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  • P1: PBU/OVY P2: PBU/OVY QC: PBU/OVY T1: PBUJWDD023-FM JWDD023-Salas-v13 October 24, 2006 17:25

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    … may be sitting right in your classroom! Everyone of your students has the potential to make adifference. And realizing that potential startsright here, in your course.

    When students succeed in your course—whenthey stay on-task and make the breakthroughthat turns confusion into confidence—they areempowered to realize the possibilities forgreatness that lie within each of them. We know your goal is to createan environment where students reach their full potential and experiencethe exhilaration of academic success that will last them a lifetime.WileyPLUS can help you reach that goal.

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    FOR INSTRUCTORS

    FOR STUDENTS

    of students surveyed said it made them better prepared for tests.76%

    “It has been a great help,and I believe it has helpedme to achieve a bettergrade.”

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    **Based on a survey of 972 student users of WileyPLUS

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    TENTHEDITION

    SALAS

    HILLE

    ETGEN

    CALCULUSONE AND SEVERAL VARIABLES

    JOHN WILEY & SONS, INC.

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    In fond remembrance ofEinar Hille

    ACQUISITIONS EDITOR Mary KittellPUBLISHER Laurie RosatoneMARKETING MANAGER Amy SellEDITORIAL ASSISTANT Danielle AmicoMARKETING ASSISTANT Tara MartinhoSENIOR PRODUCTION EDITOR Sandra DumasDESIGNER Hope MillerSENIOR MEDIA EDITOR Stefanie LiebmanPRODUCTION MANAGEMENT Suzanne IngraoFREELANCE DEVELOPMENTAL EDITOR Anne Scanlan-RohrerCOVER IMAGE Steven Puetzer/Masterfile

    This book was set in New Times Roman by Techbooks, Inc. and printed and bound by Courier(Westford). Thecover was printed by Courier(Westford).

    This book is printed on acid-free paper. ∞©Copyright © 2007 John Wiley & Sons, Inc. All rights reserved. No part of this publication may be reproduced,stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying,recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United StatesCopyright Act, without either the prior written permission of the Publisher, or authorization through payment ofthe appropriate per-copy fee to the Copyright Clearance Center, Inc. 222 Rosewood Drive, Danvers, MA 01923,website www.copyright.com. Requests to the Publisher for permission should be addressed to the PermissionsDepartment, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030-5774, (201)748-6011, fax (201)748-6008, website http://www.wiley.com/go/permissions.

    To order books or for customer service please, call 1-800-CALL WILEY (225-5945).

    Library of Congress Cataloging-in-Publication Data

    Salas, Saturnino L.Calculus—10th ed/Saturnino Salas, Einar Hille, Garrett Etgen.

    ISBN-13 978-0471-69804-3ISBN-10 0-471-69804-0

    Printed in the United States of America10 9 8 7 6 5 4 3 2 1

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    PREFACE

    This text is devoted to the study of single and multivariable calculus. While applicationsfrom the sciences, engineering, and economics are often used to motivate or illustratemathematical ideas, the emphasis is on the three basic concepts of calculus: limit,derivative, and integral.

    This edition is the result of a collaborative effort with S.L. Salas, who scrutinizedevery sentence for possible improvement in precision and readability. His gift for writingand his uncompromising standards of mathematical accuracy and clarity illuminate thebeauty of the subject while increasing its accessibility to students. It has been a pleasurefor me to work with him.

    FEATURES OF THE TENTH EDITION

    Precision and Clarity

    The emphasis is on mathematical exposition; the topics are treated in a clear andunderstandable manner. Mathematical statements are careful and precise; the basicconcepts and important points are not obscured by excess verbiage.

    Balance of Theory and Applications

    Problems drawn from the physical sciences are often used to introduce basic conceptsin calculus. In turn, the concepts and methods of calculus are applied to a variety ofproblems in the sciences, engineering, business, and the social sciences through textexamples and exercises. Because the presentation is flexible, instructors can vary thebalance of theory and applications according to the needs of their students.

    Accessibility

    This text is designed to be completely accessible to the beginning calculus student with-out sacrificing appropriate mathematical rigor. The important theorems are explained

    vii

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    viii ■ PREFACE

    and proved, and the mathematical techniques are justified. These may be covered oromitted according to the theoretical level desired in the course.

    Visualization

    The importance of visualization cannot be over-emphasized in developing students’understanding of mathematical concepts. For that reason, over 1200 illustrations ac-company the text examples and exercise sets.

    Technology

    The technology component of the text has been strengthened by revising existing exer-cises and by developing new exercises. Well over half of the exercise sets have problemsrequiring either a graphing utility or a computer algebra system (CAS). Technologyexercises are designed to illustrate or expand upon the material developed within thesections.

    Projects

    Projects with an emphasis on problem solving offer students the opportunity to investi-gate a variety of special topics that supplement the text material. The projects typicallyrequire an approach that involves both theory and applications, including the use oftechnology. Many of the projects are suitable for group-learning activities.

    Early Coverage of Differential Equations

    Differential equations are formally introduced in Chapter 7 in connection with applica-tions to exponential growth and decay. First-order linear equations, separable equations,and second linear equations with constant coefficients, plus a variety of applications,are treated in a separate chapter immediately following the techniques of integrationmaterial in Chapter 8.

    CHANGES IN CONTENT AND ORGANIZATIONIn our effort to produce an even more effective text, we consulted with the users of theNinth Edition and with other calculus instructors. Our primary goals in preparing theTenth Edition were the following:

    1. Improve the exposition. As noted above, every topic has been examined for possibleimprovement in the clarity and accuracy of its presentation. Essentially every sectionin the text underwent some revision; a number of sections and subsections werecompletely rewritten.

    2. Improve the illustrative examples. Many of the existing examples have been mod-ified to enhance students’ understanding of the material. New examples have beenadded to sections that were rewritten or substantially revised.

    3. Revise the exercise sets. Every exercise set was examined for balance between drillproblems, midlevel problems, and more challenging applications and conceptualproblems. In many instances, the number of routine problems was reduced and newmidlevel to challenging problems were added.

    Specific changes made to achieve these goals and meet the needs of today’s studentsand instructors include:

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    PREFACE ■ ix

    Comprehensive Chapter-End Review Exercise Sets

    The Skill Mastery Review Exercise Sets introduced in the Ninth Edition have beenexpanded into chapter-end exercise sets. Each chapter concludes with a comprehensiveset of problems designed to test and to re-enforce students’ understanding of basicconcepts and methods developed within the chapter. These review exercise sets averageover 50 problems per set.

    Precalculus Review (Chapter 1)

    The content of this chapter—inequalities, basic analytic geometry, the function conceptand the elementary functions—is unchanged. However, much of the material has beenrewritten and simplified.

    Limits (Chapter 2)

    The approach to limits is unchanged, but many of the explanations have been revised.The illustrative examples throughout the chapter have been modified, and new exampleshave been added.

    Differentiation and Applications (Chapters 3 and 4)

    There are some significant changes in the organization of this material. Realizing thatour treatments of linear motion, rates of change per unit time, and the Newton-Raphsonmethod depended on an understanding of increasing/decreasing functions and the con-cavity of graphs, we moved these topics from Chapter 3 (the derivative) to Chapter 4(applications of the derivative). Thus, Chapter 3 is now a shorter chapter which focusessolely on the derivative and the processes of differentiation, and Chapter 4 is expandedto encompass all of the standard applications of the derivative—curve-sketching, opti-mization, linear motion, rates of change, and approximation. As in all previous editions,Chapter 4 begins with the mean-value theorem as the theoretical basis for all the appli-cations.

    Integration and Applications (Chapters 5 and 6)

    In a brief introductory section, area and distance are used to motivate the definiteintegral in Chapter 5. While the definition of the definite integral is based on upper andlower sums, the connection with Riemann sums is also given. Explanations, examples,and exercises throughout Chapters 5 and 6 have been modified, but the content andorganization remain as in the Ninth Edition.

    The Transcendental Functions, Techniques of Integration(Chapters 7 and 8)

    The coverage of the inverse trigonometric functions (Chapter 7) has been reducedslightly. The treatment of powers of the trigonometric functions (Chapter 8) has beencompletely rewritten. The optional sections on first-order linear differential equationsand separable differential equations have been moved to Chapter 9, the new chapter ondifferential equations.

    Some Differential Equations (Chapter 9)

    This new chapter is a brief introduction to differential equations and their applications.In addition to the coverage of first-order linear equations and separable equations noted

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    x ■ PREFACE

    above, we have moved the section on second-order linear homogeneous equations withconstant coefficients from the Ninth Edition’s Chapter 18 to this chapter.

    Sequences and Series (Chapters 11 and 12)

    Efforts were made to reduce the overall length of these chapters through rewritingand eliminating peripheral material. Eliminating extraneous problems reduced severalexercise sets. Some notations and terminology have been modified to be consistent withcommon usage.

    Vectors and Vector Calculus (Chapters 13 and 14)

    The introduction to vectors in three-dimensional space has been completely rewrit-ten and reduced from two sections to one. The parallel discussion of vectors in two-and three-dimensional space has been eliminated—the primary focus is on three-dimensional space. The treatments of the dot product, the cross product, lines andplanes in Chapter 13, and vector calculus in Chapter 14 are unchanged.

    Functions of Several Variables, Gradients, Extreme Values(Chapters 15 and 16); Multiple Integrals, Line and Surface Integrals(Chapters 16 and 17)

    The basic content and organization of the material in these four chapters remain as in theninth edition. Improvements have been made in the exposition, examples, illustrations,and exercises.

    Differential Equations (Chapter 19)

    This chapter continues the study of differential equations begun in Chapter 9. Thesections on Bernoulli, homogeneous and exact equations have been rewritten, andelementary numerical methods are now covered in a separate section. The section onsecond-order linear nonhomogeneous equations picks up from the treatment of linearhomogeneous equations in the new Chapter 9. The applications section—vibratingmechanical systems—is unchanged.

    SUPPLEMENTSAn Instructor’s Solutions Manual, ISBN 0470127309, includes solutions for all prob-

    lems in the text.

    A Student Solutions Manual, ISBN 0470105534, includes solutions for selected prob-lems in the text.

    A Companion Web site, www.wiley.com/college/salas, provides a wealth of resourcesfor students and instructors, including:

    • PowerPoint Slides for important ideas and graphics for study and note taking.

