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CALCULUS WITH ANALYTIC GEOMETRY SECOND EDITION GEORGE F. SIMMONS Colorado Springs Colorado College, THE McGRAW-HILL COMPANIES, INC. New York St. Louis San Francisco Auckland Bogota Caracas Lisbon London Madrid Mexico City Milan Montreal New Delhi San Juan Singapore Sydney Tokyo Toronto

McGraw-Hill zz A Division ofTheMcGraw-HillCompanies CALCULUS WITH ANALYTIC GEOMETRY © 1996, 1985 by The McGraw-Hill Companies, Inc. All rights reserved. Copyright Printed in the United States of America. Except as permitted under the United States Copyright Act of 1976, no part of this publication a data base or retrieval may be reproduced or distributed system, without prior written permission in any form or by any means, or stored in of the publisher. This book is printed on acid-free paper. l 2 3 4 5 6 7 8 9 0 DOW DOW 9 0 9 8 7 6 5 ISBN 0-07-057642-4 This book was set in Times Roman by York Graphic Services, The editors were Jack Shira, Maggie Lanzillo, the designer was Joan E. O'Connor; the production R. R. Donnelley & Sons Company was printer and binder. was Denise L. Puryear. supervisor Inc. and Ty McConnell; Library of Congress Cataloging-in-Publication Data Simmons, George Finlay, (date). Calculus with analytic geometry/by George F. Simmons.-2nd ed. p. cm. Includes bibliographical references ISBN 0-07-057642-4 I. Calculus. 2. Geometry, Analytic. 199 and index. I. Title. QA303.S5547 515'.15-dc20 95-38343 I TERNATIONAL EDITION Copyright© 1996. Exclusive rights export. This book cannot be re-exported Hill. The International Edition is not available by The McGraw-Hill Companies, Inc. for manufacture and from the country to which it is consigned by McGraw in North America. When ordering this title, use ISBN 0-07-114 716-0.

ABOUT THE AUTHOR George F. Simmons has the usual academic degrees (CalTech, Chicago, Yale) and taught at several colleges and universities orado College in 1962. He is also the author of Introduction Modern Analysis (McGraw-Hill, 1963), Differential and Historical Notes (McGraw-Hill, ematics in a Nutshell (Janson Publications, and Memorable Mathematics (McGraw-Hill, before joining the faculty of Col to Topology and 1972, 2nd edition 1991), Precalculus Math 1992). 1981 ), and Calculus Gems: Brief Lives Equations with Applications When not working or talking or eating or drinking or cooking, Professor Sim mons is likely to be traveling (Western and Southern Europe, Turkey, Israel, Egypt, Russia, China, Southeast Asia), trout fishing (Rocky Mountain states), playing pocket billiards, raphy, science, and enough thrillers his personal heroes is the older friend who once said to him, "I should probably spend about an hour a week revising my to achieve enjoyment without guilt). or reading (literature, biography and autobiog opinions." history, One of

For My Grandson Nicky without young people to continue to wonder and care and study and learn, it's all over. With all humility, I think "Whatsoever nitely more important than the vain attempt to love one's neighbor as oneself. to hit a bird on the wing, you must have all your will ing about yourself, be living in your eye on that bird. Every achievement thy hand findeth to do, do it with thy might" infi If you want in a focus; you must not be think you must and, equally, you must not be thinking about your neighbor; is a bird on the wing. Oliver Wendell Holmes If you bring forth what is within you, what you bring forth will save you. If you do not bring forth what is within you, what you do not bring forth will destroy you. Jesus. The Gospel of Thomas in the Nag Hammadi manuscripts The more I work and practice, the luckier I seem to get. Gary Player (professional golfer) A witty chess master once said that the difference between a master and a beginning chess player is that the beginner has everything clearly fixed in mind, while to the master every thing is a mystery. N. la. Vilenkin Marshall's Generalized Iceberg Theorem: Seven-eighths of everything can't be seen. Everything should be made as simple as possible, but not simpler. Albert Einstein

