Evaluate the integral. (use c for the constant of integration.) e4θ sin(5θ) dθ

TO JULIE –Jon TO ALEXA AND COLTON –Colin

Publisher: Terri Ward Developmental Editors: Tony Palermino, Katrina Wilhelm Marketing Manager: Cara LeClair Market Development Manager: Shannon Howard Executive Media Editor: Laura Judge Associate Editor: Marie Dripchak Editorial Assistant: Victoria Garvey Director of Editing, Design, and Media Production: Tracey Kuehn Managing Editor: Lisa Kinne Project Editor: Kerry O’Shaughnessy Production Manager: Paul Rohloff Cover and Text Designer: Blake Logan Illustration Coordinator: Janice Donnola Illustrations: Network Graphics and Techsetters, Inc. Photo Editors: Eileen Liang, Christine Buese Photo Researcher: Eileen Liang Composition: John Rogosich/Techsetters, Inc. Printing and Binding: RR Donnelley Cover photo: ayzek/Shutterstock

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ABOUT THE AUTHORS

COLIN ADAMS

C olin Adams is the Thomas T. Read professor of Mathematics at Williams College,where he has taught since 1985. Colin received his undergraduate degree from MIT and his PhD from the University of Wisconsin. His research is in the area of knot theory and low-dimensional topology. He has held various grants to support his research, and written numerous research articles.

Colin is the author or co-author of The Knot Book, How toAce Calculus: The Streetwise Guide, How to Ace the Rest of Calculus: The Streetwise Guide, Riot at the Calc Exam and Other Mathematically Bent Stories, Why Knot?, Introduction to Topology: Pure and Applied, and Zombies & Calculus. He co-wrote and appears in the videos “The Great Pi vs. E Debate” and “Derivative vs. Integral: the Final Smackdown.”

He is a recipient of the Haimo National Distinguished Teaching Award from the Mathematical Association of America (MAA) in 1998, an MAA Polya Lecturer for 1998- 2000, a Sigma Xi Distinguished Lecturer for 2000-2002, and the recipient of the Robert Foster Cherry Teaching Award in 2003.

Colin has two children and one slightly crazy dog, who is great at providing the entertainment.

JON ROGAWSKI

A s a successful teacher for more than 30 years, Jon Rogawski listened and learnedmuch from his own students. These valuable lessons made an impact on his thinking, his writing, and his shaping of a calculus text.

Jon Rogawski received his undergraduate and master’s degrees in mathematics si- multaneously from Yale University, and he earned his PhD in mathematics from Princeton University, where he studied under Robert Langlands. Before joining the Department of Mathematics at UCLAin 1986, where he was a full professor, he held teaching and visiting positions at the Institute for Advanced Study, the University of Bonn, and the University of Paris at Jussieu and Orsay.

Jon’s areas of interest were number theory, automorphic forms, and harmonic analy- sis on semisimple groups. He published numerous research articles in leading mathemat- ics journals, including the research monograph Automorphic Representations of Unitary Groups in Three Variables (Princeton University Press). He was the recipient of a Sloan Fellowship and an editor of the Pacific Journal of Mathematics and the Transactions of the AMS.

Sadly, Jon Rogawski passed away in September 2011. Jon’s commitment to present- ing the beauty of calculus and the important role it plays in students’ understanding of the wider world is the legacy that lives on in each new edition of Calculus.

CONTENTS CALCULUS Early Transcendentals

Chapter 1 PRECALCULUS REVIEW 1

1.1 Real Numbers, Functions, and Graphs 1 1.2 Linear and Quadratic Functions 12 1.3 The Basic Classes of Functions 19 1.4 Trigonometric Functions 23 1.5 Inverse Functions 32 1.6 Exponential and Logarithmic Functions 40 1.7 Technology: Calculators and Computers 48

Chapter Review Exercises 53

Chapter 2 LIMITS 55

2.1 Limits, Rates of Change, and Tangent Lines 55 2.2 Limits: A Numerical and Graphical Approach 63 2.3 Basic Limit Laws 72 2.4 Limits and Continuity 75 2.5 Evaluating Limits Algebraically 84 2.6 Trigonometric Limits 89 2.7 Limits at Infinity 94 2.8 Intermediate Value Theorem 100 2.9 The Formal Definition of a Limit 103

Chapter Review Exercises 110

Chapter 3 DIFFERENTIATION 113

3.1 Definition of the Derivative 113 3.2 The Derivative as a Function 121 3.3 Product and Quotient Rules 135 3.4 Rates of Change 142 3.5 Higher Derivatives 151 3.6 Trigonometric Functions 156 3.7 The Chain Rule 159 3.8 Implicit Differentiation 167 3.9 Derivatives of General Exponential and Logarithmic

Functions 175 3.10 Related Rates 182

Chapter Review Exercises 189

Chapter 4 APPLICATIONS OF THE DERIVATIVE 193

4.1 Linear Approximation and Applications 193 4.2 Extreme Values 200 4.3 The Mean Value Theorem and Monotonicity 210 4.4 The Shape of a Graph 217 4.5 L’Hôpital’s Rule 224 4.6 Graph Sketching and Asymptotes 231 4.7 Applied Optimization 239 4.8 Newton’s Method 251

Chapter Review Exercises 256

Chapter 5 THE INTEGRAL 259

5.1 Approximating and Computing Area 259 5.2 The Definite Integral 272 5.3 The Indefinite Integral 281 5.4 The Fundamental Theorem of Calculus, Part I 288 5.5 The Fundamental Theorem of Calculus, Part II 294 5.6 Net Change as the Integral of a Rate of Change 300 5.7 Substitution Method 306 5.8 Further Transcendental Functions 313 5.9 Exponential Growth and Decay 318

Chapter Review Exercises 328

Chapter 6 APPLICATIONS OF THE INTEGRAL 333

6.1 Area Between Two Curves 333 6.2 Setting Up Integrals: Volume, Density, Average Value 341 6.3 Volumes of Revolution 351 6.4 The Method of Cylindrical Shells 359 6.5 Work and Energy 365

Chapter Review Exercises 371

Chapter 7 TECHNIQUES OF INTEGRATION 373

7.1 Integration by Parts 373 7.2 Trigonometric Integrals 379 7.3 Trigonometric Substitution 386 7.4 Integrals Involving Hyperbolic and Inverse Hyperbolic

Functions 392 7.5 The Method of Partial Fractions 398 7.6 Strategies for Integration 407 7.7 Improper Integrals 414 7.8 Probability and Integration 425 7.9 Numerical Integration 431

Chapter Review Exercises 440

Chapter 8 FURTHER APPLICATIONS OF THE INTEGRAL AND TAYLOR POLYNOMIALS 443

8.1 Arc Length and Surface Area 443 8.2 Fluid Pressure and Force 450 8.3 Center of Mass 456 8.4 Taylor Polynomials 465

Chapter Review Exercises 476

Chapter 9 INTRODUCTION TO DIFFERENTIAL EQUATIONS 479

9.1 Solving Differential Equations 479 9.2 Models Involving y′ = k(y − b) 487

iv

CONTENTS v

9.3 Graphical and Numerical Methods 492 9.4 The Logistic Equation 500 9.5 First-Order Linear Equations 504

Chapter Review Exercises 510

Chapter 10 INFINITE SERIES 513

10.1 Sequences 513 10.2 Summing an Infinite Series 523 10.3 Convergence of Series with Positive Terms 534 10.4 Absolute and Conditional Convergence 543 10.5 The Ratio and Root Tests and Strategies for

Choosing Tests 548 10.6 Power Series 553 10.7 Taylor Series 563

Chapter Review Exercises 575

Chapter 11 PARAMETRIC EQUATIONS, POLAR COORDINATES, AND CONIC SECTIONS 579

11.1 Parametric Equations 579 11.2 Arc Length and Speed 590 11.3 Polar Coordinates 596 11.4 Area and Arc Length in Polar Coordinates 604 11.5 Conic Sections 609

Chapter Review Exercises 622

Chapter 12 VECTOR GEOMETRY 625

12.1 Vectors in the Plane 625 12.2 Vectors in Three Dimensions 635 12.3 Dot Product and the Angle Between Two Vectors 645 12.4 The Cross Product 653 12.5 Planes in 3-Space 664 12.6 A Survey of Quadric Surfaces 670 12.7 Cylindrical and Spherical Coordinates 678

Chapter Review Exercises 685

Chapter 13 CALCULUS OF VECTOR-VALUED FUNCTIONS 689

13.1 Vector-Valued Functions 689 13.2 Calculus of Vector-Valued Functions 697 13.3 Arc Length and Speed 706 13.4 Curvature 711 13.5 Motion in 3-Space 722 13.6 Planetary Motion According to Kepler and Newton 731

Chapter Review Exercises 737

Chapter 14 DIFFERENTIATION IN SEVERAL VARIABLES 739

14.1 Functions of Two or More Variables 739 14.2 Limits and Continuity in Several Variables 750 14.3 Partial Derivatives 757 14.4 Differentiability and Tangent Planes 767

14.5 The Gradient and Directional Derivatives 774 14.6 The Chain Rule 787 14.7 Optimization in Several Variables 795 14.8 Lagrange Multipliers: Optimizing with a Constraint 809

Chapter Review Exercises 818

Chapter 15 MULTIPLE INTEGRATION 821

15.1 Integration in Two Variables 821 15.2 Double Integrals over More General Regions 832 15.3 Triple Integrals 845 15.4 Integration in Polar, Cylindrical, and Spherical

Coordinates 856 15.5 Applications of Multiple Integrals 866 15.6 Change of Variables 878

Chapter Review Exercises 891

Chapter 16 LINE AND SURFACE INTEGRALS 895

16.1 Vector Fields 895 16.2 Line Integrals 905 16.3 Conservative Vector Fields 919 16.4 Parametrized Surfaces and Surface Integrals 930 16.5 Surface Integrals of Vector Fields 944

Chapter Review Exercises 954

Chapter 17 FUNDAMENTAL THEOREMS OF VECTOR ANALYSIS 957

17.1 Green’s Theorem 957 17.2 Stokes’ Theorem 971 17.3 Divergence Theorem 981

Chapter Review Exercises 993

APPENDICES A1 A. The Language of Mathematics A1 B. Properties of Real Numbers A7 C. Induction and the Binomial Theorem A12 D. Additional Proofs A16

ANSWERS TO ODD-NUMBERED EXERCISES ANS1

REFERENCES R1

INDEX I1

Additional content can be accessed online via LaunchPad:

ADDITIONAL PROOFS

• L’Hôpital’s Rule • Error Bounds for Numerical Integration • Comparison Test for Improper Integrals

ADDITIONAL CONTENT

• Second Order Differential Equations • Complex Numbers

PREFACE

ABOUT CALCULUS

On Teaching Mathematics I consider myself very lucky to have a career as a teacher and practitioner of mathematics. When I was young, I decided I wanted to be a writer. I loved telling stories. But I was also good at math, and, once in college, it didn’t take me long to become enamored with it. I loved the fact that success in mathematics does not depend on your presentation skills or your interpersonal relationships. You are either right or you are wrong and there is little subjective evaluation involved. And I loved the satisfaction of coming up with a solution. That intensified when I started solving problems that were open research questions that had previously remained unsolved.

So, I became a professor of mathematics. And I soon realized that teaching mathe- matics is about telling a story. The goal is to explain to students in an intriguing manner, at the right pace, and in as clear a way as possible, how mathematics works and what it can do for you. I find mathematics immensely beautiful. I want students to feel that way, too.

On Writing a Calculus Text I had always thought I might write a calculus text. But that is a daunting task. These days, calculus books average over a thousand pages. And I would need to convince myself that I had something to offer that was different enough from what already appears in the existing books. Then, I was approached about writing the third edition of Jon Rogawski’s calculus book. Here was a book for which I already had great respect. Jon’s vision of what a calculus book should be fit very closely with my own. Jon believed that as math teachers, how we say it is as important as what we say. Although he insisted on rigor at all times, he also wanted a book that was written in plain English, a book that could be read and that would entice students to read further and learn more. Moreover, Jon strived to create a text in which exposition, graphics, and layout would work together to enhance all facets of a student’s calculus experience.

In writing his book, Jon paid special attention to certain aspects of the text:

1. Clear, accessible exposition that anticipates and addresses student difficulties. 2. Layout and figures that communicate the flow of ideas. 3. Highlighted features that emphasize concepts and mathematical reasoning: Conceptual Insight, Graphical Insight, Assumptions Matter, Reminder, and Historical Perspective.

4. A rich collection of examples and exercises of graduated difficulty that teach basic skills, problem-solving techniques, reinforce conceptual understanding, and motivate cal- culus through interesting applications. Each section also contains exercises that develop additional insights and challenge students to further develop their skills.

Coming into the project of creating the third edition, I was somewhat apprehensive. Here was an already excellent book that had attained the goals set for it by its author. First and foremost, I wanted to be sure that I did it no harm. On the other hand, I have been teaching calculus now for 30 years, and in that time, I have come to some conclusions about what does and does not work well for students.

As a mathematician, I want to make sure that the theorems, proofs, arguments and development are correct. There is no place in mathematics for sloppiness of any kind. As a teacher, I want the material to be accessible. The book should not be written at the mathematical level of the instructor. Students should be able to use the book to learn the material, with the help of their instructor. Working from the high standard that Jon set, I have tried hard to maintain the level of quality of the previous edition while making the changes that I believe will bring the book to the next level.

vi

PREFACE vii

Placement of Taylor Polynomials Taylor polynomials appear in Chapter 8, before infinite series in Chapter 10. The goal here is to present Taylor polynomials as a natural extension of linear approximation. When teaching infinite series, the primary focus is on convergence, a topic that many students find challenging. By the time we have covered the basic convergence tests and studied the convergence of power series, students are ready to tackle the issues involved in representing a function by its Taylor series. They can then rely on their previous work with Taylor polynomials and the error bound from Chapter 8. However, the section on Taylor polynomials is written so that you can cover this topic together with the materials on infinite series if this order is preferred.

Careful, Precise Development W. H. Freeman is committed to high quality and precise textbooks and supplements. From this project’s inception and throughout its development and production, quality and precision have been given significant priority. We have in place unparalleled procedures to ensure the accuracy of the text:

• Exercises and Examples • Exposition • Figures • Editing • Composition

Together, these procedures far exceed prior industry standards to safeguard the quality and precision of a calculus textbook.

New to the Third Edition There are a variety of changes that have been implemented in this edition. Following are some of the most important.

MORE FOCUS ON CONCEPTS The emphasis has been shifted to focus less on the memo- rization of specific formulas, and more on understanding the underlying concepts. Memo- rization can never be completely avoided, but it is in no way the crux of calculus. Students will remember how to apply a procedure or technique if they see the logical progression that generates it. And they then understand the underlying concepts rather than seeing the topic as a black box in which you insert numbers. Specific examples include:

• (Section 1.2) Removed the general formula for the completion of a square and instead, emphasized the method so students need not memorize the formula.

• (Section 7.2) Changed the methods for evaluating trigonometric integrals to focus on techniques to apply rather than formulas to memorize.

• (Chapter 9) Discouraged the memorization of solutions of specific types of differ- ential equations and instead, encouraged the use of methods of solution.

• (Section 12.2) Decreased number of formulas for parametrizing a line from two to one, as the second can easily be derived from the first.

• (Section 12.6) De-emphasized the memorization of the various formulas for quadric surfaces. Instead, moved the focus to slicing with planes to find curves and using those to determine the shape of the surface. These methods will be useful regardless of the type of surface it is.

• (Section 14.4) Decreased the number of essential formulas for linear approximation of functions of two variables from four to two, providing the background to derive the others from these.

CHANGES IN NOTATION There are numerous notational changes. Some were made to bring the notation more into line with standard usage in mathematics and other fields in which mathematics is applied. Some were implemented to make it easier for students to remember the meaning of the notation. Some were made to help make the corresponding concepts that are represented more transparent. Specific examples include:

viii PREFACE

• (Section 4.6) Presented a new notation for graphing that gives the signs of the first and second derivative and then simple symbols (slanted up and down arrows and up and down u’s) to help the student keep track of when the graph is increasing or decreasing and concave up or concave down over the given interval.

• (Section 7.1) Simplified the notation for integration by parts and provided a visual method for remembering it.

