Calculus Textbook PDF by Ron Larson and Bruce Edwards (2017)

Index of Applications Engineering and Physical Explorer 18, 745 Muzzle velocity, 761 Sciences Explorer 55, 698 Navigation, 699, 761 F...

Index of Applications Engineering and Physical Explorer 18, 745 Muzzle velocity, 761 Sciences Explorer 55, 698 Navigation, 699, 761 Falling object, 311, 434, 437 Newton’s Law of Cooling, 419, 422 Acceleration, 128, 132, 160, 162, 180, Ferris wheel, 870 Newton’s Law of Gravitation, 1045 257, 910 Field strength, 548 Newton’s Law of Universal Gravitation, Air pressure, 439 Flight control, 159 487, 492, 854 Air traffic control, 158, 749, 854 Flow rate, 290, 294, 307, 351, 1109 Oblateness of Saturn, 473 Aircraft glide path, 197 Fluid force, 506, 507, 508, 509, 510, 512, Ohm’s Law, 241 Angle of elevation, 155, 159, 160 514, 546, 549 Oil leak, 294 Angular rate of change, 381 Force, 293, 509, 774, 775, 785, 786 Orbit Angular speed, 38, 381 Free-falling object, 73, 95 of Earth, 698 Apparent temperature, 903 Frictional force, 862, 866, 868 of the moon, 690 Archimedes’ Principle, 514 Fuel efficiency, 581 of a satellite, 698, 731, 870 Architecture, 698 Gauss’s Law, 1107, 1109 Orbital speed, 854 Asteroid Apollo, 742 Geography, 807, 818 Parabolic reflector, 688 Atmospheric pressure and altitude, 323, Gravitational fields, 1045 Particle motion, 132, 291, 294, 295, 698, 349, 955 Gravitational force, 581 717, 827, 835, 837, 844, 853, 854, Automobile aerodynamics, 30 Halley’s Comet, 698, 741 865 Average speed, 44, 93 Hanging power cables, 393, 397 Path Average temperature, 988, 1038 Harmonic motion, 142, 162, 349 of a ball, 706, 842 Average velocity, 116 Heat equation, 901 of a baseball, 709, 841, 842, 843, 864 Beam deflection, 697 Heat flux, 1127 of a bomb, 843, 869 Beam strength, 226 Heat transfer, 332 of a football, 843 Boyle’s Law, 493, 512 Heat-seeking particle, 925 of a projectile, 186, 716, 842, 843, 968 Braking load, 778 Heat-seeking path, 930 of a shot, 843 Breaking strength of a steel cable, 360 Height Pendulum, 142, 241, 910 Bridge design, 698 of a Ferris wheel, 40 Planetary motion, 745 Building design, 453, 571, 1012, 1039, 1068 of a man, 581 Planetary orbits, 691 Cable tension, 761, 769, 818 rate of change of, 157 Power, 173, 910 Carbon dating, 421 Highway design, 173, 197, 870 Producing a machine part, 463 Center of mass, 504 Honeycomb, 173 Projectile motion, 164, 241, 679, 709, Centripetal acceleration, 854 Hooke’s Law, 487, 491, 512 761, 840, 842, 843, 851, 853, 854, Centripetal force, 854 Hydraulics, 1005 864, 868, 869, 917, 968 Centroid, 502, 503, 527 Hyperbolic detection system, 695 Psychrometer, 844 Charles’s Law, 78 Hyperbolic mirror, 699 Radioactive decay, 352, 417, 421, 429, 439 Chemical mixture problem, 435, 437 Ideal Gas Law, 883, 903, 918 Rectilinear motion, 257 Chemical reaction, 430, 558, 966 Illumination, 226, 245 Refraction of light, 963 Circular motion, 844, 852 Inductance, 910 Resultant force, 758, 760, 761 Comet Hale-Bopp, 745 Kepler’s Laws, 741, 742, 866 Resultant velocity, 758 Construction, 158, 769 Kinetic and potential energy, 1075, 1078 Ripples in a pond, 29, 153 Cooling superconducting magnets with Law of Conservation of Energy, 1075 Rotary engine, 747 liquid helium, 78 Length Satellite antenna, 746 Cycloidal motion, 844, 853 of a cable, 477, 481 Satellites, 131 Dissolving chlorine, 85 of Gateway Arch, 482 Sending a space module into orbit, 488, 575 Doppler effect, 142 of pursuit, 484 Solar collector, 697 Einstein’s Special Theory of Relativity and of a stream, 483 Sound intensity, 44, 323, 422 Newton’s First Law of Motion, 207 of warblers, 584 Specific gravity of water, 198 Electric circuit, 371, 414, 434, 437 Linear vs. angular speed, 160, 162 Speed of sound, 286 Electric force, 492, Load supports, 769 Surveying, 241, 565 Electric force fields, 1045 Lunar gravity, 257 Suspension bridge, 484 Electric potential, 882 Machine design, 159 Temperature, 18, 180, 208, 322, 340, Electrical charge, 1109 Machine part, 471 413, 963 Electrical resistance, 189, 910 Magnetic field of Earth, 1054 at which water boils, 323 Electricity, 159, 307 Mass, 1059, 1065, 1066 normal daily maximum in Chicago, 142 Electromagnetic theory, 581 on the surface of Earth, 494 Temperature distribution, 882, 902, 925, Electronically controlled thermostat, 29 Mechanical design, 453, 797 930, 967 Emptying a tank of oil, 489 Meteorology, 883 Theory of Relativity, 93 Engine design, 1067 Motion of a liquid, 1122, 1123, 1126 Topography, 875, 929, 930 Engine efficiency, 207 Motion of a spring, 531 Torque, 783, 785, 816 Escape velocity, 98, 257 Moving ladder, 93, 158 Torricelli’s Law, 441, 442 Explorer 1, 698 Moving shadow, 159, 160, 162, 164 Tossing bales, 843 (continued on back inside cover)  11e Ron Larson The Pennsylvania State University The Behrend College Bruce Edwards University of Florida Australia Brazil Mexico Singapore United Kingdom United States  Calculus, Eleventh Edition © 2018, 2014 Cengage Learning Ron Larson, Bruce Edwards ALL RIGHTS RESERVED. 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For valuable information on pricing, previous editions, changes to current editions, and alternate formats, please visit www.cengage.com/highered to search by ISBN#, author, title, or keyword for materials in your areas of interest. Important Notice: Media content referenced within the product description or the product text may not be available in the eBook version. Contents P Preparation for Calculus 1 P.1 Graphs and Models 2 P.2 Linear Models and Rates of Change 10 P.3 Functions and Their Graphs 19 P.4 Review of Trigonometric Functions 31 Review Exercises 41 P.S. Problem Solving 43 1 Limits and Their Properties 45 1.1 A Preview of Calculus 46 1.2 Finding Limits Graphically and Numerically 52 1.3 Evaluating Limits Analytically 63 1.4 Continuity and One-Sided Limits 74 1.5 Infinite Limits 87 Section Project: Graphs and Limits of Trigonometric Functions 94 Review Exercises 95 P.S. Problem Solving 97 2 Differentiation 99 2.1 The Derivative and the Tangent Line Problem 100 2.2 Basic Differentiation Rules and Rates of Change 110 2.3 Product and Quotient Rules and Higher-Order Derivatives 122 2.4 The Chain Rule 133 2.5 Implicit Differentiation 144 Section Project: Optical Illusions 151 2.6 Related Rates 152 Review Exercises 161 P.S. Problem