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  1. 1. Based on the original work by George B. Thomas, Jr.Massachusetts Institute of Technology as revised by Maurice D. WeirNaval Postgraduate School Joel Hass University of California, DavisTHOMAS CALCULUSEARLY TRANSCENDENTALS Twelfth Edition7001_ThomasET_FM_SE_pi-xvi.qxd 11/3/09 3:18 PM Page i
  2. 2. Editor-in-Chief: Deirdre Lynch Senior Acquisitions Editor:William Hoffman Senior Project Editor: Rachel S. Reeve AssociateEditor: Caroline Celano Associate Project Editor: Leah GoldbergSenior Managing Editor: Karen Wernholm Senior Production ProjectManager: Sheila Spinney Senior Design Supervisor: Andrea NixDigital Assets Manager: Marianne Groth Media Producer: Lin MahoneySoftware Development: Mary Durnwald and Bob Carroll ExecutiveMarketing Manager: Jeff Weidenaar Marketing Assistant: Kendra BassiSenior Author Support/Technology Specialist: Joe Vetere SeniorPrepress Supervisor: Caroline Fell Manufacturing Manager: EvelynBeaton Production Coordinator: Kathy Diamond Composition: NesbittGraphics, Inc. Illustrations: Karen Heyt, IllustraTech CoverDesign: Rokusek Design Cover image: Forest Edge, Hokuto, Hokkaido,Japan 2004 Michael Kenna About the cover: The cover image of a treeline on a snow-swept landscape, by the photographer Michael Kenna,was taken in Hokkaido, Japan. The artist was not thinking ofcalculus when he composed the image, but rather, of a visual haikuconsisting of a few elements that would spark the viewersimagination. Similarly, the minimal design of this text allows thecentral ideas of calculus developed in this book to unfold toignite the learners imagination. For permission to use copyrightedmaterial, grateful acknowledgment is made to the copyright holderson page C-1, which is hereby made part of this copyright page. Manyof the designations used by manufacturers and sellers todistinguish their products are claimed as trademarks. Where thosedesignations appear in this book, and Addison-Wesley was aware of atrademark claim, the designa- tions have been printed in initialcaps or all caps. Library of Congress Cataloging-in-PublicationData Weir, Maurice D. Thomas calculus : early transcendentals /Maurice D. Weir, Joel Hass, George B. Thomas.12th ed. p. cmIncludes index. ISBN 978-0-321-58876-0 1. CalculusTextbooks. 2.Geometry, AnalyticTextbooks. I. Hass, Joel. II. Thomas, George B.(George Brinton), 19142006. III. Title IV. Title: Calculus.QA303.2.W45 2009 515dc22 2009023070 Copyright 2010, 2006, 2001Pearson Education, Inc. All rights reserved. No part of thispublication may be reproduced, stored in a retrieval system, ortransmitted, in any form or by any means, electronic, mechanical,photocopying, recording, or otherwise, without the prior writtenpermission of the publisher. Printed in the United States ofAmerica. For information on obtaining permission for use ofmaterial in this work, please submit a written request to PearsonEducation, Inc., Rights and Contracts Department, 501 BoylstonStreet, Suite 900, Boston, MA 02116, fax your request to617-848-7047, or e-mail athttp://www.pearsoned.com/legal/permissions.htm. 1 2 3 4 5 6 7 8 910CRK12 11 10 09 ISBN-10: 0-321-58876-2 www.pearsoned.com ISBN-13:978-0-321-58876-0 7001_ThomasET_FM_SE_pi-xvi.qxd 11/3/09 3:18 PMPage ii
  3. 3. iii Preface ix 1 Functions 1 1.1 Functions and Their Graphs1 1.2 Combining Functions; Shifting and Scaling Graphs 14 1.3Trigonometric Functions 22 1.4 Graphing with Calculators andComputers 30 1.5 Exponential Functions 34 1.6 Inverse Functions andLogarithms 40 QUESTIONS TO GUIDE YOUR REVIEW 52 PRACTICE EXERCISES53 ADDITIONAL AND ADVANCED EXERCISES 55 2 Limits and Continuity 582.1 Rates of Change and Tangents to Curves 58 2.2 Limit of aFunction and Limit Laws 65 2.3 The Precise Definition of a Limit 762.4 One-Sided Limits 85 2.5 Continuity 92 2.6 Limits InvolvingInfinity; Asymptotes of Graphs 103 QUESTIONS TO GUIDE YOUR REVIEW116 PRACTICE EXERCISES 117 ADDITIONAL AND ADVANCED EXERCISES 119 3Differentiation 122 3.1 Tangents and the Derivative at a Point 1223.2 The Derivative as a Function 126 CONTENTS7001_ThomasET_FM_SE_pi-xvi.qxd 11/3/09 3:18 PM Page iii
  4. 4. 3.3 Differentiation Rules 135 3.4 The Derivative as a Rateof Change 145 3.5 Derivatives of Trigonometric Functions 155 3.6The Chain Rule 162 3.7 Implicit Differentiation 170 3.8 Derivativesof Inverse Functions and Logarithms 176 3.9 Inverse TrigonometricFunctions 186 3.10 Related Rates 192 3.11 Linearization andDifferentials 201 QUESTIONS TO GUIDE YOUR REVIEW 212 PRACTICEEXERCISES 213 ADDITIONAL AND ADVANCED EXERCISES 218 4 Applicationsof Derivatives 222 4.1 Extreme Values of Functions 222 4.2 The MeanValue Theorem 230 4.3 Monotonic Functions and the First DerivativeTest 238 4.4 Concavity and Curve Sketching 243 4.5 IndeterminateForms and LHpitals Rule 254 4.6 Applied Optimization 263 4.7Newtons Method 274 4.8 Antiderivatives 279 QUESTIONS TO GUIDE YOURREVIEW 289 PRACTICE EXERCISES 289 ADDITIONAL AND ADVANCED EXERCISES293 5 Integration 297 5.1 Area and Estimating with Finite Sums 2975.2 Sigma Notation and Limits of Finite Sums 307 5.3 The DefiniteIntegral 313 5.4 The Fundamental Theorem of Calculus 325 5.5Indefinite Integrals and the Substitution Method 336 5.6Substitution and Area Between Curves 344 QUESTIONS TO GUIDE YOURREVIEW 354 PRACTICE EXERCISES 354 ADDITIONAL AND ADVANCED EXERCISES358 6 Applications of Definite Integrals 363 6.1 Volumes UsingCross-Sections 363 6.2 Volumes Using Cylindrical Shells 374 6.3 ArcLength 382 iv Contents 7001_ThomasET_FM_SE_pi-xvi.qxd 11/3/09 3:18PM Page iv
  5. 5. 6.4 Areas of Surfaces of Revolution 388 6.5 Work and FluidForces 393 6.6 Moments and Centers of Mass 402 QUESTIONS TO GUIDEYOUR REVIEW 413 PRACTICE EXERCISES 413 ADDITIONAL AND ADVANCEDEXERCISES 415 7 Integrals and Transcendental Functions 417 7.1 TheLogarithm Defined as an Integral 417 7.2 Exponential Change andSeparable Differential Equations 427 7.3 Hyperbolic Functions 4367.4 Relative Rates of Growth 444 QUESTIONS TO GUIDE YOUR REVIEW 450PRACTICE EXERCISES 450 ADDITIONAL AND ADVANCED EXERCISES 451 8Techniques of Integration 453 8.1 Integration by Parts 454 8.2Trigonometric Integrals 462 8.3 Trigonometric Substitutions 467 8.4Integration of Rational Functions by Partial Fractions 471 8.5Integral Tables and Computer Algebra Systems 481 8.6 NumericalIntegration 486 8.7 Improper Integrals 496 QUESTIONS TO GUIDE YOURREVIEW 507 PRACTICE EXERCISES 507 ADDITIONAL AND ADVANCED EXERCISES509 9 First-Order Differential Equations 514 9.1 Solutions, SlopeFields, and Eulers Method 514 9.2 First-Order Linear Equations 5229.3 Applications 528 9.4 Graphical Solutions of AutonomousEquations 534 9.5 Systems of Equations and Phase Planes 541QUESTIONS TO GUIDE YOUR REVIEW 547 PRACTICE EXERCISES 547ADDITIONAL AND ADVANCED EXERCISES 548 10 Infinite Sequences andSeries 550 10.1 Sequences 550 10.2 Infinite Series 562 Contents v7001_ThomasET_FM_SE_pi-xvi.qxd 11/3/09 3:18 PM Page v
  6. 6. 10.3 The Integral Test 571 10.4 Comparison Tests 576 10.5The Ratio and Root Tests 581 10.6 Alternating Series, Absolute andConditional Convergence 586 10.7 Power Series 593 10.8 Taylor andMaclaurin Series 602 10.9 Convergence of Taylor Series 607 10.10The Binomial Series and Applications of Taylor Series 614 QUESTIONSTO GUIDE YOUR REVIEW 623 PRACTICE EXERCISES 623 ADDITIONAL ANDADVANCED EXERCISES 625 11 Parametric Equations and PolarCoordinates 628 11.1 Parametrizations of Plane Curves 628 11.2Calculus with Parametric Curves 636 11.3 Polar Coordinates 645 11.4Graphing in Polar Coordinates 649 11.5 Areas and Lengths in PolarCoordinates 653 11.6 Conic Sections 657 11.7 Conics in PolarCoordinates 666 QUESTIONS TO GUIDE YOUR REVIEW 672 PRACTICEEXERCISES 673 ADDITIONAL AND ADVANCED EXERCISES 675 12 Vectors andthe Geometry of Space 678 12.1 Three-Dimensional Coordinate Systems678 12.2 Vectors 683 12.3 The Dot Product 692 12.4 The CrossProduct 700 12.5 Lines and Planes in Space 706 12.6 Cylinders andQuadric Surfaces 714 QUESTIONS TO GUIDE YOUR REVIEW 719 PRACTICEEXERCISES 720 ADDITIONAL AND ADVANCED EXERCISES 722 13Vector-Valued Functions and Motion in Space 725 13.1 Curves inSpace and Their Tangents 725 13.2 Integrals of Vector Functions;Projectile Motion 733 13.3 Arc Length in Space 742 13.4 Curvatureand Normal Vectors of a Curve 746 13.5 Tangential and NormalComponents of Acceleration 752 13.6 Velocity and Acceleration inPolar Coordinates 757 vi Contents 7001_ThomasET_FM_SE_pi-xvi.qxd11/3/09 3:18 PM Page vi
  7. 7. QUESTIONS TO GUIDE YOUR REVIEW 760 PRACTICE EXERCISES 761ADDITIONAL AND ADVANCED EXERCISES 763 14 Partial Derivatives 76514.1 Functions of Several Variables 765 14.2 Limits and Continuityin Higher Dimensions 773 14.3 Partial Derivatives 782 14.4 TheChain Rule 793 14.5 Directional Derivatives and Gradient Vectors802 14.6 Tangent Planes and Differentials 809 14.7 Extreme Valuesand Saddle Points 820 14.8 Lagrange Multipliers 829 14.9 TaylorsFormula for Two Variables 838 14.10 Partial Derivatives withConstrained Variables 842 QUESTIONS TO GUIDE YOUR REVIEW 847PRACTICE EXERCISES 847 ADDITIONAL AND ADVANCED EXERCISES 851 15Multiple Integrals 854 15.1 Double and Iterated Integrals overRectangles 854 15.2 Double Integrals over General Regions 859 15.3Area by Double Integration 868 15.4 Double Integrals in Polar Form871 15.5 Triple Integrals in Rectangular Coordinates 877 15.6Moments and Centers of Mass 886 15.7 Triple Integrals inCylindrical and Spherical Coordinates 893 15.8 Substitutions inMultiple Integrals 905 QUESTIONS TO GUIDE YOUR REVIEW 914 PRACTICEEXERCISES 914 ADDITIONAL AND ADVANCED EXERCISES 916 16 Integrationin Vector Fields 919 16.1 Line Integrals 919 16.2 Vector Fields andLine Integrals: Work, Circulation, and Flux 925 16.3 PathIndependence, Conservative Fields, and Potential Functions 938 16.4Greens Theorem in the Plane 949 16.5 Surfaces and Area 961 16.6Surface Integrals 971 Contents vii 7001_ThomasET_FM_SE_pi-xvi.qxd11/3/09 3:18 PM Page vii
  8. 8. 16.7 StokesTheorem 980 16.8 The Divergence Theorem and aUnified Theory 990 QUESTIONS TO GUIDE YOUR REVIEW 1001 PRACTICEEXERCISES 1001 ADDITIONAL AND ADVANCED EXERCISES 1004 17Second-Order Differential Equations online 17.1 Second-Order LinearEquations 17.2 Nonhom*ogeneous Linear Equations 17.3 Applications17.4 Euler Equations 17.5 Power Series Solutions Appendices AP-1A.1 Real Numbers and the Real Line AP-1 A.2 Mathematical InductionAP-6 A.3 Lines, Circles, and Parabolas AP-10 A.4 Proofs of LimitTheorems AP-18 A.5 Commonly Occurring Limits AP-21 A.6 Theory ofthe Real Numbers AP-23 A.7 Complex Numbers AP-25 A.8 TheDistributive Law for Vector Cross Products AP-35 A.9 The MixedDerivative Theorem and the Increment Theorem AP-36 Answers toOdd-Numbered Exercises A-1 Index I-1 Credits C-1 A Brief Table ofIntegrals T-1 viii Contents 7001_ThomasET_FM_SE_pi-xvi.qxd 11/3/093:18 PM Page viii