• (Chapter 10) Changed names of the various tests for convergence/divergence of infinite series to evoke the usage of the test and thereby make it easier for students to remember them.

• (Chapters 13–17) Rather than using c(t) for a path, we consistently switched to the vector-valued function r(t). This also allowed us to replace ds with dr as a differential, which means there is less likely to be confusion with ds, dS and dS.

MORE EXPLANATIONS OF DERIVATIONS Occasionally, in the previous edition, a result was given and verified, without motivating where the derivation came from. I believe it is important for students to understand how someone might come up with a particular result, thereby helping them to picture how they might themselves one day be able to derive results.

• (Section 14.4) Developed the equation of the tangent plane in a manner that makes geometric sense.

• (Section 14.5) Included a proof of the fact the gradient of a function f of three variables is orthogonal to the surfaces that are the level sets of f .

• (Section 14.8) Gave an intuitive explanation for why the Method of Lagrange Multipliers works.

• (Section 15.5) Developed the center of mass formulas by first discussing the one- dimensional case of a seesaw.

REORDERING AND ADDING TOPICS There were some specific rearrangements among the sections and additions. These include:

• A subsection on piecewise-defined functions has been added to Section 1.3. • The section on implicit differentiation in Chapter 3 (previously Section 3.10) has

been moved up to become Section 3.8 and has absorbed the previous Section 3.8 (in- verse functions) so that implicit differentiation can be applied to derive the various derivatives as necessary.

• The section on indefinite integrals (previously Section 4.9) has been moved from Chapter 4 (Applications of the Derivative) to Chapter 5 (The Integral). This is a more natural placement for it.

• A new section on choosing from amongst the various methods of integration has been added to Chapter 7.

• A subsection on choosing the appropriate convergence/divergence test has been added to Section 10.5.

• An explanation of how to find indefinite limits using power series has been added to Section 10.6.

• The definitions of divergence and curl have been moved from Chapter 17 to Section 16.1. This allows us to utilize them at an appropriate earlier point in the text.

• A list all of the different types of integrals that have been introduced in Chapter 16 has been added to Section 16.5.

• A subsection on the Vector Form of Green’s Theorem has been added to Section 17.1.

NEW EXAMPLES, FIGURES, AND EXERCISES Numerous examples and accompanying figures have been added to clarify concepts. A variety of exercises have also been added throughout the text, particularly where new applications are available or further conceptual development is advantageous. Figures marked with a icon have been made dynamic and can be accessed via LaunchPad. A selection of these figures also includes brief tutorial videos explaining the concepts at work.

ONLINE HOMEWORK OPTIONS ix

SUPPLEMENTS

For Instructors Instructor’s Solutions Manual Contains worked-out solutions to all exercises in the text.

Test Bank Computerized (CD-ROM), ISBN:1-3190-0939-5 Includes a comprehensive set of multiple-choice test items.

Instructor’s Resource Manual Provides sample course outlines, suggested class time, key points, lecture material, discussion topics, class activities, work- sheets, projects, and questions to accompany the Dynamic Fig- ures.

For Students Student Solutions Manual Single Variable ISBN: 1-4641-7188-2 Multivariable ISBN: 1-4641-7189-0 Contains worked-out solutions to all odd-numbered exercises in the text.

Software Manuals Maple™ and Mathematica® software manuals serve as basic introductions to popular mathematical software options.

ONLINE HOMEWORK OPTIONS

Our new course space, LaunchPad, combines an interactive e-Book with high-quality multimedia content and ready-made assessment options, including LearningCurve adap- tive quizzing. Pre-built, curated units are easy to assign or adapt with your own material, such as readings, videos, quizzes, discussion groups, and more. LaunchPad includes a gradebook that provides a clear window on performance for your whole class, for individ- ual students, and for individual assignments. While a streamlined interface helps students focus on what’s due next, social commenting tools let them engage, make connections, and learn from each other. Use LaunchPad on its own or integrate it with your school’s learning management system so your class is always on the same page. Contact your rep to make sure you have access.

Assets integrated into LaunchPad include:

Interactive e-Book: Every LaunchPad e-Book comes with powerful study tools for stu- dents, video and multimedia content, and easy customization for instructors. Students can search, highlight, and bookmark, making it easier to study and access key content. And instructors can make sure their class gets just the book they want to deliver: customize and rearrange chapters, add and share notes and discussions, and link to quizzes, activities, and other resources.

LearningCurve provides students and instructors with powerful adaptive quizzing, a game-like format, direct links to the e-Book, and instant feedback. The quizzing system features questions tailored specifically to the text and adapts to students’ responses, pro- viding material at different difficulty levels and topics based on student performance.

Dynamic Figures: Over 250 figures from the text have been recreated in a new interactive format for students and instructors to manipulate and explore, making the visual aspects and dimensions of calculus concepts easier to grasp. Brief tutorial videos accompany selected figures and explain the concepts at work.

CalcClips: These whiteboard tutorials provide animated and narrated step-by-step solu- tions to exercises that are based on key problems in the text.

SolutionMaster offers an easy-to-use Web-based version of the instructor’s solutions, allowing instructors to generate a solution file for any set of homework exercises.

x FEATURES

www.webassign.net/freeman.com WebAssign Premium integrates the book’s exercises into the world’s most popular and trusted online homework system, making it easy to assign algorithmically generated homework and quizzes. Algorithmic exercises offer the instructor optional algorith- mic solutions. WebAssign Premium also offers access to resources, including Dynamic Figures, CalcClips whiteboard tutorials, and a “Show My Work” feature. In addition, WebAssign Premium is available with a fully customizable e-Book option.

webwork.maa.org W. H. Freeman offers thousands of algorithmically generated questions (with full solu- tions) through this free, open-source online homework system created at the University of Rochester. Adopters also have access to a shared national library test bank with thou- sands of additional questions, including 2,500 problem sets matched to the book’s table of contents.

FEATURES

CONCEPTUAL INSIGHT Leibniz notation is widely used for several reasons. First, it re- minds us that the derivative df/dx, although not itself a ratio, is in fact a limit of ratios

. Second, the notation specifies the independent variable. This is useful when variables other than x are used. For example, if the independent variable is t , we write df/dt . Third, we often think of d/dx as an “operator” that performs differentiation on functions. In other words, we apply the operator d/dx to f to obtain the derivative df/dx. We will see other advantages of Leibniz notation when we discuss the Chain Rule in Section 3.7.

Ch. 3, p. 123

Conceptual Insights encourage students to develop a conceptual understanding of calculus by explaining important ideas clearly but informally.

GRAPHICAL INSIGHT Can we visualize the rate represented by f (x)? The second derivative is the rate at which f (x) is changing, so f (x) is large if the slopes of the tangent lines change rapidly, as in Figure 3(A). Similarly, f (x) is small if the slopes of the tangent lines change slowly—in this case, the curve is relatively flat, as in Figure 3(B). If f is a linear function [Figure 3(C)], then the tangent line does not change at all and f (x) = 0. Thus, f (x) measures the “bending” or concavity of the graph.

(A) Large second derivative: Tangent lines turn rapidly.

(B) Smaller second derivative: Tangent lines turn slowly.

(C) Second derivative is zero: Tangent line does not change.

FIGURE 3 Ch. 3, p. 153

Graphical Insights enhance students’ visual understanding by making the crucial connections between graphical properties and the underlying concepts.

FEATURES xi

EXAMPLE 3 Evaluate sin2 x dx.

Solution We could apply the reduction formula Eq. (5) from the last section. However, instead, we apply a method that does not rely on knowing that formula. We utilize the trigonometric identity called the double angle formula sin2 x = 12 (1 − cos 2x). Then

sin2 x dx = 1 2 (1 − cos 2x) dx = x

2 − sin 2x

4 + C

Using the trigonometric identities in the margin, we can also integrate cos2 x, obtain- ing the following:REMINDER Useful Identities:

sin2 x = 1 2 (1 − cos 2x)

cos2 x = 1 2 (1 + cos 2x)

sin 2x = 2 sin x cos x

cos 2x = cos2 x − sin2 x

sin2 x dx = x 2

− sin 2x 4

+ C = x 2

− 1 2

sin x cos x + C 1

cos2 x dx = x 2

+ sin 2x 4

+ C = x 2

+ 1 2

sin x cos x + C 2

Ch. 7, p. 380

Reminders are margin notes that link the current discussion to important concepts introduced earlier in the text to give students a quick review and make connections with related ideas.

EXAMPLE 1 Use L’Hôpital’s Rule to evaluate lim x→2

x3 − 8 x4 + 2x − 20 .

Solution Let f (x) = x3 − 8 and g(x) = x4 + 2x − 20. Both f and g are differentiable and f (x)/g(x) is indeterminate of type 0/0 at a = 2 because f (2) = g(2) = 0:

• Numerator: f (2) = 23 − 1 = 0 • Denominator: g(2) = 24 + 2(2) − 20 = 0

Furthermore, g (x) = 4x3 + 2 is nonzero near x = 2, so L’Hôpital’s Rule applies. We may replace the numerator and denominator by their derivatives to obtain

CAUTION When using L’Hôpital’s Rule, be sure to take the derivative of the numerator and denominator separately:

lim x→a

f (x)

g(x) = lim

x→a f (x)

g (x)

Do not differentiate the quotient function y = f (x)/g(x).

lim x→2

x3 − 8 x4 + 2x − 2 = limx→2

(x3 − 8) (x4 + 2x − 2)

L’Hôpital’s Rule

= lim x→2

3x2

4x3 + 2 = 3(22)

4(23) + 2 = 12 34

= 6 17

Ch. 4, p. 224

Caution Notes warn students of common pitfalls they may encounter in understanding the material.

Historical Perspectives are brief vignettes that place key discoveries and conceptual advances in their historical context. They give students a glimpse into some of the accomplishments of great mathematicians and an appreciation for their significance.

HISTORICAL

PERSPECTIVE

(Mechanics Magazine London, 1824)

Geometric series were used as early as the third century bce by Archimedes in a brilliant argu- ment for determining the area S of a “parabolic segment” (shaded region in Figure 3). Given two points A and C on a parabola, there is a point B between A and C where the tangent line is paral- lel to AC (apparently, Archimedes was aware of the Mean Value Theorem more than 2000 years before the invention of calculus). Let T be the area of triangle ABC. Archimedes proved that if D is chosen in a similar fashion relative to AB and E is chosen relative to BC, then

1 4 T = Area( ADB) + Area( BEC) 6

This construction of triangles can be continued. The next step would be to construct the four tri- angles on the segments AD, DB, BE, EC, of

total area 14 2 T . Then construct eight triangles

of total area 14 3 T , etc. In this way, we obtain in-

finitely many triangles that completely fill up the parabolic segment. By the formula for the sum of a geometric series, we get

S = T + 1 4 T + 1

16 T + · · · = T

∞

n=0

1 4n

= 4 3 T

For this and many other achievements, Archi-

medes is ranked together with Newton and Gauss as one of the greatest scientists of all time.

The modern study of infinite series began in the seventeenth century with Newton, Leib- niz, and their contemporaries. The divergence

of ∞

n=1 1/n (called the harmonic series) was

known to the medieval scholar Nicole d’Oresme (1323–1382), but his proof was lost for cen- turies, and the result was rediscovered on more than one occasion. It was also known that the

sum of the reciprocal squares ∞

n=1 1/n2 con-

verges, and in the 1640s, the Italian Pietro Men- goli put forward the challenge of finding its sum. Despite the efforts of the best mathematicians of the day, including Leibniz and the Bernoulli brothers Jakob and Johann, the problem resisted solution for nearly a century. In 1735 the great master Leonhard Euler (at the time, 28 years old) astonished his contemporaries by proving that

1

12 + 1

22 + 1

32 + 1

42 + 1

52 + 1

62 + · · · = π

2

6

This formula, surprising in itself, plays a role in a variety of mathematical fields. A theorem from number theory states that two whole num- bers, chosen randomly, have no common factor with probability 6/π2 ≈ 0.6 (the reciprocal of Euler’s result). On the other hand, Euler’s re- sult and its generalizations appear in the field of statistical mechanics.

Ch. 10, p. 530

xii ACKNOWLEDGMENTS

Assumptions Matter uses short explanations and well-chosen counterexamples to help students appreciate why hypotheses are needed in theorems.

EXAMPLE 3 Assumptions Matter Show that the Product Law cannot be applied to lim x→0

f (x)g(x) if f (x) = x and g(x) = x−1.

Solution For all x = 0, we have f (x)g(x) = x · x−1 = 1, so the limit of the product exists:

lim x→0

f (x)g(x) = lim x→0

1 = 1

However, lim x→0

x−1 does not exist because g(x) = x−1 approaches ∞ as x → 0+ and it approaches −∞ as x → 0−. Therefore, the Product Law cannot be applied and its conclusion does not hold:

lim x→0

f (x) lim x→0

g(x) = lim x→0

x lim x→0

x−1

Does not exist Ch. 2, p. 74

Section Summaries summarize a section’s key points in a concise and useful way and emphasize for students what is most important in each section.

Section Exercise Sets offer a comprehensive set of exercises closely coordinated with the text. These exercises vary in difficulty from routine, to moderate, to more challenging. Also included are icons indicating problems that require the student to give a written

response or require the use of technology .

Chapter Review Exercises offer a comprehensive set of exercises closely coordinated with the chapter material to provide additional problems for self-study or assignments.

ACKNOWLEDGMENTS Colin Adams and W. H. Freeman and Company are grateful to the many instructors from across the United States and Canada who have offered comments that assisted in the development and refinement of this book. These contributions included class testing, manuscript reviewing, problems reviewing, and participating in surveys about the book and general course needs.

ALABAMA Tammy Potter, Gadsden State Community College; David Dempsey, Jacksonville State University; Edwin Smith, Jacksonville State University; Jeff Dodd, Jacksonville State University; Douglas Bailer, Northeast Alabama Community College; Michael Hicks, Shelton State Community College; Patricia C. Eiland, Troy University, Montgomery Campus; Chadia Affane Aji, Tuskegee University; James L. Wang, The University of Alabama; Stephen Brick, University of South Alabama; Jo- erg Feldvoss, University of South Alabama ALASKA Mark A. Fitch, University of Alaska Anchorage; Kamal Narang, University of Alaska An- chorage; Alexei Rybkin, University of Alaska Fairbanks; Martin Getz, University of Alaska Fairbanks ARIZONA Stefania Tracogna, Ari- zona State University; Bruno Welfert, Arizona State University; Light Bryant, Arizona Western College; Daniel Russow, Arizona Western Col- lege; Jennifer Jameson, Coconino College; George Cole, Mesa Com- munity College; David Schultz, Mesa Community College; Michael Bezusko, Pima Community College, Desert Vista Campus; Garry Car- penter, Pima Community College, Northwest Campus; Paul Flasch, Pima County Community College; Jessica Knapp, Pima Community College, Northwest Campus; Roger Werbylo, Pima County Community College; Katie Louchart, Northern Arizona University; Janet McShane, North- ern Arizona University; Donna M. Krawczyk, The University of Ari- zona ARKANSAS Deborah Parker, Arkansas Northeastern College;

J. Michael Hall, Arkansas State University; Kevin Cornelius, Ouachita Baptist University; Hyungkoo Mark Park, Southern Arkansas Univer- sity; Katherine Pinzon, University of Arkansas at Fort Smith; Denise LeGrand, University of Arkansas at Little Rock; John Annulis, University of Arkansas at Monticello; Erin Haller, University of Arkansas, Fayet- teville; Shannon Dingman, University of Arkansas, Fayetteville; Daniel J. Arrigo, University of Central Arkansas CALIFORNIA Michael S. Gagliardo, California Lutheran University; Harvey Greenwald, Califor- nia Polytechnic State University, San Luis Obispo; Charles Hale, Cali- fornia Polytechnic State University; John Hagen, California Polytechnic State University, San Luis Obispo; Donald Hartig, California Polytech- nic State University, San Luis Obispo; Colleen Margarita Kirk, California Polytechnic State University, San Luis Obispo; Lawrence Sze, California Polytechnic State University, San Luis Obispo; Raymond Terry, Califor- nia Polytechnic State University, San Luis Obispo; James R. McKinney, California State Polytechnic University, Pomona; Robin Wilson, Cali- fornia State Polytechnic University, Pomona; Charles Lam, California State University, Bakersfield ; David McKay, California State University, Long Beach; Melvin Lax, California State University, Long Beach; Wal- lace A. Etterbeek, California State University, Sacramento; Mohamed Al- lali, Chapman University; George Rhys, College of the Canyons; Janice Hector, DeAnza College; Isabelle Saber, Glendale Community College;