Solving 163 3 Applications of Differentiation 165 3.1 Extrema on an Interval 166 3.2 Rolle’s Theorem and the Mean Value Theorem 174 3.3 Increasing and Decreasing Functions and the First Derivative Test 181 Section Project: Even Fourth-Degree Polynomials 190 3.4 Concavity and the Second Derivative Test 191 3.5 Limits at Infinity 199 3.6 A Summary of Curve Sketching 209 3.7 Optimization Problems 219 Section Project: Minimum Time 228 3.8 Newton’s Method 229 3.9 Differentials 235 Review Exercises 242 P.S. Problem Solving 245 iii iv Contents 4 Integration 247 4.1 Antiderivatives and Indefinite Integration 248 4.2 Area 258 4.3 Riemann Sums and Definite Integrals 270 4.4 The Fundamental Theorem of Calculus 281 Section Project: Demonstrating the Fundamental Theorem 295 4.5 Integration by Substitution 296 Review Exercises 309 P.S. Problem Solving 311 Logarithmic, Exponential, and 5 Other Transcendental Functions 313 5.1 The Natural Logarithmic Function: Differentiation 314 5.2 The Natural Logarithmic Function: Integration 324 5.3 Inverse Functions 333 5.4 Exponential Functions: Differentiation and Integration 342 5.5 Bases Other than e and Applications 352 Section Project: Using Graphing Utilities to Estimate Slope 361 5.6 Indeterminate Forms and L’Hôpital’s Rule 362 5.7 Inverse Trigonometric Functions: Differentiation 373 5.8 Inverse Trigonometric Functions: Integration 382 5.9 Hyperbolic Functions 390 Section Project: Mercator Map 399 Review Exercises 400 P.S. Problem Solving 403 6 Differential Equations 405 6.1 Slope Fields and Euler’s Method 406 6.2 Growth and Decay 415 6.3 Separation of Variables and the Logistic Equation 423 6.4 First-Order Linear Differential Equations 432 Section Project: Weight Loss 438 Review Exercises 439 P.S. Problem Solving 441 7 Applications of Integration 443 7.1 Area of a Region Between Two Curves 444 7.2 Volume: The Disk Method 454 7.3 Volume: The Shell Method 465 Section Project: Saturn 473 7.4 Arc Length and Surfaces of Revolution 474 7.5 Work 485 Section Project: Pyramid of Khufu 493 7.6 Moments, Centers of Mass, and Centroids 494 7.7 Fluid Pressure and Fluid Force 505 Review Exercises 511 P.S. Problem Solving 513  Contents v 8 Integration Techniques and Improper Integrals 515 8.1 Basic Integration Rules 516 8.2 Integration by Parts 523 8.3 Trigonometric Integrals 532 Section Project: The Wallis Product 540 8.4 Trigonometric Substitution 541 8.5 Partial Fractions 550 8.6 Numerical Integration 559 8.7 Integration by Tables and Other Integration Techniques 566 8.8 Improper Integrals 572 Review Exercises 583 P.S. Problem Solving 585 9 Infinite Series 587 9.1 Sequences 588 9.2 Series and Convergence 599 Section Project: Cantor’s Disappearing Table 608 9.3 The Integral Test and p-Series 609 Section Project: The Harmonic Series 615 9.4 Comparisons of Series 616 9.5 Alternating Series 623 9.6 The Ratio and Root Tests 631 9.7 Taylor Polynomials and Approximations 640 9.8 Power Series 651 9.9 Representation of Functions by Power Series 661 9.10 Taylor and Maclaurin Series 668 Review Exercises 680 P.S. Problem Solving 683 Conics, Parametric Equations, and 10 Polar Coordinates 685 10.1 Conics and Calculus 686 10.2 Plane Curves and Parametric Equations 700 Section Project: Cycloids 709 10.3 Parametric Equations and Calculus 710 10.4 Polar Coordinates and Polar Graphs 719 Section Project: Cassini Oval 728 10.5 Area and Arc Length in Polar Coordinates 729 10.6 Polar Equations of Conics and Kepler’s Laws 738 Review Exercises 746 P.S. Problem Solving 749 vi Contents 11 Vectors and the Geometry of Space 751 11.1 Vectors in the Plane 752 11.2 Space Coordinates and Vectors in Space 762 11.3 The Dot Product of Two Vectors 770 11.4 The Cross Product of Two Vectors in Space 779 11.5 Lines and Planes in Space 787 Section Project: Distances in Space 797 11.6 Surfaces in Space 798 11.7 Cylindrical and Spherical Coordinates 808 Review Exercises 815 P.S. Problem Solving 817 12 Vector-Valued Functions 819 12.1 Vector-Valued Functions 820 Section Project: Witch of Agnesi 827 12.2 Differentiation and Integration of Vector-Valued Functions 828 12.3 Velocity and Acceleration 836 12.4 Tangent Vectors and Normal Vectors 845 12.5 Arc Length and Curvature 855 Review Exercises 867 P.S. Problem Solving 869 13 Functions of Several Variables 871 13.1 Introduction to Functions of Several Variables 872 13.2 Limits and Continuity 884 13.3 Partial Derivatives 894 13.4 Differentials 904 13.5 Chain Rules for Functions of Several Variables 911 13.6 Directional Derivatives and Gradients 919 13.7 Tangent Planes and Normal Lines 931 Section Project: Wildflowers 939 13.8 Extrema of Functions of Two Variables 940 13.9 Applications of Extrema 948 Section Project: Building a Pipeline 955 13.10 Lagrange Multipliers 956 Review Exercises 964 P.S. Problem Solving 967 14 Multiple Integration 969 14.1 Iterated Integrals and Area in the Plane 970 14.2 Double Integrals and Volume 978 14.3 Change of Variables: Polar Coordinates 990 14.4 Center of Mass and Moments of Inertia 998 Section Project: Center of Pressure on a Sail 1005 14.5 Surface Area 1006 Section Project: Surface Area in Polar Coordinates 1012 14.6 Triple Integrals and Applications 1013 14.7 Triple Integrals in Other Coordinates 1024 Section Project: Wrinkled and Bumpy Spheres 1030 14.8 Change of Variables: Jacobians 1031 Review Exercises 1038 P.S. Problem Solving 1041  Contents vii 15 Vector Analysis 1043 15.1 Vector Fields 1044 15.2 Line Integrals 1055 15.3 Conservative Vector Fields and Independence of Path 1069 15.4 Green’s Theorem 1079 Section Project: Hyperbolic and Trigonometric Functions 1087 15.5 Parametric Surfaces 1088 15.6 Surface Integrals 1098 Section Project: Hyperboloid of One Sheet 1109 15.7 Divergence Theorem 1110 15.8 Stokes’s Theorem 1118 Review Exercises 1124 P.S. Problem Solving 1127 16 Additional Topics in Differential Equations (Online)* 16.1 Exact First-Order Equations 16.2 Second-Order Homogeneous Linear Equations 16.3 Second-Order Nonhomogeneous Linear Equations Section Project: Parachute Jump 16.4 Series Solutions of Differential Equations Review Exercises P.S. Problem Solving Appendices Appendix A: Proofs of Selected Theorems A2 Appendix B: Integration Tables A3 Appendix C: Precalculus Review (Online)* Appendix D: Rotation and the General Second-Degree