SCHAUMS OUTLINE OF THEORY AND PROBLEMS OF DIFFERENTIAL AND INTEGRAL CALCULUS Third Edition 0 FRANK AYRES, JR,P h.D. Formerly Professor and Head Department of Mathematics Dickinson College and ELLIOTT MENDELSON,P h.D. Professor of Mathematics Queens College 0 SCHAUM’S OUTLINE SERIES McGRAW-HILL New York St. Louis San Francisco Auckland Bogota Caracas Lisbon London Madrid Mexico City Milan Montreal New Delhi San Juan Singapore Sydney Tokyo Toronto FRANK AYRES, Jr., Ph.D., was formerly Professor and Head of the Department of Mathematics at Dickinson College, Carlisle, Pennsyl- vania. He is the author of eight Schaum’s Outlines, including TRI- GONOMETRY, DIFFERENTIAL EQUATIONS, FIRST YEAR COL- LEGE MATH, and MATRICES. ELLIOTT MENDELSON, Ph.D. , is Professor of Mathematics at Queens College. He is the author of Schaum’s Outlines of BEGINNING CAL- CULUS and BOOLEAN ALGEBRA AND SWITCHING CIRCUITS. Schaum’s Outline of Theory and Problems of CALCULUS Copyright 0 1990, 1962 by The McGraw-Hill Companies, Inc. All Rights Reserved. Printed in the United States of America. Except as permitted under the Copyright Act of 1976, no part of this pub- lication may be reproduced or distributed in any form or by any means, or stored in a data base or retrieval system, without the prior written permission of the publisher. 9 10 11 12 13 14 15 16 17 18 19 20 BAW BAW 9 8 7 6 ISBN 0-07-002bb2-9 Sponsoring Editor, David Beckwith Production Supervisor, Leroy Young Editing Supervisor, Meg Tobin Library of Congress Catabghg-io-PubkationD ata Ayres, Frank, Schaum’s outline of theory and problems of differential and integral calculus / Frank Ayres, Jr. and Elliott Mendelson. -- 3rd ed . p. cm. -- (Schaum’s outline series) ISBN 0-07-002662-9 1. Calculus-- Outlines, syllabi, etc. 2. Calculus-- Problems, exercises, etc. 1. Mendelson, Elliott. 11. Title. QA303.A% 1990 5 15- -dc20 89- 13068 CIP McGraw -Hill A Division of The McGraw-HitlC ompanies- This third edition of the well-known calculus review book by Frank Ayres, Jr., has been thoroughly revised and includes many new features. Here are some of the more significant changes: 1. Analytic geometry, knowledge of which was presupposed in the first two editions, is now treated in detail from the beginning. Chapters 1 through 5 are completely new and introduce the reader to the basic ideas and results. 2. Exponential and logarithmic functions are now treated in two places. They are first discussed briefly in Chapter 14, in the classical manner of earlier editions. Then, in Chapter 40, they are introduced and studied rigorously as is now customary in calculus courses. A thorough treatment of exponential growth and decay also is included in that chapter. 3. Terminology, notation, and standards of rigor have been brought up to date. This is especially true in connection with limits, continuity, the chain rule, and the derivative tests for extreme values. 4. Definitions of the trigonometric functions and information about the important trigonometric identities have been provided. 5. The chapter on curve tracing has been thoroughly revised, with the emphasis shifted from singular points to examples that occur more frequently in current calculus courses. The purpose and method of the original text have nonetheless been pre- served. In particular, the direct and concise exposition typical of the Schaum Outline Series has been retained. The basic aim is to offer to students a collection of carefully solved problems that are representative of those they will encounter in elementary calculus courses (generally, the first two or three semesters of a calculus sequence). Moreover, since all fundamental concepts are defined and the most important theorems are proved, this book may be used as a text for a regular calculus course, in both colleges and secondary schools. Each chapter begins