ACKNOWLEDGMENTS xiii

Peter Stathis, Glendale Community College; Douglas B. Lloyd, Golden West College; Thomas Scardina, Golden West College; Kristin Hartford, Long Beach City College; Eduardo Arismendi-Pardi, Orange Coast Col- lege; Mitchell Alves, Orange Coast College; Yenkanh Vu, Orange Coast College; Yan Tian, Palomar College; Donna E. Nordstrom, Pasadena City College; Don L. Hancock, Pepperdine University; Kevin Iga, Pep- perdine University; Adolfo J. Rumbos, Pomona College; Virginia May, Sacramento City College; Carlos de la Lama, San Diego City College; Matthias Beck, San Francisco State University; Arek Goetz, San Fran- cisco State University; Nick Bykov, San Joaquin Delta College; Eleanor Lang Kendrick, San Jose City College; Elizabeth Hodes, Santa Barbara City College; William Konya, Santa Monica College; John Kennedy, Santa Monica College; Peter Lee, Santa Monica College; Richard Salome, Scotts Valley High School; Norman Feldman, Sonoma State University; Elaine McDonald, Sonoma State University; John D. Eggers, University of California, San Diego; Adam Bowers, University of California, San Diego; Bruno Nachtergaele, University of California, Davis; Boumedi- ene Hamzi, University of California, Davis; Olga Radko, University of California, Los Angeles; Richard Leborne, University of California, San Diego; Peter Stevenhagen, University of California, San Diego; Jeffrey Stopple, University of California, Santa Barbara; Guofang Wei, Uni- versity of California, Santa Barbara; Rick A. Simon, University of La Verne; Alexander E. Koonce, University of Redlands; Mohamad A. Al- wash, West Los Angeles College; Calder Daenzer, University of California, Berkeley; Jude Thaddeus Socrates, Pasadena City College; Cheuk Ying Lam, California State University Bakersfield ; Borislava Gutarts, Califor- nia State University, Los Angeles; Daniel Rogalski, University of Cali- fornia, San Diego; Don Hartig, California Polytechnic State University; Anne Voth, Palomar College; Jay Wiestling, Palomar College; Lindsey Bramlett-Smith, Santa Barbara City College; Dennis Morrow, College of the Canyons; Sydney Shanks, College of the Canyons; Bob Tolar, College of the Canyons; Gene W. Majors, Fullerton College; Robert Diaz, Fuller- ton College; Gregory Nguyen, Fullerton College; Paul Sjoberg, Fullerton College; Deborah Ritchie, Moorpark College; Maya Rahnamaie, Moor- park College; Kathy Fink, Moorpark College; Christine Cole, Moor- park College; K. Di Passero, Moorpark College; Sid Kolpas, Glendale Community College; Miriam Castrconde, Irvine Valley College; Ilkner Erbas-White, Irvine Valley College; Corey Manchester, Grossmont Col- lege; Donald Murray, Santa Monica College; Barbara McGee, Cuesta College; Marie Larsen, Cuesta College; Joe Vasta, Cuesta College; Mike Kinter, Cuesta College; Mark Turner, Cuesta College; G. Lewis, Cuesta College; Daniel Kleinfelter, College of the Desert; Esmeralda Medrano, Citrus College; James Swatzel, Citrus College; Mark Littrell, Rio Hondo College; Rich Zucker, Irvine Valley College; Cindy Torigison, Palomar College; Craig Chamberline, Palomar College; Lindsey Lang, Diablo Valley College; Sam Needham, Diablo Valley College; Dan Bach, Dia- blo Valley College; Ted Nirgiotis, Diablo Valley College; Monte Collazo, Diablo Valley College; Tina Levy, Diablo Valley College; Mona Pan- chal, East Los Angeles College; Ron Sandvick, San Diego Mesa College; Larry Handa, West Valley College; Frederick Utter, Santa Rose Junior College; Farshod Mosh, DeAnza College; Doli Bambhania, DeAnza Col- lege; Charles Klein, DeAnza College; Tammi Marshall, Cauyamaca Col- lege; Inwon Leu, Cauyamaca College; Michael Moretti, Bakersfield Col- lege; Janet Tarjan, Bakersfield College; Hoat Le, San Diego City College; Richard Fielding, Southwestern College; Shannon Gracey, Southwestern College; Janet Mazzarella, Southwestern College; Christina Soderlund, California Lutheran University; Rudy Gonzalez, Citrus College; Robert Crise, Crafton Hills College; Joseph Kazimir, East Los Angeles College; Randall Rogers, Fullerton College; Peter Bouzar, Golden West College; Linda Ternes, Golden West College; Hsiao-Ling Liu, Los Angeles Trade Tech Community College; Yu-Chung Chang-Hou, Pasadena City College; Guillermo Alvarez, San Diego City College; Ken Kuniyuki, San Diego Mesa College; Laleh Howard, San Diego Mesa College; Sharareh Ma- sooman, Santa Barbara City College; Jared Hersh, Santa Barbara City College; Betty Wong, Santa Monica College; Brian Rodas, Santa Monica College; Veasna Chiek, Riverside City College COLORADO Tony

Weathers, Adams State College; Erica Johnson, Arapahoe Community College; Karen Walters, Arapahoe Community College; Joshua D. Lai- son, Colorado College; G. Gustave Greivel, Colorado School of Mines; Holly Eklund, Colorado School of the Mines; Mike Nicholas, Colorado School of the Mines; Jim Thomas, Colorado State University; Eleanor Storey, Front Range Community College; Larry Johnson, Metropolitan State College of Denver; Carol Kuper, Morgan Community College; Larry A. Pontaski, Pueblo Community College; Terry Chen Reeves, Red Rocks Community College; Debra S. Carney, Colorado School of the Mines; Louis A. Talman, Metropolitan State College of Denver; Mary A. Nel- son, University of Colorado at Boulder; J. Kyle Pula, University of Den- ver; Jon Von Stroh, University of Denver; Sharon Butz, University of Denver; Daniel Daly, University of Denver; Tracy Lawrence, Arapa- hoe Community College; Shawna Mahan, University of Colorado Den- ver; Adam Norris, University of Colorado at Boulder; Anca Radulescu, University of Colorado at Boulder; Mike Kawai, University of Colorado Denver; Janet Barnett, Colorado State University–Pueblo; Byron Hur- ley, Colorado State University–Pueblo; Jonathan Portiz, Colorado State University–Pueblo; Bill Emerson, Metropolitan State College of Denver; Suzanne Caulk, Regis University;Anton Dzhamay, University of Northern Colorado CONNECTICUT Jeffrey McGowan, Central Connecticut State University; Ivan Gotchev, Central Connecticut State University; CharlesWaiveris, Central Connecticut State University; Christopher Ham- mond, Connecticut College; Anthony Y. Aidoo, Eastern Connecticut State University; Kim Ward, Eastern Connecticut State University; Joan W. Weiss, Fairfield University; Theresa M. Sandifer, Southern Connecti- cut State University; Cristian Rios, Trinity College; Melanie Stein, Trinity College; Steven Orszag, Yale University DELAWARE Patrick F. Mw- erinde, University of Delaware DISTRICT OF COLUMBIA Jef- frey Hakim, American University; Joshua M. Lansky, American Univer- sity; James A. Nickerson, Gallaudet University FLORIDA Gregory Spradlin, Embry-Riddle University at Daytona Beach; Daniela Popova, Florida Atlantic University; Abbas Zadegan, Florida International Uni- versity; Gerardo Aladro, Florida International University; Gregory Hen- derson, Hillsborough Community College; Pam Crawford, Jacksonville University; Penny Morris, Polk Community College; George Schultz, St. Petersburg College; Jimmy Chang, St. Petersburg College; Carolyn Kistner, St. Petersburg College; Aida Kadic-Galeb, The University of Tampa; Constance Schober, University of Central Florida; S. Roy Choud- hury, University of Central Florida; Kurt Overhiser, Valencia Commu- nity College; Jiongmin Yong, University of Central Florida; Giray Okten, The Florida State University; Frederick Hoffman, Florida Atlantic Uni- versity; Thomas Beatty, Florida Gulf Coast University; Witny Librun, Palm Beach Community College North; Joe Castillo, Broward County College; Joann Lewin, Edison College; Donald Ransford, Edison Col- lege; Scott Berthiaume, Edison College; Alexander Ambrioso, Hillsbor- ough Community College; Jane Golden, Hillsborough Community Col- lege; Susan Hiatt, Polk Community College–Lakeland Campus; Li Zhou, Polk Community College–Winter Haven Campus; Heather Edwards, Semi- nole Community College; Benjamin Landon, Daytona State College; Tony Malaret, Seminole Community College; Lane Vosbury, Seminole Commu- nity College; William Rickman, Seminole Community College; Cheryl Cantwell, Seminole Community College; Michael Schramm, Indian River State College; Janette Campbell, Palm Beach Community College–Lake Worth; Kwai-Lee Chui, University of Florida; Shu-Jen Huang, Univer- sity of Florida GEORGIA Christian Barrientos, Clayton State Uni- versity; Thomas T. Morley, Georgia Institute of Technology; Doron Lu- binsky, Georgia Institute of Technology; Ralph Wildy, Georgia Military College; Shahram Nazari, Georgia Perimeter College; Alice Eiko Pierce, Georgia Perimeter College, Clarkson Campus; Susan Nelson, Georgia Perimeter College, Clarkson Campus; Laurene Fausett, Georgia South- ern University; Scott N. Kersey, Georgia Southern University; Jimmy L. Solomon, Georgia Southern University; Allen G. Fuller, Gordon Col- lege; Marwan Zabdawi, Gordon College; Carolyn A. Yackel, Mercer Uni- versity; Blane Hollingsworth, Middle Georgia State College; Shahryar Heydari, Piedmont College; Dan Kannan, The University of Georgia; June

xiv ACKNOWLEDGMENTS

Jones, Middle Georgia State College; Abdelkrim Brania, Morehouse Col- lege; Ying Wang, Augusta State University; James M. Benedict, Augusta State University; Kouong Law, Georgia Perimeter College; Rob Williams, Georgia Perimeter College; Alvina Atkinson, Georgia Gwinnett Col- lege; Amy Erickson, Georgia Gwinnett College HAWAII Shuguang Li, University of Hawaii at Hilo; Raina B. Ivanova, University of Hawaii at Hilo IDAHO Uwe Kaiser, Boise State University; Charles Kerr, Boise State University; Zach Teitler, Boise State University; Otis Kenny, Boise State University; Alex Feldman, Boise State University; Doug Bul- lock, Boise State University; Brian Dietel, Lewis-Clark State College; Ed Korntved, Northwest Nazarene University; Cynthia Piez, University of Idaho ILLINOIS Chris Morin, Blackburn College; Alberto L. Del- gado, Bradley University; John Haverhals, Bradley University; Herbert E. Kasube, Bradley University; Marvin Doubet, Lake Forest College; MarvinA. Gordon, Lake Forest Graduate School of Management; Richard J. Maher, Loyola University Chicago; Joseph H. Mayne, Loyola University Chicago; Marian Gidea, Northeastern Illinois University; John M.Alongi, Northwestern University; Miguel Angel Lerma, Northwestern Univer- sity; Mehmet Dik, Rockford College; Tammy Voepel, Southern Illinois University Edwardsville; Rahim G. Karimpour, Southern Illinois Univer- sity; Thomas Smith, University of Chicago; Laura DeMarco, University of Illinois; Evangelos Kobotis, University of Illinois at Chicago; Jennifer McNeilly, University of Illinois at Urbana-Champaign; Timur Oikhberg, University of Illinois at Urbana-Champaign; Manouchehr Azad, Harper College; Minhua Liu, Harper College; Mary Hill, College of DuPage; Arthur N. DiVito, Harold Washington College INDIANA Vania Mas- cioni, Ball State University; Julie A. Killingbeck, Ball State University; Kathie Freed, Butler University; Zhixin Wu, DePauw University; John P. Boardman, Franklin College; Robert N. Talbert, Franklin College; Robin Symonds, Indiana University Kokomo; Henry L. Wyzinski, In- diana University Northwest; Melvin Royer, Indiana Wesleyan Univer- sity; Gail P. Greene, Indiana Wesleyan University; David L. Finn, Rose- Hulman Institute of Technology; Chong Keat Arthur Lim, University of Notre Dame IOWA Nasser Dastrange, Buena Vista University; Mark A. Mills, Central College; Karen Ernst, Hawkeye Community College; Richard Mason, Indian Hills Community College; Robert S. Keller, Lo- ras College; Eric Robert Westlund, Luther College; Weimin Han, The University of Iowa KANSAS Timothy W. Flood, Pittsburg State Uni- versity; Sarah Cook, Washburn University; Kevin E. Charlwood, Wash- burn University; Conrad Uwe, Cowley County Community College; David N. Yetter, Kansas State University KENTUCKY Alex M. McAllister, Center College; Sandy Spears, Jefferson Community & Technical College; Leanne Faulkner, Kentucky Wesleyan College; Donald O. Clayton, Madis- onville Community College; Thomas Riedel, University of Louisville; Manabendra Das, University of Louisville; Lee Larson, University of Louisville; Jens E. Harlander, Western Kentucky University; Philip Mc- Cartney, Northern Kentucky University; Andy Long, Northern Kentucky University; Omer Yayenie, Murray State University; Donald Krug, North- ern Kentucky University LOUISIANA William Forrest, Baton Rouge Community College; Paul Wayne Britt, Louisiana State University; Galen Turner, Louisiana Tech University; Randall Wills, Southeastern Louisiana University; Kent Neuerburg, Southeastern Louisiana University; Guoli Ding, Louisiana State University; Julia Ledet, Louisiana State Univer- sity; Brent Strunk, University of Louisiana at Monroe MAINE An- drew Knightly, The University of Maine; Sergey Lvin, The University of Maine; Joel W. Irish, University of Southern Maine; Laurie Woodman, University of Southern Maine; David M. Bradley, The University of Maine; William O. Bray, The University of Maine MARYLAND Leonid Stern, Towson University; Jacob Kogan, University of Maryland Balti- more County; Mark E. Williams, University of Maryland Eastern Shore; Austin A. Lobo, Washington College; Supawan Lertskrai, Harford Com- munity College; Fary Sami, Harford Community College;Andrew Bulleri, Howard Community College MASSACHUSETTS Sean McGrath, Algonquin Regional High School; Norton Starr, Amherst College; Re- nato Mirollo, Boston College; Emma Previato, Boston University; Laura K Gross, Bridgewater State University; Richard H. Stout, Gordon Col-

