR DERIVATIVES 62 Chapter 11 ROLLE'S THEOREM, THE MEAN VALUE THEOREM, AND THE SIGN OF THE DERIVATIVE 69 Chapter 12 HIGHER-ORDER DERIVATIVES AND IMPLICIT DIFFERENTIATION 75 Chapter 13 MAXIMA AND MINIMA 81 Chapter 14 RELATED RATES 88 Chapter 15 CURVE SKETCHING (GRAPHS) 100 Chapter 16 APPLIED MAXIMUM AND MINIMUM PROBLEMS 118 Chapter 17 RECTILINEAR MOTION 133 Chapter 18 APPROXIMATION BY DIFFERENTIALS 138 Chapter 19 ANTIDERIVATIVES (INDEFINITE INTEGRALS) 142 Chapter 20 THE DEFINITE INTEGRAL AND THE FUNDAMENTAL THEOREM OF CALCULUS 152 Chapter 21 AREA AND ARC LENGTH 163 Chapter 22 VOLUME 173 Chapter 23 THE NATURAL LOGARITHM 185 Chapter 24 EXPONENTIAL FUNCTIONS 195 Chapter 25 L'HOPITAL'S RULE 208 Chapter 26 EXPONENTIAL GROWTH AND DECAY 215 iii iv CONTENTS Chapter 27 INVERSE TRIGONOMETRIC FUNCTIONS 220 Chapter 28 INTEGRATION BY PARTS 232 Chapter 29 TRIGONOMETRIC INTEGRANDS AND SUBSTITUTIONS 238 Chapter 30 INTEGRATION OF RATIONAL FUNCTIONS: THE METHOD OF PARTIAL FRACTIONS 245 Chapter 31 INTEGRALS FOR SURFACE AREA, WORK, CENTROIDS 253 Surface Area of a Solid of Revolution / Work / Centroid of a Planar Region / Chapter 32 IMPROPER INTEGRALS 260 Chapter 33 PLANAR VECTORS 268 Chapter 34 PARAMETRIC EQUATIONS, VECTOR FUNCTIONS, CURVILINEAR MOTION 274 Parametric Equations of Plane Curves / Vector-Valued Functions / Chapter 35 POLAR COORDINATES 289 Chapter 36 INFINITE SEQUENCES 305 Chapter 37 INFINITE SERIES 312 Chapter 38 POWER SERIES 326 Chapter 39 TAYLOR AND MACLAURIN SERIES 340 Chapter 40 VECTORS IN SPACE. LINES AND PLANES 347 Chapter 41 FUNCTIONS OF SEVERAL VARIABLES 361 Multivariate Functions and Their Graphs / Cylindrical and Spherical Coordinates / Chapter 42 PARTIAL DERIVATIVES 376 Chapter 43 DIRECTIONAL DERIVATIVES AND THE GRADIENT. EXTREME VALUES 392 Chapter 44 MULTIPLE INTEGRALS AND THEIR APPLICATIONS 405 Chapter 45 VECTOR FUNCTIONS IN SPACE. DIVERGENCE AND CURL. LINE INTEGRALS 425 Chapter 46 DIFFERENTIAL EQUATIONS 431 INDEX 443 To the Student This collection of solved problems covers elementary and intermediate calculus, and much of advanced calculus. We have aimed at presenting the broadest range of problems that you are likely to encounter—the old chestnuts, all the current standard types, and some not so standard. Each chapter begins with very elementary problems. Their difficulty usually increases as the chapter pro- gresses, but there is no uniform pattern. It is assumed that you have available a calculus textbook, including tables for the trigonometric, logarith- mic, and exponential functions. Our ordering of the chapters follows the customary order found in many textbooks, but as no two textbooks have exactly the same sequence of topics, you must expect an occasional discrepancy from the order followed in your course. The printed solution that immediately follows a problem statement gives you all the details of one way to solve the problem. You might wish to delay consulting that solution until you have outlined an attack in your own mind. You might even disdain to read it until, with pencil and paper, you have solved the problem yourself (or failed gloriously). Used thus, 3000 Solved Problems in Calculus can almost serve as a supple- ment to any course in calculus, or even as an independent refresher course. V This page intentionally left blank HAPTER 1 nequalities 1.1 Solve 3 + 2*<7. 2x < 4 [Subtract 3 from both sides. This is equivalent to adding -3 to both sides.] Answer x<2 [Divide both sides by 2. This is equivalent to multiplying by 5.] In interval notation, the solution is the set (—°°, 2). 1.2 Solve 5 - 3* < 5x + 2. 5-3>x<5x + 2, 5<8* + 2 [Add 3x to both sides.], 3<8* [Subtract 2 from both sides.] Answer 1 <x [Divide both sides by 8.] In interval notation, the solution is the set (|,°°). 1.3 Solve -7<2x + 5<9. -7 < 2* + 5 < 9, -12 < 2x < 4 [Subtract 5 from all terms.] Answer — 6 < x < 2 [Divide by 2.] In interval notation, the solution is the set (—6,2). 1.4 Solve 3<4x-l<5. 3<4x-l<5, 4<4x<6 [Add 1 to all terms.] Answer 1 s x < \ [Divide by 4.] In interval notation, the solution is the set [1, |). 1.5 Solve 4<-2x + 5<7. 4<-2x + 5<7, -K-2jc<2 [Subtracts.] Answer \ >*>-! [Divide by -2. Since -2 is negative, we must reverse the inequalities.] In interval notation, the solution is the set [-1, |). 1.6 Solve 5 < \x. + 1 s 6. 5<|x + l<6, 4<|*s5 [Subtract 1.] Answer 12<^sl5 [Multiply by 3.] In interval notation, the solution is the set [12,15]. 1.7 Solve 2/jc<3. x may be positive or negative. Case 1. x>0. 2/x<3. 2<3x [Multiply by AC.], |<jc [Divide by 3.] Case 2. x<0. 2/x<3. 2>3x [Multiply by jr. Reverse the inequality.], |>jc [Divide by 3.] Notice that this condition |>x is satisfied whenever jc<0. Hence, in the case where x<0, the inequality is satisfied by all such x. Answer f < x or x < 0. As shown in Fig. 1-1, the solution is the union of the intervals (1,«) and (—°°, 0). Fig. 1-1 1.8 Solve We cannot simply multiply both sides by x - 3, because we do not know whether x - 3 is positive or negative. Case 1. x-3>0 [This is equivalent to x>3.] Multiplying the given inequality (1) by the positive quantity x-3 preserves the inequality: * + 4<2;t-6, 4<x-6 [Subtract jr.], 10<x [Add 6.] Thus, when x>3, the given inequality holds when and only when x>10. Case 2. x-3<0 [This is equivalent to x<3]. Multiplying the given inequality (1) by the negative quantity x — 3 reverses the inequality: * + 4>2*-6, 4>x-6 [Subtract*.], 10>x [Add 6.] Thus, when x<3, the inequality 1 2 CHAPTER 1 (1) holds when and only when x < 10. But x < 3 implies x < 10, and, therefore, the inequality (1) holds for all x<3. Answer *>10 or x<3. As shown in Fig. 1-2, the solution is the union of the intervals (10, oo) and (~»,3). Fig. 1-2 1.9 Solve 1. x + 5>0 [This is equivalent to x>-5.]. We multiply the inequality (1) by x + 5. x<I Case 1. x + 5>0 [This is equivalent to x>-5.]. We multiply the inequality (1) by x + 5. x< x + 5, 0<5 [Subtract x.] This is always true. So, (1) holds throughout this case, that is, wheneverx + 5, 0<5 [Subtract x.] This is always true. So, (1) holds throughout this case, that is, whenever x>-5. Case 2. x + 5<0 [This is equivalent to x<-5.]. We multiply the inequality (1) by x + 5. The inequality is reversed, since we are multiplying by a negative number. x>x + 5, 0>5 [Subtract*.] Butinequality is reversed, since we are multiplying by a negative number. x>x + 5, 0>5 [Subtract*.] But 0 > 5 is false. Hence, the inequality (1) does not hold at all in this case. Answer x > -5. In interval notation, the solution is the set (-5, °°). 1.10 Solve Case 1. x + 3>0 [This is equivalent to jc>-3.]. Multiply the inequality (1) by x + 3. x-7> 2x + 6, -7>x+6 [Subtract x.], -13>x [Subtract 6.] But x<-13 is always false when *>-3. Hence, this case yields no solutions. Case 2. x + 3<0 [This is equivalent to x<— 3.]. Multiply the inequality (1) by x + 3. Since x + 3 is negative, the inequality is reversed. x-7<2x + 6, —7<x + 6 [Subtract x.] ~\3<x [Subtract 6.] Thus, when x < —3, the inequality (1) holds when and only when *>-13. Answer —13 < x < —3. In interval notation, the solution is the set (—13, —3). 1.11 Solve (2jt-3)/(3;t-5)>3. Case 1. 3A.-5>0 [This is equivalent to *>§.]. 2x-3>9x-l5 [Multiply by 3jf-5.], -3> 7x-15 [Subtract 2x.], I2>7x [Add 15.], T a* [Divide by 7.] So, when x>f, the solutions must satisfy x<". Case 2. 3x-5<0 [This is equivalent to x<|.]. 2* - 3 < 9* - 15 [Multiply by 3*-5. Reverse the inequality.], -3<7jr-15 [Subtract 2*.], 12 < 7x [Add 15.], ^ s x [Divide by 7.] Thus, when x< f, the solutions must satisfy x^ !f. This is impossible. Hence, this case yields no solutions. Answer f < x s -y. In interval notation, the solution is the set (§, ^]. 