Equation (Online)* Appendix E: Complex Numbers (Online)* Appendix F: Business and Economic Applications (Online)* Appendix G: Fitting Models to Data (Online)* Answers to All Odd-Numbered Exercises A7 Index A121 *Available at the text-specific website www.cengagebrain.com  Preface Welcome to Calculus, Eleventh Edition. We are excited to offer you a new edition with even more resources that will help you understand and master calculus. This textbook includes features and resources that continue to make Calculus a valuable learning tool for students and a trustworthy teaching tool for instructors. Calculus provides the clear instruction, precise mathematics, and thorough coverage that you expect for your course. Additionally, this new edition provides you with free access to three companion websites: CalcView.com––video solutions to selected exercises CalcChat.com––worked-out solutions to odd-numbered exercises and access to online tutors LarsonCalculus.com––companion website with resources to supplement your learning These websites will help enhance and reinforce your understanding of the material presented in this text and prepare you for future mathematics courses. CalcView® and CalcChat® are also available as free mobile apps. Features NEW ® The website CalcView.com contains video solutions of selected exercises. Watch instructors progress step-by-step through solutions, providing guidance to help you solve the exercises. The CalcView mobile app is available for free at the Apple® App Store® or Google Play™ store. The app features an embedded QR Code® reader that can be used to scan the on-page codes and go directly to the videos. You can also access the videos at CalcView.com. UPDATED ® In each exercise set, be sure to notice the reference to CalcChat.com. This website provides free step-by-step solutions to all odd-numbered exercises in many of our textbooks. Additionally, you can chat with a tutor, at no charge, during the hours posted at the site. For over 14 years, hundreds of thousands of students have visited this site for help. The CalcChat mobile app is also available as a free download at the Apple® App Store® or Google Play™ store and features an embedded QR Code® reader. App Store is a service mark of Apple Inc. Google Play is a trademark of Google Inc. QR Code is a registered trademark of Denso Wave Incorporated. viii  Preface ix REVISED LarsonCalculus.com All companion website features have been updated based on this revision. Watch videos explaining concepts or proofs from the book, explore examples, view three-dimensional graphs, download articles from math journals, and much more. NEW Conceptual Exercises The Concept Check exercises and Exploring Concepts exercises appear in each section. These exercises will help you develop a deeper and clearer knowledge of calculus. Work through these exercises to build and strengthen your understanding of the calculus concepts and to prepare you for the rest of the section exercises. REVISED Exercise Sets The exercise sets have been carefully and extensively examined to ensure they are rigorous and relevant and to include topics our users have suggested. The exercises are organized and titled so you can better see the connections between examples and exercises. Multi-step, real-life exercises reinforce problem-solving skills and mastery of concepts by giving you the opportunity to apply the concepts in real-life situations. REVISED Section Projects Projects appear in selected sections and encourage you to explore applications related to the topics you are studying. We have added new projects, revised others, and kept some of our favorites. All of these projects provide an interesting and engaging way for you and other students to work and investigate ideas collaboratively. Table of Contents Changes Based on market research and feedback from users, we have made several changes to the table of contents. We added a review of trigonometric functions (Section P.4) to Chapter P. To cut back on the length of the text, we moved previous Section P.4 Fitting Models to Data (now Appendix G in the Eleventh Edition) to the text-specific website at CengageBrain.com. To provide more flexibility to the order of coverage of calculus topics, Section 3.5 Limits at Infinity was revised so that it can be covered after Section 1.5 Infinite Limits. As a result of this revision, some exercises moved from Section 3.5 to Section 3.6 A Summary of Curve Sketching. We moved Section 4.6 Numerical Integration to Section 8.6. We moved Section 8.7 Indeterminate Forms and L’Hôpital’s Rule to Section 5.6. Chapter Opener Each Chapter Opener highlights real-life applications used in the examples and exercises. x Preface Section Objectives A bulleted list of learning objectives provides 166 Chapter 3 Applications of Differentiation you with the opportunity to preview what will 3.1 Extrema on an Interval be presented in the upcoming section. Understand the definition of extrema of a function on an interval. Understand the definition of relative extrema of a function on an open interval. Theorems Find extrema on a closed interval. Extrema of a Function Theorems provide the conceptual framework In calculus, much effort is devoted to determining the behavior of a function f on an for calculus. Theorems are clearly stated and interval I. Does f have a maximum value on I? Does it have a minimum value? Where is the function increasing? Where is it decreasing? In this chapter, you will learn separated from the rest of the text by boxes how derivatives can be used to answer these questions. You will also see why these questions are important in real-life applications. for quick visual reference. Key proofs often y follow the theorem and can be found at 5 (2, 5) Maximum Definition of Extrema Let f be defined on an interval I containing c. LarsonCalculus.com. 4 f(x) = x 2 + 1 1. f (c) is the minimum of f on I when f (c) ≤ f (x) for all x in I. 3 2. f (c) is the maximum of f on I when f (c) ≥ f (x) for all x in I. 2 Definitions (0, 1) Minimum The minimum and maximum of a function on an interval are the extreme values, or extrema (the singular form of extrema is extremum), of the function on the interval. The minimum and maximum of a function on an interval are As with theorems, definitions are clearly stated −1 1 2 3 x also called the absolute minimum and absolute maximum, or the global minimum and global maximum, on the interval. Extrema can occur at interior using precise, formal wording and are separated (a) f is continuous, [−1, 2] is closed. points or endpoints of an interval (see Figure 3.1). Extrema that occur at the endpoints are called endpoint extrema. from the text by boxes for quick visual reference. 