  9. 9. We have significantly revised this edition of ThomasCalculus: Early Transcendentals to meet the changing needs oftodays instructors and students. The result is a book with moreexamples, more mid-level exercises, more figures, better conceptualflow, and increased clarity and precision. As with previouseditions, this new edition provides a modern intro- duction tocalculus that supports conceptual understanding but retains theessential ele- ments of a traditional course. These enhancementsare closely tied to an expanded version of MyMathLab for this text(discussed further on), providing additional support for stu- dentsand flexibility for instructors. In this twelfth edition earlytranscendentals version, we introduce the basic transcen- dentalfunctions in Chapter 1. After reviewing the basic trigonometricfunctions, we pres- ent the family of exponential functions usingan algebraic and graphical approach, with the natural exponentialdescribed as a particular member of this family. Logarithms arethen defined as the inverse functions of the exponentials, and wealso discuss briefly the inverse trigonometric functions. We fullyincorporate these functions throughout our de- velopments oflimits, derivatives, and integrals in the next five chapters of thebook, in- cluding the examples and exercises. This approach givesstudents the opportunity to work early with exponential andlogarithmic functions in combinations with polynomials, ra- tionaland algebraic functions, and trigonometric functions as they learnthe concepts, oper- ations, and applications of single-variablecalculus. Later, in Chapter 7, we revisit the defi- nition oftranscendental functions, now giving a more rigorous presentation.Here we define the natural logarithm function as an integral withthe natural exponential as its inverse. Many of our students wereexposed to the terminology and computational aspects of calculusduring high school. Despite this familiarity, students algebra andtrigonometry skills often hinder their success in the collegecalculus sequence. With this text, we have sought to balance thestudents prior experience with calculus with the algebraic skillde- velopment they may still need, all without undermining orderailing their confidence. We have taken care to provide enoughreview material, fully stepped-out solutions, and exer- cises tosupport complete understanding for students of all levels. Weencourage students to think beyond memorizing formulas and togeneralize con- cepts as they are introduced. Our hope is thatafter taking calculus, students will be confi- dent in theirproblem-solving and reasoning abilities. Mastering a beautifulsubject with practical applications to the world is its own reward,but the real gift is the ability to think and generalize. We intendthis book to provide support and encouragement for both. Changesfor the Twelfth Edition CONTENT In preparing this edition we havemaintained the basic structure of the Table of Contents from theeleventh edition, yet we have paid attention to requests by currentusers and reviewers to postpone the introduction of parametricequations until we present polar coordinates. We have made numerousrevisions to most of the chapters, detailed as follows: ix PREFACE7001_ThomasET_FM_SE_pi-xvi.qxd 11/3/09 3:18 PM Page ix
  10. 10. Functions We condensed this chapter to focus on reviewingfunction concepts and in- troducing the transcendental functions.Prerequisite material covering real numbers, in- tervals,increments, straight lines, distances, circles, and parabolas ispresented in Ap- pendices 13. Limits To improve the flow of thischapter, we combined the ideas of limits involving infinity andtheir associations with asymptotes to the graphs of functions,placing them together in the final section of Chapter 3.Differentiation While we use rates of change and tangents to curvesas motivation for studying the limit concept, we now merge thederivative concept into a single chapter. We reorganized andincreased the number of related rates examples, and we added newexamples and exercises on graphing rational functions. LHpitalsRule is presented as an application section, consistent with ourearly coverage of the transcendental functions. Antiderivatives andIntegration We maintain the organization of the eleventh edition inplacing antiderivatives as the final topic of Chapter 4, coveringapplications of derivatives. Our focus is on recovering a functionfrom its derivative as the solution to the simplest type offirst-order differential equation. Integrals, as limits of Riemannsums, motivated primarily by the problem of finding the areas ofgeneral regions with curved boundaries, are a new topic forming thesubstance of Chapter 5. After carefully developing the integralconcept, we turn our attention to its evaluation and connection toantiderivatives captured in the Fundamental Theorem of Calculus.The ensuing ap- plications then define the various geometric ideasof area, volume, lengths of paths, and centroids, all as limits ofRiemann sums giving definite integrals, which can be evalu- ated byfinding an antiderivative of the integrand. We return later to thetopic of solving more complicated first-order differentialequations. Differential Equations Some universities prefer thatthis subject be treated in a course separate from calculus.Although we do cover solutions to separable differential equationswhen treating exponential growth and decay applications in Chapter7 on integrals and transcendental functions, we organize the bulkof our material into two chapters (which may be omitted for thecalculus sequence). We give an introductory treatment of first-order differential equations in Chapter 9, including a new sectionon systems and phase planes, with applications to thecompetitive-hunter and predator-prey models. We present anintroduction to second-order differential equations in Chapter 17,which is in- cluded in MyMathLab as well as the ThomasCalculus:Early Transcendentals Web site, www.pearsonhighered.com/thomas.Series We retain the organizational structure and content of theeleventh edition for the topics of sequences and series. We haveadded several new figures and exercises to the various sections,and we revised some of the proofs related to convergence of powerse- ries in order to improve the accessibility of the material forstudents. The request stated by one of our users as, anything youcan do to make this material easier for students will be welcomedby our faculty, drove our thinking for revisions to this chapter.Parametric Equations Several users requested that we move thistopic into Chapter 11, where we also cover polar coordinates andconic sections. We have done this, realiz- ing that manydepartments choose to cover these topics at the beginning ofCalculus III, in preparation for their coverage of vectors andmultivariable calculus. Vector-Valued Functions We streamlined thetopics in this chapter to place more em- phasis on the conceptualideas supporting the later material on partial derivatives, thegradient vector, and line integrals. We condensed the discussionsof the Frenet frame and Keplers three laws of planetary motion.Multivariable Calculus We have further enhanced the art in thesechapters, and we have added many new figures, examples, andexercises. We reorganized the opening material on double integrals,and we combined the applications of double and triple integrals tomasses and moments into a single section covering both two- andthree- dimensional cases. This reorganization allows for betterflow of the key mathematical concepts, together with theirproperties and computational aspects. As with the x Preface7001_ThomasET_FM_SE_pi-xvi.qxd 11/3/09 3:18 PM Page x
  11. 11. eleventh edition, we continue to make the connections ofmultivariable ideas with their single-variable analogues studiedearlier in the book. Vector Fields We devoted considerable effortto improving the clarity and mathemati- cal precision of ourtreatment of vector integral calculus, including many additionalex- amples, figures, and exercises. Important theorems and resultsare stated more clearly and completely, together with enhancedexplanations of their hypotheses and mathe- matical consequences.The area of a surface is now organized into a single section, andsurfaces defined implicitly or explicitly are treated as specialcases of the more general parametric representation. Surfaceintegrals and their applications then follow as a sep- aratesection. Stokes Theorem and the Divergence Theorem are stillpresented as gen- eralizations of Greens Theorem to threedimensions. EXERCISES AND EXAMPLES We know that the exercises andexamples are critical com- ponents in learning calculus. Because ofthis importance, we have updated, improved, and increased thenumber of exercises in nearly every section of the book. There areover 700 new exercises in this edition. We continue ourorganization and grouping of exercises by topic as in earliereditions, progressing from computational problems to applied andtheo- retical problems. Exercises requiring the use of computersoftware systems (such as Maple or Mathematica ) are placed at theend of each exercise section, labeled Com- puter Explorations. Mostof the applied exercises have a subheading to indicate the kind ofapplication addressed in the problem. Many sections include newexamples to clarify or deepen the meaning of the topic be- ingdiscussed and to help students understand its mathematicalconsequences or applica- tions to science and engineering. At thesame time, we have removed examples that were a repetition ofmaterial already presented. ART Because of their importance tolearning calculus, we have continued to improve exist- ing figuresin Thomas Calculus: Early Transcendentals, and we have created asignificant number of new ones. We continue to use colorconsistently and pedagogically to enhance the conceptual idea thatis being illustrated. We have also taken a fresh look at all of thefigure captions, paying considerable attention to clarity andprecision in short statements. FIGURE 2.50, page 104 The geometricFIGURE 16.9, page 926 A surface in a explanation of a finite limitas . space occupied by a moving fluid. MYMATHLAB AND MATHXL Theincreasing use of and demand for online homework systems has driventhe changes to MyMathLab and MathXL for Thomas Calculus: x : ; q zx y x y No matter what positive number is, the graph enters thisband at x and stays. 1 y M 1 N 1 y 0 No matter what positive numberis, the graph enters this band at x and stays. 1 y 1 x Preface xi7001_ThomasET_FM_SE_pi-xvi.qxd 11/3/09 3:18 PM Page xi
  12. 12. Early Transcendentals. The MyMathLab course now includessignificantly more exer- cises of all types. Continuing FeaturesRIGOR The level of rigor is consistent with that of earliereditions. We continue to distin- guish between formal and informaldiscussions and to point out their differences. We think startingwith a more intuitive, less formal, approach helps studentsunderstand a new or diffi- cult concept so they can then appreciateits full mathematical precision and outcomes. We pay attention todefining ideas carefully and to proving theorems appropriate forcalculus students, while mentioning deeper or subtler issues theywould study in a more advanced course. Our organization anddistinctions between informal and formal discussions give theinstructor a de- gree of flexibility in the amount and depth ofcoverage of the various topics. For example, while we do not provethe Intermediate Value Theorem or the Extreme Value Theorem forcontinu- ous functions on , we do state these theorems precisely,illustrate their meanings in numerous examples, and use them toprove other important results. Furthermore, for those in- structorswho desire greater depth of coverage, in Appendix 6 we discuss thereliance of the validity of these theorems on the completeness ofthe real numbers. WRITING EXERCISES Writing exercises placedthroughout the text ask students to ex- plore and explain a varietyof calculus concepts and applications. In addition, the end of eachchapter contains a list of questions for students to review andsummarize what they have learned. Many of these exercises make goodwriting assignments. END-OF-CHAPTER REVIEWS AND PROJECTS Inaddition to problems appearing after each section, each chapterculminates with review questions, practice exercises covering theentire chapter, and a series of Additional and Advanced Exercisesserving to include more challenging or synthesizing problems. Mostchapters also include descriptions of several TechnologyApplication Projects that can be worked by individual students orgroups of students over a longer period of time. These projectsrequire the use of a com- puter running Mathematica or Maple andadditional material that is available over the Internet atwww.pearsonhighered.com/thomas and in MyMathLab. WRITING ANDAPPLICATIONS As always, this text continues to be easy to read,conversa- tional, and mathematically rich. Each new topic ismotivated by clear, easy-to-understand examples and is thenreinforced by its application to real-world problems of immediatein- terest to students. A hallmark of this book has been theapplication of calculus to science and engineering. These appliedproblems have been updated, improved, and extended con- tinuallyover the last several editions. TECHNOLOGY In a course using thetext, technology can be incorporated according to the taste of theinstructor. Each section contains exercises requiring the use oftechnology; these are marked with a if suitable for calculator orcomputer use, or they are labeled Computer Explorations if acomputer algebra system (CAS, such as Maple or Mathe- matica) isrequired. Text Versions THOMAS CALCULUS: EARLY TRANSCENDENTALS,Twelfth Edition Complete (Chapters 116), ISBN 0-321-58876-2 |978-0-321-58876-0 Single Variable Calculus (Chapters 111),0-321-62883-7 | 978-0-321-62883-1 Multivariable Calculus (Chapters1016), ISBN 0-321-64369-0 | 978-0-321-64369-8 T a x b xii Preface7001_ThomasET_FM_SE_pi-xvi.qxd 4/7/10 10:13 AM Page xii