    • Online Review Quizzes to enable students to test their knowledge of key concepts.For further review diagnostic feedback is provided that refers to pertinent sections ofthe text.

    • Animations comprise a series of interactive Java applets that allow students to explorethe geometric significance of many major concepts of Calculus.

    • Algebra and Trigonometry Refreshers is a self-paced, guided review of key algebraand trigonometry topics that are essential for mastering calculus.

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    PREFACE ■ xi

    • Personal Response System Questions provide a convenient source of questions touse with a variety of personal response systems.

    • Printed Test Bank contains static tests which can be printed for quick tests.

    • Computerized Test Bank includes questions from the printed test bank with algo-rithmically generated problems.

    WILEYPLUS

    Expect More from Your Classroom Technology

    This text is supported by WileyPLUS—a powerful and highly integrated suite of teach-ing and learning resources designed to bridge the gap between what happens in theclassroom and what happens at home. WileyPLUS includes a complete online versionof the text, algorithmically generated exercises, all of the text supplements, plus courseand homework management tools, in one easy-to-use website.

    Organized Around the Everyday Activities You Perform in Class,WileyPLUS Helps You:

    Prepare and present: WileyPLUS lets you create class presentations quickly andeasily using a wealth of Wiley-provided resources, including an online version of thetextbook, PowerPoint slides, and more. You can adapt this content to meet the needsof your course.

    Create assignments: WileyPLUS enables you to automate the process of assigningand grading homework or quizzes. You can use algorithmically generated problemsfrom the text’s accompanying test bank, or write your own.

    Track student progress: An instructor’s grade book allows you to analyze individualand overall class results to determine students’ progress and level of understanding.

    Promote strong problem-solving skills: WileyPLUS can link homework problems tothe relevant section of the online text, providing students with context-sensitive help.WileyPLUS also features mastery problems that promote conceptual understandingof key topics and video walkthroughs of example problems.

    Provide numerous practice opportunities: Algorithmically generated problems pro-vide unlimited self-practice opportunities for students, as well as problems for home-work and testing.

    Support varied learning styles: WileyPLUS includes the entire text in digital format,enhanced with varied problem types to support the array of different student learningstyles in today’s classroom.

    Administer your course: You can easily integrate WileyPLUS with another coursemanagement system, grade books, or other resources you are using in your class,enabling you to build your course your way.

    WileyPLUS Includes A Wealth of Instructor and Student Resources:

    Student Solutions Manual: Includes worked-out solutions for all odd-numbered prob-lems and study tips.

    Instructor’s Solutions Manual: Presents worked out solutions to all problems.

    PowerPoint Lecture Notes: In each section of the book a corresponding set of lecturenotes and worked out examples are presented as PowerPoint slides that are tied tothe examples in the text.

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    xii ■ PREFACE

    View an online demo at www.wiley.com/college/wileyplus or contact your localWiley representative for more details.

    The Wiley Faculty Network—Where Faculty Connect

    The Wiley Faculty Network is a faculty-to-faculty network promoting the effective useof technology to enrich the teaching experience. The Wiley Faculty Network facilitatesthe exchange of best practices, connects teachers with technology, and helps to enhanceinstructional efficiency and effectiveness. The network provides technology trainingand tutorials, including WileyPLUS training, online seminars, peer-to-peer exchangesof experiences and ideas, personalized consulting, and sharing of resources.

    Connect with a Colleague

    Wiley Faculty Network mentors are faculty like you, from educational institutionsaround the country, who are passionate about enhancing instructional efficiency andeffectiveness through best practices. You can engage a faculty mentor in an onlineconversation at www.wherefacultyconnect.com.

    Connect with the Wiley Faculty Network

    Web: www.wherefacultyconnect.comPhone: 1-866-FACULTY

    ACKNOWLEDGMENTSThe revision of a text of this magnitude and stature requires a lot of encouragementand help. I was fortunate to have an ample supply of both from many sources. Thepresent book owes much to the people who contributed to the first nine editions, mostrecently: Omar Adawi, Parkland College; Mihaly Bakonyi, Georgia State University;Edward B. Curtis, University of Washington; Boris A. Datskovsky, Temple University;Kathy Davis University of Texas-Austin; Dennis DeTurck, University of Pennsylva-nia; John R. Durbin, University of Texas-Austin; Ronald Gentle, Eastern WashingtonUniversity; Robert W. Ghrist, Georgia Institute of Technology; Charles H. Giffen, Uni-versity of Virginia–Charlottesville; Michael Kinyon, Indiana University-South Bend;Susan J. Lamon, Marquette University; Peter A. Lappan, Michigan State University;Nicholas Macri, Temple University; James Martino, Johns Hopkins University; JamesR. McKinney, California State Polytechnic University-Pomona; Jeff Morgan, Texas A& M University; Peter J. Mucha, Georgia Institute of Technology; Elvira Munoz-Garcia,University of California, Los Angeles; Ralph W. Oberste-Vorth, University of SouthFlorida; Charles Odion, Houston Community College; Charles Peters, University ofHouston; Clifford S. Queen, Lehigh University; J. Terry Wilson, San Jacinto CollegeCentral and Yang Wang, Georgia Institute of Technology. I am deeply indebted to allof them.

    The reviewers and contributors to the Tenth Edition supplied detailed criticismsand valuable suggestions. I offer my sincere appreciation to the following individuals:

    Omar Adawi, Parkland College

    Ulrich Albrecht, Auburn University

    Joseph Borzellino, California Polytechnic State University, San Luis Obispo

    Michael R. Colvin, The University of St. Thomas

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    PREFACE ■ xiii

    James Dare, Indiana-Purdue University at Fort Wayne

    Nasser Dastrange, Buena Vista University

    David Dorman, Middlebury College

    Martin E. Flashman, Humboldt State University/Occidental College

    David Frank, University of Minnesota

    Melanie Fulton, High Point University

    Isobel Gaensler, Georgia State University

    Frieda Ganter, West Hills College, Lemoore

    Murli M. Gupta, George Washington University

    Aida Kadic-Galeb, University of Tampa

    Mohammad Ghomi, Georgia Institute of Technology

    Robert W. Ghrist, University of Illinois, Urbana-Champaign

    Semion Gutman, University of Oklahoma

    Rahim G. Karimpour, Southern Illinois University- Edwardsville

    Robert Keller, Loras College

    Kevin P. Knudson, Mississippi State University

    Ashok Kumar, Valdosta State University

    Jeff Leader, Rose-Hulman Institute of Technology

    Xin Li, University of Central Florida

    Doron Lubinsky, Georgia Institute of Technology

    Edward T. Migliore, Monterey Peninsula College/University of California,Santa Cruz

    Maya Mukherjee, Morehouse College

    Sanjay Mundkur, Kennesaw State University

    Michael M. Neumann, Mississippi State University

    Charles Odion, Houston Community College

    Dan Ostrov, Santa Clara University

    Shahrokh Parvini, San Diego Mesa College

    Chuang Peng, Morehouse College

    Kanishka Perera, Florida Institute of Technology

    Denise Reid, Valdosta State University

    Paul Seeburger, Monroe Community College

    Constance Schober, University of Central Florida

    Peter Schumer, Middlebury College

    Kimberly Shockey

    Henry Smith, River Parishes Community College

    James Thomas, Colorado State University

    James L. Wang, University of Alabama

    Ying Wang, Augusta State University

    Charles Waters, Minnesota State University, Mankato

    Mark Woodard, Furman University

    Yan Wu, Georgia Southern University

    Dekang Xu, University of Houston

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    xiv ■ PREFACE

    I am especially grateful to Paul Lorczak and Neil Wigley, who carefully read therevised material. They provided many corrections and helpful comments. I would liketo thank Bill Ardis for his advice and guidance in the creation of the new technologyexercises in the text.

    I am deeply indebted to the editorial staff at John Wiley & Sons. Everyone in-volved in this project has been encouraging, helpful, and thoroughly professional atevery stage. In particular, Laurie Rosatone, Publisher, Mary Kittell, AcquisitionsEditor, Anne Scanlan-Rohrer, Freelance Developmental Editor and Danielle Amico,Editorial Assistant, provided organization and support when I needed it and proddingwhen prodding was required. Special thanks go to Sandra Dumas, Production Editor,and Suzanne Ingrao, Production Coordinator, who were patient and understanding asthey guided the project through the production stages; Hope Miller, Designer, whosecreativity produced the attractive interior design as well as the cover and StefanieLiebman, Senior Media Editor, who has produced valuable media resources to supportmy text.

    Finally, I want to acknowledge the contributions of my wife, Charlotte; without hercontinued support, I could not have completed this work.

    Garret J. Etgen

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    CONTENTS

    CHAPTER 1 PRECALCULUS REVIEW 1

    1.1 What is Calculus? 11.2 Review of Elementary Mathematics 31.3 Review of Inequalities 111.4 Coordinate Plane; Analytic Geometry 171.5 Functions 241.6 The Elementary Functions 321.7 Combinations of Functions 411.8 A Note on Mathematical Proof; Mathematical Induction 47

    Girth

    Leng

    th

    y

    x

    L

    c c + δc – δ

    f

    L – ε

    L + ε

    CHAPTER 2 LIMITS AND CONTINUITY 53

    2.1 The Limit Process (An Intuitive Introduction) 532.2 Definition of Limit 642.3 Some Limit Theorems 732.4 Continuity 822.5 The Pinching Theorem; Trigonometric Limits 912.6 Two Basic Theorems 97

    Project 2.6 The Bisection Method for Finding the Roots of f (x) = 0 102

    xv

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    xvi ■ CONTENTS

    CHAPTER 3 THE DERIVATIVE; THE PROCESS OF DIFFERENTIATION 105

    3.1 The Derivative 1053.2 Some Differentiation Formulas 1153.3 The d/dx Notation; Derivatives of Higher Order 1243.4 The Derivative As A Rate of Change 1303.5 The Chain Rule 1333.6 Differentiating The Trigonometric Functions 1423.7 Implicit Differentiation; Rational Powers 147

    CHAPTER 4 THE MEAN-VALUE THEOREM; APPLICATIONSOF THE FIRST AND SECOND DERIVATIVES 154

    4.1 The Mean-Value Theorem 1544.2 Increasing and Decreasing Functions 1604.3 Local Extreme Values 1674.4 Endpoint Extreme Values; Absolute Extreme Values 1744.5 Some Max-Min Problems 182