1.12 Solve (2*-3)/(3*-5)>3. Remember that a product is positive when and only when both factors have the same sign. Casel. Jt-2>0 and x + 3>0. Then x>2 and jt>—3. But these are equivalent to x>2 alone, since x>2 im-and x + 3>0. Then x>2 and jt>—3. But these are equivalent to x>2 alone, since x>2 im- plies x>-3. Case 2. *-2<0 and A: + 3<0. Then x<2 and jc<—3, which are equivalent to x<—3, since x<-3 implies x<2. Answer x > 2 or x < -3. In interval notation, this is the union of (2, °°) and (—<», —3). 1.13 Solve Problem 1.12 by considering the sign of the function f(x) = (x — 2)(x + 3). Refer to Fig. 1-3. To the left of x = — 3, both x-2 and x + 3 are negative and /(*) is positive. As one passes through x - — 3, the factor x - 3 changes sign and, therefore, f(x) becomes negative. f(x) remains negative until we pass through x = 2, where the factor x — 2 changes sign and f(x) becomes and then remains positive. Thus, f(x) is positive for x < — 3 and for x > 2. Answer Fig. 1-3 INEQUALITIES 3 1.14 Solve (x-l)(x + 4)<0. The key points of the function g(x) = (x - l)(x + 4) are x = — 4 and x = l (see Fig. 1-4). To the left of x = -4, both x — 1 and x + 4 are negative and, therefore, g(x) is positive. As we pass through x = — 4, jr + 4 changes sign and g(x) becomes negative. When we pass through * = 1, A: - 1 changes sign and g(x) becomes and then remains positive. Thus, (x - \)(x + 4) is negative for -4 < x < 1. Answer Fig. 1-4 Fig. 1-5 1.15 Solve x2 - 6x + 5 > 0. Factor: x2 -6x + 5 = (x - l)(x - 5). Let h(x) = (x - \)(x - 5). To the left of x = 1 (see Fig. 1-5), both .* - 1 and jc - 5 are negative and, therefore, h(x) is positive. When we pass through x = \, x-\ changes sign and h(x) becomes negative. When we move further to the right and pass through x = 5, x — 5 changes sign and h(x) becomes positive again. Thus, h(x) is positive for x < 1 and for x>5. Answer x > 5 or x < 1. This is the union of the intervals (5, °°) and (—°°, 1). 1.16 Solve x2 + Ix - 8 < 0. Factor: x2 + Ix - 8 = (x + &)(x - 1), and refer to Fig. 1-6. For jc<-8, both x + 8 and x-l are negative and, therefore, F(x) = (x + 8)(x - 1) is positive. When we pass through x = -8, x + 8 changes sign and, therefore, so does F(x). But when we later pass through x = l, x-l changes sign and F(x) changes back to being positive. Thus, F(x) is negative for -8 < x < 1. Answer Fig. 1-6 Fig. 1-7 1.17 Solve 5x - 2x2 > 0. Factor: 5x - 2x2 = x(5 - 2x), and refer to Fig. 1-7. The key points for the function G(x) = x(5 - 2x) are x = 0 and *=|. For x<Q, 5-2x is positive and, therefore, G(x) is negative. As we pass through x = 0, x changes sign and. therefore, G(x) becomes positive. When we pass through x= |, 5 — 2x changes sign and, therefore, G(x) changes back to being negative. Thus, G(x) is positive when and only when 0 < x < |. Answer 1.18 Solve (Jt-l)2(* + 4)<0. (x — I)2 is always positive except when x = 1 (when it is 0). So, the only solutions occur when * + 4<0 and jc^l. Answer x<— 4 [In interval notation, (—=°, — 4).] 1.19 Solve x(x-l)(x + l)>0. The key points for H(x) = x(x - l)(x + 1) are x = 0, x = l, and jc=-l (see Fig. 1-8). For x to the left of — 1, x, x — 1, and x + 1 all are negative and, therefore, H(x) is negative. As we pass through x = — 1, x + 1 changes sign and, therefore, so does H(x). When we later pass through x = 0, x changes sign and, therefore, H(x) becomes negative again. Finally, when we pass through x = l, x-\ changes sign and H(x) becomes and remains positive. Therefore, H(x) is positive when and only when — 1 < A: < 0 or x>\. Answer

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Schaums Outline of 3000 Solved Problems in Calculus pdf free download. This collection of solved problems covers elementary and intermediate calculus, and much of advanced calculus. We have aimed at presenting the broadest range of problems that you are likely to encounter—the old chestnuts, all the current standard types, and some not so standard. Each chapter begins with very elementary problems. Their difficulty usually increases as the chapter progresses, but there is no uniform pattern.

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444 INDEX Circles (Cont.): Continuity of functions (Cont.): parametric equations of, 34.1, multivariate functions, 41.60 to Bessel's equation, 38.70, 38.71, 34.17 46.81 perimeter of, 16.46 41.63 in polar coordinates, 35.7 to Binomial series, 38.31, 38.78, 35.9,35.13, 35.14, 35.17, at a point, 7.1 to 7.23, 9.46 38.103 to 38.106, 38.108 35.18, 35.41, 35.66 to 35.69, removable discontinuities, 7.5, 35.87, 35.88, 44.33, 44.38, Binormal vectors, 45.17 44.42, 44.44 7.9 Bounded infinite sequences, 36.47, on the right, 7.18, 7.19 radius of, 4.1 to 4.6, 4.10 to Rolle's theorem and, 11.1 to 36.62 4.15,4.23 Boyle's law, 14.39 11.9, 11.27, 11.31, 11.38, Cardioids, 35.30, 35.34 to 35.36, tangents to, 4.15, 4.19 to 4.22, 11.47, 11.49 12.46 Contour maps (level curves), 41.32 35.44, 35.49, 35.55, 35.66, to 41.43 35.67, 35.69, 35.73, 35.78, triangles inscribed in, 16.61 Convergence: 35.84, 35.91, 35.102,44.33, Circumference of circles, 27.64, infinite sequences, 36.19 to 44.38, 44.81 36.45, 36.47, 36.52, 36.62 Cartesian coordinates (see 34.32 infinite series, 37.1 to 37.3, 37.5, Rectangular coordinates) Collinear points, 40.15 37.10to37.16, 37.23, 37.24, Cauchy-Riemann equations, 42.54, Completing the square, 4.6 to 4.8, 37.27 to 37.56, 37.64, 37.69 42.92 to 37.116 Cauchy's inequality, 33.28, 33.29, 4.10,4.11,4.13, 4.16, 27.51, integrals, 32.1 to 32.10, 32.23 to 43.68 27.53, 29.29, 29.39, 30.13, 32.27, 32.29, 32.31, 32.41 to Celsius temperature scale, 3.71 34.14,40.11 32.44, 32.48, 32.53, 32.54 Center of circles, 4.1 to 4.6, 4.10 Components of vectors, 33.4, 33.5 power series, 38.1 to 38.33, to 4.15, 4.23 Composite functions, 9.1 to 9.6, 38.67, 38.68,38.109to Center of mass (see Centroids) 9.24 to 9.34, 9.46, 10.19 38.114 Central angles, 10.1, 10.5, 10.6 Compound interest, 26.8 to 26.14, vector, 34.43 Centroids (center of mass): 26.40 to 26.42 Coordinate systems: by integration, 31.24 to 31.35, Concave functions, 15.1 to 15.5, Cartesian (see Rectangular 35.72, 35.73, 35.88, 44.80 to 15.16, 15.17, 15.20, 15.21, coordinates) 44.85, 44.89 to 44.92 15.23 to 15.29, 15.39 to cylindrical, 41.67 to 41.85 right triangle, 31.28, 31.34 15.41, 15.43, 15.44, 15.48, polar (see Polar coordinates) Chain rule: 15.52, 34.29, 34.30 rectangular (see Rectangular for derivatives, 9.4 to 9.23, 9.35 Cones: coordinates) to 9.44, 9.48, 9.49, 10.19 to circumscribed about spheres, spherical, 41.85 to 41.98 10.23, 10.25 to 10.29, 12.1, 16.42 Coulomb's law, 31.22 12.2, 12.10, 12.29to 12.31, cylinders inscribed in, 16.43 Critical numbers: 13.34, 13.36, 42.64 to 42.66, equations of, 41.75, 41.77, 41.87 absolute extrema and, 13.9 to 42.68 to 42.73, 42.81 to frustrum of, 22.49 13.13, 13.15to 13.17, 13.21, 42.91, 42.96 graphs of, 41.8, 41.10,41.16 13.23 to 13.34, 13.36, 15.31 for vector functions, 34.61 surface area of, 16.9, 31.14, relative extrema and, 13.3 to Circles: 44.58 13.8, 13.14, 13.18to 13.20, angles inscribed in, 33.12 volume of, 14.6, 14.18, 14.29, 15.6 to 15.15, 15.17 to 15.22, arc length and, 10.5, 10.6, 27.64, 14.38, 16.9, 16.42, 22.2, 15.31 to 15.35, 15.39, 15.41 34.32 31.34,42.86,43.67,44.41, to 15.46, 15.48to 15.51, area of, 14.8, 14.14, 16.48 44.51 15.53, 15.54 center of, 4.1 to 4.6, 4.10 to Continuity of functions: Cross product of vectors, 40.40 to 4.15,4.23 composite functions, 9.46 40.43, 40.45 to 40.68, 45.11, circumference of, 27.64, 34.32 definite integrals, 20.42, 20.59 45.12, 45.17 to 45.20, 45.22, as contours (level curves), 41.32 differentiable functions, 8.25, 45.23, 45.28, 45.36, 45.37, curvilinear motion and, 34.74 to 8.26 45.39 34.76 generalized mean value theorem Cube roots: equations of, 4.1 to 4.6, 4.9, and, 11.38, 11.39 approximation of, 18.4, 18.8, 4.10,4.12,4.13,4.17to intermediate value theorem and, 18.17, 18.19, 18.21 4.20, 4.22, 4.25 to 4.30 7.20 to 7.23, 11.28, 11.32, in limits, 6.50, 6.51 graphs of, 5.3, 34.1, 35.7 to 35.9, 11.37, 11.42 of real numbers, 5.84 