CONTENTS Preface to the Instructor To the Student xiii XXlll PART I 1 NUMBERS, FUNCTIONS, AND GRAPHS I. l Introduction 1.2 The Real Line and Coordinate Plane. Pythagoras 1.3 Slopes and Equations of Straight 1.4 Circles and Parabolas. Descartes l.5 The Concept of a Function l.6 Graphs of Functions l.7 Introductory Trigonometry. Lines and Fermat The Functions sin 8 and cos 8 Review: Definitions, Concepts, Methods Additional Problems 2 THE DERIVATIVE OF A FUNCTION THE COMPUTATION OF 3 DERIVATIVES 2.1 What is Calculus? The Problem of Tangents 2.2 How to Calculate 2.3 The Definition 2.4 Velocity and Rates of Change. Newton and Leibniz 2.5 The Concept of a Limit. Two Trigonometric Limits 2.6 Continuous Functions. The Mean Value Theorem and the Slope of the Tangent of the Derivative Other Theorems Review: Definitions, Concepts, Methods Additional Problems of Polynomials 3.1 Derivatives Rules 3.2 The Product and Quotient 3.3 Composite Functions and the Chain Rule 3.4 Some Trigonometric 3.5 Implicit 3.6 Derivatives of Higher Order Derivatives Functions and Fractional Exponents Review: Concepts, Formulas, Methods Additional Problems 4 APPLICATIONS OF DERIVATIVES and Decreasing Functions. Maxima and Minima 4.1 Increasing 4.2 Concavity and Points of Inflection 4.3 Applied Maximum and Minimum Problems 4.4 More Maximum-Minimum Problems. Reflection and Refraction 4.5 Related Rates vii 1 2 11 15 22 30 37 46 47 51 53 58 62 68 74 81 81 83 88 92 98 102 107 111 111 115 120 123 131 139

Vlll CONTENTS 4.6 Newton's Method for Solving 4.7 (Optional) Applications Review: Concepts, Additional 143 146 to Economics. Marginal Analysis 1 56 156 Methods Problems Equations 5 5.1 Introduction INDEFINITE INTEGRALS 5.2 Differentials AND DIFFERENTIAL 5.3 Indefinite and Tangent Line Approximations by Substitution Integrals. Integration EQUATIONS 5.4 Differential 5.5 Motion under Gravity. Equations. Separation Escape Ve locity of Variables Review: Concepts, Additional Problems Methods DEFINITE INTEGRALS 6.2 The Problem of Areas 6 6.1 Introduction 163 163 170 178 and Black Holes 1 8 1 188 188 190 191 194 Riemann 197 Special Integrals. Sums 203 206 2 1 3 217 217 2 1 8 222 225 231 236 240 244 252 254 254 257 6.3 The S igma Notation and Certain 6.4 The Area under a Curve. Definite 6.5 The Computation 6.6 The Fundamental 6.7 Properties of Definite Integrals of Areas as Limits Theorem of Calculus Methods Review: Concepts, Additional Appendix: Problems The Lunes of Hippocrates The Intuitive APPLICATIONS OF 7.2 The Area between Two 7 7.1 Introduction. INTEGRATION 7.3 Volumes: 7.4 Volumes: 7.6 The Area of a Surface 7.7 Work and Energy 7.8 Hydrostatic 7.5 Arc Length The Disk Method The Method of Cylindrical Shells of Revolution Curves Force Meaning of I ntegration 221 Review: Concepts, Additional Appendix: Methods Problems Archimedes and the Volume of a Sphere PART II 8 8.1 Introduction LOGAHITHM IT TCTIONS 8.3 The Number e and the EXPONENTIAL AND 8.2 Review of Exponents y = ex 8.4 The Natural 8.5 Applications. 8.6 More Applicati and Logarithms Function Function y = In x. Euler 260 261 264 269 Decay 277 Growth and Radioactive 283 etc. 287 288 Logarithm Population ons. Inhibited Formulas Population Growth, Concepts, Review: Additional Problems 9 9.1 Review of Trigonometry FUNCTIONS 9.3 The Integrals TRIGONOMETIUC 9.2 The Derivatives of the Sine and Cosine of the Sine and Cosine. 292 301 The Needle Problem 306