  13. 13. The early transcendentals version of Thomas Calculusintroduces and integrates transcen- dental functions (such asinverse trigonometric, exponential, and logarithmic functions) intothe exposition, examples, and exercises of the early chaptersalongside the algebraic functions. The Multivariable book forThomas Calculus: Early Transcendentals is the same text asThomasCalculus, Multivariable. THOMAS CALCULUS, Twelfth EditionComplete (Chapters 116), ISBN 0-321-58799-5 | 978-0-321-58799-2Single Variable Calculus (Chapters 111), ISBN 0-321-63742-9 |978-0-321-63742-0 Multivariable Calculus (Chapters 1016), ISBN0-321-64369-0 | 978-0-321-64369-8 Instructors EditionsThomasCalculus: Early Transcendentals, ISBN 0-321-62718-0 |978-0-321-62718-6 ThomasCalculus, ISBN 0-321-60075-4 |978-0-321-60075-2 In addition to including all of the answerspresent in the student editions, the Instructors Editions includeeven-numbered answers for Chapters 16. University Calculus (EarlyTranscendentals) University Calculus: Alternate Edition (LateTranscendentals) University Calculus: Elements with EarlyTranscendentals The University Calculus texts are based on ThomasCalculus and feature a streamlined presentation of the contents ofthe calculus course. For more information about these titles, visitwww.pearsonhighered.com. Print Supplements INSTRUCTORS SOLUTIONSMANUAL Single Variable Calculus (Chapters 111), ISBN 0-321-62717-2| 978-0-321-62717-9 Multivariable Calculus (Chapters 1016), ISBN0-321-60072-X | 978-0-321-60072-1 The Instructors Solutions Manualby William Ardis, Collin County Community College, containscomplete worked-out solutions to all of the exercises in ThomasCalculus: Early Transcendentals. STUDENTS SOLUTIONS MANUAL SingleVariable Calculus (Chapters 111), ISBN 0-321-65692-X |978-0-321-65692-6 Multivariable Calculus (Chapters 1016), ISBN0-321-60071-1 | 978-0-321-60071-4 The Students Solutions Manual byWilliam Ardis, Collin County Community College, is designed for thestudent and contains carefully worked-out solutions to all the odd-numbered exercises in ThomasCalculus: Early Transcendentals.JUST-IN-TIME ALGEBRA AND TRIGONOMETRY FOR EARLY TRANSCENDENTALSCALCULUS, Third Edition ISBN 0-321-32050-6 | 978-0-321-32050-6Sharp algebra and trigonometry skills are critical to masteringcalculus, and Just-in-Time Algebra and Trigonometry for EarlyTranscendentals Calculus by Guntram Mueller and Ronald I. Brent isdesigned to bolster these skills while students study calculus. Asstu- dents make their way through calculus, this text is with themevery step of the way, show- ing them the necessary algebra ortrigonometry topics and pointing out potential problem spots. Theeasy-to-use table of contents has algebra and trigonometry topicsarranged in the order in which students will need them as theystudy calculus. CALCULUS REVIEW CARDS The Calculus Review Cards(one for Single Variable and another for Multivariable) are astudent resource containing important formulas, functions,definitions, and theorems that correspond precisely to the ThomasCalculus series. These cards can work as a reference for completinghomework assignments or as an aid in studying, and are availablebundled with a new text. Contact your Pearson sales representativefor more information. Preface xiii 7001_ThomasET_FM_SE_pi-xvi.qxd4/7/10 10:13 AM Page xiii
  14. 14. Media and Online Supplements TECHNOLOGY RESOURCE MANUALSMaple Manual by James Stapleton, North Carolina State UniversityMathematica Manual by Marie Vanisko, Carroll College TI-GraphingCalculator Manual by Elaine McDonald-Newman, Sonoma StateUniversity These manuals cover Maple 13, Mathematica 7, and theTI-83 Plus/TI-84 Plus and TI-89, respectively. Each manual providesdetailed guidance for integrating a specific software package orgraphing calculator throughout the course, including syntax andcommands. These manuals are available to qualified instructorsthrough the Thomas Calculus: Early Transcendentals Web site,www.pearsonhighered.com/thomas, and MyMathLab. WEB SITEwww.pearsonhighered.com/thomas The Thomas Calculus: EarlyTranscendentals Web site contains the chapter on Second- OrderDifferential Equations, including odd-numbered answers, andprovides the expanded historical biographies and essays referencedin the text. Also available is a collection of Maple andMathematica modules, the Technology Resource Manuals, and theTechnologyApplica- tion Projects, which can be used as projects byindividual students or groups of students. MyMathLab Online Course(access code required) MyMathLab is a text-specific, easilycustomizable online course that integrates interactive multimediainstruction with textbook content. MyMathLab gives you the toolsyou need to deliver all or a portion of your course online, whetheryour students are in a lab setting or working from home.Interactive homework exercises, correlated to your textbook at theobjective level, are algorithmically generated for unlimitedpractice and mastery. Most exercises are free- response and provideguided solutions, sample problems, and learning aids for extrahelp. Getting Ready chapter includes hundreds of exercises thataddress prerequisite skills in algebra and trigonometry. Eachstudent can receive remediation for just those skills he or sheneeds help with. Personalized Study Plan, generated when studentscomplete a test or quiz, indicates which topics have been masteredand links to tutorial exercises for topics students have notmastered. Multimedia learning aids, such as video lectures, Javaapplets, animations, and a complete multimedia textbook, helpstudents independently improve their understand- ing andperformance. Assessment Manager lets you create online homework,quizzes, and tests that are automatically graded. Select just theright mix of questions from the MyMathLab exer- cise bank andinstructor-created custom exercises. Gradebook, designedspecifically for mathematics and statistics, automatically tracksstudents results and gives you control over how to calculate finalgrades. You can also add offline (paper-and-pencil) grades to thegradebook. MathXL Exercise Builder allows you to create static andalgorithmic exercises for your online assignments. You can use thelibrary of sample exercises as an easy starting point. PearsonTutor Center (www.pearsontutorservices.com) access is automaticallyin- cluded with MyMathLab. The Tutor Center is staffed by qualifiedmath instructors who provide textbook-specific tutoring forstudents via toll-free phone, fax, email, and in- teractive Websessions. xiv Preface 7001_ThomasET_FM_SE_pi-xvi.qxd 11/3/09 3:18PM Page xiv
  15. 15. MyMathLab is powered by CourseCompass, Pearson Educationsonline teaching and learning environment, and by MathXL, our onlinehomework, tutorial, and assessment system. MyMathLab is availableto qualified adopters. For more information, visitwww.mymathlab.com or contact your Pearson sales representative.Video Lectures with Optional Captioning The Video Lectures withOptional Captioning feature an engaging team of mathematics in-structors who present comprehensive coverage of topics in the text.The lecturers pres- entations include examples and exercises fromthe text and support an approach that em- phasizes visualizationand problem solving. Available only through MyMathLab and MathXL.MathXL Online Course (access code required) MathXL is an onlinehomework, tutorial, and assessment system that accompanies Pearsonstextbooks in mathematics or statistics. Interactive homeworkexercises, correlated to your textbook at the objective level, arealgorithmically generated for unlimited practice and mastery. Mostexercises are free- response and provide guided solutions, sampleproblems, and learning aids for extra help. Getting Ready chapterincludes hundreds of exercises that address prerequisite skills inalgebra and trigonometry. Each student can receive remediation forjust those skills he or she needs help with. Personalized StudyPlan, generated when students complete a test or quiz, indicateswhich topics have been mastered and links to tutorial exercises fortopics students have not mastered. Multimedia learning aids, suchas video lectures, Java applets, and animations, help studentsindependently improve their understanding and performance.Gradebook, designed specifically for mathematics and statistics,automatically tracks students results and gives you control overhow to calculate final grades. MathXL Exercise Builder allows youto create static and algorithmic exercises for your onlineassignments.You can use the library of sample exercises as an easystarting point. Assessment Manager lets you create online homework,quizzes, and tests that are automatically graded. Select just theright mix of questions from the MathXL exercise bank, orinstructor-created custom exercises. MathXL is available toqualified adopters. For more information, visit our Web site atwww.mathxl.com, or contact your Pearson sales representative.TestGen TestGen (www.pearsonhighered.com/testgen) enablesinstructors to build, edit, print, and administer tests using acomputerized bank of questions developed to cover all the ob-jectives of the text. TestGen is algorithmically based, allowinginstructors to create multi- ple but equivalent versions of thesame question or test with the click of a button. Instruc- tors canalso modify test bank questions or add new questions. Tests can beprinted or administered online. The software and testbank areavailable for download from Pearson Educations online catalog.PowerPoint Lecture Slides These classroom presentation slides aregeared specifically to the sequence and philosophy of theThomasCalculus series. Key graphics from the book are included tohelp bring the concepts alive in the classroom.These files areavailable to qualified instructors through the Pearson InstructorResource Center, www.pearsonhighered/irc, and MyMathLab. Preface xv7001_ThomasET_FM_SE_pi-xvi.qxd 11/3/09 3:18 PM Page xv
  16. 16. Acknowledgments We would like to express our thanks to thepeople who made many valuable contributions to this edition as itdeveloped through its various stages: Accuracy Checkers BlaiseDeSesa Paul Lorczak Kathleen Pellissier Lauri Semarne Sarah StreettHolly Zullo Reviewers for the Twelfth Edition Meighan Dillon,Southern Polytechnic State University Anne Dougherty, University ofColorado Said Fariabi, San Antonio College Klaus Fischer, GeorgeMason University Tim Flood, Pittsburg State University Rick Ford,California State UniversityChico Robert Gardner, East TennesseeState University Christopher Heil, Georgia Institute of TechnologyJoshua Brandon Holden, Rose-Hulman Institute of TechnologyAlexander Hulpke, Colorado State University Jacqueline Jensen, SamHouston State University Jennifer M. Johnson, Princeton UniversityHideaki Kaneko, Old Dominion University Przemo Kranz, University ofMississippi Xin Li, University of Central Florida Maura Mast,University of MassachusettsBoston Val Mohanakumar, HillsboroughCommunity CollegeDale Mabry Campus Aaron Montgomery, CentralWashington University Christopher M. Pavone, California StateUniversity at Chico Cynthia Piez, University of Idaho BrookeQuinlan, Hillsborough Community CollegeDale Mabry Campus Rebecca A.Segal, Virginia Commonwealth University Andrew V. Sills, GeorgiaSouthern University Alex Smith, University of WisconsinEau ClaireMark A. Smith, Miami University Donald Solomon, University ofWisconsinMilwaukee John Sullivan, Black Hawk College Maria Terrell,Cornell University Blake Thornton, Washington University in St.Louis David Walnut, George Mason University Adrian Wilson,University of Montevallo Bobby Winters, Pittsburg State UniversityDennis Wortman, University of MassachusettsBoston xvi Preface7001_ThomasET_FM_SE_pi-xvi.qxd 11/3/09 3:18 PM Page xvi