lege; Matthew P. Leingang, Harvard University; Suellen Robinson, North Shore Community College; Walter Stone, North Shore Community Col- lege; Barbara Loud, Regis College; Andrew B. Perry, Springfield College; Tawanda Gwena, Tufts University; Gary Simundza, Wentworth Institute of Technology; Mikhail Chkhenkeli, Western New England College; David Daniels, Western New England College; Alan Gorfin, Western New Eng- land College; Saeed Ghahramani, Western New England College; Julian Fleron, Westfield State College; Maria Fung, Worchester State University; Brigitte Servatius, Worcester Polytechnic Institute; John Goulet, Worces- ter Polytechnic Institute; Alexander Martsinkovsky, Northeastern Uni- versity; Marie Clote, Boston College; Alexander Kastner, Williams Col- lege; Margaret Peard, Williams College; Mihai Stoiciu, Williams College MICHIGAN Mark E. Bollman, Albion College; Jim Chesla, Grand Rapids Community College; Jeanne Wald, Michigan State University; Al- lan A. Struthers, Michigan Technological University; Debra Pharo, North- western Michigan College; Anna Maria Spagnuolo, Oakland University; Diana Faoro, Romeo Senior High School; Andrew Strowe, University of Michigan–Dearborn; Daniel Stephen Drucker, Wayne State University; Christopher Cartwright, Lawrence Technological University; Jay Treiman, Western Michigan University MINNESOTA Bruce Bordwell, Anoka- Ramsey Community College; Robert Dobrow, Carleton College; Jessie K. Lenarz, Concordia College–Moorhead Minnesota; Bill Tomhave, Con- cordia College; David L. Frank, University of Minnesota; Steven I. Sper- ber, University of Minnesota; Jeffrey T. McLean, University of St. Thomas; Chehrzad Shakiban, University of St. Thomas; Melissa Loe, University of St. Thomas; Nick Christopher Fiala, St. Cloud State University; Vic- tor Padron, Normandale Community College; Mark Ahrens, Normandale Community College; Gerry Naughton, Century Community College; Car- rie Naughton, Inver Hills Community College MISSISSIPPI Vivien G. Miller, Mississippi State University; Ted Dobson, Mississippi State Uni- versity; Len Miller, Mississippi State University; Tristan Denley, The Uni- versity of Mississippi MISSOURI Robert Robertson, Drury Univer- sity; Gregory A. Mitchell, Metropolitan Community College–Penn Valley; Charles N. Curtis, Missouri Southern State University; Vivek Narayanan, Moberly Area Community College; Russell Blyth, Saint Louis University; Julianne Rainbolt, Saint Louis University; Blake Thornton, Saint Louis University; Kevin W. Hopkins, Southwest Baptist University; Joe Howe, St. Charles Community College; Wanda Long, St. Charles Community College;Andrew Stephan, St. Charles Community College MONTANA Kelly Cline, Carroll College; Veronica Baker, Montana State University, Bozeman; Richard C. Swanson, Montana State University; Thomas Hayes- McGoff, Montana State University; Nikolaus Vonessen, The University of Montana NEBRASKA Edward G. Reinke Jr., Concordia University; Judith Downey, University of Nebraska at Omaha NEVADA Jennifer Gorman, College of Southern Nevada; Jonathan Pearsall, College of South- ern Nevada; Rohan Dalpatadu, University of Nevada, Las Vegas; Paul Ai- zley, University of Nevada, Las Vegas NEW HAMPSHIRE Richard Jardine, Keene State College; Michael Cullinane, Keene State College; Roberta Kieronski, University of New Hampshire at Manchester; Erik Van Erp, Dartmouth College NEW JERSEY Paul S. Rossi, College of Saint Elizabeth; Mark Galit, Essex County College; Katarzyna Potocka, Ramapo College of New Jersey; Nora S. Thornber, Raritan Valley Com- munity College; Abdulkadir Hassen, Rowan University; Olcay Ilicasu, Rowan University; Avraham Soffer, Rutgers, The State University of New Jersey; Chengwen Wang, Rutgers, The State University of New Jersey; Shabnam Beheshti, Rutgers University, The State University of New Jer- sey; Stephen J. Greenfield, Rutgers, The State University of New Jersey; John T. Saccoman, Seton Hall University; Lawrence E. Levine, Stevens Institute of Technology; Jana Gevertz, The College of New Jersey; Barry Burd, Drew University; Penny Luczak, Camden County College; John Climent, Cecil Community College; Kristyanna Erickson, Cecil Commu- nity College; Eric Compton, Brookdale Community College; John Atsu- Swanzy, Atlantic Cape Community College NEW MEXICO Kevin Leith, Central New Mexico Community College; David Blankenbaker, Central New Mexico Community College; Joseph Lakey, New Mexico State University; Kees Onneweer, University of New Mexico; Jurg Bolli,

ACKNOWLEDGMENTS xv

The University of New Mexico NEW YORK Robert C. Williams, Al- fred University; Timmy G. Bremer, Broome Community College State University of New York; Joaquin O. Carbonara, Buffalo State College; Robin Sue Sanders, Buffalo State College; Daniel Cunningham, Buffalo State College; Rose Marie Castner, Canisius College; Sharon L. Sullivan, Catawba College; Fabio Nironi, Columbia University; Camil Muscalu, Cornell University; Maria S. Terrell, Cornell University; Margaret Mulli- gan, Dominican College of Blauvelt; Robert Andersen, Farmingdale State University of New York; Leonard Nissim, Fordham University; Jennifer Roche, Hobart and William Smith Colleges; James E. Carpenter, Iona Col- lege; Peter Shenkin, John Jay College of Criminal Justice/CUNY ; Gordon Crandall, LaGuardia Community College/CUNY ; Gilbert Traub, Maritime College, State University of New York; Paul E. Seeburger, Monroe Commu- nity College Brighton Campus; Abraham S. Mantell, Nassau Community College; Daniel D. Birmajer, Nazareth College; Sybil G. Shaver, Pace Uni- versity; Margaret Kiehl, Rensselaer Polytechnic Institute; Carl V. Lutzer, Rochester Institute of Technology; Michael A. Radin, Rochester Institute of Technology; Hossein Shahmohamad, Rochester Institute of Technology; Thomas Rousseau, Siena College; Jason Hofstein, Siena College; Leon E. Gerber, St. Johns University; Christopher Bishop, Stony Brook Univer- sity; James Fulton, Suffolk County Community College; John G. Michaels, SUNY Brockport; Howard J. Skogman, SUNY Brockport; Cristina Ba- cuta, SUNY Cortland ; Jean Harper, SUNY Fredonia; David Hobby, SUNY New Paltz; Kelly Black, Union College; Thomas W. Cusick, University at Buffalo/The State University of New York; Gino Biondini, University at Buffalo/The State University of New York; Robert Koehler, University at Buffalo/The State University of New York; Donald Larson, University of Rochester; Robert Thompson, Hunter College; Ed Grossman, The City College of New York NORTH CAROLINA Jeffrey Clark, Elon Uni- versity; William L. Burgin, Gaston College; Manouchehr H. Misaghian, Johnson C. Smith University; Legunchim L. Emmanwori, North Carolina A&T State University; Drew Pasteur, North Carolina State University; Demetrio Labate, North Carolina State University; Mohammad Kazemi, The University of North Carolina at Charlotte; Richard Carmichael, Wake Forest University; Gretchen Wilke Whipple, Warren Wilson College; John Russell Taylor, University of North Carolina at Charlotte; Mark Ellis, Piedmont Community College NORTH DAKOTA Jim Coykendall, North Dakota State University; Anthony J. Bevelacqua, The University of North Dakota; Richard P. Millspaugh, The University of North Dakota; Thomas Gilsdorf, The University of North Dakota; Michele Iiams, The University of North Dakota; Mohammad Khavanin, University of North Dakota OHIO Christopher Butler, Case Western Reserve University; Pamela Pierce, The College of Wooster; Barbara H. Margolius, Cleveland State University; Tzu-Yi Alan Yang, Columbus State Community College; Greg S. Goodhart, Columbus State Community College; Kelly C. Stady, Cuyahoga Community College; Brian T. Van Pelt, Cuyahoga Commu- nity College; David Robert Ericson, Miami University; Frederick S. Gass, Miami University; Thomas Stacklin, Ohio Dominican University; Vitaly Bergelson, The Ohio State University; Robert Knight, Ohio University; John R. Pather, Ohio University, Eastern Campus; Teresa Contenza, Ot- terbein College; Ali Hajjafar, The University of Akron; Jianping Zhu, The University of Akron; Ian Clough, University of Cincinnati Clermont Col- lege; Atif Abueida, University of Dayton; Judith McCrory, The Univer- sity at Findlay; Thomas Smotzer, Youngstown State University; Angela Spalsbury, Youngstown State University; James Osterburg, The University of Cincinnati; Mihaela A. Poplicher, University of Cincinnati; Frederick Thulin, University of Illinois at Chicago; Weimin Han, The Ohio State Uni- versity; Crichton Ogle, The Ohio State University; Jackie Miller, The Ohio State University; Walter Mackey, Owens Community College; Jonathan Baker, Columbus State Community College OKLAHOMA Christo- pher Francisco, Oklahoma State University; Michael McClendon, Univer- sity of Central Oklahoma; Teri Jo Murphy, The University of Oklahoma; Kimberly Adams, University of Tulsa; Shirley Pomeranz, University of Tulsa OREGON Lorna TenEyck, Chemeketa Community College; Angela Martinek, Linn-Benton Community College; Filix Maisch, Oregon State University; Tevian Dray, Oregon State University; Mark Ferguson,

Chemekata Community College; Andrew Flight, Portland State Univer- sity; Austina Fong, Portland State University; Jeanette R. Palmiter, Port- land State University PENNSYLVANIA John B. Polhill, Bloomsburg University of Pennsylvania; Russell C. Walker, Carnegie Mellon Univer- sity; Jon A. Beal, Clarion University of Pennsylvania; Kathleen Kane, Community College of Allegheny County; David A. Santos, Community College of Philadelphia; David S. Richeson, Dickinson College; Chris- tine Marie Cedzo, Gannon University; Monica Pierri-Galvao, Gannon University; John H. Ellison, Grove City College; Gary L. Thompson, Grove City College; Dale McIntyre, Grove City College; Dennis Ben- choff, Harrisburg Area Community College; William A. Drumin, King’s College; Denise Reboli, King’s College; Chawne Kimber, Lafayette Col- lege; Elizabeth McMahon, Lafayette College; Lorenzo Traldi, Lafayette College; David L. Johnson, Lehigh University; Matthew Hyatt, Lehigh University; Zia Uddin, Lock Haven University of Pennsylvania; Donna A. Dietz, Mansfield University of Pennsylvania; Samuel Wilcock, Mes- siah College; Richard R. Kern, Montgomery County Community College; Michael Fraboni, Moravian College; Neena T. Chopra, The Pennsylva- nia State University; Boris A. Datskovsky, Temple University; Dennis M. DeTurck, University of Pennsylvania; Jacob Burbea, University of Pittsburgh; Mohammed Yahdi, Ursinus College; Timothy Feeman, Vil- lanova University; Douglas Norton, Villanova University; Robert Styer, Villanova University; Michael J. Fisher, West Chester University of Penn- sylvania; Peter Brooksbank, Bucknell University; Emily Dryden, Bucknell University; Larry Friesen, Butler County Community College; Lisa An- gelo, Bucks County College; Elaine Fitt, Bucks County College; Pauline Chow, Harrisburg Area Community College; Diane Benner, Harrisburg Area Community College; Emily B. Dryden, Bucknell University; Erica Chauvet, Waynesburg University RHODE ISLAND Thomas F. Ban- choff, Brown University; Yajni Warnapala-Yehiya, Roger Williams Uni- versity; Carol Gibbons, Salve Regina University; Joe Allen, Community College of Rhode Island ; Michael Latina, Community College of Rhode Island SOUTH CAROLINA Stanley O. Perrine, Charleston South- ern University; Joan Hoffacker, Clemson University; Constance C. Ed- wards, Coastal Carolina University; Thomas L. Fitzkee, Francis Mar- ion University; Richard West, Francis Marion University; John Harris, Furman University; Douglas B. Meade, University of South Carolina; GeorgeAndroulakis, University of South Carolina;Art Mark, University of South Carolina Aiken; Sherry Biggers, Clemson University; Mary Zachary Krohn, Clemson University; Andrew Incognito, Coastal Carolina Univer- sity; Deanna Caveny, College of Charleston SOUTH DAKOTA Dan Kemp, South Dakota State University TENNESSEE Andrew Miller, Belmont University; Arthur A. Yanushka, Christian Brothers University; Laurie Plunk Dishman, Cumberland University; Maria Siopsis, Maryville College; Beth Long, Pellissippi State Technical Community College; Ju- dith Fethe, Pellissippi State Technical Community College;Andrzej Gutek, Tennessee Technological University; Sabine Le Borne, Tennessee Tech- nological University; Richard Le Borne, Tennessee Technological Uni- versity; Maria F. Bothelho, University of Memphis; Roberto Triggiani, University of Memphis; Jim Conant, The University of Tennessee; Pavlos Tzermias, The University of Tennessee; Luis Renato Abib Finotti, Uni- versity of Tennessee, Knoxville; Jennifer Fowler, University of Tennessee, Knoxville; Jo Ann W. Staples, Vanderbilt University; Dave Vinson, Pellis- sippi State Community College; Jonathan Lamb, Pellissippi State Com- munity College TEXAS Sally Haas, Angelina College; Karl Havlak, Angelo State University; Michael Huff, Austin Community College; John M. Davis, Baylor University; Scott Wilde, Baylor University and The Uni- versity of Texas at Arlington; Rob Eby, Blinn College; Tim Sever, Hous- ton Community College–Central; Ernest Lowery, Houston Community College–Northwest; Brian Loft, Sam Houston State University; Jianzhong Wang, Sam Houston State University; Shirley Davis, South Plains Col- lege; Todd M. Steckler, South Texas College; Mary E. Wagner-Krankel, St. Mary’s University; Elise Z. Price, Tarrant County College, Southeast Campus; David Price, Tarrant County College, Southeast Campus; Run- chang Lin, Texas A&M University; Michael Stecher, Texas A&M Univer- sity; Philip B. Yasskin, Texas A&M University; Brock Williams, Texas

xvi ACKNOWLEDGMENTS

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Central Washington University; Patrick Averbeck, Edmonds Community College; Tana Knudson, Heritage University; Kelly Brooks, Pierce Col- lege; Shana P. Calaway, Shoreline Community College; Abel Gage, Skagit Valley College; Scott MacDonald, Tacoma Community College; Jason Preszler, University of Puget Sound ; Martha A. Gady, Whitworth Col- lege; Wayne L. Neidhardt, Edmonds Community College; Simrat Ghu- man, Bellevue College; Jeff Eldridge, Edmonds Community College; Kris Kissel, Green River Community College; Laura Moore-Mueller, Green River Community College; David Stacy, Bellevue College; Eric Schultz, Walla Walla Community College; Julianne Sachs, Walla Walla Community College WEST VIRGINIA David Cusick, Marshall Uni- versity; Ralph Oberste-Vorth, Marshall University; Suda Kunyosying, Shepard University; Nicholas Martin, Shepherd University; Rajeev Ra- jaram, Shepherd University; Xiaohong Zhang, West Virginia State Uni- versity; Sam B. Nadler, West Virginia University WYOMING Clau- dia Stewart, Casper College; Pete Wildman, Casper College; Charles Newberg, Western Wyoming Community College; Lynne Ipina, Univer- sity of Wyoming; John Spitler, University of Wyoming WISCON- SIN Erik R. Tou, Carthage College; Paul Bankston, Marquette Uni- versity; Jane Nichols, Milwaukee School of Engineering; Yvonne Yaz, Milwaukee School of Engineering; Simei Tong, University of Wisconsin– Eau Claire; Terry Nyman, University of Wisconsin–Fox Valley; Robert L. Wilson, University of Wisconsin–Madison; Dietrich A. Uhlen- brock, University of Wisconsin–Madison; Paul Milewski, University of Wisconsin–Madison; Donald Solomon, University of Wisconsin– Milwaukee; Kandasamy Muthuvel, University of Wisconsin–Oshkosh; Sheryl Wills, University of Wisconsin–Platteville; Kathy A. Tomlinson, University of Wisconsin–River Falls; Cynthia L. McCabe, University of Wisconsin–Stevens Point; Matthew Welz, University of Wisconsin– Stevens Point; Joy Becker, University of Wisconsin-Stout; Jeganathan Sriskandarajah , Madison Area Tech College; Wayne Sigelko, Madison Area Tech College CANADA Don St. Jean, George Brown College; Robert Dawson, St. Mary’s University; Len Bos, University of Calgary; Tony Ware, University of Calgary; Peter David Papez, University of Cal- gary; John O’Conner, Grant MacEwan University; Michael P. Lamoureux, University of Calgary; Yousry Elsabrouty, University of Calgary; Darja Kalajdzievska, University of Manitoba; Andrew Skelton, University of Guelph; Douglas Farenick, University of Regina

T he creation of this third edition could not have happened without the help of many people. First, I want to thank the individualswhom I have worked with at W. H. Freeman. Terri Ward and Ruth Baruth convinced me that I should take on this project, and I am grateful to them for their support and their confidence in my ability to tackle it. Throughout this process, Terri has been a huge help. I can always count on her to keep this train on track. Katrina Wilhelm has also been an amazing resource. She brings calm competence and organizational skills that constantly impress me. Tony Palermino has provided expert editorial help throughout the process. He is incredibly knowledgeable about all aspects of mathematics textbooks and has an eye for the details that make a book work. Kerry O’Shaughnessy kept the production process moving forward in a timely manner without ever resorting to threats. John Rogosich was the superb compositor. Patti Brecht handled the copyediting in an expert manner. My thanks are also due to W. H. Freeman’s superb production team: Janice Donnola, Eileen Liang, Blake Logan, Paul Rohloff, and to Ron Weickart at Network Graphics for his skilled and creative execution of the art program.