CONTENTS IX of the Other Four Functions 9.4 9.5 9.6 9.7 Trigonometric The Derivatives The Inverse Simple Harmonic (Optional) Review: Additional Motion. Hyperbolic Definitions, Problems Formulas Functions Functions The Pendulum for Dealing Fractions Trigonometric Integrals tions Introduction. The Basic Formulas The Method of Substitution Certain Trigonometric Substitu Completing the Square The Method of Partial Integration by Parts A Mixed B ag. Strategy Miscellaneous Numerical Review: Additional Appendix Integration. Simpson's Rule E = l · l · ± · ± · .§. • .§. • • • I: The Catenary, or Curve of a Hanging Appendix 2 · Wallis's Product Problems f=l-t+t-t+ ... Appendix 3: How Leibniz . 3 3 5 2 1 Discovered His Formula with Integrals Formulas, Methods 5 7 Types of Chain System The Center of Mass of a Discrete Centroids The Theorems Moment of Inertia Review: Additional Problems Definitions, Concepts of Pappus 3 1 0 3 1 3 3 1 9 324 330 330 334 337 340 344 348 351 357 362 369 375 375 378 380 382 384 386 391 393 396 396 398 400 404 409 414 424 424 427 432 439 445 451 455 461 465 470 470 10 IO.I METHODS OF 1 0.2 INTEGRATION 1 0.3 1 0.4 1 0.5 10.6 10.7 10.8 1 0.9 11 I I. I FlJRTHEH APPUCATIOl\S OF 1 1.2 INTEGRATION 1 1.3 1 1.4 12 1 2.1 INDETERMINATE FORMS 1 2.2 AND IMPROPER INTEGRALS 1 2.3 1 2.4 1 2.5 The Mean Value Theorem Revisit ed te Form 010. L'Hospital 's Rule Introduction. The Indetermina Other Indeterminate Forms Improper The Normal Distribution. Review: Definitions, Problems Additional I ntegrals Gauss Concepts 13 1 3. 1 INFINITE SERIES OF 1 3.2 CONSTANTS 1 3.3 1 3.4 1 3.5 1 3.6 1 3.7 1 3.8 of Convergent Series Terms. Comparison Tests Sequences and Divergent Series Properties of Nonnegative What Is an Infinite Series ? Convergent Convergent General Series Test. Euler's The Integral The Ratio Test and Root Test The Alternating Review: Additional Definitions, Problems Concepts, Tests Constant Series Test. Absolute Convergence

x CONTENTS 1: Euler's Discovery of the Formula L] :2 = :2 476 2: More about 3: The Series Irrational L l!Pn of the Reciprocals of 1T ls Irrational 478 the Primes 480 Appendix Appendix Appendix Numbers. 14 1 4. 1 Introducti on POWER SERIES 1 4.2 The Interval 1 4.3 Differentiation 1 4.4 Taylor Series 1 4.5 Computations 14.6 Applications 14.7 (Optional) 14.8 (Optional) of Convergence and Integrat and Taylor's Formula Using Taylor's Formula to Differential Equations Operations Complex Numbers and Euler's on Power Series Formula ion of Power Series Formulas, Methods Review: Concepts, Additional Appendix: Problems The Bernoulli Discover ies of Euler Numbers and Some Wondetful PART III 15 1 5.1 Introduction. Sections of a Cone CONIC SECTIONS 1 5.2 Another Look at Circles and Parabolas 1 5.3 Ellipses 1 5.4 Hyperbolas 1 5.5 The Focus-Directrix-Eccentricity Definitions 1 5.6 (Optional) Equations. Rotation of Axes Second-Degree Review: Additional Definitions, Problems Properties 16 1 6. 1 The Polar Coordinate System POLAR COORDINATES 1 6.2 More Graphs of Polar Equations 1 6.3 Polar Equations of Circles, Conics, and Spirals 1 6.4 Arc Length and Tangent Lines 1 6.5 Areas in Polar Coordinates Review: Concepts, Additional Formulas Problems 483 484 489 494 504 509 5 14 521 523 523 525 529 5 3 1 535 543 550 552 557 558 560 564 569 575 580 583 583 17 1 7.1 Parametric PARAMETRIC EQUATIONS. 1 7.2 The Cycloid VECTORS IN THE PLANE 1 7.3 Vector Algebra. 1 7.4 Derivatives 1 7.5 Curvature 1 7.6 Tangential 1 7.7 Kepler's Equations of Curves and Other Similar Curves The Unit Vectors i and j of Vector Functions. Velocity 586 592 600 and Acceleration 605 and the Unit Normal Vector and Normal Components of Acceleration 6 1 5 Laws and Newton's Law of Gravitation 619 627 Review: Concepts, Additional 627 Problem 629 Appendix: Formulas Problems Bernoulli's of the Brachistochrone Solution 6 11

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