35.13, 35.14,35.17, 35.18, over an interval, 7.24 to 7.27 Cubes: 35.41, 35.66, 35.67, 35.69, on the left, 7.18, 7.19 diagonals of, 14.45, 40.37, 35.87, 44.33, 44.37, 44.38, mean value theorem and, 11.10 40.38 44.42, 44.44 to 11.17, 11.30, 11.33 to difference of, 5.84 intersections of, 4.24 to 4.27, 11.36, 11.38 to 11.41, 11.43, edges of, 40.38 12.27 11.45, 11.46 Cubic functions: Definite integrals (Cont.): INDEX 0 445 as contours (level curves), 41.34 powers or roots, 20.9, 20.12 to Derivatives (Cont.): graphs of, 5.10, 5.19, 41.34 20.14, 20.16 to 20.19, 20.23, 20.25 to 20.32, 20.35 to higher-order, 12.1 to 12.13, Curl, 45.28 to 45.31, 45.33, 45.34, 20.38, 20.40, 20.57, 20.64, 12.15, 12.19 to 12.23, 12.25, 45.36 to 45.38 20.71,20.72, 20.75 to 20.78, 12.26, 12.29 to 12.37, 12.47, 20.91,20.92,23.61, 23.62 12.48, 13.1, 13.3 to 13.7, Curvature (K),34.94 to 34.101, trigonometric functions, 20.10, 13.20, 13.21, 39.30 to 39.32, 34.103,35.108,35.109,45.21 20.11,20.15, 20.20 to 20.22, 39.43 to 45.23 20.24, 20.39, 20.49, 20.60, 20.62, 20.63, 20.79 to 20.84, hyperbolic functions, 24.95, Curves (see Graphs) 28.32 to 28.34, 28.53, 29.17, 24.96, 24.99, 24.100 Curvilinear motion, 34.43 to 29.18,29.30,29.33, 32.13, 32.31, 32.32, 32.44, 32.45, implicit differentiation, 12.2, 34.108 32.51, 32.53,32.59 12.11 to 12.22, 12.24, 12.34, Cusps in graphs, 15.44 12.35, 12.38 to 12.48 Cycloids: Degenerate spirals, 35.111 Degree measure of angles, 10.2 to implicit partial differentiation, acceleration vector for, 34.86 42.13 to 42.18 arch of, 34.39 10.7 curvature (K) of, 34.101 intermediate value theorem and, equations of, 34.20 Delta-(A-)definition of derivatives, 11.28, 11.32, 11.37, 11.42 Cylinders: 8.1 to 8.4, 8.11,8.17,8.21, equations of, 41.26, 41.27, 41.68, 8.22, 8.24 to 8.31, 8.39, 8.45 inverse trigonometric functions, to 8.47, 10.17, 10.44 27.2, 27.4, 27.22 to 27.38, 41.83,41.93,41.95,41.97 27.58 to 27.60, 27.65, 27.69, graphs of, 41.1,41.2 Derivatives: 27.70, 27.72 to 27.79 inscribed in cones, 16.43 absolute value in, 9.47, 9.48, surface area of, 16.22, 42.89 13.34, 13.36 Laplace transform of, 32.60 volume of, 14.2, 14.48, 16.7, approximation of (see L'Hopital's rule and, 25.1 to Approximation, by 16.8, 16.16, 16.18, 16.43, differentials) 25.53, 32.4, 32.6 to 32.8, 43.67, 44.70 chain rule for, 9.4 to 9.23, 9.35 32.20, 32.36, 36.15, 36.20, Cylindrical coordinates, 41.67 to to 9.44, 9.48, 9.49, 10.19 to 37.47, 37.108, 38.15, 38.24, 41.85 10.23, 10.25 to 10.29, 12.1, 41.56,41.57 12.2, 12.10, 12.29 to 12.31, mean value theorem and, 11.10 Decay (see Exponential growth 13.34, 13.36, 42.64 to 42.66, to 11.17, 11.30, 11.33 to and decay) 42.68 to 42.73, 42.81 to 11.36, 11.38 to 11.41, 11.43, 42.91,42.% 11.45, 11.46 Decreasing functions, 11.19 to of definite integrals, 20.42 to natural logarithms, 23.1 to 23.9 11.26 20.47,20.51,20.73, 20.74, partial, 42.1 to 42.126 20.87 polynomials, 8.5, 8.6, 8.12, 8.32, Decreasing infinite sequences, delta-(A-)defmition of, 8.1 to 8.4, 12.23 36.58, 36.61 to 36.63 8.11,8.17,8.21,8.22,8.24 product rule for, 8.7, 8.8, 8.40, to 8.31, 8.39, 8.45 to 8.47, 8.41,8.48,9.10,9.16,9.40, Definite integrals: 10.17, 10.44 12.2, 12.20, 12.32 approximating sums for, 20.1, differentiable functions and, quotient rule for, 8.7, 8.9, 8.10, 20.2, 20.5 8.21, 8.25, 8.26, 8.30, 8.31, 8.49,8.50,9.9, 9.13,9.21, convergence of, 32.1 to 32.10, 8.43 to 8.47, 9.18, 9.45, 9.38, 9.41, 9.43, 10.24, 12.1, 32.23 to 32.27, 32.29, 32.31, 9.49 12.11 to 12.13, 12.19, 12.33 32.41 to 32.44, 32.48, 32.53, directional, 43.1 to 43.13, 43.15 Rolle's theorem and, 11.1 to 32.54 to 43.17 11.9, 11.27, 11.31, 11.38, derivatives of, 20.42 to 20.47, exponential functions, 24.7 to 11.47, 11.49 20.51, 20.73, 20.74, 20.87 24.16, 24.30 to 24.39, 24.51, second derivative test for exponential functions, 24.45 to 24.52, 24.64, 24.77, 24.88 to relative extrema, 13.1, 13.3 24.49, 24.55, 24.76, 24.92, 24.90 to 13.7, 13.20, 13.21, 15.1 to 24.93,32.6,32.7,32.12to first derivative test for relative 15.9, 15.12, 15.13, 15.16to 32.14, 32.17 to 32.22, 32.38, extrema, 13.2, 13.6 to 13.8, 15.18, 15.20to 15.29, 15.32, 32.39, 32.58 13.18, 13.20, 13.29, 15.10, 15.33, 15.35, 15.37, 15.39 to inverse trigonometric functions, 15.11, 15.14, 15.15, 15.19, 15.56 27.61 to 27.64, 27.81,27.82, 15.34, 15.37, 15.44, 15.45, sum rule for, 8.7, 9.11 32.54 15.49 trigonometric functions, 10.17 to Laplace transforms, 32.57 to generalized mean value theorem 10.29, 10.36 to 10.43 32.60 and, 11.38, 11.39 (See also Antiderivatives) natural logarithms, 23.10 to Determinants, Hessian, 43.23 23.22, 23.53, 23.68, 23.69, Diagonals: 23.79, 23.80, 28.25 to 28.31, cube, 14.45, 40.37, 40.38 32.4, 32.5, 32.8, 32.10, parallelogram, 33.26, 33.30 32.15,32.16,32.36,32.37, rhombus, 33.32 32.50 446 0 INDEX Disks, segments of, 29.43 Escape velocity, 19.95, 19.96 Difference of cubes, 5.84 Distance: Estimation (see Approximation) Differentiable functions, 8.21, 8.25, Euler's constant (\"/), 37.110 between lines, 40.113, 40.114, Euler's theorem for homogeneous 8.26, 8.30, 8.31, 8.43 to 43.31, 43.38 8.47, 9.18, 9.45, 9.49, 42.96, functions, 42.74 to 42.77 42.97 between planes, 40.91, 40.92 Even functions, 5.38, 5.40, 5.41, Differential equations: between points, 40.1 to 40.4 Bernoulli's equation, 46.40 to from points to lines, 33.9, 33.10, 5.48, 5.53, 5.55, 5.56, 5.90, 46.44 5.91, 5.93, 8.46, 8.47, 15.31, Bessel's equation, 38.70, 38.71, 33.34,40.14,40.20,40.24, 15.36, 15.46, 20.50, 38.73, 46.81 40.34, 40.77 to 40.79 38.93 exact, 46.25, 46.26 from points to planes, 40.44, Exact differential equations, 46.25, fourth-order, 46.76 to 46.78 40.88, 40.89 46.26 integrating factors for, 46.27 to Divergence, 45.25 to 45.27, 45.30 Exponential functions: 46.33, 46.35, 46.36, 46.45 to 45.32, 45.34, 45.35, 45.37, absolute extrema of, 24.50, 24.57 Legendre's equation, 46.80 45.39, 45.47 to 45.49 to 24.62, 24.73 nth order, 46.56 Domain: antiderivatives (indefinite orthogonal trajectories and, of definition, 41.64 to 41.66 integrals) and, 24.17 to 46.22 to 46.24 of functions, 5.1 to 5.19, 5.24 to 24.29, 24.54, 28.1 to 28.3, partial (see Partial differential 5.31, 5.34 to 5.37, 9.16, 28.9, 28.13,28.20, 28.42, equations) 27.21, 41.64 to 41.66 28.43, 28.50 partial fraction decomposition Doomsday equation, 26.31 defined, 11.48 and, 46.22 Dot product of vectors, 33.4, 33.5, definite integrals and, 24.45 to power series solutions for, 38.49, 33.9, 33.10,33.12, 33.18 to 24.49, 24.55, 24.76, 24.92, 38.56, 38.66 33.20, 33.28 to 33.35, 33.37 24.93, 32.6, 32.7, 32.12 to predator-prey system, 46.70 to 33.41, 40.33 to 40.35, 32.14, 32.17 to 32.22, 32.38, Riccati equation, 46.43, 46.44 40.39, 40.44, 40.45, 40.47, 32.39, 32.58 second-order homogeneous, 40.49, 40.55 to 40.58, 40.60 derivatives of, 24.7 to 24.16, 46.48 to 46.55, 46.72 to 40.67, 43.1 to 43.7, 43.10 24.30 to 24.39, 24.51,24.52, second-order nonhomogeneous, to 43.12, 43.16, 43.17, 43.66, 24.64, 24.77, 24.88 to 24.90 45.57 to 45.68 45.9, 45.13 to 45.15, 45.35, graphs of, 24.57 to 24.63, 24.84 separable variables in, 19.88 to 45.37,45.39,45.41,45.43, to 24.86, 24.91, 24.94, 25.56, 19.94,46.1 to 46.21, 46.34, 45.45 to 45.49 25.57 46.37 to 46.39, 46.46, 46.47 Double integrals, 44.1 to 44.4, 44.6 hyperbolic functions and, 24.95 substitutions in, 46.17 to 46.21, to 44.46, 44.53 to 44.58, to 24.111 46.34 44.74, 44.75, 44.79 to 44.82, Laplace transform of, 32.58 third-order, 46.73 to 46.75, 46.79 44.86, 44.87, 44.89, 44.90 limits and, 24.74, 24.75, 24.78 to variation of parameters and, 24.81,24.83 46.65 to 46.69 Ellipses: Maclaurin series and, 39.1, 39.7, Differentials, approximation by area of, 20.72,41.22 39.18, 39.21, 39.24, 39.25, (see Approximation, by as contours (level curves), 41.37 39.32, 39.35, 39.44, 39.50 differentials) equations of, 41.12 power series and, 38.39 to 38.42, Differentiation: graphs of, 16.20, 