    Project 4.5 Flight Paths of Birds 190

    4.6 Concavity and Points of Inflection 1904.7 Vertical and Horizontal Asymptotes; Vertical Tangents and Cusps 1954.8 Some Curve Sketching 2014.9 Velocity and Acceleration; Speed 209

    Project 4.9A Angular Velocity; Uniform Circular Motion 217

    Project 4.9B Energy of a Falling Body (Near the Surface of the Earth) 217

    4.10 Related Rates of Change per Unit Time 2184.11 Differentials 223

    Project 4.11 Marginal Cost, Marginal Revenue, Marginal Profit 228

    4.12 Newton-Raphson Approximations 229

    a c b

    l

    (a, f (a))

    (b, f (b))

    tangent

    CHAPTER 5 INTEGRATION 234

    5.1 An Area Problem; a Speed-Distance Problem 2345.2 The Definite Integral of a Continuous Function 237

    5.3 The Function f (x) =∫ x

    af (t) dt 246

    5.4 The Fundamental Theorem of Integral Calculus 2545.5 Some Area Problems 260

    Project 5.5 Integrability; Integrating Discontinuous Functions 266

    5.6 Indefinite Integrals 2685.7 Working Back from the Chain Rule; the u-Substitution 2745.8 Additional Properties of the Definite Integral 2815.9 Mean-Value Theorems for Integrals; Average Value of a Function 285

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    CONTENTS ■ xvii

    CHAPTER 6 SOME APPLICATIONS OF THE INTEGRAL 292

    6.1 More on Area 2926.2 Volume by Parallel Cross Sections; Disks and Washers 2966.3 Volume by the Shell Method 3066.4 The Centroid of a Region; Pappus’s Theorem on Volumes 312

    Project 6.4 Centroid of a Solid of Revolution 319

    6.5 The Notion of Work 319*6.6 Fluid Force 327

    y

    x

    g(y)

    CHAPTER 7 THE TRANSCENDENTAL FUNCTIONS 333

    7.1 One-to-One Functions; Inverses 3337.2 The Logarithm Function, Part I 3427.3 The Logarithm Function, Part II 3477.4 The Exponential Function 356

    Project 7.4 Some Rational Bounds for the Number e 364

    7.5 Arbitrary Powers; Other Bases 3647.6 Exponential Growth and Decay 3707.7 The Inverse Trigonometric Functions 378

    Project 7.7 Refraction 387

    7.8 The Hyperbolic Sine and Cosine 388*7.9 The Other Hyperbolic Functions 392

    f –1

    x

    y

    f –1

    f

    f

    1

    –1 1

    –1

    CHAPTER 8 TECHNIQUES OF INTEGRATION 398

    8.1 Integral Tables and Review 3988.2 Integration by Parts 402

    Project 8.2 Sine Waves y = sin nx and Cosine Waves y = cos nx 4108.3 Powers and Products of Trigonometric Functions 4118.4 Integrals Featuring

    √a2 − x2, √a2 + x2, √x2 − a2 417

    8.5 Rational Functions; Partial Fractions 422*8.6 Some Rationalizing Substitutions 4308.7 Numerical Integration 433

    f

    x

    y

    3

    2

    1

    −1

    −2

    −3

    −4 −2 2 4x

    CHAPTER 9 SOME DIFFERENTIAL EQUATIONS 443

    9.1 First-Order Linear Equations 4449.2 Integral Curves; Separable Equations 451

    Project 9.2 Orthogonal Trajectories 458

    9.3 The Equation y′′ + ay′ + by = 0 459

    ∗Denotes optional section.

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    xviii ■ CONTENTS

    CHAPTER 10 THE CONIC SECTIONS; POLAR COORDINATES;PARAMETRIC EQUATIONS 469

    10.1 Geometry of Parabola, Ellipse, Hyperbola 46910.2 Polar Coordinates 47810.3 Sketching Curves in Polar Coordinates 484

    Project 10.3 Parabola, Ellipse, Hyperbola in Polar Coordinates 491

    10.4 Area in Polar Coordinates 49210.5 Curves Given Parametrically 496

    Project 10.5 Parabolic Trajectories 503

    10.6 Tangents to Curves Given Parametrically 50310.7 Arc Length and Speed 50910.8 The Area of A Surface of Revolution; The Centroid of a Curve; Pappus’s Theorem

    on Surface Area 517

    Project 10.8 The Cycloid 525

    y

    x

    CHAPTER 11 SEQUENCES; INDETERMINATE FORMS;IMPROPER INTEGRALS 528

    11.1 The Least Upper Bound Axiom 52811.2 Sequences of Real Numbers 53211.3 Limit of a Sequence 538

    Project 11.3 Sequences and the Newton-Raphson Method 547

    11.4 Some Important Limits 55011.5 The Indeterminate Form (0/0) 55411.6 The Indeterminate Form (∞/∞); Other Indeterminate Forms 56011.7 Improper Integrals 565

    y

    x

    CHAPTER 12 INFINITE SERIES 575

    12.1 Sigma Notation 57512.2 Infinite Series 57712.3 The Integral Test; Basic Comparison, Limit Comparison 58512.4 The Root Test; the Ratio Test 59312.5 Absolute Convergence and Conditional Convergence; Alternating Series 59712.6 Taylor Polynomials in x ; Taylor Series in x 60212.7 Taylor Polynomials and Taylor Series in x − a 61312.8 Power Series 61612.9 Differentiation and Integration of Power Series 623

    Project 12.9A The Binomial Series 633

    Project 12.9B Estimating π 634

    y

    x

    6

    –3

    P0(x)

    P1(x)

    f (x) = e x

    5

    4

    3

    2

    1

    –2 –1 1 2 3

    P2(x)

    P3(x)

    P2

    P3 P1

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    CONTENTS ■ xix

    CHAPTER 13 VECTORS IN THREE-DIMENSIONAL SPACE 638

    13.1 Rectangular Space Coordinates 63813.2 Vectors in Three-Dimensional Space 64413.3 The Dot Product 653

    Project 13.3 Work 663

    13.4 The Cross Product 66313.5 Lines 67113.6 Planes 679

    Project 13.6 Some Geometry by Vector Methods 688

    13.7 Higher Dimensions 689

    P1

    P3

    P2

    P

    CHAPTER 14 VECTOR CALCULUS 692

    14.1 Limit, Continuity, Vector Derivative 69414.2 The Rules of Differentiation 70114.3 Curves 70514.4 Arc Length 714

    Project 14.4 More General Changes of Parameter 721

    14.5 Curvilinear Motion; Curvature 723Project 14.5A Transition Curves 732

    Project 14.5B The Frenet Formulas 733

    14.6 Vector Calculus in Mechanics 733*14.7 Planetary Motion 741

    y

    x

    z

    CHAPTER 15 FUNCTIONS OF SEVERAL VARIABLES 748

    15.1 Elementary Examples 74815.2 A Brief Catalogue of the Quadric Surfaces; Projections 75115.3 Graphs; Level Curves and Level Surfaces 758

    Project 15.3 Level Curves and Surfaces 766

    15.4 Partial Derivatives 76715.5 Open and Closed Sets 77415.6 Limits and Continuity; Equality of Mixed Partials 777

    Project 15.6 Partial Differential Equations 785y

    x

    z

    (0, 1, 0)

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    xx ■ CONTENTS

    CHAPTER 16 GRADIENTS; EXTREME VALUES; DIFFERENTIALS 788

    16.1 Differentiability and Gradient 78816.2 Gradients and Directional Derivatives 79616.3 The Mean-Value Theorem; the Chain Rule 80516.4 The Gradient as a Normal; Tangent Lines and Tangent Planes 81816.5 Local Extreme Values 82816.6 Absolute Extreme Values 83616.7 Maxima and Minima with Side Conditions 841

    Project 16.7 Maxima and Minima with Two Side Conditions 849

    16.8 Differentials 84916.9 Reconstructing a Function from Its Gradient 855

    z

    y

    1

    1

    CHAPTER 17 DOUBLE AND TRIPLE INTEGRALS 864

    17.1 Multiple-Sigma Notation 86417.2 Double Integrals 86717.3 The Evaluation of Double Integrals by Repeated Integrals 87817.4 The Double Integral as the Limit of Riemann Sums; Polar Coordinates 88817.5 Further Applications of the Double Integral 89517.6 Triple Integrals 90217.7 Reduction to Repeated Integrals 90717.8 Cylindrical Coordinates 91617.9 The Triple Integral as the Limit of Riemann Sums; Spherical Coordinates 92217.10 Jacobians; Changing Variables in Multiple Integration 930

    Project 17.10 Generalized Polar Coordinates 935

    z

    y

    x

    CHAPTER 18 LINE INTEGRALS AND SURFACE INTEGRALS 938

    18.1 Line Integrals 93818.2 The Fundamental Theorem for Line Integrals 94618.3 Work-Energy Formula; Conservation of Mechanical Energy 95118.4 Another Notation for Line Integrals; Line Integrals with Respect to Arc Length 95418.5 Green’s Theorem 95918.6 Parametrized Surfaces; Surface Area 96918.7 Surface Integrals 98018.8 The Vector Differential Operator ∇ 98918.9 The Divergence Theorem 995

    Project 18.9 Static Charges 1000

    18.10 Stokes’s Theorem 1001

    T1 T2

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    CONTENTS ■ xxi

    CHAPTER 19 ADDITIONAL DIFFERENTIAL EQUATIONS 1010

    19.1 Bernoulli Equations; Homogeneous Equations 101019.2 Exact Differential Equations; Integrating Factors 101319.3 Numerical Methods 1018

    Project 19.3 Direction Fields 1022

    19.4 The Equation y′′ + ay′ + by = φ(x) 102219.5 Mechanical Vibrations 1030 0

    x

    x

    yy

    –1–2–3 1 2 3–1.5

    –1

    –0.50

    0.51

    1.5

    APPENDIX A SOME ADDITIONAL TOPICS A-1

    A.1 Rotation of Axes; Eliminating the xy-Term A-1A.2 Determinants A-3

    APPENDIX B SOME ADDITIONAL PROOFS A-8

    B.1 The Intermediate-Value Theorem A-8B.2 Boundedness; Extreme-Value Theorem A-9B.3 Inverses A-10B.4 The Integrability of Continuous Functions A-11B.5 The Integral as the Limit of Riemann Sums A-14

    ANSWERS TO ODD-NUMBERED EXERCISES A-15

    Index I-1

    Table of Integrals Inside Covers

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    xxii

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    CHAPTER

    1PRECALCULUS REVIEW

    In this chapter we gather together for reference and review those parts of elementarymathematics that are necessary for the study of calculus. We assume that you arefamiliar with most of this material and that you don’t require detailed explanations. Butfirst a few words about the nature of calculus and a brief outline of the history of thesubject.