5 y Not a maximum 4 f(x) = x 2 + 1 A function need not have a minimum or a maximum on an interval. For instance, in Explorations 3 Figures 3.1(a) and (b), you can see that the function f (x) = x2 + 1 has both a minimum and a maximum on the closed interval [−1, 2] but does not have a maximum on the open interval (−1, 2). Moreover, in Figure 3.1(c), you can see that continuity (or the Explorations provide unique challenges to 2 lack of it) can affect the existence of an extremum on the interval. This suggests the Minimum theorem below. (Although the Extreme Value Theorem is intuitively plausible, a proof study concepts that have not yet been formally (0, 1) x of this theorem is not within the scope of this text.) −1 1 2 3 covered in the text. They allow you to learn by (b) f is continuous, (−1, 2) is open. THEOREM 3.1 The Extreme Value Theorem discovery and introduce topics related to ones y If f is continuous on a closed interval [a, b], then f has both a minimum and a presently being studied. Exploring topics in this 5 (2, 5) Maximum maximum on the interval. 4 way encourages you to think outside the box. 3 g(x) = x 2 + 1, x ≠ 0 2, x=0 2 Exploration Finding Minimum and Maximum Values The Extreme Value Theorem (like Remarks Not a minimum x the Intermediate Value Theorem) is an existence theorem because it tells of the existence of minimum and maximum values but does not show how to find These hints and tips reinforce or expand upon −1 1 2 3 these values. Use the minimum and maximum features of a graphing utility to find the extrema of each function. In each case, do you think the x-values are (c) g is not continuous, [−1, 2] is closed. concepts, help you learn how to study Figure 3.1 exact or approximate? Explain your reasoning. a. f (x) = x2 − 4x + 5 on the closed interval [−1, 3] mathematics, caution you about common errors, b. f (x) = x3 − 2x2 − 3x − 2 on the closed interval [−1, 3] address special cases, or show alternative or additional steps to a solution of an example. How Do You See It? Exercise 9781337275347_0301.indd 166 9/15/16 12:48 PM The How Do You See It? exercise in each section presents a problem that you will solve by visual inspection using the concepts learned in the lesson. This exercise is excellent for classroom discussion or test preparation. Applications Carefully chosen applied exercises and examples are included throughout to address the question, “When will I use this?” These applications are pulled from diverse sources, such as current events, world data, industry trends, and more, and relate to a wide range of interests. Understanding where calculus is (or can be) used promotes fuller understanding of the material. Historical Notes and Biographies Historical Notes provide you with background information on the foundations of calculus. The Biographies introduce you to the people who created and contributed to calculus. Technology Throughout the book, technology boxes show you how to use technology to solve problems and explore concepts of calculus. These tips also point out some pitfalls of using technology. Putnam Exam Challenges Putnam Exam questions appear in selected sections. These actual Putnam Exam questions will challenge you and push the limits of your understanding of calculus. Student Resources Student Solutions Manual for Calculus of a Single Variable ISBN-13: 978-1-337-27538-5 Student Solutions Manual for Multivariable Calculus ISBN-13: 978-1-337-27539-2 Need a leg up on your homework or help to prepare for an exam? The Student Solutions Manuals contain worked-out solutions for all odd-numbered exercises in Calculus of a Single Variable 11e (Chapters P–10 of Calculus 11e) and Multivariable Calculus 11e (Chapters 11–16 of Calculus 11e). These manuals are great resources to help you understand how to solve those tough problems. CengageBrain.com To access additional course materials, please visit www.cengagebrain.com. At the CengageBrain.com home page, search for the ISBN of your title (from the back cover of your book) using the search box at the top of the page. This will take you to the product page where these resources can be found. MindTap for Mathematics MindTap® provides you with the tools you need to better manage your limited time––you can complete assignments whenever and wherever you are ready to learn with course material specifically customized for you by your instructor and streamlined in one proven, easy-to-use interface. With an array of tools and apps––from note taking to flashcards––you’ll get a true understanding of course concepts, helping you to achieve better grades and setting the groundwork for your future courses. This access code entitles you to 3 terms of usage. Enhanced WebAssign® Enhanced WebAssign (assigned by the instructor) provides you with instant feedback on homework assignments. This online homework system is easy to use and includes helpful links to textbook sections, video examples, and problem-specific tutorials. xi  Instructor Resources Complete Solutions Manual for Calculus of a Single Variable, Vol. 1 ISBN-13: 978-1-337-27540-8 Complete Solutions Manual for Calculus of a Single Variable, Vol. 2 ISBN-13: 978-1-337-27541-5 Complete Solutions Manual for Multivariable Calculus ISBN-13: 978-1-337-27542-2 The Complete Solutions Manuals contain worked-out solutions to all exercises in the text. They are posted on the instructor companion website. Instructor’s Resource Guide (on instructor companion site) This robust manual contains an abundance of instructor resources keyed to the textbook at the section and chapter level, including section objectives, teaching tips, and chapter projects. Cengage Learning Testing Powered by Cognero (login.cengage.com) CLT is a flexible online system that allows you to author, edit, and manage test bank content; create multiple test versions in an instant; and deliver tests from your LMS, your classroom, or wherever