Many faculty gave critical feedback on the second edition, and their names appear above. I am deeply grateful to them. I do want to particularly thank all of the advisory board members who gave me feedback month after month. Maria Shea Terrell continually sent me excellent unsolicited feedback until I asked to have her on the board. Then it became solicited. The accuracy reviewers, John Alongi, CK Cheung, Kwai-Lee Chui, John Davis, John Eggers, Stephen Greenfield, Roger Lipsett, Vivek Narayanan, and Olga Radko, helped to bring the final version into the form in which it now appears. You think you have found the errors, but you have not.

I also want to thank my colleagues in the Mathematics and Statistics Department at Williams College. I have always known I am incredibly lucky to be a member of this department. There are so many interesting projects and clever pedagogical ideas coming out of the department that it motivates me just because I am trying to keep up.

I would further like to thank my students. Their enthusiasm is what makes teaching fun. I enjoy coming to work every day, and they are what make it such a pleasure.

Finally, I want to thank my two children, Alexa and Colton. They are the ones who keep me grounded, who remind me what works and what doesn’t in the real world. This book is dedicated to them.

Colin Adams

Functions that yield the amount of seismic

activity as a function of time help scientists to

predict volcanic eruptions and earthquakes.

(Douglas Peebles/Science Source)

1 PRECALCULUS REVIEW

C alculus builds on the foundation of algebra, analytic geometry, and trigonometry. Inthis chapter, therefore, we review some concepts, facts, and formulas from precalculus that are used throughout the text. In the last section, we discuss ways in which technology can be used to enhance your visual understanding of functions and their properties.

1.1 Real Numbers, Functions, and Graphs We begin with a short discussion of real numbers. This gives us the opportunity to recall some basic properties and standard notation.

A real number is a number represented by a decimal or “decimal expansion.” There are three types of decimal expansions: finite, repeating, and infinite but nonrepeating. For example,

3 8

= 0.375, 1 7

= 0.142857142857 . . . = 0.142857

π = 3.141592653589793 . . . The number 38 is represented by a finite decimal, whereas

1 7 is represented by a repeatingor

periodic decimal. The bar over 142857 indicates that this sequence repeats indefinitely. The decimal expansion of π is infinite but nonrepeating.

The set of all real numbers is denoted by a boldface R. When there is no risk of confusion, we refer to a real number simply as a number. We also use the standard symbol ∈ for the phrase “belongs to.” Thus,

a ∈ R reads “a belongs to R” The set of integers is commonly denoted by the letter Z (this choice comes from theAdditional properties of real numbers are

discussed in Appendix B. German word Zahl, meaning “number”). Thus, Z = {. . . , −2, −1, 0, 1, 2, . . . }. A whole number is a nonnegative integer—that is, one of the numbers 0, 1, 2, . . . .

A real number is called rational if it can be represented by a fraction p/q, where p and q are integers with q ̸= 0. The set of rational numbers is denoted Q (for “quotient”). Numbers that are not rational, such as π and

√ 2, are called irrational.

We can tell whether a number is rational from its decimal expansion: Rational numbers have finite or repeating decimal expansions, and irrational numbers have infinite, non- repeating decimal expansions. Furthermore, the decimal expansion of a number is unique, apart from the following exception: Every finite decimal is equal to an infinite decimal in which the digit 9 repeats. For example,

1 = 0.999 . . . , 3 8

= 0.375 = 0.374999 . . . , 47 20

= 2.35 = 2.34999 . . .

We visualize real numbers as points on a line (Figure 1). For this reason, real numbers are often referred to as points. The point corresponding to 0 is called the origin.

−2 −1 0 21 FIGURE 1 The set of real numbers represented as a line.

The absolute value of a real number a, denoted |a|, is defined by (Figure 2) 0a

|a|

FIGURE 2 |a| is the distance from a to the origin.

|a| = distance from the origin = {

a if a ≥ 0 −a if a < 0

For example, |1.2| = 1.2 and |−8.35| = 8.35. The absolute value satisfies

|a| = |−a|, |ab| = |a| |b|

1

2 C H A P T E R 1 PRECALCULUS REVIEW

The distance between two real numbers a and b is |b − a|, which is the length of the line segment joining a and b (Figure 3).

a b

|b − a|

−2 −1 0 21 FIGURE 3 The distance from a to b is |b − a|.

Two real numbers a and b are close to each other if |b − a| is small, and this is the case if their decimal expansions agree to many places. More precisely, if the decimal expansions of a and b agree to k places (to the right of the decimal point), then the distance |b − a| is at most 10−k . Thus, the distance between a = 3.1415 and b = 3.1478 is at most 10−2 because a and b agree to two places. In fact, the distance is exactly |3.1478 − 3.1415| = 0.0063.

Beware that |a + b| is not equal to |a| + |b| unless a and b have the same sign or at least one of a and b is zero. If they have opposite signs, cancellation occurs in the sum a + b, and |a + b| < |a| + |b|. For example, |2 + 5| = |2| + |5| but |−2 + 5| = 3, which is less than |−2| + |5| = 7. In any case, |a + b| is never larger than |a| + |b| and this gives us the simple but important triangle inequality:

|a + b| ≤ |a| + |b| 1

We use standard notation for intervals. Given real numbers a < b, there are four intervals with endpoints a and b (Figure 4). They all have length b − a but differ accord- ing to which endpoints are included.

Closed interval [a, b] (endpoints included)

a b Open interval (a, b) (endpoints excluded)

a b Half-open interval [a, b)

a b Half-open interval (a, b]

a b

FIGURE 4 The four intervals with endpoints a and b.

The closed interval [a, b] is the set of all real numbers x such that a ≤ x ≤ b:

[a, b] = {x ∈ R : a ≤ x ≤ b}

We usually write this more simply as {x : a ≤ x ≤ b}, it being understood that x belongs to R. The open and half-open intervals are the sets

The notation (2, 3) could mean the open interval {x : 2 < x < 3} or it could mean the point in the xy-plane with x = 2 and y = 3. In general, the meaning will be apparent from the context. (a, b) = {x : a < x < b}︸ ︷︷ ︸

Open interval (endpoints excluded)

, [a, b) = {x : a ≤ x < b}︸ ︷︷ ︸ Half-open interval

, (a, b] = {x : a < x ≤ b}︸ ︷︷ ︸ Half-open interval

The infinite interval (−∞, ∞) is the entire real line R.Ahalf-infinite interval is closed if it contains its finite endpoint and is open otherwise (Figure 5):

[a, ∞) = {x : a ≤ x}, (−∞, b] = {x : x ≤ b}

[a, ∞) a

(−∞, b] b

FIGURE 5 Closed half-infinite intervals.

Open and closed intervals may be described by inequalities. For example, the interval

0 r

|x| < r

−r FIGURE 6 The interval (−r, r) = {x : |x| < r}.

(−r, r) is described by the inequality |x| < r (Figure 6):

|x| < r ⇔ −r < x < r ⇔ x ∈ (−r, r) 2

More generally, for an interval symmetric about the value c (Figure 7),

c c + rc − r

r r

FIGURE 7 (a, b) = (c − r, c + r), where

c = a + b 2

, r = b − a 2

|x − c| < r ⇔ c − r < x < c + r ⇔ x ∈ (c − r, c + r) 3

Closed intervals are similar, with < replaced by ≤. We refer to r as the radius and to c as the midpoint or center. The intervals (a, b) and [a, b] have midpoint c = 12 (a + b) and radius r = 12 (b − a) (Figure 7).

S E C T I O N 1.1 Real Numbers, Functions, and Graphs 3

EXAMPLE 1 Describe [7, 13] using inequalities. Solution The midpoint of the interval [7, 13] is c = 12 (7 + 13) = 10 and its radius is r = 12 (13 − 7) = 3 (Figure 8). Therefore,137

3 3

10

FIGURE 8 The interval [7, 13] is described by |x − 10| ≤ 3.

[7, 13] = { x ∈ R : |x − 10| ≤ 3

}

EXAMPLE 2 Describe the set S = { x :

∣∣ 1 2x − 3

∣∣ > 4 }

in terms of intervals.

Solution It is easier to consider the opposite inequality ∣∣ 1

2x − 3 ∣∣ ≤ 4 first. By (2),

In Example 2 we use the notation ∪ to denote “union”: The union A ∪ B of sets A and B consists of all elements that belong to either A or B (or to both).

∣∣∣∣ 1 2 x − 3

∣∣∣∣ ≤ 4 ⇔ −4 ≤ 1 2 x − 3 ≤ 4

−1 ≤ 1 2 x ≤ 7 (add 3)

−2 ≤ x ≤ 14 (multiply by 2) Thus,

∣∣ 1 2x − 3

∣∣ ≤ 4 is satisfied when x belongs to [−2, 14]. The set S is the complement,−2 0 14 FIGURE 9 The set S =

{ x :

∣∣ 1 2x − 3

∣∣ > 4 } .

consisting of all numbers x not in [−2, 14]. We can describe S as the union of two intervals: S = (−∞, −2) ∪ (14, ∞) (Figure 9).

Graphing Graphing is a basic tool in calculus, as it is in algebra and trigonometry. Recall that rect- angular (or Cartesian) coordinates in the plane are defined by choosing two perpendicular axes, the x-axis and the y-axis.To a pair of numbers (a, b) we associate the point P located

The term “Cartesian” refers to the French philosopher and mathematician René Descartes (1596–1650), whose Latin name was Cartesius. He is credited (along with Pierre de Fermat) with the invention of analytic geometry. In his great work La Géométrie, Descartes used the letters x, y, z for unknowns and a, b, c for constants, a convention that has been followed ever since.

at the intersection of the line perpendicular to the x-axis at a and the line perpendicular to the y-axis at b [Figure 10(A)]. The numbers a and b are the x- and y-coordinates of P . The x-coordinate is sometimes called the “abscissa” and the y-coordinate the “ordinate.” The origin is the point with coordinates (0, 0).

xx

b

aa

yy

21−1−2 −1

−2

2

1

P = (a, b)

(A) (B)FIGURE 10 Rectangular coordinate system.

The axes divide the plane into four quadrants labeled I–IV, determined by the signs of the coordinates [Figure 10(B)]. For example, quadrant III consists of points (x, y) such that x < 0 and y < 0.

The distance d between two points P1 = (x1, y1) and P2 = (x2, y2) is computed

d

x1

P1 = (x1, y1)

P2 = (x2, y2)

x2

y1

y2

|y2 − y1|

|x2 − x1|

x

y

FIGURE 11 Distance d is given by the distance formula.

using the Pythagorean Theorem. In Figure 11, we see that P1P2 is the hypotenuse of a right triangle with sides a = |x2 − x1| and b = |y2 − y1|. Therefore,

d2 = a2 + b2 = (x2 − x1)2 + (y2 − y1)2

We obtain the distance formula by taking square roots.

Distance Formula The distance between P1 = (x1, y1) and P2 = (x2, y2) is equal to

d = √

(x2 − x1)2 + (y2 − y1)2

4 C H A P T E R 1 PRECALCULUS REVIEW

Once we have the distance formula, we can derive the equation of a circle of radius r and center (a, b) (Figure 12). A point (x, y) lies on this circle if the distance from (x, y)

a

(a, b)

(x, y)

r

b

x

y

FIGURE 12 Circle with equation (x − a)2 + (y − b)2 = r2.

to (a, b) is r: √

(x − a)2 + (y − b)2 = r Squaring both sides, we obtain the standard equation of the circle:

(x − a)2 + (y − b)2 = r2

We now review some definitions and notation concerning functions.

DEFINITION A function f from a set D to a set Y is a rule that assigns, to each element x in D, a unique element y = f (x) in Y . We write

f : D → Y

The set D, called the domain of f , is the set of “allowable inputs.” For x ∈ D, f (x) is called the value of f at x (Figure 13). The range R of f is the subset of Y consisting of all values f (x):

R = {y ∈ Y : f (x) = y for some x ∈ D} Informally, we think of f as a “machine” that produces an output y for every input xA function f : D → Y is also called a

“map.” The sets D and Y can be arbitrary. For example, we can define a map from the set of living people to the set of whole numbers by mapping each person to his or her year of birth. The range of this map is the set of years in which a living person was born. In multivariable calculus, the domain might be a set of points in the two-dimensional plane and the range a set of numbers, points, or vectors.

in the domain D (Figure 14).

f (x)x

Domain D Y

f

FIGURE 13 A function assigns an element f (x) in Y to each x ∈ D.

f (x) Output

x Input

Machine “f ”

FIGURE 14 Think of f as a “machine” that takes the input x and produces the output f (x).

The first part of this text deals with numerical functions f , where both the domain and the range are sets of real numbers. We refer to such a function as f and its value at x as f (x). The letter x is used often to denote the independent variablethat can take on any value in the domain D. We write y = f (x) and refer to y as the dependent variable (because its value depends on the choice of x).

When f is defined by a formula, its natural domain is the set of real numbers x for which the formula is meaningful. For example, the function f (x) =

√ 9 − x has domain

D = {x : x ≤ 9} because √

9 − x is defined if 9 − x ≥ 0. Here are some other examples of domains and ranges:

f (x) Domain D Range R

x2 R {y : y ≥ 0} cos x R {y : −1 ≤ y ≤ 1}

1 x + 1 {x : x ̸= −1} {y : y ̸= 0}

The graph of a function y = f (x) is obtained by plotting the points (a, f (a)) for a in the domain D (Figure 15). If you start at x = a on the x-axis, move up to the graph

x

y = f (x)

Zero of f

f (a) (a, f (a))

a c

y

FIGURE 15

and then over to the y-axis, you arrive at the value f (a). The absolute value |f (a)| is the distance from the graph to the x-axis.

A zero or root of a function f is a number c such that f (c) = 0. The zeros are the values of x where the graph intersects the x-axis.

In Chapter 4, we will use calculus to sketch and analyze graphs. At this stage, to sketch a graph by hand, we can make a table of function values, plot the corresponding points (including any zeros), and connect them by a smooth curve.

S E C T I O N 1.1 Real Numbers, Functions, and Graphs 5

EXAMPLE 3 Find the roots and sketch the graph of f (x) = x3 − 2x. Solution First, we solve

x3 − 2x = x(x2 − 2) = 0

The roots of f are x = 0 and x = ± √

2. To sketch the graph, we plot the roots and a few values listed in Table 1 and join them by a curve (Figure 16).

TABLE 1

x x3 − 2x

−2 −4 −1 1

0 0 1 −1 2 4

−1−2

−4

−1

4

1

2

1

2

2− x

y

FIGURE 16 Graph of f (x) = x3 − 2x.

Functions arising in applications are not always given by formulas. For example, data collected from observation or experiment define functions for which there may be no exact formula. Such functions can be displayed either graphically or by a table of values. Figure 17 and Table 2 display data collected by biologist Julian Huxley (1887–1975) in a study of the antler weight W of male red deer as a function of age t . We will see that many of the tools from calculus can be applied to functions constructed from data in this way.

Antler weight W (kg)

0 20 4 6 8 10 12

Age t (years)

1

2

3

4

5

6

7

8

FIGURE 17 Male red deer shed their antlers every winter and regrow them in the spring. This graph shows average antler weight as a function of age.

TABLE 2

t (years) W (kg) t (years) W (kg)

1 0.48 7 5.34 2 1.59 8 5.62 3 2.66 9 6.18 4 3.68 10 6.81 5 4.35 11 6.21 6 4.92 12 6.1

We can graph not just functions but, more generally, any equation relating y and x.

−1 1

(1, 1)

(1, −1)

x

y

1

−1

FIGURE 18 Graph of 4y2 − x3 = 3. This graph fails the Vertical Line Test, so it is not the graph of a function.

Figure 18 shows the graph of the equation 4y2 − x3 = 3; it consists of all pairs (x, y) satisfying the equation. This curve is not the graph of a function because some x-values are associated with two y-values. For example, x = 1 is associated with y = ±1. A curve is the graph of a function if and only if it passes the Vertical Line Test; that is, every vertical line x = a intersects the curve in at most one point.