    ■ 1.1 WHAT IS CALCULUS?To a Roman in the days of the empire, a “calculus” was a pebble used in counting andgambling. Centuries later, “calculare” came to mean “to calculate,” “to compute,” “tofigure out.” For our purposes, calculus is elementary mathematics (algebra, geometry,trigonometry) enhanced by the limit process.

    Calculus takes ideas from elementary mathematics and extends them to a moregeneral situation. Some examples are on pages 2 and 3. On the left-hand side you willfind an idea from elementary mathematics; on the right, this same idea as extended bycalculus.

    It is fitting to say something about the history of calculus. The origins can be tracedback to ancient Greece. The ancient Greeks raised many questions (often paradoxical)about tangents, motion, area, the infinitely small, the infinitely large—questions thattoday are clarified and answered by calculus. Here and there the Greeks themselvesprovided answers (some very elegant), but mostly they provided only questions.

    Elementary Mathematics Calculus

    slope of a line slope of a curvey = mx + b y = f (x)

    (Table continues)

    1

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    2 ■ CHAPTER 1 PRECALCULUS REVIEW

    tangent line to tangent line to a morea circle general curve

    area of a region bounded area of a region boundedby line segments by curves

    length of a line segment length of a curve

    volume of volume of a solida rectangular solid with a curved boundary

    motion along a straight motion along a curvedline with constant velocity path with varying velocity

    work done by work done bya constant force a varying force

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    1.2 REVIEW OF ELEMENTARY MATHEMATICS ■ 3

    mass of an object mass of an objectof constant density of varying density

    center of a sphere center of gravity ofa more general solid

    After the Greeks, progress was slow. Communication was limited, and each scholarwas obliged to start almost from scratch. Over the centuries, some ingenious solutionsto calculus-type problems were devised, but no general techniques were put forth.Progress was impeded by the lack of a convenient notation. Algebra, founded in theninth century by Arab scholars, was not fully systematized until the sixteenth century.Then, in the seventeenth century, Descartes established analytic geometry, and the stagewas set.

    The actual invention of calculus is credited to Sir Isaac Newton (1642–1727),an Englishman, and to Gottfried Wilhelm Leibniz (1646–1716), a German. Newton’sinvention is one of the few good turns that the great plague did mankind. The plagueforced the closing of Cambridge University in 1665, and young Isaac Newton of TrinityCollege returned to his home in Lincolnshire for eighteen months of meditation, outof which grew his method of fluxions, his theory of gravitation, and his theory of light.The method of fluxions is what concerns us here. A treatise with this title was writtenby Newton in 1672, but it remained unpublished until 1736, nine years after his death.The new method (calculus to us) was first announced in 1687, but in vague generalterms without symbolism, formulas, or applications. Newton himself seemed reluctantto publish anything tangible about his new method, and it is not surprising that itsdevelopment on the Continent, in spite of a late start, soon overtook Newton and wentbeyond him.

    Leibniz started his work in 1673, eight years after Newton. In 1675 he initiated thebasic modern notation: dx and

    ∫. His first publications appeared in 1684 and 1686. These

    made little stir in Germany, but the two brothers Bernoulli of Basel (Switzerland) tookup the ideas and added profusely to them. From 1690 onward, calculus grew rapidly andreached roughly its present state in about a hundred years. Certain theoretical subtletieswere not fully resolved until the twentieth century.

    ■ 1.2 REVIEW OF ELEMENTARY MATHEMATICSIn this section we review the terminology, notation, and formulas of elementary math-ematics.

    Sets

    A set is a collection of distinct objects. The objects in a set are called the elements ormembers of the set. We will denote sets by capital letters A, B, C, . . . and use lowercaseletters a, b, c, . . . to denote the elements.

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    4 ■ CHAPTER 1 PRECALCULUS REVIEW

    For a collection of objects to be a set it must be well-defined; that is, given anyobject x, it must be possible to determine with certainty whether or not x is an elementof the set. Thus the collection of all even numbers, the collection of all lines parallelto a given line l, the solutions of the equation x2 = 9 are all sets. The collection of allintelligent adults is not a set. It’s not clear who should be included.

    Notions and Notation

    the object x is in the set A x ∈ Athe object x is not in the set A x /∈ A

    the set of all x which satisfy property P {x : P}({x : x2 = 9} = {−3, 3})

    A is a subset of B, A is contained in B A ⊆ BB contains A B ⊇ A

    the union of A and B A ∪ B(A ∪ B = {x : x ∈ A or x ∈ B})

    the intersection of A and B A ∩ B(A ∩ B = {x : x ∈ A and x ∈ B})

    the empty set ∅These are the only notions from set theory that you will need at this point.

    Real Numbers

    Classification

    positive integers† 1, 2, 3, . . .integers 0, 1, −1, 2, −2, 3, −3, . . .

    rational numbers p/q, with p, q integers, q �= 0;for example, 5/2, −19/7, −4/1 = −4

    irrational numbers real numbers that are not rational;for example

    √2, 3

    √7, π

    Decimal Representation

    Each real number can be expressed as a decimal. To express a rational number p/q asa decimal, we divide the denominator q into the numerator p. The resulting decimaleither terminates or repeats:

    3

    5= 0.6, 27

    20= 1.35, 43

    8= 5.375

    are terminating decimals;

    2

    3= 0.6666 · · · = 0.6, 15

    11= 1.363636 · · · = 1.36, and

    116

    37= 3.135135 · · · = 3.135

    are repeating decimals. (The bar over the sequence of digits indicates that the sequencerepeats indefinitely.) The converse is also true; namely, every terminating or repeatingdecimal represents a rational number.

    †Also called natural numbers.

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    1.2 REVIEW OF ELEMENTARY MATHEMATICS ■ 5

    The decimal expansion of an irrational number can neither terminate nor repeat.The expansions

    √2 = 1.414213562 · · · and π = 3.141592653 · · ·

    do not terminate and do not develop any repeating pattern.If we stop the decimal expansion of a given number at a certain decimal place,

    then the result is a rational number that approximates the given number. For instance,1.414 = 1414/1000 is a rational number approximation to

    √2 and 3.14 = 314/100 is

    a rational number approximation to π . More accurate approximations can be obtainedby using more decimal places from the expansions.

    The Number Line (Coordinate Line, Real Line)

    On a horizontal line we choose a point O. We call this point the origin and assign to itcoordinate 0. Now we choose a point U to the right of O and assign to it coordinate 1.See Figure 1.2.1. The distance between O and U determines a scale (a unit length). Wego on as follows: the point a units to the right of O is assigned coordinate a; the pointa units to the left of O is assigned coordinate −a.

    In this manner we establish a one-to-one correspondence between the points of a

    U

    1

    O

    0

    Figure 1.2.1

    line and the numbers of the real number system. Figure 1.2.2 shows some real numbersrepresented as points on the number line. Positive numbers appear to the right of 0,negative numbers to the left of 0.

    –2 –1 0 1 2 34–7

    41

    23

    2–√2 √5

    π

    Figure 1.2.2

    Order Properties

    (i) Either a < b, b < a, or a = b. (trichotomy)(ii) If a < b and b < c, then a < c.

    (iii) If a < b, then a + c < b + c for all real numbers c.(iv) If a < b and c > 0, then ac < bc.

    (v) If a < b and c < 0, then ac > bc.

    (Techniques for solving inequalities are reviewed in Section 1.3.)

    Density

    Between any two real numbers there are infinitely many rational numbers and infinitelymany irrational numbers. In particular, there is no smallest positive real number.

    Absolute Value

    |a| ={

    a, if a ≥ 0−a, if a < 0.

    other characterizations |a| = max{a, −a}; |a| = √a2.geometric interpretation |a| = distance between a and 0;

    |a − c| = distance between a and c.

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    6 ■ CHAPTER 1 PRECALCULUS REVIEW

    properties (i) |a| = 0 iff a = 0.†(ii) | − a| = |a|.

    (iii) |ab| = |a||b|.(iv) |a + b| ≤ |a| + |b|. (the triangle inequality)††(v) ||a| − |b|| ≤ |a − b|. (a variant of the triangle inequality)

    (vi) |a|2 = |a2| = a2.

    Techniques for solving inequalities that feature absolute value are reviewed inSection 1.3.

    Intervals

    Suppose that a < b. The open interval (a, b) is the set of all numbers between a and b:

    (a, b) = {x : a < x < b}.ba

    The closed interval [a, b] is the open interval (a, b) together with the endpoints aand b:

    [a, b] = {x : a ≤ x ≤ b}.ba

    There are seven other types of intervals:

    (a, b] = {x : a < x ≤ b},ba

    [a, b) = {x : a ≤ x ≤ b},ba

    (a, ∞) = {x : a < x},a

    [a, ∞) = {x : a ≤ x},a

    (−∞, b) = {x : x < b},b

    (−∞, b] = {x : x ≤ b},b

    (−∞, ∞) = the set of real numbers.

    Interval notation is easy to remember: we use a square bracket to include an end-point and a parenthesis to exclude it. On a number line, inclusion is indicated by a soliddot, exclusion by an open dot. The symbols ∞ and −∞, read “infinity” and “negativeinfinity” (or “minus infinity”), do not represent real numbers. In the intervals listedabove, the symbol ∞ is used to indicate that the interval extends indefinitely in the pos-itive direction; the symbol −∞ is used to indicate that the interval extends indefinitelyin the negative direction.

    Open and Closed

    Any interval that contains no endpoints is called open: (a, b), (a, ∞), (−∞, b),(−∞, ∞) are open. Any interval that contains each of its endpoints (there may beone or two) is called closed: [a, b], [a, ∞), (−∞, b] are closed. The intervals (a, b]and [a, b) are called half-open (half-closed): (a, b] is open on the left and closed on theright; [a, b) is closed on the left and open on the right. Points of an interval that are notendpoints are called interior points of the interval.

    †By “iff” we mean “if and only if.” This expression is used so often in mathematics that it’s convenient tohave an abbreviation for it.††The absolute value of the sum of two numbers cannot exceed the sum of their absolute values. This isanalogous to the fact that in a triangle the length of one side cannot exceed the sum of the lengths of theother two sides.