you want. This is available online via www.cengage.com/login. Instructor Companion Site Everything you need for your course in one place! This collection of book-specific lecture and class tools is available online via www.cengage.com/login. Access and download PowerPoint® presentations, images, instructor’s manual, and more. Test Bank (on instructor companion site) The Test Bank contains text-specific multiple-choice and free-response test forms. MindTap for Mathematics MindTap® is the digital learning solution that helps you engage and transform today’s students into critical thinkers. Through paths of dynamic assignments and applications that you can personalize, real-time course analytics, and an accessible reader, MindTap helps you turn cookie cutter into cutting edge, apathy into engagement, and memorizers into higher-level thinkers. Enhanced WebAssign® Exclusively from Cengage Learning, Enhanced WebAssign combines the exceptional mathematics content that you know and love with the most powerful online homework solution, WebAssign. Enhanced WebAssign engages students with immediate feedback, rich tutorial content, and interactive, fully customizable e-books (YouBook), helping students to develop a deeper conceptual understanding of their subject matter. Quick Prep and Just In Time exercises provide opportunities for students to review prerequisite skills and content, both at the start of the course and at the beginning of each section. Flexible assignment options give instructors the ability to release assignments conditionally on the basis of students’ prerequisite assignment scores. Visit us at www.cengage.com/ewa to learn more. xii Acknowledgments We would like to thank the many people who have helped us at various stages of Calculus over the last 43 years. Their encouragement, criticisms, and suggestions have been invaluable. Reviewers Stan Adamski, Owens Community College; Tilak de Alwis; Darry Andrews; Alexander Arhangelskii, Ohio University; Seth G. Armstrong, Southern Utah University; Jim Ball, Indiana State University; Denis Bell, University of Northern Florida; Marcelle Bessman, Jacksonville University; Abraham Biggs, Broward Community College; Jesse Blosser, Eastern Mennonite School; Linda A. Bolte, Eastern Washington University; James Braselton, Georgia Southern University; Harvey Braverman, Middlesex County College; Mark Brittenham, University of Nebraska; Tim Chappell, Penn Valley Community College; Mingxiang Chen, North Carolina A&T State University; Oiyin Pauline Chow, Harrisburg Area Community College; Julie M. Clark, Hollins University; P.S. Crooke, Vanderbilt University; Jim Dotzler, Nassau Community College; Murray Eisenberg, University of Massachusetts at Amherst; Donna Flint, South Dakota State University; Michael Frantz, University of La Verne; David French, Tidewater Community College; Sudhir Goel, Valdosta State University; Arek Goetz, San Francisco State University; Donna J. Gorton, Butler County Community College; John Gosselin, University of Georgia; Arran Hamm; Shahryar Heydari, Piedmont College; Guy Hogan, Norfolk State University; Marcia Kleinz, Atlantic Cape Community College; Ashok Kumar, Valdosta State University; Kevin J. Leith, Albuquerque Community College; Maxine Lifshitz, Friends Academy; Douglas B. Meade, University of South Carolina; Bill Meisel, Florida State College at Jacksonville; Shahrooz Moosavizadeh; Teri Murphy, University of Oklahoma; Darren Narayan, Rochester Institute of Technology; Susan A. Natale, The Ursuline School, NY; Martha Nega, Georgia Perimeter College; Sam Pearsall, Los Angeles Pierce College; Terence H. Perciante, Wheaton College; James Pommersheim, Reed College; Laura Ritter, Southern Polytechnic State University; Leland E. Rogers, Pepperdine University; Paul Seeburger, Monroe Community College; Edith A. Silver, Mercer County Community College; Howard Speier, Chandler-Gilbert Community College; Desmond Stephens, Florida A&M University; Jianzhong Su, University of Texas at Arlington; Patrick Ward, Illinois Central College; Chia-Lin Wu, Richard Stockton College of New Jersey; Diane M. Zych, Erie Community College Many thanks to Robert Hostetler, The Behrend College, The Pennsylvania State University, and David Heyd, The Behrend College, The Pennsylvania State University, for their significant contributions to previous editions of this text. We would also like to thank the staff at Larson Texts, Inc., who assisted in preparing the manuscript, rendering the art package, typesetting, and proofreading the pages and supplements. On a personal level, we are grateful to our wives, Deanna Gilbert Larson and Consuelo Edwards, for their love, patience, and support. Also, a special note of thanks goes out to R. Scott O’Neil. If you have suggestions for improving this text, please feel free to write to us. Over the years we have received many useful comments from both instructors and students, and we value these very much. Ron Larson Bruce Edwards xiii  P Preparation for Calculus P.1 Graphs and Models P.2 Linear Models and Rates of Change P.3 Functions and Their Graphs P.4 Review of Trigonometric Functions Automobile Aerodynamics (Exercise 95, p. 30) Ferris Wheel (Exercise 74, p. 40) Conveyor Design (Exercise 26, p. 16) Cell Phone Subscribers (Exercise 68, p. 9) Modeling Carbon Dioxide Concentration (Example 6, p. 7) Clockwise from top left, iStockphoto.com/EdStock; DR-Media/Shutterstock.com; 1 ChrisMilesPhoto/Shutterstock.com; Gavriel Jecan/Terra/Corbis; wandee007/Shutterstock.com 2 Chapter P Preparation for Calculus P.1 Graphs and Models Sketch the graph of an equation. Find the intercepts of a graph. Test a graph for symmetry with respect to an axis and the origin. Find the points of intersection of two graphs. Interpret mathematical models for real-life data. The Graph of an Equation In 1637, the French mathematician René Descartes revolutionized the study of mathematics by combining its two major fields—algebra and geometry. With Descartes’s coordinate plane, geometric concepts could be formulated analytically and algebraic concepts could be viewed graphically. The power of this approach was such that within a century of its introduction, much of calculus had been developed. The same approach can be followed in your study of calculus. That is, by viewing