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 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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 ACCURACY, ADEQUACY OR COMPLETENESS OF OR RESULTS TO BE OBTAINED FROM USING THE WORK, INCLUDING ANY INFORMATION THAT CAN BE ACCESSED THROUGH THE WORK VIA HYPERLINK OR OTHERWISE, AND EXPRESSLY DISCLAIM ANY WARRANTY, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO IMPLIED WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE McGraw-Hill and its licensors 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 INEQUALITIES Chapter ABSOLUTE VALUE Chapter LINES Chapter CIRCLES 19 Chapter FUNCTIONS AND THEIR GRAPHS 23 Chapter LIMITS 35 Chapter CONTINUITY 43 Chapter THE DERIVATIVE 49 Chapter 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 INTEGRALS FOR SURFACE AREA, WORK, CENTROIDS 253 Chapter 31 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 FUNCTIONS OF SEVERAL VARIABLES 361 Chapter 41 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 DIFFERENTIAL EQUATIONS 431 INDEX 443 Chapter 46 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 progresses, but there is no uniform pattern It is assumed that you have available a calculus textbook, including tables for the trigonometric, logarithmic, 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 supplement to any course in calculus, or even as an independent refresher course V This page intentionally left blank HAPTER nequalities 1.1 Solve + 2*