  17. 17. 1 1 FUNCTIONS OVERVIEW Functions are fundamental to thestudy of calculus. In this chapter we review what functions are andhow they are pictured as graphs, how they are combined and trans-formed, and ways they can be classified. We review thetrigonometric functions, and we discuss misrepresentations that canoccur when using calculators and computers to obtain a functionsgraph. We also discuss inverse, exponential, and logarithmicfunctions. The real number system, Cartesian coordinates, straightlines, parabolas, and circles are re- viewed in the Appendices. 1.1Functions and Their Graphs Functions are a tool for describing thereal world in mathematical terms. A function can be represented byan equation, a graph, a numerical table, or a verbal description;we will use all four representations throughout this book. Thissection reviews these function ideas. Functions; Domain and RangeThe temperature at which water boils depends on the elevation abovesea level (the boiling point drops as you ascend). The interestpaid on a cash investment depends on the length of time theinvestment is held. The area of a circle depends on the radius ofthe circle. The distance an object travels at constant speed alonga straight-line path depends on the elapsed time. In each case, thevalue of one variable quantity, say y, depends on the value ofanother variable quantity, which we might call x. We say that y isa function of x and write this symbolically as In this notation,the symbol represents the function, the letter x is the independentvari- able representing the input value of , and y is the dependentvariable or output value of at x. y = (x) (y equals of x). FPODEFINITION A function from a set D to a set Y is a rule thatassigns a unique (single) element to each element x H D.sxd H Y Theset D of all possible input values is called the domain of thefunction. The set of all values of (x) as x varies throughout D iscalled the range of the function. The range may not include everyelement in the set Y. The domain and range of a function can be anysets of objects, but often in calculus they are sets of realnumbers interpreted as points of a coordinate line. (In Chapters1316, we will encounter functions for which the elements of thesets are points in the coordinate plane or in space.)7001_AWLThomas_ch01p001-057.qxd 10/1/09 2:23 PM Page 1
  18. 18. Often a function is given by a formula that describes howto calculate the output value from the input variable. Forinstance, the equation is a rule that calculates the area A of acircle from its radius r (so r, interpreted as a length, can onlybe positive in this formula). When we define a function with aformula and the domain is not stated explicitly or restricted bycontext, the domain is assumed to be the largest set of realx-values for which the formula gives real y-values, the so-callednatural domain. If we want to restrict the domain in some way, wemust say so. The domain of is the en- tire set of real numbers. Torestrict the domain of the function to, say, positive values of x,we would write Changing the domain to which we apply a formulausually changes the range as well. The range of is The range of isthe set of all numbers ob- tained by squaring numbers greater thanor equal to 2. In set notation (see Appendix 1), the range is or orWhen the range of a function is a set of real numbers, the functionis said to be real- valued. The domains and ranges of manyreal-valued functions of a real variable are inter- vals orcombinations of intervals. The intervals may be open, closed, orhalf open, and may be finite or infinite. The range of a functionis not always easy to find. A function is like a machine thatproduces an output value (x) in its range whenever we feed it aninput value x from its domain (Figure 1.1).The function keys on acalculator give an example of a function as a machine. Forinstance, the key on a calculator gives an out- put value (thesquare root) whenever you enter a nonnegative number x and pressthe key. A function can also be pictured as an arrow diagram(Figure 1.2). Each arrow associ- ates an element of the domain Dwith a unique or single element in the set Y. In Figure 1.2, thearrows indicate that (a) is associated with a, (x) is associatedwith x, and so on. Notice that a function can have the same valueat two different input elements in the domain (as occurs with (a)in Figure 1.2), but each input element x is assigned a singleoutput value (x). EXAMPLE 1 Lets verify the natural domains andassociated ranges of some simple functions. The domains in eachcase are the values of x for which the formula makes sense.Function Domain (x) Range ( y) [0, 1] Solution The formula gives areal y-value for any real number x, so the domain is The range ofis because the square of any real number is nonnegative and everynonnegative number y is the square of its own square root, for Theformula gives a real y-value for every x except For consistency inthe rules of arithmetic, we cannot divide any number by zero. Therange of the set of reciprocals of all nonzero real numbers, is theset of all nonzero real numbers, since That is, for the number isthe input assigned to the output value y. The formula gives a realy-value only if The range of is because every nonnegative number issome numbers square root (namely, it is the square root of its ownsquare). In the quantity cannot be negative. That is, or Theformula gives real y-values for all The range of is the set of allnonnegative numbers. [0, qd,14 - xx 4.x 4. 4 - x 0,4 - xy = 14 - x,[0, qd y = 1xx 0.y = 1x x = 1>yy Z 0y = 1>(1>y). y =1>x, x = 0.y = 1>x y 0.y = A 2yB2 [0, qdy = x2 s- q, qd. y =x2 [-1, 1]y = 21 - x2 [0, qds- q, 4]y = 24 - x [0, qd[0, qdy = 2xs- q, 0d s0, qds - q, 0d s0, qdy = 1>x [0, qds - q, qdy = x2 2x2x [4, qd.5y y 465x2 x 26 y = x2 , x 2,[0, qd.y = x2 y = x2 , x 70. y = x2 y = sxd A = pr2 2 Chapter 1: Functions Input (domain)Output (range) x f(x)f FIGURE 1.1 A diagram showing a function as akind of machine. x a f(a) f(x) D domain set Y set containing therange FIGURE 1.2 A function from a set D to a set Y assigns aunique element of Y to each element in D.7001_AWLThomas_ch01p001-057.qxd 10/1/09 2:23 PM Page 2
  19. 19. 1.1 Functions and Their Graphs 3 The formula gives a realy-value for every x in the closed interval from to 1. Outside thisdomain, is negative and its square root is not a real number. Thevalues of vary from 0 to 1 on the given domain, and the squareroots of these values do the same. The range of is [0, 1]. Graphsof Functions If is a function with domain D, its graph consists ofthe points in the Cartesian plane whose coordinates are theinput-output pairs for . In set notation, the graph is The graph ofthe function is the set of points with coordinates (x, y) for whichIts graph is the straight line sketched in Figure 1.3. The graph ofa function is a useful picture of its behavior. If (x, y) is apoint on the graph, then is the height of the graph above the pointx. The height may be posi- tive or negative, depending on the signof (Figure 1.4).sxd y = sxd y = x + 2. sxd = x + 2 5sx, sxdd x HD6. 21 - x2 1 - x2 1 - x2 -1 y = 21 - x2 x y 2 0 2 y x 2 FIGURE 1.3The graph of is the set of points (x, y) for which y has the valuex + 2. sxd = x + 2 y x 0 1 2 x f(x) (x, y) f(1) f(2) FIGURE 1.4 If(x, y) lies on the graph of , then the value is the height of thegraph above the point x (or below x if (x) is negative). y = sxdEXAMPLE 2 Graph the function over the interval Solution Make atable of xy-pairs that satisfy the equation . Plot the points (x,y) whose coordinates appear in the table, and draw a smooth curve(labeled with its equation) through the plotted points (see Figure1.5). How do we know that the graph of doesnt look like one ofthese curves?y = x2 y = x2 [-2, 2].y = x2 x 4 1 0 0 1 1 2 4 9 4 3 2-1 -2 y x 2 y x2 ? x y y x2 ? x y 0 1 212 1 2 3 4 (2, 4) (1, 1) (1,1) (2, 4) 3 2 9 4 , x y y x2 FIGURE 1.5 Graph of the function inExample 2. 7001_AWLThomas_ch01p001-057.qxd 10/1/09 2:23 PM Page3
  20. 20. 4 Chapter 1: Functions To find out, we could plot morepoints. But how would we then connect them? The basic questionstill remains: How do we know for sure what the graph looks likebe- tween the points we plot? Calculus answers this question, as wewill see in Chapter 4. Meanwhile we will have to settle forplotting points and connecting them as best we can. Representing aFunction Numerically We have seen how a function may be representedalgebraically by a formula (the area function) and visually by agraph (Example 2). Another way to represent a function isnumerically, through a table of values. Numerical representationsare often used by engi- neers and scientists. From an appropriatetable of values, a graph of the function can be obtained using themethod illustrated in Example 2, possibly with the aid of acomputer. The graph consisting of only the points in the table iscalled a scatterplot. EXAMPLE 3 Musical notes are pressure waves inthe air. The data in Table 1.1 give recorded pressure displacementversus time in seconds of a musical note produced by a tuning fork.The table provides a representation of the pressure function overtime. If we first make a scatterplot and then connect approximatelythe data points (t, p) from the table, we obtain the graph shown inFigure 1.6. The Vertical Line Test for a Function Not every curvein the coordinate plane can be the graph of a function. A functioncan have only one value for each x in its domain, so no verticalline can intersect the graph of a function more than once. If a isin the domain of the function , then the vertical line willintersect the graph of at the single point . A circle cannot be thegraph of a function since some vertical lines intersect the circletwice. The circle in Figure 1.7a, however, does contain the graphsof two functions of x: the upper semicircle defined by the functionand the lower semicircle defined by the function (Figures 1.7b and1.7c).g(x) = - 21 - x2 (x) = 21 - x2 (a, (a))x = a (x) TABLE 1.1Tuning fork data Time Pressure Time Pressure 0.00091 0.00362 0.2170.00108 0.200 0.00379 0.480 0.00125 0.480 0.00398 0.681 0.001440.693 0.00416 0.810 0.00162 0.816 0.00435 0.827 0.00180 0.8440.00453 0.749 0.00198 0.771 0.00471 0.581 0.00216 0.603 0.004890.346 0.00234 0.368 0.00507 0.077 0.00253 0.099 0.00525 0.002710.00543 0.00289 0.00562 0.00307 0.00579 0.00325 0.00598 0.00344-0.041 -0.035-0.248 -0.248-0.348 -0.354-0.309 -0.320-0.141 -0.164-0.080 0.6 0.4 0.2 0.2 0.4 0.6 0.8 1.0 t (sec) p (pressure) 0.0010.002 0.004 0.0060.003 0.005 Data FIGURE 1.6 A smooth curve throughthe plotted points gives a graph of the pressure functionrepresented by Table 1.1 (Example 3).7001_AWLThomas_ch01p001-057.qxd 10/1/09 2:23 PM Page 4
  21. 21. 1.1 Functions and Their Graphs 5 2 1 0 1 2 1 2 x y y x y x2y 1 y f(x) FIGURE 1.9 To graph the function shown here, we applydifferent formulas to different parts of its domain (Example 4). y= sxd x y x y x y x y 3 2 1 0 1 2 3 1 2 3 FIGURE 1.8 The absolutevalue function has domain and range [0, qd. s- q, qd 1 10 x y (a)x2 y2 1 1 10 x y 1 1 0 x y (b) y 1 x2 (c) y 1 x2 FIGURE 1.7 (a) Thecircle is not the graph of a function; it fails the vertical linetest. (b) The upper semicircle is the graph of a function (c) Thelower semicircle is the graph of a function gsxd = - 21 - x2 . sxd= 21 - x2 . Piecewise-Defined Functions Sometimes a function isdescribed by using different formulas on different parts of itsdomain. One example is the absolute value function whose graph isgiven in Figure 1.8. The right-hand side of the equation means thatthe function equals x if , and equals if Here are some otherexamples. EXAMPLE 4 The function is defined on the entire real linebut has values given by different formulas depending on theposition of x. The values of are given by when when and when Thefunction, however, is just one function whose domain is the entireset of real numbers (Figure 1.9). EXAMPLE 5 The function whosevalue at any number x is the greatest integer less than or equal tox is called the greatest integer function or the integer floorfunction. It is denoted . Figure 1.10 shows the graph. Observe thatEXAMPLE 6 The function whose value at any number x is the smallestinteger greater than or equal to x is called the least integerfunction or the integer ceiling function. It is denoted Figure 1.11shows the graph. For positive values of x, this function mightrepresent, for example, the cost of parking x hours in a parkinglot which charges $1 for each hour or part of an hour.