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    1.2 REVIEW OF ELEMENTARY MATHEMATICS ■ 7

    Boundedness

    A set S of real numbers is said to be:

    (i) Bounded above if there exists a real number M such that

    x ≤ M for all x ∈ S;such a number M is called an upper bound for S.

    (ii) Bounded below if there exists a real number m such that

    m ≤ x for all x ∈ S;such a number m is called a lower bound for S.

    (iii) Bounded if it is bounded above and below.†

    Note that if M is an upper bound for S, then any number greater than M is also anupper bound for S, and if m is a lower bound for S, than any number less than m is alsoa lower bound for S.

    Examples The intervals (−∞, 2] and (−∞, 2) are both bounded above by 2 (andby every number greater than 2), but these sets are not bounded below. The set ofpositive integers {1, 2, 3, . . .} is bounded below by 1 (and by every number less than 1),but the set is not bounded above; there being no number M greater than or equal to allpositive integers, the set has no upper bound. All finite sets of numbers are bounded—(bounded below by the least element and bounded above by the greatest). Finally, theset of all integers, {· · · ,−3, −2, −1, 0, 1, 2, 3, · · ·}, is unbounded in both directions; itis unbounded above and unbounded below. ❏

    Factorials

    Let n be a positive integer. By n factorial, denoted n!, we mean the product of theintegers from n down to 1:

    n! = n(n − 1)(n − 2) · · · 3 · 2 · 1.In particular

    1! = 1, 2! = 2 · 1 = 2, 3! = 3 · 2 · 1 = 6, 4! = 4 · 3 · 2 · 1 = 24, and so on.For convenience we define 0! = 1.

    Algebra

    Powers and Roots

    a real, p a positive integer a1 = a, a p =p factors︷ ︸︸ ︷

    a · a · · · · · aa �= 0 : a0 = 1, a−p = 1/a p

    laws of exponents a p+q = a paq , a p−q = a pa−q , (aq )p = a pqa real, q odd a1/q , called the qth root of a, is the number b such

    that bq = aa nonnegative, q even a1/q is the nonnegative number b such that bq = a

    notation a1/q can be written q√

    a (a1/2 is written√

    a)

    rational exponents a p/q = (a1/q )p

    †In defining bounded above, bounded below, and bounded we used the conditional “if,” not “iff.” We couldhave used “iff,” but that would have been unnecessary. Definitions are by their very nature “iff” statements.

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    8 ■ CHAPTER 1 PRECALCULUS REVIEW

    Examples

    20 = 1, 21 = 1, 22 = 2 · 2 = 4, 23 = 2 · 2 · 2 = 8, and so on25+3 = 25 · 23 = 32 · 8 = 256, 23−5 = 2−2 = 1/22 = 1/4

    (22)3 = 23·2 = 26 = 64, (23)2 = 22·3 = 26 = 6481/3 = 2, (−8)1/3 = −2, 161/2 = √16 = 4, 161/4 = 2

    85/3 = (81/3)5 = 25 = 32, 8−5/3 = (81/3)−5 = 2−5 = 1/25 = 1/32 ❏

    Basic Formulas

    (a + b)2 = a2 + 2ab + b2(a − b)2 = a2 − 2ab + b2(a + b)3 = a3 + 3a2b + 3ab2 + b3(a − b)3 = a3 − 3a2b + 3ab2 − b3a2 − b2 = (a − b)(a + b)a3 − b3 = (a − b)(a2 + ab + b2)a4 − b4 = (a − b)(a3 + a2b + ab2 + b3)

    More generally:

    an − bn = (a − b)(an−1 + an−2b + · · · + abn−2 + bn−1)

    Quadratic Equations

    The roots of a quadratic equation

    ax2 + bx + c = 0 with a �= 0are given by the general quadratic formula

    r = −b ±√

    b2 − 4ac2a

    .

    If b2 − 4ac > 0, the equation has two real roots; if b2 − 4ac = 0, the equation has onereal root; if b2 − 4ac < 0, the equation has no real roots, but it has two complex roots.

    Geometry

    Elementary Figures

    Triangle Equilateral Triangle

    h

    b s

    ss

    area = 12 bh area =14

    √3 s2

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    1.2 REVIEW OF ELEMENTARY MATHEMATICS ■ 9

    Rectangle Rectangular Solid

    w

    l

    h w

    l

    area = lwperimeter = 2l + 2w volume = lwhdiagonal =

    √l2 + w2 surface area = 2lw + 2lh + 2wh

    Square Cube

    x

    x x

    x

    x

    area = x2

    perimeter = 4x volume = x3

    diagonal = x√

    2 surface area = 6x2

    Circle Sphere

    rr

    area = πr2 volume = 43πr3

    circumference = 2πr surface area = 4πr2

    Sector of a Circle: radius r, central angle θ measured in radians (see Section 1.6).

    arc length

    r

    θr

    θ

    arc length = rθ area = 12r2θ

    (Table continues)

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    10 ■ CHAPTER 1 PRECALCULUS REVIEW

    Right Circular Cylinder Right Circular Cone

    r

    h h

    r

    volume = πr2h volume = 13πr2h

    lateral area = 2πrh slant height =√

    r2 + h2total surface area = 2πr2 + 2πrh lateral area = πr√r2 + h2

    total surface area = πr2 + πr√r2 + h2

    EXERCISES 1.2

    Exercises 1–10. Is the number rational or irrational?

    1. 177 . 2. −6.3. 2.131313 . . . = 2.13. 4. √2 − 3.5. 0. 6. π − 2.7. 3

    √8. 8. 0.125.

    9. −√9. 10. (√2 − √3) (√2 + √3)Exercises 11–16. Replace the symbol ∗ by , or = to makethe statement true.

    11. 34 ∗ 0.75. 12. 0.33 ∗ 13 .13.

    √2 ∗ 1.414. 14. 4 ∗ √16.

    15. − 27 ∗ −0.285714. 16. π ∗ 227 .Exercises 17–23. Evaluate

    17. |6|. 18. | − 4|.19. | − 3 − 7|. 20. | − 5| − |8|.21. | − 5| + | − 8|. 22. |2 − π |.23. |5 − √5|.Exercises 24–33. Indicate on a number line the numbers x thatsatisfy the condition.

    24. x ≥ 3 25. x ≤ − 32 .26. −2 ≤ x ≤ 3. 27. x2 < 16.28. x2 ≥ 16. 29. |x | ≤ 0.30. x2 ≥ 0. 31. |x − 4| ≤ 2.32. |x + 1| > 3. 33. |x + 3| ≤ 0.Exercises 34–40. Sketch the set on a number line.

    34. [3, ∞). 35. (−∞, 2).36. (−4, 3]. 37. [−2, 3] ∪ [1, 5].38.

    [−3, 32) ∩ ( 32 , 52

    ]. 39. (−∞, −1) ∪ (−2, ∞).

    40. (−∞, 2) ∩ [3, ∞).Exercises 41–47. State whether the set is bounded above,bounded below, bounded. If a set is bounded above, give anupper bound; if it is bounded below, give a lower bound; if it isbounded, give an upper bound and a lower bound.

    41. {0, 1, 2, 3, 4}. 42. {0, −1, −2, −3, . . .}.43. The set of even integers.

    44. {x : x ≤ 4}. 45. {x : x2 > 3}.46.

    {n−1

    n : n = 1, 2, 3 . . .}.

    47. The set of rational numbers less than√

    2.

    Exercises 48–50.

    48. Order the following numbers and place them on a numberline: 3

    √π, 2

    √π ,

    √2, 3π , π3.

    49. Let x0 = 2 and define xn =17 + 2x3n−1

    3x2n−1for n =

    1, 2, 3, 4, . . . Find at least five values for xn . Is the setS = {x0, x1, x2, . . . , xn, . . .} bounded above, bounded be-low, bounded? If so, give a lower bound and/or an upperbound for S. If n is a large positive integer, what is the ap-proximate value of xn?

    50. Rework Exercise 49 with x0 = 3 and xn =231 + 4x5n−1

    5x4n−1Exercises 51–56. Write the expression in factored form.

    51. x2 − 10x + 25. 52. 9x2 − 4.53. 8x6 + 64. 54. 27x3 − 8.55. 4x2 + 12x + 9. 56. 4x4 + 4x2 + 1.Exercises 57–64. Find the real roots of the equation.

    57. x2 − x − 2 = 0. 58. x2 − 9 = 0.59. x2 − 6x + 9 = 0. 60. 2x2 − 5x − 3 = 0.61. x2 − 2x + 2 = 0. 62. x2 + 8x + 16 = 0.63. x2 + 4x + 13 = 0. 64. x2 − 2x + 5 = 0.Exercises 65–69. Evaluate.

    65. 5!. 66.5!

    8!.

    67.8!

    3!5!. 68.

    9!

    3!6!.

    69.7!

    0!7!.

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    1.3 REVIEW OF INEQUALITIES ■ 11

    70. Show that the sum of two rational numbers is a rational num-ber.

    71. Show that the sum of a rational number and an irrationalnumber is irrational.

    72. Show that the product of two rational numbers is a rationalnumber.

    73. Is the product of a rational number and an irrational numbernecessarily rational? necessarily irrational?

    74. Show by example that the sum of two irrational numbers (a)can be rational; (b) can be irrational. Do the same for theproduct of two irrational numbers.

    75. Prove that√

    2 is irrational. HINT: Assume that√

    2 = p/qwith the fraction written in lowest terms. Square both sidesof this equation and argue that both p and q must be divisibleby 2.

    76. Prove that√

    3 is irrational.

    77. Let S be the set of all rectangles with perimeter P. Show thatthe square is the element of S with largest area.

    78. Show that if a circle and a square have the same perime-ter, then the circle has the larger area. Given that a circle and arectangle have the same perimeter, which has the larger area?

    The following mathematical tidbit was first seen by one of theauthors many years ago in Granville, Longley, and Smith, Ele-ments of Calculus, now a Wiley book.

    79. Theorem (a phony one): 1 = 2.

    PROOF (a phony one): Let a and b be real numbers, bothdifferent from 0. Suppose now that a = b. Then

    ab = b2

    ab − a2 = b2 − a2

    a(b − a) = (b + a)(b − a)a = b + a.

    Since a = b, we havea = 2a.

    Division by a, which by assumption is not 0, gives

    1 = 2. ❏What is wrong with this argument?