We are often interested in whether a function is increasing or decreasing. Roughly speaking, a function f is increasing if its graph goes up as we move to the right and is decreasing if its graph goes down [Figures 19(A) and (B)]. More precisely, we define the notion of increase/decrease on an open interval.

A function f is:

• increasing on (a, b) if f (x1) < f (x2) for all x1, x2 ∈ (a, b) such that x1 < x2. • decreasing on (a, b) if f (x1) > f (x2) for all x1, x2 ∈ (a, b) such that x1 < x2.

6 C H A P T E R 1 PRECALCULUS REVIEW

We say that f is monotonic if it is either increasing or decreasing. In Figure 19(C), the function is not monotonic because it is neither increasing nor decreasing for all x.

A function f is called nondecreasing if f (x1) ≤ f (x2) for x1 < x2 (defined by ≤ rather than a strict inequality <). Nonincreasing functions are defined similarly. Function (D) in Figure 19 is nondecreasing, but it is not increasing on the intervals where the graph is horizontal. Function (E) is increasing everywhere even though it levels off momentarily.

(A) Increasing (C)(B) Decreasing Decreasing on (a, b) but not decreasing everywhere

(D) Nondecreasing but not increasing

(E) Increasing

x

y

x

y

x

y

a b x

y

x

y

FIGURE 19 Another important property is parity, which refers to whether a function is even or

odd:

• f is even if f (−x) = f (x) • f is odd if f (−x) = −f (x)

The graphs of functions with even or odd parity have a special symmetry:

• Even function: Graph is symmetric about the y-axis. This means that if P = (a, b) lies on the graph, then so does Q = (−a, b) [Figure 20(A)].

• Odd function: Graph is symmetric with respect to the origin. This means that if P = (a, b) lies on the graph, then so does Q = (−a, −b) [Figure 20(B)].

Many functions are neither even nor odd [Figure 20(C)].

(A) Even function: f (−x) = f (x) Graph is symmetric about the y-axis.

(B) Odd function: f (−x) = − f (x) Graph is symmetric about the origin.

(C) Neither even nor odd

(a, b)

(a, b)

(−a, b) b

a−a (−a, −b)

b

a

−a

−b

x x x

y

y

y

FIGURE 20

EXAMPLE 4 Determine whether the function is even, odd, or neither.

(a) f (x) = x4 (b) g(x) = x−1 (c) h(x) = x2 + x Solution

(a) f (−x) = (−x)4 = x4. Thus, f (x) = f (−x), and f is even. (b) g(−x) = (−x)−1 = −x−1. Thus, g(−x) = −g(x), and g is odd. (c) h(−x) = (−x)2 + (−x) = x2 − x. We see that h(−x) is not equal to h(x) or to −h(x) = −x2 − x. Therefore, h is neither even nor odd.

EXAMPLE 5 Using Symmetry Sketch the graph of f (x) = 1 x2 + 1 .

Solution The function f is positive [f (x) > 0] and even [f (−x) = f (x)]. Therefore, the graph lies above the x-axis and is symmetric with respect to the y-axis. Furthermore,

S E C T I O N 1.1 Real Numbers, Functions, and Graphs 7

f is decreasing for x ≥ 0 (because a larger value of x makes the denominator larger). We use this information and a short table of values (Table 3) to sketch the graph (Figure 21). Note that the graph approaches the x-axis as we move to the right or left because f (x) gets closer to 0 as |x| increases.

TABLE 3

x 1

x2 + 1

0 1

±1 12 ±2 15

1

−1−2 21 x

y

f (x) = 1 x2 + 1

FIGURE 21

Two important ways of modifying a graph are translation (or shifting) and scaling. Translation consists of moving the graph horizontally or vertically:

DEFINITION Translation (Shifting)

• Vertical translation y = f (x) + c: Shifts the graph by |c| units vertically, upward if c > 0 and downward if c < 0.

• Horizontal translation y = f (x + c): Shifts the graph by |c| units horizontally, to the right if c < 0 and c units to the left if c > 0.

Figure 22 shows the effect of translating the graph of f (x) = 1/(x2 + 1) vertically and

Remember that f (x) + c and f (x + c) are different. The graph of y = f (x) + c is a vertical translation and y = f (x + c) a horizontal translation of the graph of y = f (x).

horizontally.

−1−2 21 x

y

1

2 Shift 1 unit

upward Shift 1 unit to the left

−1−2−3 1 x

y

1

2

−1−2 21 x

y

1

2

(A) y = f (x) = 1 + 1 x2 + 1

(B) y = f (x) + 1 = 1 x2 + 1

(C) y = f (x + 1) = 1 (x + 1)2 + 1

FIGURE 22

EXAMPLE 6 Figure 23(A) is the graph of f (x) = x2, and Figure 23(B) is a horizontal and vertical shift of (A). What is the equation of graph (B)?

−1−2 2 31

2

1

4

3

−1

(A) f (x) = x2 (B)

x

y

−1−2 2 31

2

1

4

3

−1

x

y

FIGURE 23

Solution Graph (B) is obtained by shifting graph (A) 1 unit to the right and 1 unit down. We can see this by observing that the point (0, 0) on the graph of f is shifted to (1, −1). Therefore, (B) is the graph of g(x) = (x − 1)2 − 1.

8 C H A P T E R 1 PRECALCULUS REVIEW

Scaling (also called dilation) consists of compressing or expanding the graph in the

y = −2 f (x)

y = f (x) 2

1

−2

−4

x

y

FIGURE 24 Negative vertical scale factor k = −2.

vertical or horizontal directions:

DEFINITION Scaling

• Vertical scaling y = kf (x): If k > 1, the graph is expanded vertically by the factor k. If 0 < k < 1, the graph is compressed vertically. When the scale factor k is negative (k < 0), the graph is also reflected across the x-axis (Figure 24).

• Horizontal scaling y = f (kx): If k > 1, the graph is compressed in the horizontal direction. If 0 < k < 1, the graph is expanded. If k < 0, then the graph is also reflected across the y-axis.

The amplitude of a function is half the difference between its greatest value and its least value, if it has both a greatest value and least value. Thus, vertical scaling changes the amplitude by the factor |k|.

EXAMPLE 7 Sketch the graphs of f (x) = sin(πx) and its dilates f (3x) and 3f (x). Solution The graph of f (x) = sin(πx) is a sine curve with period 2. It completes one cycle over every interval of length 2—see Figure 25(A). It has amplitude 1.

• The graph of f (3x) = sin(3πx) is a compressed version of y = f (x), completing three cycles instead of one over intervals of length 2 [Figure 25(B)]. It also has amplitude 1.

• The graph of y = 3f (x) = 3 sin(πx) differs from y = f (x) only in amplitude: It is expanded in the vertical direction by a factor of 3 [Figure 25(C)], so its amplitude is 3.

(C) Vertical expansion: y = 3 f (x) = 3sin(πx)

(B) Horizontal compression: y = f (3x) = sin(3πx)

1

2

3

−3

−2

−1

1

−1 2 41 3 2 41 3

(A) y = f (x) = sin(πx)

1

−1 2 41 3 xx x

y

yy

One cycle Three cycles

FIGURE 25 Horizontal and vertical scaling of f (x) = sin(πx).

1.1 SUMMARY

• Absolute value: |a| = {

a if a ≥ 0 −a if a < 0

• Triangle inequality: |a + b| ≤ |a| + |b| • Four intervals with endpoints a and b:

(a, b), [a, b], [a, b), (a, b] • Writing open and closed intervals using inequalities:

(a, b) = {x : |x − c| < r}, [a, b] = {x : |x − c| ≤ r}

where c = 12 (a + b) is the midpoint and r = 12 (b − a) is the radius.

S E C T I O N 1.1 Real Numbers, Functions, and Graphs 9

• Distance d between (x1, y1) and (x2, y2):

d = √

(x2 − x1)2 + (y2 − y1)2

• Equation of circle of radius r with center (a, b):

(x − a)2 + (y − b)2 = r2

• A zero or root of a function f is a number c such that f (c) = 0. • Vertical Line Test: A curve in the plane is the graph of a function if and only if each

vertical line x = a intersects the curve in at most one point.

•

Increasing: f (x1) < f (x2) if x1 < x2 Nondecreasing: f (x1) ≤ f (x2) if x1 < x2 Decreasing: f (x1) > f (x2) if x1 < x2 Nonincreasing: f (x1) ≥ f (x2) if x1 < x2

• Even function: f (−x) = f (x) (graph is symmetric about the y-axis). • Odd function: f (−x) = −f (x) (graph is symmetric about the origin). • Four ways to transform the graph of f :

f (x) + c Shifts graph vertically |c| units (upward if c > 0, downward if c < 0) f (x + c) Shifts graph horizontally |c| units (to the right if c < 0, to the left if c > 0) kf (x) Scales graph vertically by factor k;

if k < 0, graph is reflected across x-axis

f (kx) Scales graph horizontally by factor k (compresses if k > 1); if k < 0, graph is reflected across y-axis

1.1 EXERCISES

Preliminary Questions 1. Give an example of numbers a and b such that a < b and |a| > |b|.

2. Which numbers satisfy |a| = a? Which satisfy |a| = −a? What about |−a| = a?

3. Give an example of numbers a and b such that |a + b| < |a| + |b|.

4. Are there numbers a and b such that |a + b| > |a| + |b|?

5. What are the coordinates of the point lying at the intersection of the lines x = 9 and y = −4?

6. In which quadrant do the following points lie? (a) (1, 4) (b) (−3, 2) (c) (4, −3) (d) (−4, −1) 7. What is the radius of the circle with equation

(x − 7)2 + (y − 8)2 = 9? 8. The equation f (x) = 5 has a solution if (choose one):

(a) 5 belongs to the domain of f . (b) 5 belongs to the range of f .

9. What kind of symmetry does the graph have if f (−x) = −f (x)? 10. Is there a function that is both even and odd?

Exercises 1. Use a calculator to find a rational number r such that

|r − π2| < 10−4.

2. Which of (a)–(f) are true for a = −3 and b = 2? (a) a < b (b) |a| < |b| (c) ab > 0

(d) 3a < 3b (e) −4a < −4b (f) 1 a

< 1 b

In Exercises 3–8, express the interval in terms of an inequality involving absolute value.

3. [−2, 2] 4. (−4, 4) 5. (0, 4)

6. [−4, 0] 7. [1, 5] 8. (−2, 8)

In Exercises 9–12, write the inequality in the form a < x < b.

9. |x| < 8 10. |x − 12| < 8

11. |2x + 1| < 5 12. |3x − 4| < 2

In Exercises 13–18, express the set of numbers x satisfying the given condition as an interval.

13. |x| < 4 14. |x| ≤ 9

15. |x − 4| < 2 16. |x + 7| < 2

17. |4x − 1| ≤ 8 18. |3x + 5| < 1

10 C H A P T E R 1 PRECALCULUS REVIEW

In Exercises 19–22, describe the set as a union of finite or infinite in- tervals.

19. {x : |x − 4| > 2} 20. {x : |2x + 4| > 3}

21. {x : |x2 − 1| > 2} 22. {x : |x2 + 2x| > 2}

23. Match (a)–(f) with (i)–(vi).

(a) a > 3 (b) |a − 5| < 1 3

(c) ∣∣∣∣a −

1 3

∣∣∣∣ < 5 (d) |a| > 5

(e) |a − 4| < 3 (f) 1 ≤ a ≤ 5

(i) a lies to the right of 3.

(ii) a lies between 1 and 7.

(iii) The distance from a to 5 is less than 13 .

(iv) The distance from a to 3 is at most 2.

(v) a is less than 5 units from 13 .

(vi) a lies either to the left of −5 or to the right of 5.

24. Describe { x : x

x + 1 < 0 }

as an interval. Hint: Consider the sign

of x and x + 1 individually.

25. Describe {x : x2 + 2x < 3} as an interval. Hint: Plot y = x2 + 2x − 3.

26. Describe the set of real numbers satisfying |x − 3| = |x − 2| + 1 as a half-infinite interval.

27. Show that if a > b, and a, b ̸= 0, then b−1 > a−1, provided that a and b have the same sign. What happens if a > 0 and b < 0?

28. Which x satisfies both |x − 3| < 2 and |x − 5| < 1?

29. Show that if |a − 5| < 12 and |b − 8| < 12 , then |(a + b) − 13| < 1. Hint: Use the triangle inequality (|a + b| ≤ |a| + |b|).

30. Suppose that |x − 4| ≤ 1. (a) What is the maximum possible value of |x + 4|? (b) Show that |x2 − 16| ≤ 9.

31. Suppose that |a − 6| ≤ 2 and |b| ≤ 3. (a) What is the largest possible value of |a + b|? (b) What is the smallest possible value of |a + b|?

32. Prove that |x| − |y| ≤ |x − y|. Hint: Apply the triangle inequality to y and x − y.

33. Express r1 = 0.27 as a fraction. Hint: 100r1 − r1 is an integer. Then express r2 = 0.2666 . . . as a fraction.

34. Represent 1/7 and 4/27 as repeating decimals.

35. The text states: If the decimal expansions of numbers a and b agree to k places, then |a − b| ≤ 10−k . Show that the converse is false: For all k there are numbers a and b whose decimal expansions do not agree at all but |a − b| ≤ 10−k .

36. Plot each pair of points and compute the distance between them: (a) (1, 4) and (3, 2) (b) (2, 1) and (2, 4)

(c) (0, 0) and (−2, 3) (d) (−3, −3) and (−2, 3)

37. Find the equation of the circle with center (2, 4): (a) with radius r = 3. (b) that passes through (1, −1). 38. Find all points in the xy-plane with integer coordinates located at a distance 5 from the origin. Then find all points with integer coordinates located at a distance 5 from (2, 3).

39. Determine the domain and range of the function

f : {r, s, t, u} → {A,B,C,D,E}

defined by f (r) = A, f (s) = B, f (t) = B, f (u) = E. 40. Give an example of a function whose domain D has three elements and whose range R has two elements. Does a function exist whose do- main D has two elements and whose range R has three elements?

In Exercises 41–48, find the domain and range of the function.

41. f (x) = −x 42. g(t) = t4

43. f (x) = x3 44. g(t) = √

2 − t

45. f (x) = |x| 46. h(s) = 1 s

47. f (x) = 1 x2

48. g(t) = cos 1 t

In Exercises 49–52, determine where f is increasing.

49. f (x) = |x + 1| 50. f (x) = x3

51. f (x) = x4 52. f (x) = 1 x4 + x2 + 1

In Exercises 53–58, find the zeros of f and sketch its graph by plot- ting points. Use symmetry and increase/decrease information where appropriate.

53. f (x) = x2 − 4 54. f (x) = 2x2 − 4

55. f (x) = x3 − 4x 56. f (x) = x3

57. f (x) = 2 − x3 58. f (x) = 1 (x − 1)2 + 1

59. Which of the curves in Figure 26 is the graph of a function?

(B)

(D)(C)

(A)

x

x

x

x

y

y

y

y

FIGURE 26

60. Determine whether the function is even, odd, or neither. (a) f (x) = x5 (b) g(t) = t3 − t2

(c) F(t) = 1 t4 + t2

S E C T I O N 1.1 Real Numbers, Functions, and Graphs 11

61. Determine whether the function is even, odd, or neither.

(a) f (t) = 1 t4 + t + 1 −

1

t4 − t + 1 (b) g(t) = 2 t − 2−t

(c) G(θ) = sin θ + cos θ (d) H(θ) = sin(θ2)

62. Write f (x) = 2x4 − 5x3 + 12x2 − 3x + 4 as the sum of an even and an odd function.

63. Show that f (x) = ln (

1 − x 1 + x

) is an odd function.

64. State whether the function is increasing, decreasing, or neither. (a) Surface area of a sphere as a function of its radius

(b) Temperature at a point on the equator as a function of time

(c) Price of an airline ticket as a function of the price of oil

(d) Pressure of the gas in a piston as a function of volume

In Exercises 65–70, let f be the function shown in Figure 27.

65. Find the domain and range of f .

66. Sketch the graphs of y = f (x + 2) and y = f (x) + 2.

67. Sketch the graphs of y = f (2x), y = f ( 1

2x ) , and y = 2f (x).