with statements of definitions, principles, and theorems. These are followed by the solved problems that form the core of the book. They give step-by-step practice in applying the principles and provide derivations of some of the theorems. In choosing these problems, we have attempted to anticipate the difficulties that normally beset the beginner. Every chapter ends with a carefully selected group of supplementary problems (with answers) whose solution is essential to the effective use of this book. ELLIO~MT ENDELSON Table of Contents Chapter 1 ABSOLUTE VALUE; LINEAR COORDINATE SYSTEMS; INEQUALITIES .......................................... 1 Chapter 2 THE RECTANGULAR COORDINATE SYSTEM .............. 8 Chapter 3 LINES ............................ .................... 17 Chapter 4 CIRCLES ................................................ 31 Chapter 5 EQUATIONS AND THEIR GRAPHS ........................ 39 Chapter 6 FUNCTIONS ............................................. 52 Chapter 7 LIMITS .................................................. 58 Chapter 8 CONTINUITY ............................................ 68 Chapter 9 THE DERIVATIVE ....................................... 73 Chapter 10 RULES FOR DIFFERENTIATING FUNCTIONS ............... 79 Chapter 11 IMPLICIT DIFFERENTIATION ............................. 88 Chapter 12 TANGENTS AND NORMALS .............................. 91 Chapter 13 MAXIMUM AND MINIMUM VALUES ...................... 96 Chapter 14 APPLIED PROBLEMS INVOLVING MAXIMA AND MINIMA . . 106 Chapter 15 RECTILINEAR AND CIRCULAR MOTION .................. 112 Chapter 16 RELATED RATES ........................................ 116 Chapter 17 DIFFERENTIATION OF TRIGONOMETRIC FUNCTIONS ...... 120 Chapter 18 DIFFERENTIATION OF INVERSE TRIGONOMETRIC FUNC- TIONS .................................................. 129 Chapter 19 DIFFERENTIATION OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS ............................................. 133 Chapter 20 DIFFERENTIATION OF HYPERBOLIC FUNCTIONS .......... 141 Chapter 21 PARAMETRIC REPRESENTATION OF CURVES ............. 145 Chapter 22 CURVATURE ............................................ 148 Chapter 23 PLANE VECTORS ........................................ 155 Chapter 24 CURVILINEAR MOTION .................................. 165 Chapter 25 POLAR COORDINATES .................................. 172 Chapter 26 THE LAW OFTHE MEAN ................................. 183 Chapter 27 INDETERMINATE FORMS ................................ 190 Chapter 28 DIFFERENTIALS ......................................... 196 Chapter 29 CURVE TRACING ........................................ 201 Chapter 30 FUNDAMENTAL INTEGRATION FORMULAS ............... 206 Chapter 31 INTEGRATION BY PARTS ................................ 219 Chapter 32 TRIGONOMETRIC INTEGRALS ........................... 225 Chapter 33 TRIGONOMETRIC SUBSTITUTIONS ....................... 230 Chapter INTEGRATION BY PARTIAL FRACTIONS .................. 234 Chapter 35 MISCELLANEOUS SUBSTITUTIONS ........................ 239 Chapter 36 INTEGRATION OF HYPERBOLIC FUNCTIONS .............. 244 Chapter 37 APPLICATIONS OF INDEFINITE INTEGRALS ............... 247 Chapter 38 THE DEFINITE INTEGRAL ............................... 251 CONTENTS Chapter 39 PLANE AREAS BY INTEGRATION ........................ 260 Chapter 40 EXPONENTIAL AND LOGARITHMIC FUNCTIONS; EX- PONENTIAL GROWTH AND DECAY ....................... 268 Chapter 41 VOLUMES OF SOLIDS OF REVOLUTION ................... 272 Chapter 42 VOLUMES OF SOLIDS WITH KNOWN CROSS SECTIONS. .... 280 Chapter 43 CENTROIDS OF PLANE AREAS AND SOLIDS OF REVO- LUTION ........................................ ..... 284 Chapter 44 MOMENTS OF INERTIA OF PLANE AREAS AND SOLIDS OF REVOLUTION ........................................... 292 Chapter 45 FLUID PRESSURE ............................ 