16.41, 34.2, 38.76, 38.78 to 38.80, 38.82, implicit, 12.2, 12.11 to 12.22, 34.5, 35.16 38.97 12.24, 12.34, 12.35, 12.38 to intersections of hyperbolas with, properties of, 24.1 to 24.6, 24.35, 12.48 12.43 24.65 to 24.71 implicit partial, 42.13 to 42.18 parametric equations of, 34.2, (See also Natural logarithms) logarithmic, 23.23 to 23.26, 34.5,34.11 Exponential growth and decay: 23.58, 23.67, 24.35 to 24.39 in polar coordinates, 35.16 bacterial growth, 26.2, 26.20, Direction: rectangles inscribed in, 16.41 26.27, 26.29, 26.37 of steepest ascent and descent, tangents to, 12.42 compound interest, 26.8 to 43.2, 43.8 triangle perimeter and, 16.20 26.14, 26.40 to 26.42 of vectors, 33.3, 33.13, 33.25 decay and growth constants for, Direction angles, 40.32 Ellipsoids: 26.1 Direction cosines of vectors, 40.27 boxes inscribed in, 43.42 defined, 26.1 to 40.29, 40.32 equations of, 41.6, 41.79, 41.98 other applications, 26.3, 26.6, Directional derivatives, 43.1 to graphs of, 41.13 26.22 to 26.24, 26.31,26.33 43.13, 43.15 to 43.17 as level surfaces, 41.45 to 26.36, 26.39 Discontinuities (see Continuity of volume of, 41.22 population growth, 26.5, 26.7, functions) 26.19, 26.25,26.26,26.28, Discriminant, 30.13, 46.48 Equiangular spirals, 35.71, 35.77, 26.38 35.101, 35.112 Equilateral triangles, 14.36, 16.20, 16.51, 16.60 Exponential growth and decay Functions: INDEX Q 447 (Cont.): asymptotes of, 6.28 to 6.32, 15.17to 15.19, 15.26, 15.39 Functions (Cont.): radioactive decay, 26.15 to to 15.43, 15.47, 15.48, 15.50, maxima and minima of (see 26.18, 26.21, 26.30, 26.43 15.52, 15.58, 15.59, 15.61 Extrema of functions) average value of, 20.32, 20.33, multivariate, 41.1 to 41.66 rate of increase for, 26.4 20.40, 20.57, 20.91 odd, 5.43, 5.44, 5.46, 5.49, 5.51, Extrema (maxima and minima) of composite, 9.1 to 9.6, 9.24 to 5.52, 5.54 to 5.56, 5.90 to 9.34, 9.46, 10.19 5.92, 8.46, 8.47, 15.34, functions: concave, 15.1 to 15.5, 15.16, 15.47, 15.51, 20.48, 20.49, absolute: 15.17, 15.20, 15.21, 15.23 to 24.97, 38.74, 38.93 15.29, 15.39 to 15.41, 15.43, one-one, 5.57, 5.58, 5.60, 5.63, critical numbers and, 13.9 to 15.44, 15.48, 15.52, 34.29, 5.69 to 5.74, 5.92, 5.93, 13.13, 13.15 to 13.17, 34.30 5.100, 9.49 13.21, 13.23 to 13.34, continuity of (see Continuity of polynomial (see Polynomials) 13.36, 15.31 functions) range of, 5.1 to 5.19, 5.24 to contour maps (level curves) of, 5.31, 5.34 to 5.37, 27.21 of exponential functions, 24.50, 41.32 to 41.43 rational, integration of, 30.1 to 24.57 to 24.62, 24.73 critical numbers of (see Critical 30.33 numbers) self-inverse, 5.69, 5.74, 5.75 of multivariate functions, decreasing, 11.19 to 11.26 of several variables, 41.1 to 43.54, 43.55, 43.65 derivatives of (see Derivatives) 41.66 differentiable, 8.21, 8.25, 8.26, strictly increasing, 11.48 applications of, 16.1 to 16.61 8.30, 8.31,8.43 to 8.47, strictly positive, 11.48 Lagrange multipliers and, 43.56 9.18,9.45,9.49 trigonometric (see Trigonometric discontinuities in (see Continuity functions) to 43.64, 43.66, 43.67 of functions) relative: domain of, 5.1 to 5.19, 5.24 to Fundamental theorem of calculus, 5.31, 5.34 to 5.37, 9.16, 20.9 to 20.14 critical numbers and, 13.3 to 27.21, 41.64 to 41.66 13.8, 13.14, 13.18 to 13.20, even, 5.38, 5.40, 5.41,5.48, 5.53, Gamma function, 32.22 15.6 to 15.15, 15.17 to 5.55,5.56,5.90,5.91,5.93, Gauss' theorem, 45.49 15.22, 15.31 to 15.35, 8.46, 8.47, 15.31, 15.36, Generalized mean value theorem, 15.39, 15.41 to 15.46, 15.48 15.46, 20.50, 38.73, 38.93 to 15.51, 15.53, 15.54 exponential (see Exponential 11.38, 11.39 functions) Geometric mean, 43.51 first derivative test for, 13.2, extrema of (see Extrema of Geometric series, 37.5 to 37.9, 13.6 to 13.8, 13.18, 13.20, functions) 13.29,15.10,15.11,15.14, gamma, 32.22 37.12, 37.13, 37.25, 37.26, 15.15, 15.19,15.34,15.37, greatest integer, 5.6, 5.7, 6.5 37.30 to 37.34, 37.45, 37.48, 15.44, 15.45, 15.49 homogeneous multivariate, 42.74 37.67, 37.68, 37.73, 37.75, to 42.80 37.77,37.83,37.91,37.113 inflection points and, 13.2, hyperbolic (see Hyperbolic to 37.116, 38.5, 38.26, 38.29 13.6, 13.7, 13.18, 13.20, functions) Gradient, 43.1 to 43.22, 43.56 to 14.42 to 14.46, 15.7, 15.8, increasing, 11.18, 11.20 to 11.26, 43.64, 43.66, 43.67, 45.24, 15.10 to 15.19, 15.32 to 11.29, 11.48, 15.52 45.32, 45.33, 45.35, 45.36, 15.35, 15.48 to 15.51, inflection points of, 13.2, 13.6, 45.39, 45.45 15.57, 15.61 13.7, 13.18, 13.20, 15.2, Graphs: 15.3, 15.5, 15.7, 15.8, 15.10 absolute extrema in, 15.31, 15.61 of multivariate functions, 43.22 to 15.19, 15.21, 15.25, 15.26, asymptotes in, 6.28 to 6.32, to 43.53 15.32 to 15.35, 15.37, 15.40, 15.17 to 15.19, 15.26, 15.39 15.42 to 15.46, 15.48 to to 15.43, 15.47, 15.48, 15.50, second derivative test for, 13.1, 15.51, 15.53, 15.57, 15.61 15.52, 15.58, 15.59, 15.61 13.3 to 13.7, 13.20, 13.21, integrals of (see Integrals) cardioids, 35.30, 35.34 to 35.36, 15.1 to 15.9, 15.12, 15.13, inverse, 5.69 to 5.74, 5.92, 5.93, 35.44, 35.66, 35.67, 35.69, 15.16to 15.18, 15.20 to 5.100,9.49 44.33, 44.38, 44.81 15.29, 15.32, 15.33, 15.35, level surfaces of, 41.44 to 41.47 circles, 5.3, 34.1, 35.7 to 35.9, 15.37, 15.39 to 15.56 limits of (see Limits) 35.13, 35.14, 35.17, 35.18, logarithmic (see Natural 35.41, 35.66, 35.67, 35.69, Factorization of polynomials, 5.76 logarithms) 35.87, 44.33, 44.37, 44.38, to 5.86 44.42, 44.44 cones, 41.8, 41.10, 41.16 Fahrenheit temperature scale, 3.71 cubic functions, 5.10, 5.19, 41.34 First derivative test for relative cusps in, 15.44 cylinders, 41.1, 41.2 extrema, 13.2, 13.6 to 13.8, 13.18, 13.20, 13.29, 15.10, 15.11, 15.14, 15.15, 15.19, 15.34, 15.37, 15.44, 15.45, 15.49 (See also Second derivative test for relative extrema) Frequency of trigonometric functions, 10.12 Frustrum of cones, 22.49 448 Q INDEX Half-life of radioactive materials, Indefinite integrals (see 26.15 to 26.18, 26.21,26.30, Antiderivatives) Graphs (Cont.): 26.43 ellipses, 16.20, 16.41, 34.2, 34.5, Induction, mathematical, 2.18, 35.16 Harmonic series, 37.2, 37.3, 37.27, 20.3, 20.65, 24.89, 24.90, ellipsoids, 41.13 37.28, 38.6 32.22, 43.20 exponential functions, 24.57 to 24.63, 24.84 to 24.86, 24.91, Hessian determinant, 43.23 Inequalities: 24.94, 25.56, 25.57 Higher-order derivatives, 12.1 to absolute value in, 2.1 to 2.36 greatest integer function, 5.6, Cauchy's, 33.28, 33.29, 43.68 5.7, 6.5 12.13, 12.15, 12.19 to 12.23, introduced, 1.1 to 1.25 hyperbolas, 5.4, 5.8, 5.9, 16.3, 12.25, 12.26, 12.29 to 12.37, for natural logarithms, 23.41, 34.6, 34.8, 34.15, 34.16, 12.47, 12.48, 13.1, 13.3 to 23.42, 23.60, 23.77 41.19 13.7, 13.20, 13.21, 39.30 to triangle, 2.18, 2.19, 2.35, 6.13, hyperbolic functions, 24.97 39.32, 39.43 33.29, 36.47, 36.49 to 36.51 hyperboloids, 41.14, 41.15, Homogeneous multivariate 41.23, 41.24 functions, 42.74 to 42.80 Inertia, moments of, 44.86 to inflection points in, 13.2, 15.11 Hooke'slaw, 31.16, 31.17 44.88 to 15.19, 15.21, 15.25, 15.26, Hyperbolas: 15.32 to 15.35, 15.37, 15.40, as contours (level curves), 41.36, Infinite sequences: 15.42 to 15.46, 15.48 to 41.40 bounded, 36.47, 36.62 15.51, 15.53, 15.57, 15.61 curvature (K) of, 34.99 convergence of, 36.19 to 36.45, inverse functions, 5.100 equations of, 41.11 36.47, 36.52, 36.62 lemniscates, 35.32, 35.37, 35.42 graphs of, 5.4, 5.8, 5.9, 16.3, 34.6, decreasing, 36.58, 36.61 to 36.63 limagons, 35.31, 35.38 to 35.40 34.8,34.15,34.16,41.19 increasing, 36.54 to 36.57, 36.59, line segments and lines, 3.16, intersections of ellipses with, 36.60 3.36,5.5,5.12, 5.14, 5.15 to 12.43 limits of, 36.1 to 36.53, 36.65 5.18,5.20,5.23,5.88,6.33, intersections of lines with, 3.81 7.2, 7.4, 7.5, 7.8, 8.43, parametric equations of, 34.6, Infinite series: 10.46, 11.49, 16.45, 34.9, 34.8, 34.12,34.15, 34.16 alternating series test for, 37.51, 34.10, 35.10 to 35.12, 35.19, Hyperbolic functions: 37.59, 37.76, 37.98, 37.100, 35.20, 40.21 