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SCHAUM'S OUTLINE SERIES Schaum's Outline of Theory and Problems of Beginning Calculus Second Edition Elliott Mendelson, Ph.D. Professor of Mathematics Queens College City University of New York To the memory of my father, Joseph, and my mother, Helen ELLIOTT MENDELSON is Professor of Mathematics at Queens College of the City University of New York. He also has taught at the University of Chicago, Columbia University, and the University of Pennsylvania, and was a member of the Society of Fellows of Harvard University. He is the author of several books, including Schaum's Outline of Boolean Algebra and Switching Circuits. His principal area of research is mathematical logic and set theory. Schaum's Outline of Theory and Problems of BEGINNING CALCULUS Copyright © 1997,1985 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 publication 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. 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 PRS PRS 9 0 1 0 9 ISBN 0-07-041733-4 Sponsoring Editor: Arthur Biderman Production Supervisor: Suzanne Rapcavage Editing Supervisor: Maureen B. Walker Library of Congress Cataloging-in-Publication Data Mendelson, Elliott. Schaum's outline of theory and problems of beginning calculus / Elliott Mendelson, -- 2nd ed. p. cm. -- (Schaum's outline series) Includes index. ISBN 0-07-041733-4 (pbk.) 1. Calculus. 2. Calculus--Problems, exercises, etc. I. Title. QA303.M387 1997 515' .076--dc21 96-39852 CIP Preface This Outline is limited to the essentials of calculus. It carefully develops, giving all steps, the principles of differentiation and integration on which the whole of calculus is built. The book is suitable for reviewing the subject, or as a self-contained text for an elementary calculus course. The author has found that many of the difficulties students encounter in calculus are due to weakness in algebra and arithmetical computation, emphasis has been placed on reviewing algebraic and arithmetical techniques whenever they are used. Every effort has been made—especially in regard to the composition of the solved problems—to ease the beginner's entry into calculus. There are also some 1500 supplementary problems (with a complete set of answers at the end of the book). High school courses in calculus can readily use this Outline. Many of the problems are adopted from questions that have appeared in the Advanced Placement Examination in Calculus, so that students will automatically receive preparation for that test. The Second Edition has been improved by the following changes: 1. A large number of problems have been added to take advantage of the availability of graphing calculators. Such problems are preceded by the notation . Solution of these problems is not necessary for comprehension of the text, so that students not having a graphing calculator will not suffer seriously from that lack (except insofar as the use of a graphing calculator enhances their understanding of the subject). 2. Treatment of several topics have been expanded: (a) Newton's Method is now the subject of a separate section. The availability of calculators makes it much easier to work out concrete problems by this method. (b) More attention and more problems are devoted to approximation techniques for integration, such as the trapezoidal rule, Simpson's rule, and the midpoint rule. (c) The chain rule now has a complete proof outlined in an exercise. 3. The exposition has been streamlined in many places and a substantial number of new problems have been added. The author wishes to thank again the editor of the First Edition, David Beckwith, as well as the editor of the Second Edition, Arthur Biderman, and the editing supervisor, Maureen Walker. ELLIOTT MENDELSON This page intentionally left blank Contents Chapter 1 1 Coordinate Systems on a Line 1.1 The Coordinates of a Point 1 1.2 Absolute Value 2 Chapter 2 8 Coordinate Systems in a Plane 2.1 The Coordinates of a Point 8 2.2 The Distance Formula 9 2.3 The Midpoint Formulas 10 Chapter 3 14 Graphs of Equations Chapter 4 24 Straight Lines 4.1 Slope 24 4.2 Equations of a Line 27 4.3 Parallel Lines 28 4.4 Perpendicular Lines 29 Chapter 5 36 Intersections of Graphs Chapter 6 41 Symmetry 6.1 Symmetry about a Line 41 6.2 Symmetry about a Point 42 Chapter 7 46 Functions and Their Graphs 7.1 The Notion of a Function 46 7.2 Intervals 48 7.3 Even and Odd Functions 50 7.4 Algebra Review: Zeros of Polynomials 51 Chapter 8 59 Limits 8.1 Introduction 59 8.2 Properties of Limits 59 8.3 Existence or Nonexistence of the Limit 61 Chapter 9 67 Special Limits 9.1 One-Sided Limits 67 9.2 Infinite Limits: Vertical Asymptotes 68 9.3 Limits at Infinity: Horizontal Asymptotes 70 Chapter 10 78 Continuity 10.1 Definition and Properties 78 10.2 One-Sided Continuity 79 10.3 Continuity over a Closed Interval 80 Chapter 11 86 The Slope of a Tangent Line Chapter 12 92 The Derivative Chapter 13 99 More on the Derivative 13.1 Differentiability and Continuity 99 13.2 Further Rules for Derivatives 100 Chapter 14 104 Maximum and Minimum Problems 14.1 Relative Extrema 104 14.2 Absolute Extrema 105 Chapter 15 116 The Chain Rule 15.1 Composite Functions 116 15.2 Differentiation of Composite Functions 117 Chapter 16 126 Implicit Differentiation Chapter 17 129 The Mean-Value Theorem and the Sign of the Derivative 17.1 Rolle's Theorem and the Mean-Value Theorem 129 17.2 The Sign of the Derivative 130 Chapter 18 136 Rectilinear Motion and Instantaneous Velocity Chapter 19 143 Instantaneous Rate of Change Chapter 20 147 Related Rates Chapter 21 155 Approximation by Differentials; Newton's Method 21.1 Estimating the Value of a Function 155 21.2 The Differential 155 21.3 Newton's Method 156 Chapter 22 161 Higher-Order Derivatives Chapter 23 167 Applications of the Second Derivative and Graph Sketching 23.1 Concavity 167 23.2 Test for Relative Extrema 169 23.3 Graph Sketching 171 Chapter 24 179 More Maximum and Minimum Problems Chapter 25 185 Angle Measure 25.1 Arc Length and Radian Measure 185 26.2 Directed Angles 186 Chapter 26 190 Sine and Cosine Functions 26.1 General Definition 190 26.2 Properties 192 Chapter 27 202 Graphs and Derivatives of Sine and Cosine Functions 27.1 Graphs 202 27.2 Derivatives 205 Chapter 28 214 The Tangent and Other Trigonometric Functions Chapter 29 221 Antiderivatives 29.1 Definition and Notation 221 29.2 Rules for Antiderivatives 222 Chapter 30 229 The Definite Integral 30.1 Sigma Notation 229 30.2 Area under a Curve 229 30.3 Properties of the Definite Integral 232 Chapter 31 238 The Fundamental Theorem of Calculus 31.1 Calculation of the Definite Integral 238 31.2 Average Value of a Function 239 31.3 Change of Variable in a Definite Integral 240