297 Chapter 46 WORK ................... .......................... 301 Chapter 47 LENGTH OF ARC ........................................ 305 Chapter 48 AREAS OF A SURFACE OF REVOLUTION ................. 309 Chapter 49 CENTROIDS AND MOMENTS OF INERTIA OF ARCS AND SURFACES OF REVOLUTION ............................. 313 Chapter 50 PLANE AREA AND CENTROID OF AN AREA IN POLAR COORDINATES .......................................... 316 Chapter 51 LENGTH AND CENTROID OF AN ARC AND AREA OF A SURFACE OF REVOLUTION IN POLAR COORDINATES ..... 321 Chapter 52 IMPROPER INTEGRALS .................................. 326 Chapter 53 INFINITE SEQUENCES AND SERIES ....................... 332 Chapter 54 TESTS FOR THE CONVERGENCE AND DIVERGENCE OF POSITIVE SERIES ........................................ 338 Chapter 55 SERIES WITH NEGATIVE TERMS ......................... 345 Chapter 56 COMPUTATIONS WITH SERIES ........................... 349 Chapter 57 POWER SERIES .......................................... 354 Chapter 58 SERIES EXPANSION OF FUNCTIONS. ...................... 360 Chapter 59 MACLAURIN'S AND TAYLOR'S FORMULAS WITH RE- MAINDERS .............................................. 367 Chapter 60 COMPUTATIONS USING POWER SERIES. .................. 371 Chapter 61 APPROXIMATE INTEGRATION ........................... 375 Chapter 62 PARTIAL DERIVATIVES .................... 380 Chapter 63 TOTAL DIFFERENTIALS AND TOTAL DERIVATIVES ....... 386 Chapter 64 IMPLICIT FUNCTIONS .................................... 394 Chapter 65 SPACE VECTORS ............. ................... 398 Chapter 66 SPACE CURVES AND SURFACE ........................ 411 Chapter 67 DIRECTIONAL DERIVATIVES; MAXIMUM AND MINIMUM VALUES ...... ....................................... 417 Chapter 68 VECTOR DIFFERENTIATION AND INTEGRATION .......... 423 Chapter 69 DOUBLE AND ITERATED INTEGRALS .................... 435 Chapter 70 CENTROIDS AND MOMENTS OF INERTIA OF PLANE AREAS 442 Chapter 71 VOLUME UNDER A SURFACE BY DOUBLE INTEGRATION 448 Chapter 72 AREA OF A CURVED SURFACE BY DOUBLE INTEGRATION 45 1 Chapter 73 TRIPLE INTEGRALS ..................................... 456 Chapter 74 MASSES OF VARIABLE DENSITY ......................... 466 Chapter 75 DIFFERENTIAL EQUATIONS ............................. 470 Chapter 76 DIFFERENTIAL EQUATIONS OF ORDER TWO ............. 476 INDEX ..... 48 1 Chapter 1 Absolute Value; Linear Coordinate Systems; Inequalities THE SET OF REAL NUMBERS consists of the rational numbers (the fractions alb, where a and b are integers) and the irrational numbers (such as fi = 1.4142 . . . and T = 3.14159 . . .), which + are not ratios of integers. Imaginary numbers, of the form x y m , w ill not be considered. Since no confusion can result, the word number will always mean real number here. THE ABSOLUTE VALUE 1x1 of a number x is defined as follows: { x if x is zero or a positive number Ixl = --x if x is a negative number For example, 131 = 1-31 = 3 and 101 = 0. In general, if x and y are any two numbers, then - 1x1 5 x 5 1x1 -XI I-xl = 1x1 and Ix - yl = ly 1x1 = lyl implies x = *y + + Ix yl 5 1x1 Iyl (Triangle inequality) (1.5) A LINEAR COORDINATE SYSTEM is a graphical representation of the real numbers as the points of a straight line. To each number corresponds one and only one point, and conversely. To set up a linear coordinate system on a given line: (1) select any point of the line as the origin (corresponding to 0); (2) choose a positive direction (indicated by an arrow); and (3) choose a fixed distance as a unit of measure. If x is a positive number, find the point corresponding to x by moving a distance of x units from the origin in the positive direction. If x is negative, find the point corresponding to x by moving a distance of 1x1 units from the origin in the negative direction. (See Fig. 1-1.) 