antiderivatives (indefinite 37.101,38.15 natural logarithms, 23.39, 23.40, integrals) and, 24.109 to approximation of, 37.57 to 37.63, 23.46, 23.47, 23.55, 23.76, 24.111 37.65 to 37.69 25.54, 25.55 derivatives of, 24.95, 24.96, binomial series, 38.31, 38.78, one-one functions, 5.100 24.99, 24.100 38.103 to 38.106, 38.108 parabolas, 5.1,5.2, 5.11,5.101, graphs of, 24.97 convergence of, 37.1 to 37.3, 8.44, 15.30, 16.2, 16.47, identities for, 24.98, 24.101 to 37.5, 37.10 to 37.16, 37.23, 34.3, 34.4,34.7, 35.15,41.19 24.108 37.24, 37.27 to 37.56, 37.64, paraboloids, 41.4, 41.5, 41.9, Maclaurin series and, 39.14 37.69 to 37.116 41.17,41.25 power series and, 38.43, 38.44, for Euler's constant (y), 37.110 planes, 40.87, 41.3, 41.18 38.95 geometric series, 37.5 to 37.9, point functions, 5.13 Hyperboloids: 37.12,37.13,37.25,37.26, relative extrema in, 13.1, 13.2, equations of, 41.7 37.30 to 37.34, 37.45, 37.48, 15.11 to 15.15, 15.17to graphs of, 41.14, 41.15, 41.23, 37.67, 37.68, 37.73, 37.75, 15.19, 15.22to 15.24, 15.31 41.24 37.77,37.83, 37.91, 37.113 to 15.35, 15.39, 15.41 to as level surfaces, 41.46 to 37.116, 38.5, 38.26,38.29 15.46, 15.48 to 15.51, 15.53, as ruled surfaces, 41.29 harmonic series, 37.2, 37.3, 15.54, 15.57, 15.61 Hypergeometric series, 38.113, 37.27, 37.28, 38.6 roses, 35.33, 35.47, 35.48, 35.93 38.115 integral test for, 37.39, 37.41, saddle surfaces, 41.19 37.49, 37.50, 37.53, 37.54, spheres, 41.31 i unit vector, 40.28, 40.40, 40.54 37.64,38.112 tractrix, 29.42 Ideal gas law, 42.23 limit comparison test for, 37.42 trigonometric functions, 10.11 to Implicit differentiation, 12.2, 12.11 to 37.47, 37.54, 37.76, 37.78, 10.13, 15.32 to 15.38, 15.51, 37.82, 37.84, 37.86, 37.88 to 15.53, 27.1, 27.3, 27.71 to 12.22, 12.24, 12.34, 12.35, 37.90, 37.96, 37.98, 37.100, 12.38 to 12.48 37.101, 37.108 Gravity, 19.95, 19.96 Implicit partial differentiation, Maclaurin (see Maclaurin series) Greatest integer function, 5.6, 5.7, 42.13 to 42.18 p-series, 37.40, 37.44, 37.47, Improper integrals, 32.1 to 32.60 37.54, 37.69, 37.74, 37.78, 6.5 Increasing functions, 11.18, 11.20 37.82, 37.85, 37.88, 37.89, Green's theorem, 45.47 to 45.49 to 11.26, 11.29, 11.48, 15.52 37.95,38.11,38.21 Growth (see Exponential growth Increasing infinite sequences, 36.54 partial sums of, 37.4, 37.5, 37.10, to 36.57, 36.59, 36.60 37.17 to 37.22, 37.24, 37.25 and decay) power (see Power series) Infinite series (Cont.): Integrals (Cont.): INDEX 0 449 ratio test for, 37.52, 37.53, 37.55, indefinite (see Antiderivatives) 37.70 to 37.72, 37.77, 37.79 integration by parts, 28.1 to Intersections (Cont.): to 37.81, 37.87, 37.97, 28.57 lines and hyperbolas, 3.81 37.102,37.107,37.109, 38.1 iterated, 44.1 to 44.5 lines and parabolas, 3.80 to 38.19, 38.21, 38.22, 38.24, line, 45.40 to 45.44 lines and planes, 40.70 38.25, 38.31, 38.33, 38.67, mass with, 44.74 to 44.79 parabolas, 12.45 38.109 to 38.113 mean-value theorem for, 20.34 to paraboloids and planes, 41.21 repeating decimals as, 37.8, 37.9, 20.37, 20.42 planes, 40.19, 40.85, 40.87, 37.26 method of partial fractions for, 40.96,40.112 root test for, 37.91 to 37.94, 30.1 to 30.33 in polar coordinates, 35.50 to 37.105,37.106,38.20,38.23 moments of inertia with, 44.86 to 35.54, 35.105 to 35.107 Taylor (see Taylor series) 44.88 supply and demand equations, telescoping of, 37.10 moments with, 31.24 to 31.32, 3.82 Zeno's paradox and, 37.32 44.80 to 44.85, 44.89 to 44.92 Inverse functions, 5.69 to 5.74, Inflection points, 13.2, 13.6, 13.7, multiple (see Multiple integrals) 5.92, 5.93, 5.100, 9.49 13.18, 13.20, 14.42to 14.46, of rational functions, 30.1 to 30.33 15.2, 15.3, 15.5, 15.7, 15.8, Simpson's rule for Inverse trigonometric functions: 15.10 to 15.19, 15.21, 15.25, approximating, 20.68, 20.69, antiderivatives (indefinite 15.26, 15.32 to 15.35, 15.37, 23.73 integrals) and, 27.39 to 27.57, 15.40, 15.42 to 15.46, 15.48 surface area with, 31.1 to 31.15, 28.4,28.21,28.54,29.40 to 15.51, 15.53, 15.57, 15.61 32.41, 35.84 to 35.86, 44.53 definite integrals and, 27.61 to to 44.58 27.64, 27.81, 27.82, 32.54 Integral test for infinite series, trapezoidal rule for derivatives of, 27.2, 27.4, 27.22 to 37.39, 37.41,37.49,37.50, approximating, 20.66, 20.67, 27.38, 27.58 to 27.60, 27.65, 37.53, 37.54, 37.64, 38.112 20.70, 23.57 27.69, 27.70, 27.72 to 27.79 trigonometric substitutions in, Maclaurin series and, 39.10, Integrals: 29.3, 29.5, 29.19 to 29.21, 39.13, 39.31 antiderivatives (see 29.23 to 29.27, 29.29, 29.30, power series for, 38.36, 38.55, Antiderivatives) 29.38 to 29.41, 29.43 to 38.107, 38.115 approximation of (see 29.45 values of, 27.5 to 27.20 Approximation, of integrals) triple, 44.5, 44.50 to 44.52, 44.59 arc length with, 21.17 to 21.21, to 44.73, 44.76 to 44.78, Isosceles trapezoids, 3.77 21.30 to 21.34, 21.44,21.45, 44.83 to 44.85, 44.88, 44.91, Isosceles triangles, 14.46, 16.20, 23.54, 23.65, 23.66, 24.93, 44.92 27.64, 29.30 to 29.33, 34.32 volume with, 22.1 to 22.58, 16.55 to 34.42, 35.76 to 35.83 23.38, 23.59, 24.46, 24.48, Iterated integrals, 44.1 to 44.5 area with, 20.8, 20.15 to 20.20, 24.49, 27.63, 28.26, 28.27, j unit vector, 40.28, 40.40, 40.54 20.72, 20.88 to 20.90, 21.1 to 28.29 to 28.32, 29.44, 31.33 k unit vector, 40.28, 40.40, 40.54 21.16, 21.22 to 21.29, 21.35 to 31.35, 32.39, 32.40,44.16 Lagrange multipliers, 43.56 to to 21.43, 21.46, 21.47, 23.36, to 44.19, 44.21 to 44.23, 23.56, 24.45, 24.47, 24.92, 44.29 to 44.32, 44.34, 44.40, 43.64, 43.66, 43.67 27.61, 27.62, 27.81, 27.82, 44.41, 44.50, 44.51, 44.59 to Lagrange's remainder, 39.17 to 28.25,28.28, 29.43, 31.24 to 44.63, 44.66 to 44.70 31.35,32.1, 32.33, 32.34, work, 31.16 to 31.23, 45.43 39.20, 39.22 to 39.25 32.38, 32.47, 32.55, 32.56, Laplace transforms, 32.57 to 32.60 35.55 to 35.71, 35.75, 44.20, Integrating factors for differential Laplace's equation, 42.52 to 42.54 44.36 to 44.38 equations, 46.27 to 46.33, Laplacian, 45.32 centroids (center of mass) with, 46.35, 46.36, 46.45 Latus rectum of parabolas, 44.80 31.24to31.35, 35.72, 35.73, Law of cosines, 14.33, 14.51, 33.41 35.88, 44.80 to 44.85, 44.89 Integration by parts, 28.1 to 28.57 Left-hand limits, 6.5, 6.33 to 6.36, to 44.92 Intermediate value theorem, 7.20 convergence of, 32.1 to 32.10, 6.39 32.23 to 32.27, 32.29, 32.31, to 7.23, 11.28, 11.32, 11.37, Legendre's equation, 46.80 32.41 to 32.44, 32.48, 32.53, 11.42 Lemniscates, 35.32, 35.37, 35.42, 32.54 Intersections: definite (see Definite integrals) circles, 4.24 to 4.27, 12.27 35.57, 35.85, 35.86 double, 44.1 to 44.4, 44.6 to cylinders and planes, 40.18 Length of vectors, 33.3, 33.13, 40.26 44.46, 44.53 to 44.58, 44.74, ellipses and hyperbolas, 12.43 Level curves (contour maps), 41.32 44.75, 44.79 to 44.82, 44.86, ellipsoids and lines, 41.20 44.87, 44.89, 44.90 lines, 3.44, 3.78, 3.79, 10.46, to 41.43 improper, 32.1 to 32.60 10.47, 40.76 Level surfaces, 41.44 to 41.47 L'Hopital's rule, 25.1 to 25.53, 32.4, 32.6 to 32.8, 32.20, 32.36, 36.15, 36.20, 37.47, 37.108, 38.15,38.24,41.56, 41.57 450 0 INDEX Line segments and lines (Cont.): Line segments and lines (Cont.): graphs of, 3.16, 3.36, 5.5, 5.12, through two points, 3.16 to 3.19 Liebniz's formula for differentiable 5.14, 5.15 to 5.18, 5.20, vector representation of, 40.69, functions, 42.96, 42.97 5.23, 5.88, 6.33, 7.2, 7.4, 40.71,40.72 7.5, 7.8, 8.43, 10.46, 11.49, vectors and, 33.7, 33.8 Limacons, 35.31, 35.38 to 35.40, 16.45,34.9, 34.10, 35.10 to 35.56, 35.59, 35.60 35.12,35.19,35.20,40.21 Logarithmic differentiation, 23.23 intersections of, 3.44, 3.78, 3.79, to 23.26, 23.58, 23.67, 24.35 Limit comparison test for infinite 10.46, 10.47, 40.76 to 24.39 series, 37.42 to 37.47, 37.54, intersections with hyperbolas, 37.76, 37.78, 37.82, 37.84, Logarithmic functions (see Natural 37.86, 37.88 to 37.90, 37.96, 3.81 logarithms) 37.98, 37.100, 37.101, 37.108 intersections with parabolas, 3.80 Logarithmic spirals, 