    ■ 1.3 REVIEW OF INEQUALITIESAll our work with inequalities is based on the order properties of the real numbers givenin Section 1.2. In this section we work with the type if inequalities that arise frequentlyin calculus, inequalities that involve a variable.

    To solve an inequality in x is to find the numbers x that satisfy the inequality. Thesenumbers constitute a set, called the solution set of the inequality.

    We solve inequalities much as we solve an equation, but there is one importantdifference. We can maintain an inequality by adding the same number to both sides,or by subtracting the same number from both sides, or by multiplying or dividing bothsides by the same positive number. But if we multiply or divide by a negative number,then the inequality is reversed:

    x − 2 < 4 gives x < 6, x + 2 < 4 gives x < 2,1

    2x < 4 gives x < 8,

    but − 12

    x < 4 gives x > −8.↑ note, the inequality is reversed

    Example 1 Solve the inequality

    −3(4 − x) ≤ 12.

    SOLUTION Multiplying both sides of the inequality by − 13 , we have4 − x ≥ −4. (the inequality has been reversed)

    Subtracting 4, we get

    −x ≥ −8.

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    12 ■ CHAPTER 1 PRECALCULUS REVIEW

    To isolate x, we multiply by −1. This givesx ≤ 8. (the inequality has been reversed again)

    The solution set is the interval (−∞, 8]. ❏8

    There are generally several ways to solve a given inequality. For example, the lastinequality could have been solved as follows:

    −3(4 − x) ≤ 12,−12 + 3x ≤ 12,

    3x ≤ 24, (we added 12)x ≤ 8. (we divided by 3)

    To solve a quadratic inequality, we try to factor the quadratic. Failing that, we cancomplete the square and go on from there. This second method always works.

    Example 2 Solve the inequality

    x2 − 4x + 3 > 0.SOLUTION Factoring the quadratic, we obtain

    (x − 1)(x − 3) > 0.The product (x − 1)(x − 3) is zero at 1 and 3. Mark these points on a number line(Figure 1.3.1). The points 1 and 3 separate three intervals:

    (−∞, 1), (1, 3), (3, ∞).

    1 3

    0 – – – – – – – – – – – – – – – –++++++++++++ ++++++++++++0

    Figure 1.3.1

    On each of these intervals the product (x − 1)(x − 3) keeps a constant sign:on (−∞, 1) [to the left of 1] sign of (x − 1)(x − 3) = (−)(−) = +;

    on (1, 3) [between 1 and 3] sign of (x − 1)(x − 3) = (+)(−) = −;on (3,∞) [to the right of 3] sign of (x − 1)(x − 3) = (+)(+) = +.

    The product (x − 1)(x − 3) is positive on the open intervals (−∞, 1) and (3, ∞). Thesolution set is the union (−∞, 1) ∪ (3, ∞). ❏

    31

    Example 3 Solve the inequality

    x2 − 2x + 5 ≤ 0.

    SOLUTION Not seeing immediately how to factor the quadratic, we use the methodthat always works: completing the square. Note that

    x2 − 2x + 5 = (x2 − 2x + 1) + 4 = (x − 1)2 + 4.This tells us that

    x2 − 2x + 5 ≥ 4 for all real x,and thus there are no numbers that satisfy the inequality we are trying to solve. To putit in terms of sets, the solution set is the empty set ∅. ❏

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    1.3 REVIEW OF INEQUALITIES ■ 13

    In practice we frequently come to expressions of the form

    (x − a1)k1 (x − a2)k2 . . . (x − an)knk1, k2, . . . , kn positive integers, a1 < a2 < · · · < an . Such an expression is zero ata1, a2, . . . , an . It is positive on those intervals where the number of negative factors iseven and negative on those intervals where the number of negative factors is odd.

    Take, for instance,

    (x + 2)(x − 1)(x − 3).This product is zero at −2, 1, 3. It is

    negative on (−∞, −2), (3 negative terms)positive on (−2, 1), (2 negative terms)negative on (1, 3), (1 negative term)positive on (3, ∞). (0 negative terms)

    See Figure 1.3.2

    –2

    0

    1

    0

    3

    0– – – ––– – – – – – – – – – – – +++++++++++++++++++++ + + +

    Figure 1.3.2

    Example 4 Solve the inequality

    (x + 3)5(x − 1)(x − 4)2 < 0.

    SOLUTION We view (x + 3)5(x − 1)(x − 4)2 as the product of three factors: (x + 3)5,(x − 1), (x − 4)2. The product is zero at −3, 1, 4. These points separate the intervals

    (−∞, −3), (−3, 1), (1, 4), (4, ∞).On each of these intervals the product keeps a constant sign:

    positive on (−∞, −3), (2 negative factors)negative on (−3, 1), (1 negative factor)positive on (1, 4), (0 negative factors)positive on (4, ∞). (0 negative factors)

    See Figure 1.3.3.

    –3

    0

    1

    0

    4

    0– – – – – – – – – – – – –+ + + + + + + + + + + + + + + + + + + + + + + + +

    Figure 1.3.3

    The solution set is the open interval (−3, 1). ❏ 1–3This approach to solving inequalities will be justified in Section 2.6

    Inequalities and Absolute Value

    Now we take up inequalities that involve absolute values. With an eye toward developingthe concept of limits (Chapter 2), we introduce two Greek letters: δ (delta) and �(epsilon).

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    14 ■ CHAPTER 1 PRECALCULUS REVIEW

    As you know, for each real number a

    |a| ={

    a if a ≥ 0.−a, if a < 0, |a| = max{a, −a}, |a| =

    √a2.(1.3.1)

    We begin with the inequality

    |x | < δwhere δ is some positive number. To say that |x | < δ is to say that x lies within δ unitsof 0 or, equivalently, that x lies between −δ and δ. Thus

    (1.3.2) |x | < δ iff − δ < x < δ.0–

    ⎜x⎜<

    δ δ

    δδ

    δ

    The solution set is the open interval (−δ, δ).To say that |x − c| < δ is to say that x lies within δ units of c or, equivalently, that

    x lies between c − δ and c + δ. Thus

    (1.3.3) |x − c| < δ iff c − δ < x < c + δ.cc – c +

    ⎜x – c⎜<

    δ δ

    δδ

    δ

    The solution set is the open interval (c − δ, c + δ).Somewhat more delicate is the inequality

    0 < |x − c| < δ.Here we have |x − c| < δ with the additional requirement that x �= c. Consequently,

    (1.3.4) 0 < |x − c| < δ iff c − δ < x < c or c < x < c + δ.

    The solution set is the union of two open intervals: (c − δ, c) ∪ (c, c + δ).The following results are an immediate consequence of what we just showed.

    |x | < 12 iff − 12 < x < 12 ; [solution set: (− 12 , 12 )]|x − 5| < 1 iff 4 < x < 6;[solution set: (4, 6)]

    0 < |x − 5| < 1 iff 4 < x < 5 or 5 < x < 6; [solution set: (4, 5) ∪ (5, 6)]

    Example 5 Solve the inequality

    |x + 2| < 3.

    SOLUTION Once we recognize that |x + 2| = |x − (−2)|, we are in familiar territory.|x − (−2)| < 3 iff − 2 − 3 < x < −2 + 3 iff − 5 < x < 1.

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    1.3 REVIEW OF INEQUALITIES ■ 15

    The solution set is the open interval (−5, 1). ❏3 3

    –2 1–5

    Example 6 Solve the inequality

    |3x − 4| < 2.

    SOLUTION Since

    |3x − 4| = ∣∣3(x − 43 )∣∣ = |3|∣∣x − 43

    ∣∣ = 3∣∣x − 43∣∣,

    the inequality can be written

    3∣∣x − 43

    ∣∣ < 2.This gives

    ∣∣x − 43∣∣ < 23 , 43 − 23 < x < 43 + 23 , 23 < x < 2.

    The solution set is the open interval ( 23 , 2).

    ALTERNATIVE SOLUTION There is usually more than one way to solve an inequality.In this case, for example, we can write

    |3x − 4| < 2as

    −2 < 3x − 4 < 2and proceed from there. Adding 4 to the inequality, we get

    2 < 3x < 6.

    Division by 3 gives the result we had before:23 < x < 2. ❏

    Let � > 0. If you think of |a| as the distance between a and 0, then

    (1.3.5) |a| > � iff a > � or a < −�. 0 �–�⎜a⎜> �

    Example 7 Solve the inequality

    |2x + 3| > 5.

    SOLUTION In general

    |a| > � iff a > � or a < −�.So here

    2x + 3 > 5 or 2x + 3 < −5.The first possibility gives 2x > 2 and thus

    x > 1.1

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    16 ■ CHAPTER 1 PRECALCULUS REVIEW

    The second possibility gives 2x < −8 and thusx < −4

    –4

    The total solution is therefore the union

    (−∞, −4) ∪ (1, ∞). ❏–4 1

    We come now to one of the fundamental inequalities of calculus: for all real numbersa and b,

    (1.3.6) |a + b| ≤ |a| + |b|.

    This is called the triangle inequality in analogy with the geometric observation that “inany triangle the length of each side is less than or equal to the sum of the lengths of theother two sides.”

    PROOF OF THE TRIANGLE INEQUALITY The key here is to think of |x | as √x2.Note first that

    (a + b)2 = a2 + 2ab + b2 ≤ |a|2 + 2|a||b| + |b|2 = (|a| + |b|)2.Comparing the extremes of the inequality and taking square roots, we have√

    (a + b)2 ≤ |a| + |b|. (Exercise 51)The result follows from observing that√

    (a + b)2 = |a + b|. ❏

    Here is a variant of the triangle inequality that also comes up in calculus: for allreal numbers a and b,

    (1.3.7)∣∣|a| − |b|∣∣ ≤ |a − b|.

    The proof is left to you as an exercise.

    EXERCISES 1.3

    Exercises 1–20. Solve the inequality and mark the solution seton a number line.

    1. 2 + 3x < 5. 2. 12 (2x + 3) < 6.3. 16x + 64 ≤ 16. 4. 3x + 5 > 14 (x − 2).5. 12 (1 + x) < 13 (1 − x). 6. 3x − 2 ≤ 1 + 6x .7. x2 − 1 < 0. 8. x2 + 9x + 20 < 0.9. x2 − x − 6 ≥ 0. 10. x2 − 4x − 5 > 0.