  22. 22. The names even and odd come from powers of x. If y is aneven power of x, as in or it is an even function of x because andIf y is an odd power of x, as in or it is an odd function of xbecause and The graph of an even function is symmetric about they-axis. Since a point (x, y) lies on the graph if and only if thepoint lies on the graph (Figure 1.12a). A reflection across they-axis leaves the graph unchanged. The graph of an odd function issymmetric about the origin. Since a point (x, y) lies on the graphif and only if the point lies on the graph (Figure 1.12b).Equivalently, a graph is symmetric about the origin if a rotationof 180 about the origin leaves the graph unchanged. Notice that thedefinitions imply that both x and must be in the domain of .EXAMPLE 8 Even function: for all x; symmetry about y-axis. Evenfunction: for all x; symmetry about y-axis (Figure 1.13a). Oddfunction: for all x; symmetry about the origin. Not odd: but Thetwo are not equal. Not even: for all (Figure 1.13b).x Z 0s -xd + 1Z x + 1 -sxd = -x - 1.s-xd = -x + 1,sxd = x + 1 s-xd = -xsxd = xs-xd2 + 1 = x2 + 1sxd = x2 + 1 s-xd2 = x2 sxd = x2 -x s-x, -yd s-xd= -sxd, s -x, yd s-xd = sxd, s-xd3 = -x3 . s-xd1 = -xy = x3 ,y = xs-xd4 = x4 .s-xd2 = x2 y = x4 ,y = x2 Increasing and DecreasingFunctions If the graph of a function climbs or rises as you movefrom left to right, we say that the function is increasing. If thegraph descends or falls as you move from left to right, thefunction is decreasing. 6 Chapter 1: Functions DEFINITIONS Let be afunction defined on an interval I and let and be any two points inI. 1. If whenever then is said to be increasing on I. 2. Ifwhenever then is said to be decreasing on I.x1 6 x2,sx2d 6 sx1d x16 x2,sx2) 7 sx1d x2x1 x y 112 2 3 2 1 1 2 3 y x y x FIGURE 1.11 Thegraph of the least integer function lies on or above the line so itprovides an integer ceiling for x (Example 6). y = x, y =
  23. 23. 1.1 Functions and Their Graphs 7 (a) (b) x y 0 1 y x2 1 yx2 x y 01 1 y x 1 y x FIGURE 1.13 (a) When we add the constant term1 to the function the resulting function is still even and itsgraph is still symmetric about the y-axis. (b) When we add theconstant term 1 to the function the resulting function is no longerodd. The symmetry about the origin is lost (Example 8). y = x + 1y= x, y = x2 + 1y = x2 , Common Functions A variety of importanttypes of functions are frequently encountered in calculus. We iden-tify and briefly describe them here. Linear Functions A function ofthe form for constants m and b, is called a linear function. Figure1.14a shows an array of lines where so these lines pass through theorigin. The function where and is called the identity function.Constant functions result when the slope (Figure 1.14b). A linearfunction with positive slope whose graph passes through the originis called a proportionality relationship. m = 0 b = 0m = 1sxd = x b= 0,sxd = mx sxd = mx + b, x y 0 1 2 1 2 y 3 2 (b) FIGURE 1.14 (a)Lines through the origin with slope m. (b) A constant function withslope m = 0. 0 x y m 3 m 2 m 1m 1 y 3x y x y 2x y x y x 1 2 m 1 2(a) DEFINITION Two variables y and x are proportional (to oneanother) if one is always a constant multiple of the other; thatis, if for some nonzero constant k. y = kx If the variable y isproportional to the reciprocal then sometimes it is said that y isinversely proportional to x (because is the multiplicative inverseof x). Power Functions A function where a is a constant, is calleda power func- tion. There are several important cases to consider.sxd = xa , 1>x 1>x, 7001_AWLThomas_ch01p001-057.qxd 10/1/092:23 PM Page 7
  24. 24. (b) The graphs of the functions and are shown in Figure1.16. Both functions are defined for all (you can never divide byzero). The graph of is the hyperbola , which approaches thecoordinate axes far from the origin. The graph of also approachesthe coordinate axes. The graph of the function is symmetric aboutthe origin; is decreasing on the intervals and . The graph of thefunction g is symmetric about the y-axis; g is increasing on anddecreasing on .s0, q)s- q, 0) s0, q) s - q, 0) y = 1>x2 xy = 1y= 1>x x Z 0 gsxd = x-2 = 1>x2 sxd = x-1 = 1>x a = -1 or a= -2. 8 Chapter 1: Functions 1 0 1 1 1 x y y x2 1 10 1 1 x y y x 110 1 1 x y y x3 1 0 1 1 1 x y y x4 1 0 1 1 1 x y y x5 FIGURE 1.15Graphs of defined for - q 6 x 6 q .sxd = xn , n = 1, 2, 3, 4, 5,(a) The graphs of for 2, 3, 4, 5, are displayed in Figure 1.15.These func- tions are defined for all real values of x. Notice thatas the power n gets larger, the curves tend to flatten toward thex-axis on the interval and also rise more steeply for Each curvepasses through the point (1, 1) and through the origin. The graphsof functions with even powers are symmetric about the y-axis; thosewith odd powers are symmetric about the origin. The even-poweredfunctions are decreasing on the interval and increasing on ; theodd-powered functions are increasing over the entire real line .s-q, q) [0, qds- q, 0] x 7 1. s -1, 1d, n = 1,sxd = xn , a = n, apositive integer. x y x y 0 1 1 0 1 1 y 1 x y 1 x2 Domain: x 0Range: y 0 Domain: x 0 Range: y 0 (a) (b) FIGURE 1.16 Graphs of thepower functions for part (a) and for part (b) .a = -2 a = -1sxd =xa (c) The functions and are the square root and cube rootfunctions, respectively. The domain of the square root function isbut the cube root function is defined for all real x. Their graphsare displayed in Figure 1.17 along with the graphs of and (Recallthat and ) Polynomials A function p is a polynomial if where n is anonnegative integer and the numbers are real constants (called thecoefficients of the polynomial). All polynomials have domain Ifthes- q, qd. a0, a1, a2, , an psxd = anxn + an-1xn-1 + + a1x + a0x2>3 = sx1>3 d2 . x3>2 = sx1>2 d3 y = x2>3 .y =x3>2 [0, qd, gsxd = x1>3 = 23 xsxd = x1>2 = 2x a = 1 2 , 13 , 3 2 , and 2 3 . 7001_AWLThomas_ch01p001-057.qxd 10/1/09 2:23 PMPage 8
  25. 25. 1.1 Functions and Their Graphs 9 y x 0 1 1 y x32 Domain:Range: 0 x 0 y y x Domain: Range: x 0 y 0 1 1 y x23 x y 0 1 1Domain: Range: 0 x 0 y y x x y Domain: Range: x y 1 1 0 3 y xFIGURE 1.17 Graphs of the power functions for and 2 3 .a = 1 2 , 13 , 3 2 ,sxd = xa leading coefficient and then n is called thedegree of the polynomial. Linear functions with are polynomials ofdegree 1. Polynomials of degree 2, usually written as are calledquadratic functions. Likewise, cubic functions are polynomials ofdegree 3. Figure 1.18 shows the graphs of three polynomials.Techniques to graph polynomials are studied in Chapter 4. psxd =ax3 + bx2 + cx + d psxd = ax2 + bx + c, m Z 0 n 7 0,an Z 0 x y 0 y2x x3 3 x2 2 1 3 (a) y x 1 1 2 2 2 4 6 8 10 12 y 8x4 14x3 9x2 11x 1(b) 1 0 1 2 16 16 x y y (x 2)4 (x 1)3 (x 1) (c) 24 2 4 4 2 2 4FIGURE 1.18 Graphs of three polynomial functions. (a) (b) (c) 2 442 2 2 4 4 x y y 2x2 3 7x 4 0 2 4 6 8 224 4 6 2 4 6 8 x y y 11x 22x3 1 5 0 1 2 1 5 10 2 x y Line y 5 3 y 5x2 8x 3 3x2 2 NOT TO SCALEFIGURE 1.19 Graphs of three rational functions. The straight redlines are called asymptotes and are not part of the graph. RationalFunctions A rational function is a quotient or ratio where p and qare polynomials. The domain of a rational function is the set ofall real x for which The graphs of several rational functions areshown in Figure 1.19.qsxd Z 0. (x) = p(x)>q(x),7001_AWLThomas_ch01p001-057.qxd 10/1/09 2:23 PM Page 9
  26. 26. Trigonometric Functions The six basic trigonometricfunctions are reviewed in Section 1.3. The graphs of the sine andcosine functions are shown in Figure 1.21. Exponential FunctionsFunctions of the form where the base is a positive constant and arecalled exponential functions. All exponential functions have domainand range , so an exponential function never assumes the value 0.We discuss exponential functions in Section 1.5. The graphs of someexponential functions are shown in Figure 1.22. s0, qds - q, qd a Z1, a 7 0sxd = ax , 10 Chapter 1: Functions Algebraic Functions Anyfunction constructed from polynomials using algebraic opera- tions(addition, subtraction, multiplication, division, and taking roots)lies within the class of algebraic functions. All rationalfunctions are algebraic, but also included are more complicatedfunctions (such as those satisfying an equation like studied inSection 3.7). Figure 1.20 displays the graphs of three algebraicfunctions. y3 - 9xy + x3 = 0, (a) 41 3 2 1 1 2 3 4 x y y x1/3 (x 4)(b) 0 y x y (x2 1)2/33 4 (c) 10 1 1 x y 5 7 y x(1 x)2/5 FIGURE 1.20Graphs of three algebraic functions. y x 1 1 2 3 (a) f(x) sin x 0 yx 1 1 2 3 2 2 (b) f(x) cos x 0 2 5 FIGURE 1.21 Graphs of the sineand cosine functions. (a) (b) y 2x y 3x y 10x 0.51 0 0.5 1 2 4 6 810 12 y x y 2x y 3x y 10x 0.51 0 0.5 1 2 4 6 8 10 12 y x FIGURE1.22 Graphs of exponential functions.7001_AWLThomas_ch01p001-057.qxd 10/1/09 2:23 PM Page 10