34.42 Limits: midpoints of, 3.33, 11.45, 20.91, Maclaurin series: addition property of, 6.13 asymptotes and (see 40.5 exponential functions and, 39.1, Asymptotes) normal, 8.16, 8.19, 8.33, 8.36, 39.7, 39.18,39,21, 39.24, cube roots in, 6.50, 6.51 39.25, 39.32, 39.35, 39.44, defined, 6.1 8.37,9.14,9.19, 10.23, 39.50 exponential functions, 24.74, 10.28, 34.27,42.117 24.75, 24.78 to 24.81, 24.83 parallel, 3.12 to 3.15, 3.23, 3.24, higher-order derivatives and, infinite sequences, 36.1 to 36.53, 3.66, 3.68, 8.23, 40.72 to 39.30 to 39.32, 39.43 36.65 40.74 left-hand, 6.5, 6.33 to 6.36, 6.39 parametric equations of, 34.9, hyperbolic functions and, L'Hopital's rule for, 25.1 to 34.10, 40.69 to 40.79, 40.85 39.14 25.53, 32.4, 32.6 to 32.8, perpendicular, 3.20 to 3.23, 3.25, 32.20,32.36,36.15,36.20, 3.26, 3.31,3.69,40.75,40.77 inverse trigonometric functions 37.47,37.108,38.15,38.24, perpendicular bisectors of, 3.35, and, 39.10, 39.13, 39.31 41.56,41.57 3.74 multivariate functions, 41.48 to planes cut by, 40.99 to 40.102, natural logarithms and, 39.5, 41.63 40.104,40.105 39.20, 39.28, 39.34, 39.40, natural logarithms, 22.43 to point-slope equations of, 3.2, 39.46, 39.50 22.45, 23.52, 23.71, 23.72 3.48 to 3.51, 10.22, 10.23, one-sided, 6.5, 6.33 to 6.36, 6.39 10.25, 10.28 powers or roots, 39.12, 39.33, polynomials, 6.2 to 6.4, 6.6 to points above and below, 3.36 to 39.39, 39.42, 39.43 6.8, 6.10, 6.11,6.12, 6.14 to 3.38 6.16, 6.17 to 6.19, 6.20 to points on, 3.83 trigonometric functions and, 6.22, 6.28, 6.32, 6.37, 6.42, in polar coordinates, 35.10 to 39.2, 39.8, 39.9, 39.15, 6.43 to 6.49 35.12, 35.19, 35.20 39.19, 39.23, 39.26, 39.27, product rule for, 36.50 rectangular equations of, 40.69, 39.29, 39.36, 39.48 right-hand, 6.5, 6.33 to 6.36, 40.71, 40.72, 40.97, 40.99, 6.39 40.101,40.113 Magnitude of vectors, 33.3, 33.13 square roots in, 6.9, 6.23 to 6.27, reflections of, 5.97 to 5.99 Mass: 6.29 to 6.31, 6.38, 6.40, slope-intercept equations of, 3.6 6.41,6.52 to 3.11, 3.20, 3.21,3.27, 3.29 center of (see Centroids) sum rule for, 36.49 to 3.31, 3.34, 3.35, 3.41,3.44, by integration, 44.74 to 44.79 trigonometric functions, 10.14 to 3.46, 3.52 to 3.60, 8.13,8.16, Mathematical induction, 2.18, 20.3, 10.16, 10.30, 10.31, 10.44, 8.36,9.13,9.14 10.48 slope of, 3.1,3.5, 3.46, 3.61 to 20.65, 24.89, 24.90, 32.22, two-dimensional vector 3.65 43.20 functions, 34.43 tangent, 4.15, 4.19 to 4.22, 8.13 to Maxima and minima of functions 8.15,8.18,8.20,8.23,8.28, (see Extrema of functions) Line integrals, 45.40 to 45.44 8.34,8.35,8.38,8.42,9.13. Mean, arithmetic and geometric, Line segments and lines: 9.19,9.23, 10.22, 10.23, 43.51 10.25, 10.28, 10.32 to 10.35, Mean value theorem: as contours (level curves), 41.33, 10.47, 11.44, 12.14, 12.16, for functions, 11.10 to 11.17, 41.39,41.41,41.43 12.18, 12.27, 12.28, 12.37, 11.30, 11.33to 11.36, 11.38 12.42 to 12.46, 15.21, 15.28, to 11.41, 11.43, 11.45, 11.46 curvature (K) of, 34.96 15.29, 19.66 to 19.70, 19.77, for integrals, 20.34 to 20.37, curvilinear motion and, 34.69, 19.83 to 19.86,21.46,24.56, 20.42 27.58,34.26,34.31,35.89to Medians of triangles, 3.33, 3.43, 34.70 35.92, 35.95, 35.98, 42.26 to 3.72, 33.23 distance between, 40.113, 42.33,42.119,45.1,45.4,45.6 Method of Lagrange multipliers, 43.56 to 43.64, 43.66, 43.67 40.114,43.31,43.38 Method of partial fractions for distance from points to, 3.45, integrals, 30.1 to 30.33 Midpoints: 3.47,4.28,4.30, 33.9,33.10, of lines, 3.33, 11.45, 20.91, 40.5 33.34 of triangles, 33.27 families of, 3.39, 3.40 Minima and maxima of functions (see Extrema of functions) Moments, by integration, 31.24 to Natural logarithms (Con?.): INDEX 0 451 31.32, 44.80 to 44.85, 44.89 Maclaurin series and, 39.5, Parabolas (Cont.): to 44.92 39.20, 39.28, 39.34, 39.40, 39.46, 39.50 intersections of lines with, 3.80 Moments of inertia, 44.86 to 44.88 power series and, 38.38, 38.50 to intersections of, 12.45 Motion: 38.53, 38.60, 38.86, 38.88, latus rectum of, 44.80 38.94, 38.99,38.100 parametric equations of, 34.3, curvilinear, 34.43 to 34.108 properties of, 23.27 to 23.34, rectilinear, 17.1 to 17.35, 19.34 23.64,24.1 to 24.6, 24.35, 34.4, 34.7 24.65 to 24.71 in polar coordinates, 35.15 to 19.39, 19.41, 19.44, 19.57, Simpson's rule approximations tangents to, 8.28 19.59, 19.72 to 19.76 for, 23.73 vertex of, 15.30, 21.13,21.15, Multiple integrals: Taylor series and, 39.6 area with, 44.20, 44.36 to 44.38 trapezoidal rule approximations 21.22,21.23 centroids (center of mass) with, for, 23.57 Paraboloids: 44.80 to 44.85, 44.89 to (See also Exponential functions) 44.92 equations for, 41.80 double integrals, 44.1 to 44.4, Newton's law of cooling, 26.22, graphs of, 41.4, 41.5, 41.9, 41.17, 44.6 to 44.46, 44.53 to 44.58, 26.39 44.74, 44.75, 44.79 to 44.82, 41.25 44.86, 44.87, 44.89, 44.90 Newton's second law of motion, as ruled surfaces, 41.30 iterated integrals, 44.1 to 44.5 19.95, 19.% Parallel lines, 3.12 to 3.15, 3.23, mass with, 44.74 to 44.79 moments of inertia with, 44.86 to Normal acceleration, 34.106 to 3.24, 3.66, 3.68, 8.23, 40.72 44.88 34.108 to 40.74 moments with, 44.80 to 44.85, Parallel planes, 40.90, 40.91 44.89 to 44.92 Normal distribution: Parallel vectors, 33.4, 33.5, 33.36, surface area with, 44.53 to 44.58 approximations for, 38.46, 39.41 33.39, 40.68 triple integrals, 44.5, 44.50 to power series for, 38.45 Parallelepipeds, 40.45, 40.49, 44.52, 44.59 to 44.73, 44.76 40.56, 40.63 to 44.78, 44.83 to 44.85, Normal lines, 8.16, 8.19,8.33, Parallelogram law for vectors, 44.88, 44.91,44.92 8.36,8.37,9.14,9.19, 10.23, 33.17 volume with, 44.16 to 44.19, 10.28, 34.27,42.117 Parallelograms: 44.21 to 44.23, 44.29 to area of, 40.43, 40.110,40.111 44.32,44.34,44.40,44.41, Normal planes, 42.120, 42.121, diagonals of, 33.26, 33.30 44.50, 44.51, 44.59 to 44.63, 45.3, 45.4, 45.6 quadrilaterals and, 3.42 44.66 to 44.70 rhombuses as, 3.76 Multivariate functions, 41.1 to 41.66 Normal vectors, 42.105 to 42.115, vertices of, 3.28, 40.109 Natural logarithms: 42.122 to 42.126, 45.19 Parametric equations, 34.1 to antiderivatives (indefinite 34.42, 40.69 to 40.79, integrals) and, 23.10 to Odd functions, 5.43, 5.44, 5.46, 5.49, 40.85 23.22, 23.53, 23.68, 23.69, 5.51,5.52, 5.54 to 5.56, 5.90 Partial derivatives, 42.1 to 42.126 23.79,23.80,28.7,28.16, to 5.92, 8.46, 8.47, 15.34, Partial differential equations: 28.19,28.22, 28.24, 28.55 to 15.47, 15.51,20.48,20.49, Cauchy-Riemann equations, 28.57 24.97, 38.74, 38.93 42.54, 42.92 approximations for, 38.52, 38.53, Laplace's equation, 42.52 to 39.28, 39.40 One-one functions, 5.57, 5.58, 42.54 defined, 23.1 5.60, 5.63, 5.69 to 5.74, wave equation, 42.55 to 42.58 definite integrals and, 23.36 to 5.92, 5.93, 5.100, 9.49 Partial fractions, 30.1 to 30.33, 23.38, 23.56, 23.57, 23.59, 46.22 23.61,23.63,23.73, 28.25 to One-sided limits, 6.5, 6.33 to 6.36, Partial sums, 37.4, 37.5, 37.10, 28.31, 32.4, 32.5, 32.8, 6.39 37.17 to 37.22, 37.24,37.25 32.10, 32.15,32.16, 32.36, Perimeter: 32.37, 32.50 Orthogonal trajectories, 46.22 to circle, 16.46 derivatives of, 23.1 to 23.9 46.24 rectangle, 16.1, 16.15, 16.29, graphs of, 23.39, 23.40, 23.46, 16.35, 16.41, 16.44, 16.46, 23.47, 23.55, 23.76, 25.54, Osculating planes, 45.18 to 45.20 16.60 25.55 p-series, 37.40, 37.44, 37.47, 37.54, triangle, 16.19, 16.20, 16.55, inequalities for, 23.41, 23.42, 16.60, 16.61 23.60, 23.77 37.69, 37.74, 37.78, 37.82, Period of trigonometric functions, limits and, 22.43 to 22.45, 23.52, 37.85, 37.88, 37.89, 37.95, 10.12, 10.13, 10.45 23.71,23.72 38.11, 38.21 Perpendicular bisectors, 3.35, Pappus's theorem for volume, 3.74 31.33 to 31.35 Perpendicular lines, 3.20 to 3.23, Parabolas: 3.25, 3.26, 3.31, 3.69, 40.75, as contours (level curves), 41.35, 40.77 41.38, 41.42 Perpendicular planes, 40.95 curvature (K) of, 34.98 graphs of, 5.1,5.2, 5.11,5.101, 8.44, 15.30, 16.2, 16.47, 34.3, 34.4,34.7,35.15,41.19 452 Q