calculus from multiple perspectives—graphically, analytically, and numerically—you will increase your understanding of core concepts. Consider the equation 3x + y = 7. The point (2, 1) is a solution point of the equation because the equation is satisfied (is true) when 2 is substituted for x and 1 is substituted for y. This equation has many other solutions, such as (1, 4) and (0, 7). To find other solutions systematically, solve the original equation for y. RENÉ DESCARTES (1596–1650) y = 7 − 3x Analytic approach Descartes made many contributions to philosophy, Then construct a table of values by substituting several values of x. science, and mathematics. The idea of representing points in the plane by pairs of real numbers x 0 1 2 3 4 and representing curves in the Numerical approach plane by equations was described y 7 4 1 −2 −5 by Descartes in his book La Géométrie, published in 1637. y See LarsonCalculus.com to read From the table, you can see that (0, 7), (1, 4), (2, 1), more of this biography. (3, −2), and (4, −5) are solutions of the original 8 (0, 7) equation 3x + y = 7. Like many equations, this 6 equation has an infinite number of solutions. The set 4 (1, 4) 3x + y = 7 of all solution points is the graph of the equation, as 2 (2, 1) shown in Figure P.1. Note that the sketch shown in x 2 4 6 8 Figure P.1 is referred to as the graph of 3x + y = 7, −2 (3, − 2) even though it really represents only a portion of the −4 (4, − 5) graph. The entire graph would extend beyond the page. −6 In this course, you will study many sketching Graphical approach: 3x + y = 7 techniques. The simplest is point plotting—that is, Figure P.1 y you plot points until the basic shape of the graph 7 seems apparent. 6 5 4 y = x2 − 2 Sketching a Graph by Point Plotting 3 To sketch the graph of y = x2 − 2, first construct a table of values. Next, plot the points 2 1 shown in the table. Then connect the points with a smooth curve, as shown in Figure x P.2. This graph is a parabola. It is one of the conics you will study in Chapter 10. −4 −3 − 2 2 3 4 x −2 −1 0 1 2 3 The parabola y = x2 − 2 y 2 −1 −2 −1 2 7 Figure P.2 Granger, NYC  P.1 Graphs and Models 3 One disadvantage of point plotting is that to get a good idea about the shape of a graph, you may need to plot many points. With only a few points, you could badly misrepresent the graph. For instance, to sketch the graph of 1 y= x (39 − 10x2 + x4) 30 you plot five points: (−3, −3), (−1, −1), (0, 0), (1, 1), and (3, 3) as shown in Figure P.3(a). From these five points, you might conclude that the graph is a line. This, however, is not correct. By plotting several more points, you can see that the graph is more complicated, as shown in Figure P.3(b). y 1 y y = 30 x (39 − 10x 2 + x 4) 3 (3, 3) 3 2 2 1 (1, 1) 1 (0, 0) x −3 −2 −1 1 2 3 x (− 1, − 1) −3 −2 −1 1 2 3 −1 Plotting only a −1 few points can −2 misrepresent a exploration graph. −2 (− 3, − 3) −3 Comparing Graphical and −3 Analytic Approaches Use a graphing utility to (a) (b) graph each equation. In each Figure P.3 case, find a viewing window that shows the important teChnoloGy Graphing an equation has been made easier by technology. Even characteristics of the graph. with technology, however, it is possible to misrepresent a graph badly. For instance, a. y = x3 − 3x2 + 2x + 5 each of the graphing utility* screens in Figure P.4 shows a portion of the graph of b. y = x3 − 3x2 + 2x + 25 y = x3 − x2 − 25. c. y = −x3 − 3x2 + 20x + 5 From the screen on the left, you might assume that the graph is a line. From the d. y = 3x3 − 40x2 + 50x − 45 screen on the right, however, you can see that the graph is not a line. So, whether e. y = − (x + 12)3 you are sketching a graph by hand or using a graphing utility, you must realize that different “viewing windows” can produce very different views of a graph. In choosing f. y = (x − 2)(x − 4)(x − 6) a viewing window, your goal is to show a view of the graph that fits well in the A purely graphical approach context of the problem. to this problem would involve a simple “guess, check, and 10 5 revise” strategy. What types of −5 5 things do you think an analytic approach might involve? For −10 10 instance, does the graph have symmetry? Does the graph have turns? If so, where are they? As you proceed through −10 −35 Chapters 1, 2, and 3 of this Graphing utility screens of y = x − x − 25 3 2 text, you will study many new Figure P.4 analytic tools that will help you analyze graphs of equations such as these. *In this text, the term graphing utility means either a graphing calculator, such as the TI-Nspire, or computer graphing software, such as Maple or Mathematica.  4 Chapter P Preparation for Calculus Intercepts of a Graph remark Some texts Two types of solution points that are especially useful in graphing an equation are denote the x-intercept as the those having zero as their x- or y-coordinate. Such points are called intercepts because x-coordinate of the point (a, 0) they are the points at which the graph intersects the x- or y-axis. The point (a, 0) is an rather than the point itself. x-intercept of the graph of an equation when it is a solution point of the equation. To Unless it is necessary to make find the x-intercepts of a graph, let y be zero and solve the equation for x. The point a distinction, when the term (0, b) is a y-intercept of the graph of an equation when it is a solution point of the intercept is used in this text, it equation. To find the y-intercepts of a graph, let x be zero and solve the equation for y. will mean either the point or It is possible for a graph to have no intercepts, or it might have several. For the coordinate. instance, consider the four graphs shown in Figure P.5. y y y y x x x x No x-intercepts Three x-intercepts One x-intercept No intercepts One y-intercept One y-intercept Two y-intercepts Figure P.5 An analytical approach, often Finding x- and y-Intercepts referred to as an "analytic approach," is a method of problem-solving or Find the x- and y-intercepts of the graph of y = x3 − 4x. decision-making that involves Solution To find the x-intercepts, let y be zero and solve for x. breaking down complex issues or challenges into smaller, more x3 − 4x = 0 Let y be zero. manageable components for examination, understanding, and x(x − 2)(x + 2) = 0 Factor. resolution. This