  27. 27. 1.1 Functions and Their Graphs 11 Logarithmic FunctionsThese are the functions where the base is a positive constant. Theyare the inverse functions of the exponential functions, and wediscuss these functions in Section 1.6. Figure 1.23 shows thegraphs of four logarithmic functions with various bases. In eachcase the domain is and the range is s- q, qd. s0, q d a Z 1sxd =loga x, 1 10 1 x y FIGURE 1.24 Graph of a catenary or hangingcable. (The Latin word catena means chain.) 1 1 1 0 x y y log3x ylog10 x y log2x y log5x FIGURE 1.23 Graphs of four logarithmicfunctions. Transcendental Functions These are functions that arenot algebraic. They include the trigonometric, inversetrigonometric, exponential, and logarithmic functions, and manyother functions as well. A particular example of a transcendentalfunction is a catenary. Its graph has the shape of a cable, like atelephone line or electric cable, strung from one support toanother and hanging freely under its own weight (Figure 1.24). Thefunction defining the graph is discussed in Section 7.3. Exercises1.1 Functions In Exercises 16, find the domain and range of eachfunction. 1. 2. 3. 4. 5. 6. In Exercises 7 and 8, which of thegraphs are graphs of functions of x, and which are not? Givereasons for your answers. 7. a. b. x y 0 x y 0 G(t) = 2 t2 - 16 std= 4 3 - t g(x) = 2x2 - 3xF(x) = 25x + 10 sxd = 1 - 2xsxd = 1 + x28. a. b. Finding Formulas for Functions 9. Express the area andperimeter of an equilateral triangle as a function of the trianglesside length x. 10. Express the side length of a square as afunction of the length d of the squares diagonal. Then express thearea as a function of the diagonal length. 11. Express the edgelength of a cube as a function of the cubes diag- onal length d.Then express the surface area and volume of the cube as a functionof the diagonal length. x y 0 x y 0 7001_AWLThomas_ch01p001-057.qxd10/1/09 2:23 PM Page 11
  28. 28. 12. A point P in the first quadrant lies on the graph ofthe function Express the coordinates of P as functions of the slopeof the line joining P to the origin. 13. Consider the point lyingon the graph of the line Let L be the distance from the point tothe origin Write L as a function of x. 14. Consider the point lyingon the graph of Let L be the distance between the points and WriteL as a function of y. Functions and Graphs Find the domain andgraph the functions in Exercises 1520. 15. 16. 17. 18. 19. 20. 21.Find the domain of 22. Find the range of 23. Graph the followingequations and explain why they are not graphs of functions of x. a.b. 24. Graph the following equations and explain why they are notgraphs of functions of x. a. b. Piecewise-Defined Functions Graphthe functions in Exercises 2528. 25. 26. 27. 28. Find a formula foreach function graphed in Exercises 2932. 29. a. b. 30. a. b. 1 x y3 21 2 1 2 3 1 (2, 1) x y 52 2 (2, 1) t y 0 2 41 2 3 x y 0 1 2 (1,1) Gsxd = e 1>x, x 6 0 x, 0 x Fsxd = e 4 - x2 , x 1 x2 + 2x, x 71 gsxd = e 1 - x, 0 x 1 2 - x, 1 6 x 2 sxd = e x, 0 x 1 2 - x, 1 6x 2 x + y = 1 x + y = 1 y2 = x2 y = x y = 2 + x2 x2 + 4 . y = x + 34 - 2x2 - 9 . Gstd = 1> t Fstd = t> t gsxd = 2-xgsxd = 2 xsxd = 1 - 2x - x2 sxd = 5 - 2x (4, 0).(x, y) 2x - 3.y =(x, y) (0,0). (x, y)2x + 4y = 5. (x, y) sxd = 2x. 12 Chapter 1: Functions 31.a. b. 32. a. b. The Greatest and Least Integer Functions 33. Forwhat values of x is a. b. 34. What real numbers x satisfy theequation 35. Does for all real x? Give reasons for your answer. 36.Graph the function Why is (x) called the integer part of x?Increasing and Decreasing Functions Graph the functions inExercises 3746. What symmetries, if any, do the graphs have?Specify the intervals over which the function is in- creasing andthe intervals where it is decreasing. 37. 38. 39. 40. 41. 42. 43.44. 45. 46. Even and Odd Functions In Exercises 4758, say whetherthe function is even, odd, or neither. Give reasons for youranswer. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. Theory andExamples 59. The variable s is proportional to t, and whenDetermine t when s = 60. t = 75.s = 25 hstd = 2 t + 1hstd = 2t + 1hstd = t3 hstd = 1 t - 1 gsxd = x x2 - 1 gsxd = 1 x2 - 1 gsxd = x4+ 3x2 - 1gsxd = x3 + x sxd = x2 + xsxd = x2 + 1 sxd = x-5 sxd = 3 y= s-xd2>3 y = -x3>2 y = -42xy = x3 >8 y = 2-xy = 2 x y = 1x y = - 1 x y = - 1 x2 y = -x3 sxd = e :x;, x 0
  29. 29. 1.1 Functions and Their Graphs 13 60. Kinetic energy Thekinetic energy K of a mass is proportional to the square of itsvelocity If joules when what is K when 61. The variables r and sare inversely proportional, and when Determine s when 62. BoylesLaw Boyles Law says that the volume V of a gas at con- stanttemperature increases whenever the pressure P decreases, so that Vand P are inversely proportional. If when then what is V when 63. Abox with an open top is to be constructed from a rectangular pieceof cardboard with dimensions 14 in. by 22 in. by cutting out equalsquares of side x at each corner and then folding up the sides asin the figure. Express the volume V of the box as a func- tion ofx. 64. The accompanying figure shows a rectangle inscribed in anisosce- les right triangle whose hypotenuse is 2 units long. a.Express the y-coordinate of P in terms of x. (You might start bywriting an equation for the line AB.) b. Express the area of therectangle in terms of x. In Exercises 65 and 66, match eachequation with its graph. Do not use a graphing device, and givereasons for your answer. 65. a. b. c. x y f g h 0 y = x10 y = x7 y= x4 x y 1 0 1x A B P(x, ?) x x x x x x x x 22 14 P = 23.4lbs>in2 ?V = 1000 in3 , P = 14.7 lbs>in2 r = 10.s = 4. r = 6y = 10 m>sec?y = 18 m>sec, K = 12,960y. 66. a. b. c. 67. a.Graph the functions and to- gether to identify the values of x forwhich b. Confirm your findings in part (a) algebraically. 68. a.Graph the functions and together to identify the values of x forwhich b. Confirm your findings in part (a) algebraically. 69. For acurve to be symmetric about the x-axis, the point (x, y) must lieon the curve if and only if the point lies on the curve. Explainwhy a curve that is symmetric about the x-axis is not the graph ofa function, unless the function is 70. Three hundred books sell for$40 each, resulting in a revenue of For each $5 increase in theprice, 25 fewer books are sold. Write the revenue R as a functionof the number x of $5 increases. 71. A pen in the shape of anisosceles right triangle with legs of length x ft and hypotenuse oflength h ft is to be built. If fencing costs $5/ft for the legs and$10/ft for the hypotenuse, write the total cost C of constructionas a function of h. 72. Industrial costs A power plant sits next toa river where the river is 800 ft wide. To lay a new cable from theplant to a location in the city 2 mi downstream on the oppositeside costs $180 per foot across the river and $100 per foot alongthe land. a. Suppose that the cable goes from the plant to a pointQ on the opposite side that is x ft from the point P directlyopposite the plant. Write a function C(x) that gives the cost oflaying the cable in terms of the distance x. b. Generate a table ofvalues to determine if the least expensive location for point Q isless than 2000 ft or greater than 2000 ft from point P. x QP Powerplant City 800 ft 2 mi NOT TO SCALE (300)($40) = $12,000. y = 0.sx, -yd 3 x - 1 6 2 x + 1 . gsxd = 2>sx + 1dsxd = 3>sx - 1d x2 7 1 + 4 x . gsxd = 1 + s4>xdsxd = x>2 x y f h g 0 y = x5 y= 5x y = 5x T T 7001_AWLThomas_ch01p001-057.qxd 10/1/09 2:23 PMPage 13
  30. 30. 14 Chapter 1: Functions 1.2 Combining Functions; Shiftingand Scaling Graphs In this section we look at the main waysfunctions are combined or transformed to form new functions. Sums,Differences, Products, and Quotients Like numbers, functions can beadded, subtracted, multiplied, and divided (except where thedenominator is zero) to produce new functions. If and g arefunctions, then for every x that belongs to the domains of both andg (that is, for ), we define functions and g by the formulas Noticethat the sign on the left-hand side of the first equationrepresents the operation of addition of functions, whereas the onthe right-hand side of the equation means addition of the realnumbers (x) and g(x). At any point of at which we can also definethe function by the formula Functions can also be multiplied byconstants: If c is a real number, then the function c is definedfor all x in the domain of by EXAMPLE 1 The functions defined bythe formulas have domains and The points common to these do- mainsare the points The following table summarizes the formulas anddomains for the various algebraic com- binations of the twofunctions. We also write for the product function g. FunctionFormula Domain [0, 1] [0, 1] [0, 1] [0, 1) (0, 1] The graph of thefunction is obtained from the graphs of and g by adding thecorresponding y-coordinates (x) and g(x) at each point as in Figure1.25. The graphs of and from Example 1 are shown in Figure 1.26. #g + g x H Dsd Dsgd, + g sx = 0 excludedd g sxd = gsxd sxd = A 1 - xxg> sx = 1 excludedd g sxd = sxd gsxd = A x 1 - x >g s #gdsxd = sxdgsxd = 2xs1 - xd # g sg - dsxd = 21 - x - 2xg - s -gdsxd = 2x - 21 - x - g [0, 1] = Dsd Dsgds + gdsxd = 2x + 21 - x +g # g [0, qd s - q, 1] = [0, 1]. Dsgd = s- q, 1].Dsd = [0, qd sxd =2x and gsxd = 21 - x scdsxd = csxd. a gbsxd = sxd gsxd swhere gsxdZ 0d. >ggsxd Z 0,Dsd Dsgd + + sgdsxd = sxdgsxd. s - gdsxd = sxd- gsxd. s + gdsxd = sxd + gsxd. + g, - g, x H Dsd Dsgd7001_AWLThomas_ch01p001-057.qxd 10/1/09 2:23 PM Page 14
  31. 31. 1.2 Combining Functions; Shifting and Scaling Graphs 15 y (f g)(x) y g(x) y f(x) f(a) g(a) f(a) g(a) a 2 0 4 6 8 y x FIGURE1.25 Graphical addition of two functions. 5 1 5 2 5 3 5 4 10 1 x y2 1 g(x) 1 x f(x) x y f g y f g FIGURE 1.26 The domain of thefunction is the intersection of the domains of and g, the interval[0, 1] on the x-axis where these domains overlap. This interval isalso the domain of the function (Example 1). # g + g CompositeFunctions Composition is another method for combining functions.DEFINITION If and g are functions, the composite function ( com-posed with g) is defined by The domain of consists of the numbers xin the domain of g for which g(x) lies in the domain of . g s gdsxd= sgsxdd. g The definition implies that can be formed when therange of g lies in the domain of . To find first find g(x) andsecond find (g(x)). Figure 1.27 pic- tures as a machine diagram andFigure 1.28 shows the composite as an arrow di- agram. g s gdsxd, gx g f f(g(x))g(x) x f(g(x)) g(x) g f f g FIGURE 1.27 Two functionscan be composed at x whenever the value of one function at x liesin the domain of the other. The composite is denoted by g. FIGURE1.28 Arrow diagram for g. To evaluate the composite function (whendefined), we find (x) first and then g((x)). The domain of is theset of numbers x in the domain of such that (x) lies in the domainof g. The functions and are usually quite different.g g g g7001_AWLThomas_ch01p001-057.qxd 10/1/09 2:23 PM Page 15