INDEX Polar coordinates (Cont.): Power series (Cont.): Perpendicular vectors, 33.4 to 33.8, polar-to-rectangular trigonometric functions and, transformations, 35.1, 35.4 38.58, 38.59, 38.63, 38.75 to 33.12, 33.18,33.19,33.21, to 35.6, 41.67 38,77, 38.79 to 38.83, 38.86, 33.31, 33.32, 33.37, 33.39, rectangular-to-polar 38.89 to 38.91, 38.96 40.36, 40.39, 40.41, 40.42, transformations, 35.1 to 40.51, 40.68 35.3,35.21 to 35.27, 41.67 Predator-prey system, 46.70 Planes: roses (curves) in, 35.33, 35.47, Principal unit normal vector, 34.81 angle between, 40.84, 40.103, 35.48, 35.58, 35.64, 35.65, 40.112 35.72, 35.93 to 34.86, 34.88,34.90, 34.91, cut by lines, 40.99 to 40.102, slope in, 35.93, 35.94, 35.96, 34.105 to 34.107, 45.16 40.104,40.105 35.97 Product rule: distance between, 40.91, 40.92 surface area calculations in, for derivatives, 8.7, 8.8, 8.40, distances from points to, 40.88, 35.84 to 35.86 8.41, 8.48, 9.10, 9.16, 9.40, 40.89 12.2, 12.20, 12.32 equations of, 40.80 to 40.83, Polynomials: for limits, 36.50 40.86,40.93,40.98,41.69, derivatives of, 8.5, 8.6, 8.12, for vector functions, 34.54, 41.78,41.82,41.96 8.32, 12.23 34.57, 34.73, 34.92 graphs of, 40.87, 41.3, 41.18 factorization of, 5.76 to 5.86 Pyramids, 22.22 intersections of, 40.19, 40.85, limits of, 6.2 to 6.4, 6.6 to 6.8, Pythagorean theorem, 4.21, 14.1, 40.87,40.96,40.112 6.10,6.11,6.12,6.14to 14.5, 40.34 as level surfaces, 41.44 6.16, 6.17 to 6.19, 6.20 to normal, 42.120, 42.121, 45.3, 6.22, 6.28, 6.32, 6.37, 6.42, Quadratic equations, discriminant 45.4, 45.6 6.43 to 6.49 for, 30.13, 46.48 osculating, 45.18 to 45.20 roots of, 5.76 to 5.82, 11.28 parallel, 40.90, 40.91 Quadrilaterals, 3.42 perpendicular, 40.95 Population growth, 26.5, 26.7, Quotient rule: tangent, 40.94, 42.31, 42.105 to 26.19, 26.25, 26.26, 26.28, 42.115, 42.122 to 42.126 26.38 for derivatives, 8.7, 8.9, 8.10, vectors and, 40.106 to 40.108 8.49, 8.50, 9.9, 9.13, 9.21, Plots (see Graphs) Position vector, 34.44 to 34.56, 9.38,9.41,9.43, 10.24, 12.1, Point-slope equations of lines, 3.2, 34.59, 34.60, 34.65 to 34.72, 12.11 to 12.13, 12.19, 12.33 3.48 to 3.51, 10.22, 10.23, 34.74 to 34.81,34.82 to 10.25, 10.28 34.96,34.100,34.101, for vector functions, 34.73 Polar coordinates: 34.103, 34.105 to 34.108, arc length calculations in, 35.76 45.1,45.2, 45.4 to 45.6, Radian measure of angles, 10.1 to to 35.83 45.12, 45.15 to 45.23 10.7 area calculations in, 35.55 to 35.71, 35.75 Power series: Radioactive decay, 26.15 to 26.18, cardioids in, 35.30, 35.34 to Abel's theorem for, 38.60 to 26.21, 26.30, 26.43 35.36, 35.44, 35.49, 35.55, 38.62 35.66, 35.67, 35.69, 35.73, Bessel functions, 38.67 to 38.69 Radius of circles, 4.1 to 4.6, 4.10 35.78, 35.84, 35.91, 35.102, convergence of, 38.1 to 38.33, to 4.15, 4.23 44.33, 44.38, 44.81 38.67, 38.68, 38.109 to centroid calculations in, 35.72, 38.114 Radius of curvature (-rho-), 34.94, 35.73, 35.88 differential equation solutions 34.95, 34.102, 34.104, 34.105 circles in, 35.7 to 35.9, 35.13, with, 38.49, 38.56, 38.66 35.14, 35.17, 35.18, 35.41, exponential functions and, 38.39 Range of functions, 5.1 to 5.19, 5.24 35.66 to 35.69, 35.87, 35.88, to 38.42, 38.76, 38.78 to to 5.31, 5.34 to 5.37, 27.21 44.33, 44.38, 44.42, 44.44 38.80, 38.82, 38.97 curvature (K) in, 35.108, hyperbolic functions and, 38.43, Rates, related, 14.1 to 14.56 35.109 38.44, 38.95 Ratio test for infinite series, 37.52, ellipses in, 35.16 hypergeometric series, 38.113, intersections in, 35.50 to 35.54, 38.115 37.53, 37.55, 37.70 to 37.72, 35.105 to 35.107 inverse trigonometric functions 37.77, 37.79 to 37.81, 37.87, lemniscates in, 35.32, 35.37, and, 38.36, 38.55,38.107, 37.97, 37.102, 37.107, 35.42, 35.57, 35.85, 35.86 38.115 37.109,38.1 to 38.19, 38.21, limacons in, 35.31, 35.38 to natural logarithms and, 38.38, 38.22, 38.24, 38.25, 38.31, 35.40, 35.56, 35.59, 35.60 38.50 to 38.53, 38.60, 38.86, 38.33, 38.67, 38.109 to lines in, 35.10 to 35.12, 35.19, 38.88, 38.94, 38.99, 38.100 38.113 35.20 normal distribution, 38.45 Rational function integration, 30.1 parabolas in, 35.15 powers or roots, 38.34, 38.35, to 30.33 38.37, 38.64, 38.65, 38.72, Rectangles: 38.105, 38.106, 38.108 area of, 14.9, 14.34, 16.1, 16.5, 16.6, 16.15, 16.21, 16.26to 16.29, 16.31, 16.44to 16.46, 16.57, 16.60 inscribed in ellipses, 16.41 perimeter of, 16.1, 16.15, 16.29, 16.35, 16.41, 16.44, 16.46, 16.60 Rectangular coordinates: Roses (curves), 35.33, 35.47, 35.48, INDEX 0 453 cylindrical coordinates and, 35.58, 35.64, 35.65, 35.72, 41.67, 41.70 to 41.74, 41.80 35.93 Spirals (Cont.): to 41.85 equiangular, 35.71, 35.77, 35.101, cylindrical-to-rectangular Ruled surfaces, 41.29, 41.30 35.112 transformations, 41.67 Saddle surface, 41.19, 41.84 logarithmic, 34.42 polar-to-rectangular Scalar projection of vectors, 33.4, transformations, 35.1, 35.4 Square roots: to 35.6, 41.67 40.33 approximation of, 18.2, 18.3, rectangular-to-cylindrical Second derivative test for relative 18.20, 18.33 transformations, 41.67 in limits, 6.9, 6.23 to 6.27, 6.29 rectangular-to-polar extrema, 13.1, 13.3 to 13.7, to 6.31, 6.38, 6.40,6.41, transformations, 35.1 to 13.20, 13.21, 15.1 to 15.9, 6.52 35.3, 35.21 to 35.27, 41.67 15.12, 15.13, 15.16to 15.18, rectangular-to-spherical 15.20to 15.29, 15.32, 15.33, Squares, completing, 4.6 to 4.8, transformations, 41.85 15.35, 15.37, 15.39 to 15.56 4.10,4.11,4.13,4.16,27.51, spherical coordinates and, 41.85, (See also First derivative test for 27.53,29.29,29.39, 30.13, 41.88 to 41.93 relative extrema) 34.14,40.11 spherical-to-rectangular Segments of disks, 29.43 transformations, 41.85 Self-inverse functions, 5.69, 5.74, Strictly increasing functions, 5.75 11.48 Rectangular form of vectors, 33.14 Sequences: to 33.16 infinite (see Infinite sequences) Strictly positive functions, 11.48 sum of cubes of integers, 20.65 Sum rule: Rectilinear motion, 17.1 to 17.35, sum of integers, 20.4 19.34 to 19.39, 19.41, 19.44, Series (see Infinite series) for derivatives, 8.7, 9.11 19.57, 19.59, 19.72 to 19.76 Simpson's rule for approximating for limits, 36.49 integrals, 20.68, 20.69, 23.73 Sums: Reflections of curves and lines, Sketches (see Graphs) of cubes of integers, 20.65 5.94 to 5.100 Slope: of integers, 20.4 of curves, 34.28 partial, 37.4, 37.5, 37.10, 37.17 Related rates, 14.1 to 14.56 of lines, 3.1,3.5, 3.46,3.61 to Relative extrema (see Extrema of 3.65 to 37.22, 37.24, 37.25 in polar coordinates, 35.93, Supply and demand equations, functions, relative) 35.94, 35.96, 35.97 Removable discontinuities, 7.5, 7.9 Slope-intercept equations of lines, 3.82, 16.32, 16.33 Repeating decimals as infinite 3.6 to 3.11, 3.20, 3.21,3.27, Surface area: 3.29 to 3.31, 3.34, 3.35, series, 37.8, 37.9, 37.26 3.41, 3.44, 3.46, 3.52 to cone, 16.9,31.14,44.58 Revolution: 3.60, 8.13, 8.16, 8.36, 9.13, cylinder, 16.22, 42.89 9.14 by integration, 31.1 to 31.15, surface of, 41.4 to 41.12 Spheres: volume of, 31.33 to 31.35 cones circumscribed about, 16.42 32.41, 35.84 to 35.86, 44.53 Rhombuses: equations of, 40.6 to 40.12, to 44.58 diagonals of, 33.32 40.16,40.22,40.23,41.76, sphere, 14.13, 14.37, 14.43, 31.2, as parallelograms, 3.76 41.86 44.57 Riccati equation, 46.43, 46.44 graphs of, 41.31 spherical cap, 31.15 Right angles, 33.12 as level surfaces, 41.47 Surface normal vectors, Right-hand limits, 6.5, 6.33 to 6.36, planes tangent to, 40.94 42.31 surface area of, 14.13, 14.37, Surfaces: 6.39 14.43, 31.2, 44.57 level, 41.44 to 41.47 Right triangles: surface area of caps of, 31.15 of revolution, 41.4 to 41.12 volume of, 14.7, 14.13, 14.35, ruled, 41.29, 41.30 area of, 40.36 14.37, 14.43, 14.48, 22.1, saddle, 41.19, 41.84 centroidof, 31.28, 31.34 44.34, 44.50 Tangent lines, 4.15, 4.19 to 4.22, perimeter of, 16.19 Spherical coordinates, 41.85 to 8.13 to 8.15, 8.18, 8.20, Pythagorean theorem for, 4.21, 41.98 8.23, 8.28, 8.34, 8.35, 8.38, Spherical segment volume, 14.41 