68. Sketch the graphs of y = f (−x) and y = −f (−x).

69. Extend the graph of f to [−4, 4] so that it is an even function.

70. Extend the graph of f to [−4, 4] so that it is an odd function.

1 2 3 4 0

1

2

3

4

x

y

FIGURE 27

71. Suppose that f has domain [4, 8] and range [2, 6]. Find the domain and range of:

(a) y = f (x) + 3 (b) y = f (x + 3) (c) y = f (3x) (d) y = 3f (x)

72. Let f (x) = x2. Sketch the graph over [−2, 2] of: (a) y = f (x + 1) (b) y = f (x) + 1 (c) y = f (5x) (d) y = 5f (x)

73. Suppose that the graph of f (x) = sin x is compressed horizontally by a factor of 2 and then shifted 5 units to the right.

(a) What is the equation for the new graph?

(b) What is the equation if you first shift by 5 and then compress by 2?

(c) Verify your answers by plotting your equations.

74. Figure 28 shows the graph of f (x) = |x| + 1. Match the functions (a)–(e) with their graphs (i)–(v).

(a) y = f (x − 1) (b) y = −f (x) (c) y = −f (x) + 2 (d) y = f (x − 1) − 2 (e) y = f (x + 1)

y = f (x) = |x| + 1 (i) (ii)

1 2 3

−1−2−3 −1 2 31

y

x 1 2 3

−1−2−3 −1 2 31

y

x 1 2 3

−1−2−3 −1 2 31

y

x

(iv) (v)(iii)

1 2 3

−1 −2 −3

−2−3 −1 2 31

y

x 1 2 3

−1 −2 −3

−2−3 −1 2 31

y

x 1 2 3

−1 −2 −3

−2−3 −1 2 31

y

x

FIGURE 28

75. Sketch the graph of y = f (2x) and y = f ( 1

2x ) , where f (x) =

|x| + 1 (Figure 28).

76. Find the function f whose graph is obtained by shifting the parabola y = x2 by 3 units to the right and 4 units down, as in Fig- ure 29.

y = f (x)

y = x2

−4

3

y

x

FIGURE 29

77. Define f (x) to be the larger of x and 2 − x. Sketch the graph of f . What are its domain and range? Express f (x) in terms of the absolute value function.

78. For each curve in Figure 30, state whether it is symmetric with respect to the y-axis, the origin, both, or neither.

(D)

(B)

(C)

(A)

yy

yy

xx

x x

FIGURE 30

79. Show that the sum of two even functions is even and the sum of two odd functions is odd.

12 C H A P T E R 1 PRECALCULUS REVIEW

80. Suppose that f and g are both odd. Which of the following func- tions are even? Which are odd? (a) y = f (x)g(x) (b) y = f (x)3

(c) y = f (x) − g(x) (d) y = f (x) g(x)

81. Prove that the only function whose graph is symmetric with respect to both the y-axis and the origin is the function f (x) = 0.

Further Insights and Challenges 82. Prove the triangle inequality (|a + b| ≤ |a| + |b|) by adding the two inequalities

−|a| ≤ a ≤ |a|, −|b| ≤ b ≤ |b|

83. Show that a fraction r = a/b in lowest terms has a finite decimal expansion if and only if

b = 2n5m for some n, m ≥ 0.

Hint: Observe that r has a finite decimal expansion when 10Nr is an integer for some N ≥ 0 (and hence b divides 10N ).

84. Let p = p1 . . . ps be an integer with digits p1, . . . , ps . Show that p

10s − 1 = 0.p1 . . . ps

Use this to find the decimal expansion of r = 211 . Note that

r = 2 11

= 18 102 − 1

85. A function f is symmetric with respect to the vertical line x = a if f (a − x) = f (a + x). (a) Draw the graph of a function that is symmetric with respect to x = 2. (b) Show that if f is symmetric with respect to x = a, then g(x) = f (x + a) is even.

86. Formulate a condition for f to be symmetric with respect to the point (a, 0) on the x-axis.

1.2 Linear and Quadratic Functions Linear functions are the simplest of all functions, and their graphs (lines) are the simplest of all curves. However, linear functions and lines play an enormously important role in calculus. For this reason, you should be thoroughly familiar with the basic properties of linear functions and the different ways of writing an equation of a line.

Let’s recall that a linear functionis a function of the form

f (x) = mx + b (m and b constants)

The graph of f is a line of slope m, and since f (0) = b, the graph intersects the y-axis at the point (0, b) (Figure 1). The number b is called the y-intercept.

The slope-intercept form of the line with slope m and y-intercept b is given by

y = mx + b

x1 x2

y -intercept

y = mx + b

m =

!y

!y

!x

!x

y2

y1

b

y

x

FIGURE 1 The slope m is the ratio “rise over run.”

We use the symbols #x and #y to denote the change (or increment) in x and y = f (x) over an interval [x1, x2] (Figure 1):

#x = x2 − x1, #y = y2 − y1 = f (x2) − f (x1)

The slope m of a line is equal to the ratio

m = #y #x

= vertical change horizontal change

= rise run

S E C T I O N 1.2 Linear and Quadratic Functions 13

This follows from the formula y = mx + b: #y

#x = y2 − y1

x2 − x1 = (mx2 + b) − (mx1 + b)

x2 − x1 = m(x2 − x1)

x2 − x1 = m

The slope m measures the rate of change of y with respect to x. In fact, by writing

#y = m#x

we see that a 1-unit increase in x (i.e., #x = 1) produces an m-unit change #y in y. For example, if m = 5, then y increases by 5 units per unit increase in x. The rate-of-change interpretation of the slope is fundamental in calculus. We discuss it in greater detail in Section 2.1.

Graphically, the slope m measures the steepness of the line y = mx + b. Figure 2(A) shows lines through a point of varying slope m. Note the following properties:

• Steepness: The larger the absolute value |m|, the steeper the line. • Positive slope: If m > 0, the line slants upward from left to right. • Negative slope: If m < 0, the line slants downward from left to right. • f (x) = mx + b is increasing if m > 0 and decreasing if m < 0. • The horizontal line y = b has slope m = 0 [Figure 2(B)]. • A vertical line has equation x = c, where c is a constant. The slope of a vertical line

is undefined. It is not possible to write the equation of a vertical line in slope-intercept form y = mx + b. A vertical line is not the graph of a function [Figure 2(B)].

125

0.5−0.5

0

(A) Lines of varying slopes through P

−1 −2 −5

y = b (slope 0)

x = c (slope undefined)

c

b P P

(B) Horizontal and vertical lines through P

yy

xx

FIGURE 2

Scale is especially important in applications because the steepness of a graph depends

CAUTION Graphs are often plotted using different scales for the x- and y-axes. This is necessary to keep the sizes of graphs within reasonable bounds. However, when the scales are different, lines do not appear with their true slopes.

on the choice of units for the x- and y-axes. We can create very different subjective impressions by changing the scale. Figure 3 shows the growth of company profits over a 4-year period. The two plots convey the same information, but the left-hand plot makes the growth look more dramatic.

100

125

150

100 20112010 2012 2013 201420112010 2012 2013 2014

125 150 175 200 225 250 275 300

FIGURE 3 Growth of company profits.

14 C H A P T E R 1 PRECALCULUS REVIEW

Slope = m

Slope = m Slope = m

(B) Perpendicular lines (A) Parallel lines

yy

xx

Slope = − 1m

FIGURE 4 Parallel and perpendicular lines.

Next, we recall the relation between the slopes of parallel and perpendicular lines (Figure 4):

• Lines of slopes m1 and m2 are parallel if and only if m1 = m2. • Lines of slopes m1 and m2 are perpendicular if and only if

m1 = − 1

m2 (or m1m2 = −1)

CONCEPTUAL INSIGHT The increments over an interval [x1, x2]:

#x = x2 − x1, #y = f (x2) − f (x1)

are defined for any function f (linear or not), but the ratio #y/#x may depend on the interval (Figure 5). The characteristic property of a linear function f (x) = mx + b is that #y/#x has the same value m for every interval. In other words, y has a constant rate of change with respect to x. We can use this property to test if two quantities are related by a linear equation.

x

y

x

Nonlinear function: the ratio y / x changes, depending

on the interval.

Linear function: the ratio y / x is the same over

all intervals.

x

y

x

y

y

FIGURE 5

EXAMPLE 1 Testing for a Linear Relationship Do the data in Table 1 suggest a linear relation between the pressure P and temperature T of a gas?

TABLE 1

Temperature (◦C) Pressure (kPa)

40 1365.80 45 1385.40 55 1424.60 70 1483.40 80 1522.60

S E C T I O N 1.2 Linear and Quadratic Functions 15

Solution We calculate #P/#T at successive data points and check whether this ratio is constant:

(T1, P1) (T2, P2) #P

#T

(40, 1365.80) (45, 1385.40) 1385.40 − 1365.80

45 − 40 = 3.92

(45, 1385.40) (55, 1424.60) 1424.60 − 1385.40

55 − 45 = 3.92

(55, 1424.60) (70, 1483.40) 1483.40 − 1424.60

70 − 55 = 3.92

(70, 1483.40) (80, 1522.60) 1522.60 − 1483.40

80 − 70 = 3.92

Because #P/#T has the constant value 3.92, the data points lie on a line with slope m = 3.92. This is confirmed in the plot in Figure 6.

Real experimental data are unlikely to reveal perfect linearity, even if the data points do essentially lie on a line. The method of “linear regression” is used to find the linear function that best fits the data.

40 60 80

1350

1400

1450

1500

1550 Pressure (kPa)

T (°C)

FIGURE 6 Line through pressure- temperature data points.

As mentioned above, it is important to be familiar with the standard ways of writing the equation of a line. The general linear equation is

ax + by = c 1

where a and b are not both zero. For b = 0, we obtain the vertical line ax = c. When b ̸= 0, we can rewrite Eq. (1) in slope-intercept form. For example, −6x + 2y = 3 can be rewritten as y = 3x + 32 .

Two other forms we will use frequently are the point-slope and point-point forms.

(a1, b1)

(a2, b2)

b2 − b1

a2 − a1

x

y

FIGURE 7 Slope of the line between P = (a1, b1) and Q = (a2, b2) is m = b2 − b1

a2 − a1 .

Given a point P = (a, b) and a slope m, the equation of the line through P with slope m is y − b = m(x − a). Similarly, the line through two distinct points P = (a1, b1) and Q = (a2, b2) has slope (Figure 7)

m = b2 − b1 a2 − a1

Therefore, we can write its equation as y − b1 = m(x − a1).

If a = 0, point-slope form becomes slope-intercept form y = mx + b.

Additional Equations for Lines

1. Point-slope form of the line through P = (a, b) with slope m:

y − b = m(x − a)

2. Point-point form of the line through P = (a1, b1) and Q = (a2, b2):

y − b1 = m(x − a1), where m = b2 − b1 a2 − a1

16 C H A P T E R 1 PRECALCULUS REVIEW

EXAMPLE 2 Line of Given Slope Through a Given Point Find the equation of the line through (9, 2) with slope − 23 .

P = (9, 2)

x + 8y = −

2

8

x

y

9 12

2 3

FIGURE 8 Line through P = (9, 2) with slope m = − 23 .

Solution In point-slope form:

y − 2 = −2 3 (x − 9)

In slope-intercept form: y = − 23 (x − 9) + 2 or y = − 23x + 8. See Figure 8.

EXAMPLE 3 Line Through Two Points Find the equation of the line through (2, 1) and (9, 5).

Solution The line has slope

m = 5 − 1 9 − 2 =

4 7

Because (2, 1) lies on the line, its equation in point-slope form is y − 1 = 47 (x − 2).

A quadratic function is a function defined by a quadratic polynomial

f (x) = ax2 + bx + c (a, b, c, constants with a ̸= 0) The graph of f is a parabola (Figure 9). The parabola opens upward if the leading coefficient a is positive and downward if a is negative. The discriminant of f (x) is the quantity

D = b2 − 4ac The roots of f are given by the quadratic formula (see Exercise 58):

Roots of f = −b ± √

b2 − 4ac 2a

= −b ± √

D

2a

The sign of D determines whether or not f has real roots (Figure 9). If D > 0, then f has two real roots, and if D = 0, it has one real root (a “double root”). If D < 0, then

√ D is

imaginary and f has no real roots.

Two real roots a > 0 and D > 0

Double root a > 0 and D = 0

No real roots a > 0 and D < 0

Two real roots a < 0 and D > 0

xxxx

yyyy

FIGURE 9 Graphs of quadratic functions f (x) = ax2 + bx + c.

When f has two real roots r1 and r2, then f (x) factors as

f (x) = a(x − r1)(x − r2) For example, f (x) = 2x2 − 3x + 1 has discriminant D = b2 − 4ac = 9 − 8 = 1 > 0, and by the quadratic formula, its roots are (3 ± 1)/4 or 1 and 12 . Therefore,

f (x) = 2x2 − 3x + 1 = 2(x − 1) (

x − 1 2

)

The technique of completing the square consists of writing a quadratic polynomial as a multiple of a square plus a constant. Then

x2 + bx + c = x2 + bx + (

b

2

)2 −

( b

2

)2 + c =

( x + b

2

)2 −

( b

2

)2 + c

If there is a constant a multiplying the x2 term, we factor that out first, as demonstrated in the following example.

S E C T I O N 1.2 Linear and Quadratic Functions 17

EXAMPLE 4 Completing the Square Complete the square for the quadratic polyno- mial f (x) = 4x2 − 12x + 3.

Cuneiform texts written on clay tablets show that the method of completing the square was known to ancient Babylonian mathematicians who lived some 4000 years ago.

Solution First factor out the leading coefficient:

4x2 − 12x + 3 = 4 (

x2 − 3x + 3 4

)

Then complete the square for the term x2 − 3x: Ignoring air resistance, a basketball follows a parabolic path (Figure 10).

FIGURE 10

x2 − 3x = x2 − 3x + (

3 2

)2 −

( 3 2

)2 =

( x − 3

2

)2 − 9

4

Therefore,

4x2 − 12x + 3 = 4 ((

x − 3 2

)2 − 9

4 + 3

4

)

= 4 (

x − 3 2

)2 − 6

The method of completing the square can be used to find the minimum or maximum value of a quadratic function.

EXAMPLE 5 Finding the Maximum of a Quadratic Function Complete the square and find the maximum value of f (x) = −x2 + 4x + 1.

5

1

2 x

y

FIGURE 11 Graph of f (x) = −x2 + 4x + 1.

Solution We have

f (x) = −(x2 − 4x − 1) = −(x2 − 4x + 4 − 4 − 1) = −((x − 2)2 − 5) = This term is ≤ 0︷ ︸︸ ︷ −(x − 2)2 + 5

Thus, f (x) ≤ 5 for all x, and the maximum value of f is f (2) = 5 (Figure 11).

1.2 SUMMARY

• A linear function is a function of the form f (x) = mx + b. • The general equation of a line is ax + by = c. The line y = c is horizontal and x = c

is vertical. • Three convenient ways of writing the equation of a nonvertical line:

– Slope-intercept form: y = mx + b (slope m and y-intercept b) – Point-slope form: y − b = m(x − a) [slope m, passes through (a, b)] – Point-point form: The line through two points P = (a1, b1) and Q = (a2, b2) has

slope m = b2 − b1 a2 − a1

and equation y − b1 = m(x − a1).

• Two lines of slopes m1 and m2 are parallel if and only if m1 = m2, and they are perpen- dicular if and only if m1 = −1/m2.

• Quadratic function: f (x) = ax2 + bx + c. The roots are x = (−b ± √

D)/2a, where D = b2 − 4ac is the discriminant. The roots are real and distinct if D > 0, there is a double root if D = 0, and there are no real roots if D < 0.

• Completing the square consists of writing a quadratic function as a multiple of a square plus a constant.

1.2 EXERCISES

Preliminary Questions 1. What is the slope of the line y = −4x − 9? 2. Are the lines y = 2x + 1 and y = −2x − 4 perpendicular? 3. When is the line ax + by = c parallel to the y-axis? To the x-axis?

4. Suppose y = 3x + 2. What is #y if x increases by 3? 5. What is the minimum of f (x) = (x + 3)2 − 4? 6. What is the result of completing the square for f (x) = x2 + 1?