  32. 32. 16 Chapter 1: Functions EXAMPLE 2 If and find (a) (b) (c)(d) Solution Composite Domain (a) (b) (c) (d) To see why the domainof notice that is defined for all real x but belongs to the domainof only if that is to say, when Notice that if and then However,the domain of is not since requires Shifting a Graph of a FunctionA common way to obtain a new function from an existing one is byadding a constant to each output of the existing function, or toits input variable. The graph of the new function is the graph ofthe original function shifted vertically or horizontally, asfollows. x 0.2xs- q, qd,[0, qd, g s gdsxd = A 2xB2 = x.gsxd =2x,sxd = x2 x -1.x + 1 0, gsxd = x + 1 g is [-1, qd, s- q, qdsggdsxd = gsgsxdd = gsxd + 1 = sx + 1d + 1 = x + 2 [0, qds dsxd =ssxdd = 2sxd = 21x = x1>4 [0, qdsg dsxd = gssxdd = sxd + 1 = 2x+ 1 [-1, qds gdsxd = sgsxdd = 2gsxd = 2x + 1 sg gdsxd.s dsxdsgdsxds gdsxd gsxd = x + 1,sxd = 2x Shift Formulas Vertical ShiftsShifts the graph of up Shifts it down Horizontal Shifts Shifts thegraph of left Shifts it right h units if h 6 0 h units if h 7 0y =sx + hd k units if k 6 0 k units if k 7 0y = sxd + k x y 2 1 2 2units 1 unit 2 2 1 0 y x2 2 y x2 y x2 1 y x2 2 FIGURE 1.29 To shiftthe graph of up (or down), we add positive (or negative) constantsto the formula for (Examples 3a and b). sxd = x2 EXAMPLE 3 (a)Adding 1 to the right-hand side of the formula to get shifts thegraph up 1 unit (Figure 1.29). (b) Adding to the right-hand side ofthe formula to get shifts the graph down 2 units (Figure 1.29). (c)Adding 3 to x in to get shifts the graph 3 units to the left(Figure 1.30). (d) Adding to x in and then adding to the result,gives and shifts the graph 2 units to the right and 1 unit down(Figure 1.31). Scaling and Reflecting a Graph of a Function Toscale the graph of a function is to stretch or compress it,vertically or hori- zontally. This is accomplished by multiplyingthe function , or the independent variable x, by an appropriateconstant c. Reflections across the coordinate axes are specialcases where c = -1. y = sxd y = x - 2 - 1-1y = x ,-2 y = sx + 3d2 y= x2 y = x2 - 2y = x2 -2 y = x2 + 1y = x27001_AWLThomas_ch01p001-057.qxd 10/1/09 2:23 PM Page 16
  33. 33. 1.2 Combining Functions; Shifting and Scaling Graphs 17 x y03 2 1 1 y (x 2)2 y x2 y (x 3)2 Add a positive constant to x. Add anegative constant to x. 4 2 2 4 6 1 1 4 x y y x 2 1 FIGURE 1.30 Toshift the graph of to the left, we add a positive constant to x(Example 3c). To shift the graph to the right, we add a negativeconstant to x. y = x2 FIGURE 1.31 Shifting the graph of units tothe right and 1 unit down (Example 3d). y = x 2 EXAMPLE 4 Here wescale and reflect the graph of (a) Vertical: Multiplying theright-hand side of by 3 to get stretches the graph vertically by afactor of 3, whereas multiplying by compresses the graph by afactor of 3 (Figure 1.32). (b) Horizontal: The graph of is ahorizontal compression of the graph of by a factor of 3, and is ahorizontal stretching by a factor of 3 (Figure 1.33). Note that soa horizontal compression may cor- respond to a vertical stretchingby a different scaling factor. Likewise, a horizontal stretchingmay correspond to a vertical compression by a different scalingfactor. (c) Reflection: The graph of is a reflection of across thex-axis, and is a reflection across the y-axis (Figure 1.34).y = 2-xy = 2xy = - 2x y = 23x = 232x y = 2x>3y = 2x y = 23x 1>3 y =32xy = 2x y = 2x. Vertical and Horizontal Scaling and ReflectingFormulas For , the graph is scaled: Stretches the graph ofvertically by a factor of c. Compresses the graph of vertically bya factor of c. Compresses the graph of horizontally by a factor ofc. Stretches the graph of horizontally by a factor of c. For , thegraph is reflected: Reflects the graph of across the x-axis.Reflects the graph of across the y-axis.y = s-xd y = -sxd c = -1 y= sx>cd y = scxd y = 1 c sxd y = csxd c 7 1 1 10 2 3 4 1 2 3 4 5x y y x y x y 3x 3 1 stretch compress 1 0 1 2 3 4 1 2 3 4 x y y 3xy x3 y x compress stretch 3 2 1 1 2 3 1 1 x y y x y x y x FIGURE1.32 Vertically stretching and compressing the graph by a factor of3 (Example 4a). y = 1x FIGURE 1.33 Horizontally stretching andcompressing the graph by a factor of 3 (Example 4b). y = 1x FIGURE1.34 Reflections of the graph across the coordinate axes (Example4c). y = 1x 7001_AWLThomas_ch01p001-057.qxd 10/1/09 2:23 PM Page17
  34. 34. 18 Chapter 1: Functions EXAMPLE 5 Given the function(Figure 1.35a), find formulas to (a) compress the graphhorizontally by a factor of 2 followed by a reflection across they-axis (Figure 1.35b). (b) compress the graph vertically by afactor of 2 followed by a reflection across the x-axis (Figure1.35c). sxd = x4 - 4x3 + 10 Solution (a) We multiply x by 2 to getthe horizontal compression, and by to give reflection across they-axis. The formula is obtained by substituting for x in theright-hand side of the equation for : (b) The formula is EllipsesAlthough they are not the graphs of functions, circles can bestretched horizontally or ver- tically in the same way as thegraphs of functions. The standard equation for a circle of radius rcentered at the origin is Substituting cx for x in the standardequation for a circle (Figure 1.36a) gives (1)c2 x2 + y2 = r2 . x2+ y2 = r2 . y = - 1 2 sxd = - 1 2 x4 + 2x3 - 5. = 16x4 + 32x3 + 10.y = s-2xd = s-2xd4 - 4s -2xd3 + 10 -2x -1 1 0 1 2 3 4 20 10 10 20 xy f(x) x4 4x3 10 (a) 2 1 0 1 20 10 10 20 x y (b) y 16x4 32x3 10 1 01 2 3 4 10 10 x y y x4 2x3 51 2 (c) FIGURE 1.35 (a) The originalgraph of f. (b) The horizontal compression of in part (a) by afactor of 2, followed by a reflection across the y-axis. (c) Thevertical compression of in part (a) by a factor of 2, followed by areflection across the x-axis (Example 5). y = sxd y = sxd x y (a)circle r r r r0 x2 y2 r2 x y (b) ellipse, 0 c 1 r 0 c2 x2 y2 r2 r cr c x y (c) ellipse, c 1 r r 0 c2 x2 y2 r2 r c r c r FIGURE 1.36Horizontal stretching or compression of a circle produces graphs ofellipses. 7001_AWLThomas_ch01p001-057.qxd 10/1/09 2:23 PM Page18
  35. 35. 1.2 Combining Functions; Shifting and Scaling Graphs 19 Ifthe graph of Equation (1) horizontally stretches the circle; if thecir- cle is compressed horizontally. In either case, the graph ofEquation (1) is an ellipse (Figure 1.36). Notice in Figure 1.36that the y-intercepts of all three graphs are always and r. InFigure 1.36b, the line segment joining the points is called themajor axis of the ellipse; the minor axis is the line segmentjoining The axes of the el- lipse are reversed in Figure 1.36c: Themajor axis is the line segment joining the points , and the minoraxis is the line segment joining the points In both cases, themajor axis is the longer line segment. If we divide both sides ofEquation (1) by we obtain (2) where and If the major axis ishorizontal; if the major axis is vertical. The center of theellipse given by Equation (2) is the origin (Figure 1.37).Substituting for x, and for y, in Equation (2) results in (3)Equation (3) is the standard equation of an ellipse with center at(h, k). The geometric definition and properties of ellipses arereviewed in Section 11.6. sx - hd2 a2 + s y - kd2 b2 = 1. y - kx -h a 6 b,a 7 b,b = r.a = r>c x2 a2 + y2 b2 = 1 r2 , s;r>c,0d.s0, ;rd s0, ;rd. s;r>c, 0d -r c 7 10 6 c 6 1, x y a b b aMajor axis Center FIGURE 1.37 Graph of the ellipse where the majoraxis is horizontal. x2 a2 + y2 b2 = 1, a 7 b, Exercises 1.2Algebraic Combinations In Exercises 1 and 2, find the domains andranges of and 1. 2. In Exercises 3 and 4, find the domains andranges of , g, , and 3. 4. Composites of Functions 5. If and findthe following. a. b. c. d. e. f. g. h. 6. If and find thefollowing. a. b. c. d. e. f. g. h. In Exercises 710, write aformula for 7. 8. hsxd = x2 gsxd = 2x - 1,(x) = 3x + 4, hsxd = 4 -xgsxd = 3x,(x) = x + 1, g h. g(g(x))((x)) g(g(2))((2)) g((x))(g(x))g((1>2))(g(1>2)) gsxd = 1>sx + 1d,sxd = x - 1 g(g(x))((x))g(g(2))((-5)) g((x))(g(x)) g((0))(g(0)) gsxd = x2 - 3,sxd = x + 5sxd = 1, gsxd = 1 + 2x sxd = 2, gsxd = x2 + 1 g>. >g sxd = 2x+ 1, gsxd = 2x - 1 sxd = x, gsxd = 2x - 1 # g. , g, + g, 9. 10. Letand Ex- press each of the functions in Exercises 11 and 12 as acomposite in- volving one or more of , g, h, and j. 11. a. b. c. d.e. f. 12. a. b. c. d. e. f. 13. Copy and complete the followingtable. g(x) (x) ( g)(x) a. ? b. 3x ? c. ? d. ? e. ? x f. ? x 1 x 1+ 1 x x x - 1 x x - 1 2x2 - 52x - 5 x + 2 2xx - 7 y = 2x3 - 3y =22x - 3 y = x - 6y = x9 y = x3>2 y = 2x - 3 y = s2x - 6d3 y =2sx - 3d3 y = 4xy = x1>4 y = 22xy = 2x - 3 jsxd = 2x.sxd = x -3, gsxd = 2x, hsxd = x3 , hsxd = 22 - xgsxd = x2 x2 + 1 ,sxd = x +2 3 - x , hsxd = 1 xgsxd = 1 x + 4 ,sxd = 2x + 1,7001_AWLThomas_ch01p001-057.qxd 10/1/09 2:23 PM Page 19