    11. 2x2 + x − 1 ≤ 0. 12. 3x2 + 4x − 4 ≥ 0.13. x(x − 1)(x − 2) > 0. 14. x(2x − 1)(3x − 5) ≤ 0.15. x3 − 2x2 + x ≥ 0. 16. x2 − 4x + 4 ≤ 0.17. x3(x − 2)(x + 3)2 < 0. 18. x2(x − 3)(x + 4)2 > 0.19. x2(x − 2)(x + 6) > 0. 20. 7x(x − 4)2 < 0.

    Exercises 21–36. Solve the inequality and express the solutionset as an interval or as the union of intervals.

    21. |x | < 2. 22. |x | ≥ 1.23. |x | > 3. 24. |x − 1| < 1.25. |x − 2| < 12 . 26. |x − 12 | < 2.27. 0 < |x | < 1. 28. 0 < |x | < 12 .29. 0 < |x − 2| < 12 . 30. 0 < |x − 12 | < 2.31. 0 < |x − 3| < 8. 32. |3x − 5| < 3.33. |2x + 1| < 14 . 34. |5x − 3| < 12 .35. |2x + 5| > 3. 36. |3x + 1| > 5.Exercises 37–42. Each of the following sets is the solution of aninequality of the form | x − c |< δ. Find c and δ.

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    1.4 COORDINATE PLANE; ANALYTIC GEOMETRY ■ 17

    37. (−3, 3). 38. (−2, 2).39. (−3, 7). 40. (0, 4).41. (−7, 3). 42. (a, b).Exercises 43–46. Determine all numbers A > 0 for which thestatement is true.

    43. If |x − 2| < 1, then |2x − 4| < A.44. If |x − 2| < A, then |2x − 4| < 3.45. If |x + 1| < A, then |3x + 3| < 4.46. If |x + 1| < 2, then |3x + 3| < A.47. Arrange the following in order :1, x,

    √x, 1/x, 1/

    √x , given

    that: (a) x > 1; (b) 0 < x < 1.

    48. Given that x > 0, compare√x

    x + 1 and√

    x + 1x + 2 .

    49. Suppose that ab > 0. Show that if a < b, then 1/b < 1/a.

    50. Given that a > 0 and b > 0, show that if a2 ≤ b2, thena ≤ b.

    51. Show that if 0 ≤ a ≤ b, then √a ≤ √b.

    52. Show that |a − b| ≤ |a| + |b| for all real numbers a and b.53. Show that

    ∣∣|a| − |b|∣∣ ≤ |a − b| for all real numbers a and b.HINT: Calculate

    ∣∣|a| − |b|∣∣2.54. Show that |a + b| = |a| + |b| iff ab ≥ 0.55. Show that

    if 0 ≤ a ≤ b, then a1 + a ≤

    b

    1 + b .

    56. Let a, b, c be nonnegative numbers. Show that

    if a ≤ b + c, then a1 + a ≤

    b

    1 + b +c

    1 + c .

    57. Show that if a and b are real numbers and a < b, thena < (a + b)/2 < b. The number (a + b)/2 is called thearithmetic mean of a and b.

    58. Given that 0 ≤ a ≤ b, show that

    a ≤√

    ab ≤ a + b2

    ≤ b.

    The number√

    ab is called the geometric mean of a and b.

    ■ 1.4 COORDINATE PLANE; ANALYTIC GEOMETRY

    Rectangular Coordinates

    The one-to-one correspondence between real numbers and points on a line can be usedto construct a coordinate system for the plane. In the plane, we draw two number linesthat are mutually perpendicular and intersect at their origins. Let O be the point ofintersection. We set one of the lines horizontally with the positive numbers to the rightof O and the other vertically with the positive numbers above O. The point O is calledthe origin, and the number lines are called the coordinate axes. The horizontal axisis usually labeled the x-axis and the vertical axis is usually labeled the y-axis. Thecoordinate axes separate four regions, which are called quadrants. The quadrants arenumbered I, II, III, IV in the counterclockwise direction starting with the upper rightquadrant. See Figure 1.4.1.

    III

    III IV

    y

    xO

    Figure 1.4.1Rectangular coordinates are assigned to points of the plane as follows (see Figure

    1.4.2.). The point on the x-axis with line coordinate a is assigned rectangular coordinates(a, 0). The point on the y-axis with line coordinate b is assigned rectangular coordinates(0, b). Thus the origin is assigned coordinates (0, 0). A point P not on one of thecoordinate axes is assigned coordinates (a, b) provided that the line l1 that passesthrough P and is parallel to the y-axis intersects the x-axis at the point with coordinates(a, 0), and the l2 that passes through P and is parallel to the x-axis intersects the y-axisat the point with coordinates (0, b).

    y

    xO

    a

    b

    c

    dQ(c, d)

    Pl2

    l1

    Figure 1.4.2

    This procedure assigns an ordered pair of real numbers to each point of the plane.Moreover, the procedure is reversible. Given any ordered pair (a, b) of real numbers,there is a unique point P in the plane with coordinates (a, b).

    To indicate P with coordinates (a, b) we write P(a, b). The number a is called thex-coordinate (the abscissa); the number b is called the y-coordinate (the ordinate). Thecoordinate system that we have defined is called a rectangular coordinate system. Itis often referred to as a Cartesian coordinate system after the French mathematicianRené Descartes (1596–1650).

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    18 ■ CHAPTER 1 PRECALCULUS REVIEW

    Distance and Midpoint Formulas

    Let P0(x0, y0) and P1(x1, y1) be points in the plane. The formula for the distanced(P0, P1) between P0 and P1 follows from the Pythagorean theorem:

    d(P0, P1) =√

    |x1 − x0|2 + |y1 − y0|2 =√

    (x1 − x0)2 + (y1 − y0)2. (Figure 1.4.3)↑ |a|2 = a2

    y

    x

    P1(x1, y1)

    P0(x0, y0)

    P(x1, y0)

    y1 – y0

    x1 – x0

    Distance : d(P0, P1) =√

    (x1 − x0)2 + (y1 − y0)2Figure 1.4.3

    Let M(x, y) be the midpoint of the line segment P0 P1. That

    x = x0 + x12

    and y = y0 + y12

    follows from the congruence of the triangles shown in Figure 1.4.4x

    y

    P0(x0, y0)

    M (x, y)P1(x1, y1)

    Midpoint: M = ( x0+x12 , y0+y12)

    Figure 1.4.4Lines

    (i) Slope Let l be the line determined by P0(x0, y0) and P1(x1, y1). If l is not vertical,then x1 �= x0 and the slope of l is given by the formula

    m = y1 − y0x1 − x0 . (Figure 1.4.5)

    With θ (as indicated in the figure) measured counterclockwise from the x-axis,

    m = tan θ.†

    The angle θ is called the inclination of l. If l is vertical, then θ = π/2 and the slope ofl is not defined.

    (ii) Intercepts If a line intersects the x-axis, it does so at some point (a, 0). We call athe x-intercept. If a line intersects the y-axis, it does so at some point (0, b). We call bthe y-intercept. Intercepts are shown in Figure 1.4.6.

    †The trigonometric functions are reviewed in Section 1.6.

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    1.4 COORDINATE PLANE; ANALYTIC GEOMETRY ■ 19

    y

    O

    P0(x0, y0)

    P1(x1, y1)

    m =y1 – y0x1 – x0

    = 0

    0 < <2

    m =y1 – y0x1 – x0

    < 0

    < <2

    m =y1 – y0x1 – x0

    > 0

    x

    y

    x

    y

    xO O

    P0(x0, y0) P1(x1, y1)

    P0(x0, y0)

    P1(x1, y1)

    = 0

    θ

    θ θ θπ ππ

    θ

    Figure 1.4.5

    y

    x

    (a, 0)

    x-intercept a y-intercept b

    (0, b)

    l

    Figure 1.4.6

    (iii) Equations

    vertical line x = a.horizontal line y = b.

    point-slope form y − y0 = m(x − x0).slope-intercept form y = mx + b. (y = b at x = 0)

    two-intercept formx

    a+ y

    b= 1. (x-intercept a; y-intercept b)

    general form Ax + By + C = 0. (A and B not both 0)

    (iv) Parallel and Perpendicular Nonvertical Lines

    parallel m1 = m2.perpendicular m1m2 = −1.

    (v) The Angle Between Two Lines The angle between two lines that meet at rightangles is π/2. Figure 1.4.7 shows two lines (l1, l2 with inclinations θ1, θ2) that intersectbut not at right angles. These lines form two angles, marked α and π − α in the figure.The smaller of these angles, the one between 0 and π/2, is called the angle between l1and l2. This angle, marked α in the figure, is readily obtained from θ1 and θ2.

    If neither l1 nor l2 is vertical, the angle α between l1 and l2 can also be obtained

    y l1

    l2

    1

    2

    π α α

    θθ

    Figure 1.4.7

    from the slopes of the lines:

    tan α = m1 − m21 + m1m2 .

    The derivation of this formula is outlined in Exercise 75 of Section 1.6.

    Example 1 Find the slope and the y-intercept of each of the following lines:

    l1 : 20x − 24y − 30 = 0, l2 : 2x − 3 = 0, l3 : 4y + 5 = 0.

    SOLUTION The equation of l1 can be written

    y = 56 x − 54 .

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    20 ■ CHAPTER 1 PRECALCULUS REVIEW

    This is in the form y = mx + b. The slope is 56 , and the y-intercept is − 54 .The equation of l2 can be written

    x = 32 .The line is vertical and the slope is not defined. Since the line does not cross the y-axis,the line has no y-intercept.

    The third equation can be written

    y = − 54 .

    The line is horizontal. The slope is 0 and the y-intercept is − 54 . The three lines aredrawn in Figure 1.4.8. ❏

    y

    x

    y = 56

    x – 54

    x = 32

    y = – 54

    3

    2

    1

    –3 –2 –1–1

    –2

    –3

    2 3

    y

    x

    3

    2

    1

    –3 –2 –1–1

    –2

    –3

    1 2 3

    y

    x

    3

    2

    1

    –3 –2 –1–1

    –2

    –3

    1 2 31

    l2

    l1

    l3

    Figure 1.4.8

    Example 2 Write an equation for the line l2 that is parallel to

    l1 : 3x − 5y + 8 = 0and passes through the point P(−3, 2).

    SOLUTION The equation for l1 can be written

    y = 35 x + 85 .

    The slope of l1 is35 . The slope of l2 must also be

    35 . (For nonvertical parallel lines, m1 = m2.)