approach is x = 0, 2, or −2 Solve for x. characterized by the systematic analysis of data, information, Because this equation has three solutions, you can conclude that the graph has three or evidence to draw x-intercepts: conclusions, make informed decisions, and gain insights. (0, 0), (2, 0), and (−2, 0). x-intercepts teChnoloGy Example 2 To find the y-intercepts, let x be zero. Doing this produces y = 0. So, the y-intercept is uses an analytic approach (0, 0). y-intercept to finding intercepts. When an analytic approach is not (See Figure P.6.) possible, you can use a graphical y approach by finding the points at which the graph intersects the y = x 3 − 4x 4 axes. Use the trace feature of a 3 graphing utility to approximate the intercepts of the graph of the equation in Example 2. Note (− 2, 0) (0, 0) (2, 0) x that your utility may have a −4 −3 −1 1 3 4 built-in program that can find −1 the x-intercepts of a graph. −2 (Your utility may call this the −3 root or zero feature.) If so, use −4 the program to find the x-intercepts of the graph of the Intercepts of a graph equation in Example 2. Figure P.6  symmetry : exact same shape on P.1 Graphs and Models 5 the two sides Symmetry of a Graph y Knowing the symmetry of a graph before attempting to sketch it is useful because you need only half as many points to sketch the graph. The three types of symmetry listed below can be used to help sketch the graphs of equations (see Figure P.7). (−x, y) (x, y) 1. A graph is symmetric with respect to the y-axis if, whenever (x, y) is a point on the x graph, then (−x, y) is also a point on the graph. This means that the portion of the graph to the left of the y-axis is a mirror image of the portion to the right of the y-axis. y-axis symmetry 2. A graph is symmetric with respect to the x-axis if, whenever (x, y) is a point on the graph, then (x, −y) is also a point on the graph. This means that the portion of the graph below the x-axis is a mirror image of the portion above the x-axis. y 3. A graph is symmetric with respect to the origin if, whenever (x, y) is a point on the graph, then (−x, −y) is also a point on the graph. This means that the graph is unchanged by a rotation of 180° about the origin. (x, y) x tests for Symmetry x-axis (x, − y) 1. The graph of an equation in x and y is symmetric with respect to the y-axis symmetry when replacing x by −x yields an equivalent equation. 2. The graph of an equation in x and y is symmetric with respect to the x-axis when replacing y by −y yields an equivalent equation. y 3. The graph of an equation in x and y is symmetric with respect to the origin when replacing x by −x and y by −y yields an equivalent equation. (x, y) x The graph of a polynomial has symmetry with respect to the y-axis when each term has an even exponent (or is a constant). For instance, the graph of (−x, − y) Origin symmetry y = 2x4 − x2 + 2 has symmetry with respect to the y-axis. Similarly, the graph of a polynomial has symmetry with respect to the origin when each term has an odd exponent, as illustrated Figure P.7 in Example 3. In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only testing for Symmetry the operations of addition, subtraction, multiplication, and positive-integer powers of Test the graph of y = 2x3 − x for symmetry with respect to (a) the y-axis and (b) the variables. An example of a polynomial of a origin. single indeterminate x is x2 − 4x + 7. Solution Fenglish : Chand Jomle-e a. y = 2x3 − x Write original equation. y y = 2x 3 − x y = 2(−x)3 − (−x) Replace x by −x. 2 y = −2x3 + x Simplify. The result is not an equivalent equation. Because replacing x by −x does not yield an equivalent equation, you can conclude 1 (1, 1) that the graph of y = 2x3 − x is not symmetric with respect to the y-axis. x b. y = 2x3 − x Write original equation. −2 −1 1 2 −y = 2(−x)3 − (−x) Replace x by −x and y by −y. (−1, − 1) −1 −y = −2x3 + x Simplify. −2 y = 2x3 − x Equivalent equation Because replacing x by −x and y by −y yields an equivalent equation, you can Origin symmetry conclude that the graph of y = 2x3 − x is symmetric with respect to the origin, as Figure P.8 shown in Figure P.8. 6 Chapter P Preparation for Calculus Using Intercepts and Symmetry to Sketch a Graph See LarsonCalculus.com for an interactive version of this type of example. Sketch the graph of x − y2 = 1. y Solution The graph is symmetric with respect to the x-axis because replacing y by x − y2 = 1 (5, 2) −y yields an equivalent equation. 2 (2, 1) x − y2 = 1 Write original equation. 1 (1, 0) x − (−y)2 = 1 Replace y by −y. x 2 3 4 5 x − y2 = 1 Equivalent equation −1 This means that the portion of the graph below the x-axis is a mirror image of the x-intercept −2 portion above the x-axis. To sketch the graph, first plot the x-intercept and the points above the x-axis. Then reflect in the x-axis to obtain the entire graph, as shown in Figure P.9 Figure P.9. teChnoloGy Graphing utilities are designed so that they most easily graph equations in which y is a function of x (see Section P.3 for a definition of function). To graph other types of equations, you need to split the graph into two or more parts or you need to use a different graphing mode. For instance, to graph the equation in Example 4, you can split it into two parts. y1 = √x − 1 Top portion of graph y2 = − √x − 1 Bottom portion of graph Intersection : a place where two or more roads, lines, Points of Intersection etc. meet or cross each other A point of intersection of the graphs of two equations is a point that satisfies both equations. You can find the point(s) of intersection of two graphs by solving their equations simultaneously. Finding Points of Intersection y Find all points of intersection of the graphs of 2 x2 − y = 3 and x − y = 1. x−y=1 1 (2, 1) Solution Begin by sketching the graphs of both equations in the same rectangular coordinate system, as shown in Figure P.10. From the figure, it appears that the graphs x have two points of intersection. You can find these two points as follows. −2 −1 1 2 −1 y = x2 − 3 Solve first equation for y. (−1, −2) −2 y=x−1 Solve second equation for y. x2 − 3 = x − 1 Equate y-values. x2 − y = 3 x2 − x − 2 = 0 Write in general form. Two points of intersection (x − 2)(x + 1) = 0 Factor. Figure P.10 x = 2 or −1 Solve for x. The corresponding