  36. 36. 20 Chapter 1: Functions 14. Copy and complete the followingtable. g(x) (x) ( g)(x) a. ? b. ? c. ? d. ? 15. Evaluate eachexpression using the given table of values x 2x x 2x x x + 1 x - 1x x 1 x - 1 22. The accompanying figure shows the graph of shiftedto two new positions. Write equations for the new graphs. 23. Matchthe equations listed in parts (a)(d) to the graphs in the ac-companying figure. a. b. c. d. 24. The accompanying figure showsthe graph of shifted to four new positions. Write an equation foreach new graph. x y (2, 3) (4, 1) (1, 4) (2, 0) (b) (c) (d) (a) y =-x2 x y Position 2 Position 1 Position 4 Position 3 4 3 2 1 0 1 2 3(2, 2) (2, 2) (3, 2) (1, 4) 1 2 3 y = sx + 3d2 - 2y = sx + 2d2 + 2y = sx - 2d2 + 2y = sx - 1d2 - 4 x y Position (a) Position (b) y x25 0 3 y = x2 x 0 1 2 (x) 1 0 1 2 g(x) 2 1 0 0-1 -2 -1-2 a. b. c. d.e. f. 16. Evaluate each expression using the functions a. b. c. d.e. f. In Exercises 17 and 18, (a) write formulas for and and findthe (b) domain and (c) range of each. 17. 18. 19. Let Find afunction so that 20. Let Find a function so that Shifting Graphs21. The accompanying figure shows the graph of shifted to two newpositions. Write equations for the new graphs. x y 7 0 4 Position(a) Position (b)y x2 y = -x2 ( g)(x) = x + 2. y = g(x)(x) = 2x3 -4. ( g)(x) = x. y = g(x)(x) = x x - 2 . (x) = x2 , g(x) = 1 - 2x(x) = 2x + 1, g(x) = 1 x g g sgs1>2ddgss0ddss2dd gsgs-1ddgss3ddsgs0dd (x) = 2 - x, g(x) = b -x, -2 x 6 0 x - 1, 0 x 2.sgs1ddgss-2ddgsgs2dd ss-1ddgss0ddsgs -1dd7001_AWLThomas_ch01p001-057.qxd 10/1/09 2:23 PM Page 20
  37. 37. 1.2 Combining Functions; Shifting and Scaling Graphs 21Exercises 2534 tell how many units and in what directions thegraphs of the given equations are to be shifted. Give an equationfor the shifted graph. Then sketch the original and shifted graphstogether, labeling each graph with its equation. 25. 26. 27. 28.29. 30. 31. 32. 33. 34. Graph the functions in Exercises 3554. 35.36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52.53. 54. 55. The accompanying figure shows the graph of a function(x) with domain [0, 2] and range [0, 1]. Find the domains andranges of the following functions, and sketch their graphs. a. b.c. d. e. f. g. h. -sx + 1d + 1s -xd sx - 1dsx + 2d -sxd2(x) sxd -1sxd + 2 x y 0 2 1 y f(x) y = 1 sx + 1d2 y = 1 x2 + 1 y = 1 x2 - 1y= 1 sx - 1d2 y = 1 x + 2 y = 1 x + 2 y = 1 x - 2y = 1 x - 2 y = sx+ 2d3>2 + 1y = 23 x - 1 - 1 y + 4 = x2>3 y = 1 - x2>3 y =sx - 8d2>3 y = sx + 1d2>3 y = 1 - 2xy = 1 + 2x - 1 y = 1 - x- 1y = x - 2 y = 29 - xy = 2x + 4 y = 1>x2 Left 2, down 1 y =1>x Up 1, right 1 y = 1 2 sx + 1d + 5 Down 5, right 1 y = 2x - 7Up 7 y = - 2x Right 3 y = 2x Left 0.81 y = x2>3 Right 1, down 1y = x3 Left 1, down 1 x2 + y2 = 25 Up 3, left 4 x2 + y2 = 49 Down3, left 2 56. The accompanying figure shows the graph of a functiong(t) with domain and range Find the domains and ranges of thefollowing functions, and sketch their graphs. a. b. c. d. e. f. g.h. Vertical and Horizontal Scaling Exercises 5766 tell by whatfactor and direction the graphs of the given functions are to bestretched or compressed. Give an equation for the stretched orcompressed graph. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. GraphingIn Exercises 6774, graph each function, not by plotting points, butby starting with the graph of one of the standard functionspresented in Figures 1.141.17 and applying an appropriatetransformation. 67. 68. 69. 70. 71. 72. 73. 74. 75. Graph thefunction 76. Graph the function Ellipses Exercises 7782 giveequations of ellipses. Put each equation in stan- dard form andsketch the ellipse. 77. 78. 79. 80. sx + 1d2 + 2y2 = 43x2 + s y -2d2 = 3 16x2 + 7y2 = 1129x2 + 25y2 = 225 y = 2 x . y = x2 - 1 . y =s -2xd2>3 y = - 23 x y = 2 x2 + 1y = 1 2x - 1 y = s1 - xd3 + 2y= sx - 1d3 + 2 y = A 1 - x 2 y = - 22x + 1 y = 1 - x3 , stretchedhorizontally by a factor of 2 y = 1 - x3 , compressed horizontallyby a factor of 3 y = 24 - x2 , compressed vertically by a factor of3 y = 24 - x2 , stretched horizontally by a factor of 2 y = 2x + 1,stretched vertically by a factor of 3 y = 2x + 1, compressedhorizontally by a factor of 4 y = 1 + 1 x2 , stretched horizontallyby a factor of 3 y = 1 + 1 x2 , compressed vertically by a factorof 2 y = x2 - 1, compressed horizontally by a factor of 2 y = x2 -1, stretched vertically by a factor of 3 -gst - 4dgs1 - td gst -2dgs-t + 2d 1 - gstdgstd + 3 -gstdgs-td t y 3 2 04 y g(t) [-3,0].[-4, 0] 7001_AWLThomas_ch01p001-057.qxd 10/1/09 2:24 PM Page21
  38. 38. 22 Chapter 1: Functions 81. 82. 83. Write an equation forthe ellipse shifted 4 units to the left and 3 units up. Sketch theellipse and identify its center and major axis. 84. Write anequation for the ellipse shifted 3 units to the right and 2 unitsdown. Sketch the ellipse and iden- tify its center and major axis.Combining Functions 85. Assume that is an even function, g is anodd function, and both and g are defined on the entire real lineWhich of the follow- ing (where defined) are even? odd? . sx2>4d + sy2 >25d = 1 sx2 >16d + sy2 >9d = 1 6 ax + 3 2 b2 + 9 ay - 1 2 b 2 = 54 3sx - 1d2 + 2s y + 2d2 = 6 a. b. c. d. e.f. g. h. i. 86. Can a function be both even and odd? Give reasonsfor your answer. 87. (Continuation of Example 1.) Graph thefunctions and together with their (a) sum, (b) product, (c) twodifferences, (d) two quotients. 88. Let and Graph and g togetherwith and g . g gsxd = x2 .sxd = x - 7 gsxd = 21 - x sxd = 2x g g ggg2 = gg2 = g>>gg T T 1.3 Trigonometric Functions Thissection reviews radian measure and the basic trigonometricfunctions. Angles Angles are measured in degrees or radians. Thenumber of radians in the central angle within a circle of radius ris defined as the number of radius units contained in the arc ssubtended by that central angle. If we denote this central angle bywhen meas- ured in radians, this means that (Figure 1.38), oru =s>r u ACB (1)s = ru (u in radians). If the circle is a unitcircle having radius , then from Figure 1.38 and Equation (1), wesee that the central angle measured in radians is just the lengthof the arc that the an- gle cuts from the unit circle. Since onecomplete revolution of the unit circle is 360 or radians, we have(2) and Table 1.2 shows the equivalence between degree and radianmeasures for some basic angles. 1 radian = 180 p (L 57.3) degreesor 1 degree = p 180 (L 0.017) radians. p radians = 180 2p u r = 1B' B s A' C A r 1 Circle of radius r Unit circl e FIGURE 1.38 Theradian measure of the central angle is the number For a unit circleof radius is the length of arc AB that central angle ACB cuts fromthe unit circle. r = 1, u u = s>r.ACB TABLE 1.2 Angles measuredin degrees and radians Degrees 0 30 45 60 90 120 135 150 180 270360 (radians) 0 2p 3p 2 p 5p 6 3p 4 2p 3 p 2 p 3 p 4 p 6 p 4 p 2 3p4 pU 4590135180 7001_AWLThomas_ch01p001-057.qxd 10/1/09 2:24 PMPage 22
  39. 39. 1.3 Trigonometric Functions 23 x y x y Positive measureInitial ray Terminal ray Terminal ray Initial ray Negative measureFIGURE 1.39 Angles in standard position in the xy-plane. x y 4 9 xy 3 x y 4 3 x y 2 5 FIGURE 1.40 Nonzero radian measures can bepositive or negative and can go beyond 2p. hypotenuse adjacentopposite sin opp hyp adj hyp cos tan opp adj csc hyp opp hyp adjsec cot adj opp FIGURE 1.41 Trigonometric ratios of an acute angle.An angle in the xy-plane is said to be in standard position if itsvertex lies at the origin and its initial ray lies along thepositive x-axis (Figure 1.39). Angles measured counter- clockwisefrom the positive x-axis are assigned positive measures; anglesmeasured clock- wise are assigned negative measures. Anglesdescribing counterclockwise rotations can go arbitrarily far beyondradi- ans or 360 . Similarly, angles describing clockwise rotationscan have negative measures of all sizes (Figure 1.40). 2p AngleConvention: Use Radians From now on, in this book it is assumedthat all angles are measured in radians unless degrees or someother unit is stated explicitly. When we talk about the angle , wemean radians (which is 60 ), not degrees. We use radians because itsimplifies many of the operations in calculus, and some results wewill obtain involving the trigonometric functions are not true whenangles are measured in degrees. The Six Basic TrigonometricFunctions You are probably familiar with defining the trigonometricfunctions of an acute angle in terms of the sides of a righttriangle (Figure 1.41). We extend this definition to obtuse andnegative angles by first placing the angle in standard position ina circle of radius r. We then define the trigonometric functions interms of the coordinates of the point P(x, y) where the anglesterminal ray intersects the circle (Figure 1.42). sine: cosecant:cosine: secant: tangent: cotangent: These extended definitionsagree with the right-triangle definitions when the angle is acute.Notice also that whenever the quotients are defined, csc u = 1 sinu sec u = 1 cos u cot u = 1 tan u tan u = sin u cos u cot u = xytan u = y x sec u = r xcos u = x r csc u = r ysin u = y rp>3p>3p>3 x y P(x, y) r rO y x FIGURE 1.42 Thetrigonometric functions of a general angle are defined in terms ofx, y, and r. u 7001_AWLThomas_ch01p001-057.qxd 10/1/09 2:24 PM Page23
  40. 40. 24 Chapter 1: Functions As you can see, and are not definedif This means they are not defined if is Similarly, and are notdefined for values of for which namely The exact values of thesetrigonometric ratios for some angles can be read from the trianglesin Figure 1.43. For instance, The CAST rule (Figure 1.44) is usefulfor remembering when the basic trigonometric func- tions arepositive or negative. For instance, from the triangle in Figure1.45, we see that sin 2p 3 = 23 2 , cos 2p 3 = - 1 2 , tan 2p 3 = -23. tan p 3 = 23tan p 6 = 1 23 tan

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