    Since l2 passes through (−3, 2) with slope 35 , we can use the point-slope formulaand write the equation as

    y − 2 = 35 (x + 3). ❏

    Example 3 Write an equation for the line that is perpendicular to

    l1 : x − 4y + 8 = 0and passes through the point P(2, −4).

    SOLUTION The equation for l1 can be written

    y = 14 x + 2.

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    1.4 COORDINATE PLANE; ANALYTIC GEOMETRY ■ 21

    The slope of l1 is14 . The slope of l2 is therefore −4. (For nonvertical perpendicular

    lines, m1m2 = −1.)Since l2 passes through (2, −4) with slope −4, we can use the point-slope formula

    and write the equation as

    y + 4 = −4(x − 2). ❏

    Example 4 Show that the lines

    l1 : 3x − 4y + 8 = 0 and l2 : 12x − 5y − 12 = 0

    intersect and find their point of intersection.

    SOLUTION The slope of l1 is34 and the slope of l2 is

    125 . Since l1 and l2 have different

    slopes, they intersect at a point.To find the point of intersection, we solve the two equations simultaneously:

    3x − 4y + 8 = 012x − 5y − 12 = 0.

    Multiplying the first equation by −4 and adding it to the second equation, we obtain11y − 44 = 0

    y = 4.

    Substituting y = 4 into either of the two given equations, we find that x = 83 . The linesintersect at the point ( 83 , 4). ❏

    Circle, Ellipse, Parabola, Hyperbola

    These curves and their remarkable properties are thoroughly discussed in Section 10.1.The information we give here suffices for our present purposes.

    Circle

    y

    x

    x2 + y2 = r2

    r

    (h, k)

    (x – h)2 + (y – k)2 = r2

    y

    x

    r

    Figure 1.4.9

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    22 ■ CHAPTER 1 PRECALCULUS REVIEW

    Ellipse

    x2

    a2

    y2

    b2+ = 1, a > b

    ab

    y

    x

    a

    b

    x2

    a2

    y2

    b2+ = 1, b > a

    x

    y

    Figure 1.4.10

    Parabola

    y = ax2, a > 0

    y = ax2 + bx + c , a > 0

    x

    y

    y = ax2 + bx + c, a < 0

    y = ax2, a < 0

    x

    y

    Figure 1.4.11

    Hyperbola

    x2

    a2

    y2

    b2– = 1

    x

    y

    y2

    a2

    x2

    b2– = 1

    x

    y

    Figure 1.4.12

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    1.4 COORDINATE PLANE; ANALYTIC GEOMETRY ■ 23

    Remark The circle, the ellipse, the parabola, and the hyperbola are known as theconic sections because each of these configurations can be obtained by slicing a “doubleright circular cone” by a suitably inclined plane. (See Figure 1.4.13.) ❏

    circle ellipse parabola hyperbola

    Figure 1.4.13

    EXERCISES 1.4

    Exercises 1–4. Find the distance between the points.

    1. P0(0, 5), P1(6, −3). 2. P0(2, 2), P1(5, 5).3. P0(5, −2), P1(−3, 2). 4. P0(2, 7), P1(−4, 7).

    Exercises 5–8. Find the midpoint of the line segment P0 P1.

    5. P0(2, 4), P1(6, 8). 6. P0(3, −1), P1(−1, 5).7. P0(2, −3), P1(7, −3). 8. P0(a, 3), P1(3, a).

    Exercises 9–14. Find the slope of the line through the points.

    9. P0(−2, 5), P1(4, 1). 10. P0(4, −3), P1(−2, −7).11. P(a, b), Q(b, a). 12. P(4, −1), Q(−3, −1).13. P(x0, 0), Q(0, y0). 14. O(0, 0), P(x0, y0).

    Exercises 15–20. Find the slope and y-intercept.

    15. y = 2x − 4. 16. 6 − 5x = 0.17. 3y = x + 6. 18. 6y − 3x + 8 = 0.19. 7x − 3y + 4 = 0. 20. y = 3.Exercises 21–24. Write an equation for the line with

    21. slope 5 and y-intercept 2.

    22. slope 5 and y-intercept −2.23. slope −5 and y-intercept 2.24. slope −5 and y-intercept −2.Exercises 25–26. Write an equation for the horizontal line3 units

    25. above the x-axis.

    26. below the x-axis.

    Exercises 27–28. Write an equation for the vertical line 3 units

    27. to the left of the y-axis.

    28. to the right of the y-axis.

    Exercises 29–34. Find an equation for the line that passesthrough the point P(2, 7) and is

    29. parallel to the x-axis.

    30. parallel to the y-axis.

    31. parallel to the line 3y − 2x + 6 = 0.32. perpendicular to the line y − 2x + 5 = 0.33. perpendicular to the line 3y − 2x + 6 = 0.34. parallel to the line y − 2x + 5 = 0.Exercises 35–38. Determine the point(s) where the line inter-sects the circle.

    35. y = x, x2 + y2 = 1.36. y = mx, x2 + y2 = 4.37. 4x + 3y = 24, x2 + y2 = 25.38. y = mx + b, x2 + y2 = b2.Exercises 39–42. Find the point where the lines intersect.

    39. l1 : 4x − y − 3 = 0, l2 : 3x − 4y + 1 = 0.40. l1 : 3x + y − 5 = 0, l2 : 7x − 10y + 27 = 0.41. l1 : 4x − y + 2 = 0, l2 : 19x + y = 0.

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    24 ■ CHAPTER 1 PRECALCULUS REVIEW

    42. l1 : 5x − 6y + 1 = 0, l2 : 8x + 5y + 2 = 0.43. Find the area of the triangle with vertices (1,−2), (−1, 3),

    (2, 4).

    44. Find the area of the triangle with vertices (−1, 1), (3, √2),(√

    2, −1).45. Determine the slope of the line that intersects the circle

    x2 + y2 = 169 only at the point (5, 12).46. Find an equation for the line which is tangent to the circle

    x2 + y2 − 2x + 6y − 15 = 0 at the point (4, 1). HINT: Aline is tangent to a circle at a point P iff it is perpendicularto the radius at P.

    47. The point P(1, −1) is on a circle centered at C(−1, 3). Findan equation for the line tangent to the circle at P.

    Exercises 48–51. Estimate the point(s) of intersection.

    48. l1 : 3x − 4y = 7, l2 : −5x + 2y = 11.49. l1 : 2.41x + 3.29y = 5, l2 : 5.13x − 4.27y = 13.50. l1 : 2x − 3y = 5, circle : x2 + y2 = 4.51. circle : x2 + y2 = 9, parabola : y = x2 − 4x + 5.Exercises 52–53. The perpendicular bisector of the line seg-ment P Q is the line which is perpendicular to P Q and passesthrough the midpoint of P Q. Find an equation for the perpen-dicular bisector of the line segment that joins the two points.

    52. P(−1, 3), Q(3, −4).53. P(1, −4), Q(4, 9).Exercises 54–56. The points are the vertices of a triangle. Statewhether the triangle is isosceles (two sides of equal length), aright triangle, both of these, or neither of these.

    54. P0(−4, 3), P1(−4, −1), P2(2, 1).55. P0(−2, 5), P1(1, 3), P2(−1, 0).56. P0(−1, 2), P1(1, 3), P2(4, 1).57. Show that the distance from the origin to the line Ax + By +

    C = 0 is given by the formula

    d(0, l) = |C |√A2 + B2 .

    58. An equilateral triangle is a triangle the three sides of whichhave the same length. Given that two of the vertices of anequilateral triangle are (0, 0) and (4, 3), find all possible lo-cations for a third vertex. How many such triangles are there?

    59. Show that the midpoint M of the hypotenuse of a right tri-angle is equidistant from the three vertices of the triangle.HINT: Introduce a coordinate system in which the sides ofthe triangle are on the coordinate axes; see the figure.

    (0, b)

    (a, 0) x

    y

    M

    60. A median of a triangle is a line segment from a vertex to themidpoint of the opposite side. Find the lengths of the medi-ans of the triangle with vertices (−1, −2), (2, 1), (4, −3).

    61. The vertices of a triangle are (1, 0), (3, 4), (−1, 6). Find thepoint(s) where the medians of this triangle intersect.

    62. Show that the medians of a triangle intersect in a single point(called the centroid of the triangle). HINT: Introduce a co-ordinate system such that one vertex is at the origin and oneside is on the positive x-axis; see the figure.

    (c, 0)

    (a, b)

    y

    x

    63. Prove that each diagonal of a parallelogram bisects the other.HINT: Introduce a coordinate system with one vertex at theorigin and one side on the positive x-axis.

    64. P1(x1, y1), P2(x2, y2), P3(x3, y3), P4(x4, y4) are the verticesof a quadrilateral. Show that the quadrilateral formed byjoining the midpoints of adjacent sides is a parallelogram.

    65. Except in scientific work, temperature is usually measuredin degrees Fahrenheit (F) or in degrees Celsius (C). The re-lation between F and C is linear. (In the equation that relatesF to C, both F and C appear to the first degree.) The freezingpoint of water in the Fahrenheit scale is 32◦F; in the Celsiusscale it is 0◦C. The boiling point of water in the Fahrenheitscale is 212◦F; in the Celsius scale it is 100◦C. Find an equa-tion that gives the Fahrenheit temperature F in terms of theCelsius temperature C. Is there a temperature at which theFahrenheit and Celsius readings are equal? If so, find it.

    66. In scientific work, temperature is measured on an absolutescale, called the Kelvin scale (after Lord Kelvin, who initi-ated this mode of temperature measurement). The relationbetween Fahrenheit temperature F and absolute temperatureK is linear. Given that K= 273◦ when F= 32◦, and K= 373◦when F = 212◦, express K in terms of F. Then use your resultin Exercise 65 to determine the connection between Celsiustemperature and absolute temperature.

    ■ 1.5 FUNCTIONSThe fundamental processes of calculus (called differentiation and integration) are pro-cesses applied to functions. To understand these processes and to be able to carry themout, you have to be comfortable working with functions. Here we review some of thebasic ideas and the nomenclature. We assume that you are familiar with all of this.

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    1.5 FUNCTIONS ■ 25

    Functions can be applied in a very general setting. At this stage, and throughout thefirst thirteen chapters of this text, we will be working with what are called real-valuedfunctions of a real variable, functions that assign real numbers to real numbers.

    Domain and Range

    Let’s suppose that D is some set of r