1 1 1 1 1 1 I I I 1 I I 1 1 I 1 1 1 1 1 I I I I I 11 1 Y -4 -3 -512 -2 -312 -1 0 1/2 1 2 3r 4 ~ Fig. 1-1 The number assigned to a point on such a line is called the coordinate of that point. We often will make no distinction between a point and its coordinate. Thus, we might refer to “the point 3” rather than to “the point with coordinate 3.” If points P, and P, on the line have coordinates x, and x, (as in Fig. 1-2), then Ix, - x2(= PIP2= distance between P, and P2 (1.6) As a special case, if x is the coordinate of a point P, then 1x1 = distance between P and the origin (1.7) 1 2 ABSOLUTE VALUE; LINEAR COORDINATE SYSTEMS; INEQUALITIES [CHAP. 1 x2 Fig. 1-2 FINITE INTER QLS. Let a and b be two points such that a < b. By the open ',zterval (a, ") we mean the set of all points between a and b, that is, the set of all x such that a < x < b. By the closed interval [a,b ] we mean the set of all points between a and b or equal to a or b, that is, the set of all x such that a Ix 5 b. (See Fig. 1-3.) The points a and b are called the endpoints of the intervals (a, b) and [a,b ]. - - * - A 4 L m W U b U b Open interval (a, b): a < x < b Closed interval [a, b]: a Ix Ib Fig. 1-3 By a huff-open interval we mean an open interval (a, b) together with one of its endpoints. There are two such intervals: [a,b ) is the set of all x such that a 5 x < b, and (a, b] is the set of all x such that a < x 5 b. For any positive number c, 1x1 5 c if and only if -c 5 x Ic 1x1 < c if and only if -c < x < c See Fig. 1-4. - - I + n 1 n * 1 W 1 W -C 0 C -C 0 C Fig. 1-4 INFINITE INTERVALS. Let a be any number. The set of all points x such that a < x is denoted by (a,3 0); the set of all points x such that a Ix is denoted by [a,0 0). Similarly, (-00, b) denotes the set of all points x such that x < b, and (-00, b] denotes the set of all x such that x 5 b. INEQUALITIES such as 2x - 3 > 0 and 5 < 3x + 10 I1 6 define intervals on a line, with respect to a given coordinate system. EXAMPLE 1: Solve 2x - 3 > 0. 2 ~ - 3 > 0 2x > 3 (Adding 3) x > (Dividing by 2) Thus, the corresponding interval is ($, 00). CHAP. 11 ABSOLUTE VALUE; LINEAR COORDINATE SYSTEMS; INEQUALITIES 3 EXAMPLE 2: Solve 5 < 3x + 10 5 16. 5<3x+10116 -5< 3x 5 6 (Subtracting 10) - ;< x 12 (Dividing by 3) Thus, the corresponding interval is (-5/3,2]. EXAMPLE 3: Solve -2x + 3 < 7. -2~+3<7 - 2x < 4 (Subtracting 3) x > -2 (Dividing by - 2) Note, in the last step, that division by a negative number reverses an inequality (as does multiplication by a negative number). Solved Problems 1. Describe and diagram the following intervals, and write their interval notation: (a) - 3 < x<5; (b) 2 1 x 5 6 ; (c) -4<x50; (d)x>5; (e)xs2; (f)3 x-458; (g) 1<5-3x<11. (a) All numbers greater than -3 and less than 5; the interval notation is (-3,5): (6) All numbers equal to or greater than 2 and less than or equal to 6; [2,6]: (c) All numbers greater than -4 and less than or equal to 0; (-4,0]: (d) All numbers greater than 5; ( 5 , ~ ) : (e) All numbers less than or equal to 2; (-W, 21: (f) 3x - 4 I8 is equivalent to 3x I1 2 and, therefore, to x 5 4. Thus, we get (-m, 41: 1< 5 - 3x < 11 -4< -3x <6 (Subtracting 5) -2 < x < (Dividing by -3; note the reversal of inequalities) Thus, we obtain (-2, $): 4 ABSOLUTE VALUE; LINEAR COORDINATE SYSTEMS; INEQUALITIES [CHAP. 1 2. Describe and diagram the intervals determined by the following inequalities: (a) 1x1 <2; (6) 1x1 > 3; (c) Ix - 31 < 1; (d) Ix - 21 < 6, where 6 > 0; (e) Ix + 21 5 3; (f)0 < Ix - 41 < 6, where 6 CO. (a) This is equivalent to -2 < x < 2, defining the open interval (-2,2): (6) This is equivalent to x >3 or x < -3, defining the union of the infinite intervals (3, a) and (-m, -3). (c) This is equivalent to saying that the distance between x and 3 is less than 1, or that 2 < x < 4, which defines the open interval (2,4): We can also note that Ix - 31 < 1 