values of y are obtained by substituting x = 2 and x = −1 into either of the original equations. Doing this produces two points of intersection: (2, 1) and (−1, −2). Points of intersection You can check the points of intersection in Example 5 by substituting into both of the original equations or by using the intersect feature of a graphing utility. Substituting : o take the place of somebody/something else; to use somebody/something instead of somebody/something else  P.1 Graphs and Models 7 Mathematical Models Real-life applications of mathematics often use equations as mathematical models. In developing a mathematical model to represent actual data, you should strive for two (often conflicting) goals––accuracy and simplicity. That is, you want the model to be simple enough to be workable, yet accurate enough to produce meaningful results. Appendix G explores these goals more completely. Comparing two mathematical models The Mauna Loa Observatory in Hawaii records the carbon dioxide concentration y (in parts per million) in Earth’s atmosphere. The January readings for various years are shown in Figure P.11. In the July 1990 issue of Scientific American, these data were used to predict the carbon dioxide level in Earth’s atmosphere in the year 2035, using the quadratic model y = 0.018t 2 + 0.70t + 316.2 Quadratic model for 1960–1990 data where t = 0 represents 1960, as shown in Figure P.11(a). The data shown in Figure P.11(b) represent the years 1980 through 2014 and can be modeled by y = 0.014t 2 + 0.66t + 320.3 Quadratic model for 1980–2014 data where t = 0 represents 1960. What was the prediction given in the Scientific American The Mauna Loa Observatory article in 1990? Given the second model for 1980 through 2014, does this prediction in Hawaii has been measuring for the year 2035 seem accurate? the increasing concentration of carbon dioxide in Earth’s y y atmosphere since 1958. 400 400 390 390 CO2 (in parts per million) CO2 (in parts per million) 380 380 370 370 360 360 350 350 340 340 330 330 320 320 310 310 t t 5 10 15 20 25 30 35 40 45 50 55 5 10 15 20 25 30 35 40 45 50 55 Year (0 ↔ 1960) Year (0 ↔ 1960) (a) (b) Figure P.11 Solution To answer the first question, substitute t = 75 (for 2035) into the first model. y = 0.018(75)2 + 0.70(75) + 316.2 = 469.95 Model for 1960–1990 data So, the prediction in the Scientific American article was that the carbon dioxide concentration in Earth’s atmosphere would reach about 470 parts per million in the year 2035. Using the model for the 1980–2014 data, the prediction for the year 2035 is y = 0.014(75)2 + 0.66(75) + 320.3 = 448.55. Model for 1980–2014 data So, based on the model for 1980–2014, it appears that the 1990 prediction was too high. The models in Example 6 were developed using a procedure called least squares regression (see Section 13.9). The older model has a correlation of r 2 ≈ 0.997, and for the newer model it is r 2 ≈ 0.999. The closer r 2 is to 1, the “better” the model. Gavriel Jecan/Terra/Corbis  Green for solved easily Yellow for solved hardly red for tried but couldn't solve 8 Chapter P Preparation for Calculus P.1 Exercises See CalcChat.com for tutorial help and worked-out solutions to odd-numbered exercises. Finding Intercepts In Exercises 19–28, find ConCept CheCk any intercepts. 1. Finding Intercepts Describe how to find the x- and y-intercepts of the graph of an equation. 19. y = 2x − 5 20. y = 4x2 + 3 2. Verifying Points of Intersection How can you check that an ordered pair is a point of intersection of 21. y = x2 + x − 2 22. y2 = x3 − 4x two graphs? 23. y = x√16 − x2 24. y = (x − 1)√x2 + 1 2 − √x x2 + 3x 25. y = 26. y = matching In Exercises 3–6, match the equation with its 5x + 1 (3x + 1)2 graph. [The graphs are labeled (a), (b), (c), and (d).] 27. x2y − x2 + 4y = 0 28. y = 2x − √x2 + 1 (a) y (b) y testing for Symmetry In Exercises 29– 40, 3 test for symmetry with respect to each axis and to 2 2 the origin. 1 1 x x 29. y = x2 − 6 30. y = 9x − x2 −1 1 −1 1 2 3 −1 −1 31. y = x − 8x 2 3 32. y = x3 + x (c) y (d) y 33. xy = 4 34. xy2 = −10 2 4 35. y = 4 − √x + 3 36. xy − √4 − x2 = 0 1 2 x x5 37. y = 38. y = x x2 + 1 4 − x2 −2 1 2 x −1 −2 −2 −2 2 39. y = x3 + x ∣ ∣ ∣∣ 40. y − x = 3 Using Intercepts and Symmetry to Sketch 3. y = − 32 x + 3 4. y = √9 − x2 a Graph In Exercises 41–56, find any intercepts 5. y = 3 − x2 6. y = x3 − x and test for symmetry. Then sketch the graph of the equation. Sketching a Graph by Point Plotting In 41. y = 2 − 3x 42. y = 23 x + 1 Exercises 7–16, sketch the graph of the equation by point plotting. 43. y = 9 − x2 44. y = 2x2 + x 45. y = x3 + 2 46. y = x3 − 4x 7. y = 12 x + 2 8. y = 5 − 2x 47. y = x√x + 5 48. y = √25 − x2 9. y = 4 − x2 10. y = (x − 3)2 49. x = y 3 50. x = y4 − 16 ∣ 11. y = x + 1 ∣ ∣∣ 12. y = x − 1 8 10 13. y = √x − 6 14. y = √x + 2 51. y = 52. y = x x2 + 1 3 1 15. y = x 16. y = x+2 53. y = 6 − x ∣∣ ∣ 54. y = 6 − x ∣ 55. 3y2 −x=9 56. x2 + 4y2 = 4 approximating Solution Points Using technology In Exercises 17 and 18, use a graphing utility to graph the Finding Points of Intersection In Exercises equation. Move the cursor along the curve to approximate the 57–62, find the points of intersection of the graphs unknown coordinate of each solution point accurate to two of the equations. decimal places. 57. x + y = 8 58. 3x − 2y = −4 17. y = √5 − x 18. y = x5 − 5x 4x − y = 7 4x + 2y = −10 (a) (2, y) (a) (−0.5, y) 59. x2 + y = 15 60. x = 3 − y2 (b) (x, 3) (b) (x, −4) −3x + y = 11 y=x−1 The symbol and a red exercise number indicates that a video solution can be seen The symbol indicates an exercise in which you are instructed to use graphing at CalcView.com. technology or a symbolic computer algebra system. The solutions of other exercises may also be facilitated by the use of appropriate technology.  P.1 Graphs and Models 9 61. x2 + y2 = 5 62. x2 + y2 = 16 69. Break-even Point Find the sales necessary to break x−y=1 x + 2y = 4 even (R = C) when the cost C of producing x units is C = 2.04x + 5600 and the revenue R from selling x units is Finding Points of Intersection Using technology In R = 3.29x. Exercises 63–66, use a graphing utility to find the points of 70. Using Solution Points For what values of k does the intersection of the graphs of the equations. Check your results graph of y2 = 4kx pass through the