8.42,9.13,9.19,9.23, 10.22, 14.1, 14.5,40.34 Spirals: 10.23, 10.25, 10.28, 10.32 to vertices of, 3.25, 3.26, 40.13 Archimedean, 35.70, 35.76, 10.35, 10.47, 11.44, 12.14, Rolle's theorem, 11.1 to 11.9, 35.103, 35.110 12.16, 12.18, 12.27, 12.28, degenerate, 35.111 12.37, 12.42 to 12.46, 15.21, 11.27, 11.31, 11.38, 11.47, 15.28, 15.29, 19.66 to 19.70, 11.49 19.77, 19.83to 19.86,21.46, Root test for infinite series, 37.91 24.56, 27.58, 34.26, 34.31, to 37.94, 37.105, 37.106, 35.89 to 35.92, 35.95, 35.98, 38.20, 38.23 42.26 to 42.33, 42.119, 45.1, Roots: 45.4, 45.6 cube (see Cube roots) Tangent planes, 42.31, 42.105 to of polynomials, 5.76 to 5.82, 11.28 42.115, 42.122 to 42.126 square (see Square roots) 454 0 INDEX Trigonometric functions (Con/.): Unit tangent vector, 34.47 to 34.50, approximation of, 18.23, 18.25, 34.81 to 34.93, 34.105 to Tangent vectors, 34.44 to 34.56, 18.26, 18.28, 18.30, 18.32, 34.107 34.59, 34.60, 34.65 to 34.72, 38.55, 38.75, 38.77, 39.27, 34.74 to 34.80, 34.89, 34.92, 39.29 Unit vectors, 33.24, 34.67, 40.28, 42.118,42.119,45.1,45.3to defined, 10.8 40.31,40.33,40.40,40.54, 45.6 definite integrals and, 20.10, 43.66 20.11,20.15, 20.20 to 20.22, Tangential acceleration, 34.106 to 20.24, 20.39, 20.49, 20.60, Vector convergence, 34.43 34.108 20.62, 20.63, 20.79 to 20.84, Vector functions: 28.32 to 28.34, 28.53, 29.17, Taylor series: 29.18,29.30,29.33,32.13, chain rule for, 34.61 Lagrange's remainder and, 39.17 32.31, 32.32, 32.44, 32.45, curl, 45.28 to 45.31, 45.33, 45.34, to 39.20, 39.22 to 39.25 32.51, 32.53, 32.59 natural logarithms, 39.6 derivatives of, 10.17 to 10.29, 45.36 to 45.38 powers or roots, 39.4, 39.11, 10.36 to 10.43, 13.7, 13.15 to divergence, 45.25 to 45.27, 45.30 39.37, 39.38, 39.47 13.17, 13.30to 13.34 trigonometric functions, 39.3, extremaof, 13.7, 13.15 to 13.17, to 45.32, 45.34, 45.35, 45.37, 39.16, 39.22, 39.45 13.30 to 13.34 45.39, 45.47 to 45.49 frequency of, 10.12 Gauss' theorem for, 45.49 Telescoping of infinite series, 37.10 graphs of, 10.11 to 10.13, 15.32 gradient, 43.1 to 43.22, 43.56 to Temperature scales, 3.71 to 15.38, 15.51, 15.53,27.1, 43.64, 43.66, 43.67, 45.24, Tetrahedrons, 22.23, 42.113 27.3, 27.71 45.32, 45.33, 45.35, 45.36, Toruses: higher-order derivatives of, 12.8 45.39, 45.45 to 12.10 Green's theorem for, 45.47 to equations of, 41.94 implicit differentiation of, 12.16 45.49 volume of, 31.33 to 12.18, 12.38 to 12.41, Laplacian, 45.32 Tractrix (curve), 29.42 12.47 product rule for, 34.54, 34.57, Trapezoidal rule for approximating inverse (see Inverse 34.73, 34.92 trigonometric functions) quotient rule for, 34.73 integrals, 20.66, 20.67, Laplace transform of, 32.59 Vector projection of vectors, 33.4, 20.70, 23.57 limits of, 10.14 to 10.16, 10.30, 33.20, 33.39, 40.33 Trapezoids, 3.77 10.31, 10.44, 10.48 Vectors: Triangle inequality, 2.18, 2.19, Maclaurin series and, 39.2, 39.8, acceleration, 34.51, 34.53, 34.56, 2.35, 6.13, 33.29, 36.47, 39.9, 39.15, 39.19, 39.23, 34.69, 34.71, 34.74 to 34.80, 36.49 to 36.51 39.26, 39.27, 39.29, 39.36, 34.83 to 34.86, 34.92, 34.106 Triangles: 39.48 to 34.108, 45.12, 45.16, altitudes of, 3.34, 3.41, 3.73 mean value theorem and, 11.34, 45.19,45.20,45.22,45.23 circles circumscribed about, 11.39, 11.40, 11.42 addition and subtraction of, 33.2, 16.61 period of, 10.12, 10.13, 10.45 33.3, 33.17, 33.21, 33.22, equilateral, 14.36, 16.20, 16.51, power series and, 38.58, 38.59, 40.30 16.60 38.63, 38.75 to 38.77, 38.79 angle between, 33.3, 33.13, isosceles, 14.46, 16.20, 16.55 to 38.83, 38.86, 38.89 to 33.33, 33.35, 33.38, 33.40, law of cosines for, 14.33, 14.51, 38.91, 38.96 40.35, 40.37, 40.38 33.41 Rolle's theorem and, 11.47 between two points, 40.25 medians of, 3.33, 3.43, 3.72, 33.23 Taylor series and, 39.3, 39.16, binormal, 45.17 midpoints of, 33.27 39.22, 39.45 Cauchy's inequality for, 33.28, perimeter of, 16.19, 16.20, 16.55, values of, 10.9, 10.10 33.29 16.60, 16.61 wavelength of, 10.12 components of, 33.4, 33.5 perpendicular bisectors of, 3.35, zeros of, 11.47 cross product of, 40.40 to 40.43, 3.74 (See also Angles) 40.45 to 40.68, 45.11,45.12, right (see Right triangles) 45.17to45.20, 45.22, 45.23, vertices of, 40.17 Trigonometric substitutions in 45.28, 45.36, 45.37, 45.39 Trigonometric functions: integrals, 29.3, 29.5, 29.19 to defined, 33.1 amplitude of, 10.13, 10.45 29.21,29.23 to 29.27, 29.29, direction cosines of, 40.27 to antiderivatives (indefinite 29.30, 29.38 to 29.41, 29.43 40.29, 40.32 integrals) and, 19.11 to to 29.45 direction of, 33.3, 33.13, 33.25 19.13, 19.15, 19.16, 19.20 to distance from points to lines 19.22, 19.31, 19.39, 19.40, Triple integrals, 44.5, 44.50 to with, 33.9, 33.10, 33.34 19.43, 19.45, 19.47, 19.48, 44.52, 44.59 to 44.73, 44.76 dot product of, 33.4, 33.5, 33.9, 19.53, 19.55, 19.56, 19.60, to 44.78, 44.83 to 44.85, 33.10, 33.12, 33.18 to 33.20, 19.78 to 19.82, 19.98 to 44.88, 44.91, 44.92 33.28 to 33.35, 33.37 to 19.100, 28.2, 28.5 to 28.12, 33.41,40.33 to 40.35, 40.39, 28.14,28.15,28.17, 28.18, 40.44, 40.45, 40.47, 40.49, 28.35 to 28.41, 28.44, 28.45, 28.49,29.1 to 29.16, 29.19 to 29.29, 29.34 to 29.37, 29.45 Vectors; dot product of (Cont.): Vectors (Cont.): INDEX Q 455 40.55 to 40.58, 40.60 to 40.67, surface normal, 42.31 43.1 to 43.7, 43.10 to 43.12, tangent, 34.44 to 34.56, 34.59, Volume (Cont.): 43.16,43.17,43.66,45.9, 34.60, 34.65 to 34.72, 34.74 to cylinder, 14.2, 14.48, 16.7, 16.8, 45.13 to 45.15, 45.35, 45.37, 34.80,34.89,34.92,42.118, 16.16, 16.18, 16.43,43.67, 45.39,45.41,45.43,45.45to 42.119, 45.1, 45.3 to 45.6 44.70 45.49 triangle inequality for, 33.29 ellipsoid, 41.22 length of, 33.3, 33.13, 40.26 unit, 33.24, 34.67, 40.28, 40.31, by integration, 22.1 to 22.58, lines and, 33.7, 33.8 40.33, 40.40, 40.54, 43.66 23.38, 23.59, 24.46, 24.48, magnitude of, 33.3, 33.13 unit tangent, 34.47 to 34.50, 34.81 24.49, 27.63, 28.26, 28.27, multiplication by scalars, 33.2, to 34.93, 34.105 to 34.107 28.29 to 28.32, 29.44, 31.33 40.30 vector projection of, 33.4, 33.20, to 31.35, 32.39, 32.40,44.16 normal, 42.105 to 42.115, 42.122 33.39, 40.33 to 44.19, 44.21 to 44.23, to 42.126, 45.19 velocity, 34.44 to 34.56, 34.59, 44.29 to 44.32, 44.34, 44.40, parallel, 33.4, 33.5, 33.36, 33.39, 34.60, 34.65 to 34.72, 34.74 44.41,44.50,44.51, 44.59 to 40.68 to 34.80, 34.89, 34.92,45.1, 44.63, 44.66 to 44.70 parallelogram law for, 33.17 45.2, 45.4 to 45.6, 45.12, Pappus's theorem for, 31.33 to perpendicular, 33.4 to 33.8, 45.15 to 45.23 31.35 33.12, 33.18,33.19, 33.21, zero, 33.11 parallelepiped, 40.45, 40.49, 33.31, 33.32, 33.37, 33.39, 40.56, 40.63 40.36,40.39,40.41,40.42, Velocity, escape, 19.95, 19.96 pyramid, 22.22 40.51, 40.68 Velocity vector, 34.44 to 34.56, of revolution, 31.33 to 31.35 planes and, 40.106 to 40.108 sphere, 14.7, 14.13, 14.35, 14.37, position, 34.44 to 34.56, 34.59, 34.59, 34.60, 34.65 to 34.72, 14.43, 14.48,22.1,44.34, 34.60, 34.65 to 34.72, 34.74 34.74 to 34.80, 34.89, 34.92, 44.50 to 34.81, 34.82 to 34.96, 45.1,45.2, 45.4 to 45.6, spherical segment, 14.41 34.100,34.101,34.103, 45.12, 45.15 to 45.23 tetrahedron, 22.23, 42.113 34.105 to 34.108, 45.1,45.2, Vertex of parabolas, 15.30, 21.13, torus, 31.33 45.4 to 45.6, 45.12, 45.15 to 21.15,21.22, 21.23 45.23 Vertices: Wave equation, 42.55 to 42.58 principal unit normal, 34.81 to parallelogram, 3.28, 40.109 Wavelength of trigonometric 34.86, 34.88, 34.90, 34.91, right triangle, 3.25, 3.26, 40.13 34.105 to 34.107, 45.16 triangle, 40.17 functions, 10.12 rectangular form of, 33.14 to Volume: Work, by integration, 31.16 to 33.16 cone, 14.6, 14.18, 14.29, 14.38, scalar projection of, 33.4, 40.33 16.9, 16.42, 22.2, 31.34, 31.23, 45.43 42.86,43.67,44.41,44.51 Wronskian, 46.65 to 46.69 cone frustrum, 22.49 Zeno's paradox, 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3000 Solved Problems in Calculus By Elliot Mendelson

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