is equivalent to - 1 < x - 3 < 1. Adding 3, we obtain 2 < x < 4. (d)T his is equivalent to saying that the distance between x and 2 is less than 6, or that 2 - 6 < x < 2 + 6, which defines the open interval (2 - 6,2 + 6). This interval is called the 6--ne ighborhood of 2: nv 1 0 2-6 2 2+6 (e) Ix + 21 < 3 is equivalent to -3 < x + 2 < 3. Subtracting 2, we obtain -5 < x < 1, which defines the open interval (-5, 1): (f)T he inequality Ix - 41 < 6 determines the interval 4 - 6 < x < 4 + 6. The additional condition 0 < Ix - 41 tells us that x # 4. Thus, we get the union of the two intervals (4 - 6,4) and (4,4 + 6). - The result is called the deleted 6-neighborhood of 4: - n n n W e 4-6 4 4+6 XI 3. Describe and diagram the intervals determined by the following inequalities: (a) 15 - 5 3; (6) 1 2-~ 3 1 <5; (c) 11 -4x(< $. (a) Since 15 - XI = Ix - 51, we have Ix - 51 I3 , which is equivalent to -3 5 x - 5 5 3. Adding 5, we get 2 Ix 5 8, which defines the open interval (2,s): (6) 12x - 31 < 5 is equivalent to -5 < 2x - 3 < 5. Adding 3, we have -2 < 2x < 8; then dividing by 2 yields - 1 < x < 4, which defines the open interval (- 1,4): - v -1 4 (c) Since 11 - 4x1 = 14x - 11, we have (4x - 11 < 4, which is equivalent to - 4 < 4x - 1 < 4. Adding 1, we get 5 < 4x < t. Dividing by 4, we obtain Q < x < i,w hich defines the interval (Q , ): CHAP. 11 ABSOLUTE VALUE; LINEAR COORDINATE SYSTEMS; INEQUALITIES 5 4. Solve the inequalities (a) 18x - 3x2 > 0, (b) (x + 3)(x - 2)(x - 4) < 0, and (x + l)L(x- 3) > 0, and diagram the solutions. Set 18x - 3x2 = 3x(6 - x) = 0, obtaining x = 0 and x = 6. We need to determine the sign of 18x - 3x‘ on each of the intervals x < 0, 0 < x < 6, and x > 6, to determine where 18x - 3x’ > 0. We note that it is negative when x < 0, and that it changes sign when we pass through 0 and 6. Hence, it is positive when and only when O<x<6: The crucial points are x = -3, x = 2, and x = 4. Note that (x + 3)(x - 2)(x - 4) is negative for x < -3 (since each of the factors is negative) and that it changes sign when we pass through each of the crucial points. Hence, it is negative for x < - 3 and for 2 < x < 4: * -3 2 4 Note that (x + 1)’ is always positive (except at x = - 1, where it is 0). Hence (x + l)*(x- 3) > 0 when and only when x - 3 > 0, that is, for x > 3: 5. Solve 13x - 71 = 8. In general, when c I0 , lul= c if and only if U = c or U = - c. Thus, we need to solve 3x - 7 = 8 and 3x-7=-8, from which wegetx=5orx=-+. + 2x 1 6. Solve -> 3. x+3 Case 2 : x + 3 > 0. Multiply by x + 3 to obtain 2x + 1 > 3x + 9, which reduces to - 8 > x. However, since x + 3 > 0, it must be that x > -3. Thus, this case yields no solutions. Case 2: x + 3 < 0. Multiply by x + 3 to obtain 2x + 1 < 3x + 9. (Note that the inequality is reversed, since we multiplied by a negative number.) This yields - 8 < x. Since x + 3 < 0, we have x < - 3. Thus, the only solutions are -8 < x < -3. I f 7. solve - 31 < 5. 2 The given inequality is equivalent to -5 < - - 3 < 5. Add 3 to obtain -2 < 2/x < 8, and divide by 2 X to get -1 < l/x<4. Case I : x > 0. Multiply by x to get -x < 1 < 4x. Then x > j and x > - 1; these two inequalities are equivalent to the single inequality x > i. Case 2: x < 0. Multiply by x to obtain -x > 1 > 4x. (Note that the inequalities have been reversed, since we multiplied by the negative number x.) Then x < and x < - 1. These two inequalities are equivalent to x < - 1. Thus, the solutions are x > 4 or x < - 1, the union of the two infinite intervals (4, M) and (-E, - 1). 8. Solve 12x - 51 I3 . Let us first solve the negation 12x - 51 < 3. The latter is equivalent to -3 < 2x - 5 < 3. Add 5 to obtain 2 < 2x < 8, and divide by 2 to obtain 1 < x < 4. Since this is the solution of the negation, the original inequality has the solution x 5 1 or x 2 4. + + I 9. Prove the triangle inequality, Ix yI 5 1x1 yl.