18 C H A P T E R 1 PRECALCULUS REVIEW

Exercises In Exercises 1–4, find the slope, the y-intercept, and the x-intercept of the line with the given equation.

1. y = 3x + 12 2. y = 4 − x

3. 4x + 9y = 3 4. y − 3 = 12 (x − 6) In Exercises 5–8, find the slope of the line.

5. y = 3x + 2 6. y = 3(x − 9) + 2 7. 3x + 4y = 12 8. 3x + 4y = −8

In Exercises 9–20, find the equation of the line with the given descrip- tion.

9. Slope 3, y-intercept 8

10. Slope −2, y-intercept 3 11. Slope 3, passes through (7, 9)

12. Slope −5, passes through (0, 0) 13. Horizontal, passes through (0, −2) 14. Passes through (−1, 4) and (2, 7) 15. Parallel to y = 3x − 4, passes through (1, 1) 16. Passes through (1, 4) and (12, −3) 17. Perpendicular to 3x + 5y = 9, passes through (2, 3) 18. Vertical, passes through (−4, 9) 19. Horizontal, passes through (8, 4)

20. Slope 3, x-intercept 6

21. Find the equation of the perpendicular bisector of the segment join- ing (1, 2) and (5, 4) (Figure 12). Hint: The midpoint Q of the segment

joining (a, b) and (c, d) is (

a + c 2

, b + d

2

) .

Q

(1, 2)

(5, 4)

Perpendicular bisector

x

y

FIGURE 12

22. Intercept-Intercept Form Show that if a, b ̸= 0, then the line with x-intercept x = a and y-intercept y = b has equation (Figure 13)

x

a + y

b = 1

b

a x

y

FIGURE 13

23. Find an equation of the line with x-intercept x = 4 and y-intercept y = 3.

24. Find y such that (3, y) lies on the line of slope m = 2 through (1, 4).

25. Determine whether there exists a constant c such that the line x + cy = 1: (a) has slope 4. (b) passes through (3, 1). (c) is horizontal. (d) is vertical.

26. Assume that the number N of concert tickets that can be sold at a price of P dollars per ticket is a linear function N(P ) for 10 ≤ P ≤ 40. Determine N(P ) (called the demand function) if N(10) = 500 and N(40) = 0. What is the decrease #N in the number of tickets sold if the price is increased by #P = 5 dollars?

27. Suppose that the number of a certain type of computer that can be sold when its price is P (in dollars) is given by a linear function N(P ). Determine N(P ) if N(1000) = 10,000 and N(1500) = 7,500. What is the change #N in the number of computers sold if the price is increased by #P = 100 dollars?

28. Suppose that the demand for Colin’s kidney pies is linear in the price P . Determine the demand function N as a function of P giving the number of pies sold when the price is P if he can sell 100 pies when the price is $5.00 and he can sell 40 pies when the price is $10.00. De- termine the revenue (N × P) for prices P = 5, 6, 7, 8, 9, 10 and then choose a price to maximize the revenue.

29. Materials expand when heated. Consider a metal rod of length L0 at temperature T0. If the temperature is changed by an amount #T , then the rod’s length approximately changes by #L = αL0#T , where α is the thermal expansion coefficient and #T is not an extreme temperature change. For steel, α = 1.24 × 10−5 ◦C−1. (a) A steel rod has length L0 = 40 cm at T0 = 40◦C. Find its length at T = 90◦C. (b) Find its length at T = 50◦C if its length at T0 = 100◦C is 65 cm. (c) Express length L as a function of T if L0 = 65 cm at T0 = 100◦C.

30. Do the points (0.5, 1), (1, 1.2), (2, 2) lie on a line?

31. Find b such that (2, −1), (3, 2), and (b, 5) lie on a line.

32. Find an expression for the velocity v as a linear function of t that matches the following data:

t (s) 0 2 4 6

v (m/s) 39.2 58.6 78 97.4

33. The period T of a pendulum is measured for pendulums of several different lengths L. Based on the following data, does T appear to be a linear function of L?

L (cm) 20 30 40 50

T (s) 0.9 1.1 1.27 1.42

34. Show that f is linear of slope m if and only if

f (x + h) − f (x) = mh (for all x and h)

That is to say, prove the following two statements: (a) f is linear of slope m implies that f (x + h) − f (x) = mh (for all x and h).

S E C T I O N 1.3 The Basic Classes of Functions 19

(b) f (x + h) − f (x) = mh (for all x and h) implies that f is linear of slope m.

35. Find the roots of the quadratic polynomials: (a) f (x) = 4x2 − 3x − 1 (b) f (x) = x2 − 2x − 1 In Exercises 36–43, complete the square and find the minimum or max- imum value of the quadratic function.

36. y = x2 + 2x + 5 37. y = x2 − 6x + 9

38. y = −9x2 + x 39. y = x2 + 6x + 2

40. y = 2x2 − 4x − 7 41. y = −4x2 + 3x + 8

42. y = 3x2 + 12x − 5 43. y = 4x − 12x2

44. Sketch the graph of y = x2 − 6x + 8 by plotting the roots and the minimum point.

45. Sketch the graph of y = x2 + 4x + 6 by plotting the minimum point, the y-intercept, and one other point.

46. If the alleles A and B of the cystic fibrosis gene occur in a popu- lation with frequencies p and 1 − p (where p is a fraction between 0 and 1), then the frequency of heterozygous carriers (carriers with both alleles) is 2p(1 − p). Which value of p gives the largest frequency of heterozygous carriers?

47. For which values of c does f (x) = x2 + cx + 1 have a double root? No real roots?

48. Let f be a quadratic function and c a constant. Which of the following statements is correct? Explain graphically. (a) There is a unique value of c such that y = f (x) − c has a double root. (b) There is a unique value of c such that y = f (x − c) has a double root.

49. Prove that x + 1x ≥ 2 for all x > 0. Hint: Consider (x1/2 − x−1/2)2. 50. Let a, b > 0. Show that the geometric mean

√ ab is not larger than

the arithmetic mean (a + b)/2. Hint: Use a variation of the hint given in Exercise 49.

51. If objects of weights x and w1 are suspended from the balance in Figure 14(A), the cross-beam is horizontal if bx = aw1. If the lengths a and b are known, we may use this equation to determine an unknown weight x by selecting w1 such that the cross-beam is horizontal. If a and b are not known precisely, we might proceed as follows. First balance x by w1 on the left as in (A). Then switch places and balance x by w2 on the right as in (B). The average x̄ = 12 (w1 + w2) gives an estimate for x. Show that x̄ is greater than or equal to the true weight x.

w1

(A) (B)

a

w2x x

b a b

FIGURE 14

52. Find numbers x and y with sum 10 and product 24. Hint: Find a quadratic polynomial satisfied by x.

53. Find a pair of numbers whose sum and product are both equal to 8.

54. Show that the parabola y = x2 consists of all points P such that d1 = d2, where d1 is the distance from P to

( 0, 14

) and d2 is the dis-

tance from P to the line y = − 14 (Figure 15).

d1

d2

P = (x, x2)

y = x2

1 4

1 4

−

x

y

FIGURE 15

Further Insights and Challenges 55. Show that if f and g are linear, then so is f + g. Is the same true of fg?

56. Show that if f and g are linear functions such that f (0) = g(0) and f (1) = g(1), then f = g.

57. Show that #y/#x for the function f (x) = x2 over the interval [x1, x2] is not a constant, but depends on the interval. Determine the exact dependence of #y/#x on x1 and x2.

58. Complete the square and use the result to derive the quadratic for- mula for the roots of ax2 + bx + c = 0.

59. Let a, c ̸= 0. Show that the roots of

ax2 + bx + c = 0 and cx2 + bx + a = 0

are reciprocals of each other.

60. Show, by completing the square, that the parabola

y = ax2 + bx + c

is congruent to y = ax2 by a vertical and horizontal translation. 61. Prove Viète’s Formulas: The quadratic polynomial with α and β as roots is x2 + bx + c, where b = −α − β and c = αβ.

1.3 The Basic Classes of Functions It would be impossible (and useless) to describe all possible functions f . Since the values of a function can be assigned arbitrarily, a function chosen at random would likely be so complicated that we could neither graph it nor describe it in any reasonable way. However, calculus makes no attempt to deal with all functions. The techniques of calculus, powerful

20 C H A P T E R 1 PRECALCULUS REVIEW

and general as they are, apply only to functions that are sufficiently “well-behaved” (we will see what well-behaved means when we study the derivative in Chapter 3). Fortunately, such functions are adequate for a vast range of applications.

Most of the functions considered in this text are constructed from the following familiar classes of well-behaved functions:

polynomials rational functions algebraic functions

exponential functions trigonometric functions

logarithmic functions inverse trigonometric functions

We shall refer to these as the basic functions.

• Polynomials: For any real number m, f (x) = xm is called the power function with exponent m. Power functions include f (x) = x3, f (x) = x−7 and f (x) = xπ . The base is the variable and the exponent is a constant. For now, we are interested in power functions with exponents that are positive integers. A polynomial is a sum of multiples of power functions with exponents that are positive integers or zero (making the term a constant in that case) (Figure 1):

5

2−2 −1 1 x

y

FIGURE 1 The polynomial y = x5 − 5x3 + 4x.

f (x) = x5 − 5x3 + 4x, g(t) = 7t6 + t3 − 3t − 1, h(x) = x9

Thus, the function f (x) = x + x−1 is not a polynomial because it includes a power x−1 with a negative exponent. The general polynomial P in the variable x may be written

P(x) = anxn + an−1xn−1 + · · · + a1x + a0 – The numbers a0, a1, . . . , an are called coefficients. – The degree of P is n (assuming that an ̸= 0). – The coefficient an is called the leading coefficient. – The domain of P is R.

• A rational function is a quotient of two polynomials (Figure 2):

5

−3

−2 1 x

y

FIGURE 2 The rational function

f (x) = x + 1 x3 − 3x + 2 .

f (x) = P(x) Q(x)

[P(x) and Q(x) polynomials]

The domain of f is the set of numbers x such that Q(x) ̸= 0. For example,

f (x) = 1 x2

domain {x : x ̸= 0}

h(t) = 7t 6 + t3 − 3t − 1

t2 − 1 domain {t : t ̸= ±1}

Every polynomial is also a rational function [with Q(x) = 1]. • An algebraic function is produced by taking sums, products, and quotients of roots

of polynomials and rational functions (Figure 3):

2−2 x

y

FIGURE 3 The algebraic function f (x) =

√ 1 + 3x2 − x4.

f (x) = √

1 + 3x2 − x4, g(t) = ( √

t − 2)−2, h(z) = z + z −5/3

5z3 − √z A number x belongs to the domain of f if each term in the formula is defined and

Any function that is not algebraic is called transcendental. Exponential and trigonometric functions are examples, as are the Bessel and gamma functions that appear in engineering and statistics. The term “transcendental” goes back to the 1670s, when it was used by Gottfried Wilhelm Leibniz (1646–1716) to describe functions of this type.

the result does not involve division by zero. For example, g(t) is defined if t ≥ 0 and

√ t ̸= 2, so the domain of g is D = {t : t ≥ 0 and t ̸= 4}. More generally,

algebraic functions are defined by polynomial equations between x and y. In this case, we say that y is implicitly defined as a function of x. For example, the equation y4 + 2x2y + x4 = 1 defines y implicitly as a function of x.

• Exponential functions: The function f (x) = bx , where b > 0, is called the expo- nential function with base b. Some examples are

f (x) = 2x, g(t) = 10t , h(x) = (

1 3

)x , p(t) = (

√ 5)t

S E C T I O N 1.3 The Basic Classes of Functions 21

Exponential functions and their inverses, the logarithmic functions, are treated in greater detail in Section 1.6.

• Trigonometric functions are functions built from sin x and cos x. These functions and their inverses are discussed in the next two sections.

Constructing New Functions Given functions f and g, we can construct new functions by forming the sum, difference, product, and quotient functions:

(f + g)(x) = f (x) + g(x), (f − g)(x) = f (x) − g(x)

(fg)(x) = f (x) g(x), (

f

g

) (x) = f (x)

g(x) (where g(x) ̸= 0)

For example, if f (x) = x2 and g(x) = sin x, then (f + g)(x) = x2 + sin x, (f − g)(x) = x2 − sin x

(fg)(x) = x2 sin x, (

f

g

) (x) = x

2

sin x

We can also multiply functions by constants. A function of the form

h(x) = c1f (x) + c2g(x) (c1, c2 constants) is called a linear combination of f and g.

Composition is another important way of constructing new functions. The compo- sition of f and g is the function f ◦ g defined by (f ◦ g)(x) = f (g(x)). The domain of f ◦ g is the set of values of x in the domain of g such that g(x) lies in the domain of f .

EXAMPLE 1 Compute the composite functions f ◦ g and g ◦ f and discuss their domains, where

f (x) = √x, g(x) = 1 − x Solution We haveExample 1 shows that the composition of

functions is not commutative: The functions f ◦ g and g ◦ f may be (and usually are) different.

(f ◦ g)(x) = f (g(x)) = f (1 − x) = √

1 − x The square root

√ 1 − x is defined if 1 − x ≥ 0 or x ≤ 1, so the domain of f ◦ g is

{x : x ≤ 1}. On the other hand, (g ◦ f )(x) = g(f (x)) = g(√x) = 1 − √x

The domain of g ◦ f is {x : x ≥ 0}.

Elementary Functions As noted above, we can produce new functions by applying the operations of addition,Inverse functions are discussed in Section

1.5. subtraction, multiplication, division, and composition. It is convenient to refer to a function constructed in this way from the basic functions listed above as an elementary function. The following functions are elementary:

f (x) = √

2x + sin x, f (x) = 10 √

x, f (x) = 1 + x −1

1 + cos x

Piecewise-Defined Functions We can also create new functions by piecing together functions defined over limited domains, obtaining piecewise-defined functions. One example we have already seen is the absolute value function defined by

|x| = {−x when x < 0 x when x ≥ 0

22 C H A P T E R 1 PRECALCULUS REVIEW

EXAMPLE 2 Given the function f , determine its domain, range, and whether or not it is increasing or decreasing for different values of x.

f (x) = {

1 when x < 0 x + 1 when x ≥ 0

Solution The function f appears in Figure 4. It is defined for all values of x so the domain

y = x + 1

y = 1

x

y

x < 0 x ≥ 0

FIGURE 4 A function defined piecewise. is all real numbers. However, for x < 0 the range is just the single value of 1, and for x ≥ 0 the range is all x ≥ 1. Hence, the range of the function is {x : x ≥ 1}. The function is neither increasing nor decreasing for x < 0; however, the function is increasing for x ≥ 0.

1.3 SUMMARY

• For m a real number, f (x) = xm is called the power function with exponent m. A poly- nomial P is a sum of multiples of xm, where m is a whole number:

P(x) = anxn + an−1xn−1 + · · · + a1x + a0 This polynomial has degree n (assuming that an ̸= 0) and an is called the leading coef- ficient.

• A rational function is a quotient P/Q of two polynomials (defined when Q(x) ̸= 0). • An algebraic function is produced by taking sums, products, and nth roots of polynomials

and rational functions. • Exponential function: f (x) = bx , where b > 0 (b is called the base). • The composite function f ◦ g is defined by (f ◦ g)(x) = f (g(x)). The domain of f ◦ g

is the set of x in the domain of g such that g(x) belongs to the domain of f . • The elementary functions are obtained by taking products, sums, differences, quotients,

and compositions of the basic functions, which include polynomials, rational functions, algebraic functions, exponential functions, trigonometric functions, logarithmic func- tions, and inverse trigonometric functions.

• Apiecewise-defined function is obtained by defining a function over two or more distinct domains.

1.3 EXERCISES

Preliminary Questions 1. Give an example of a rational function.

2. Is y = |x| a polynomial function? What about y = |x2 + 1|? 3. What is unusual about the domain of the composite function f ◦ g

for the functions f (x) = x1/2 and g(x) = −1 − |x|? 4. Is f (x) =

( 1 2 )x increasing or decreasing?

5. Give an example of a transcendental function.

Exercises In Exercises 1–12, determine the domain of the function.

1. f (x) = x1/4 2. g(t) = t2/3

3. f (x) = x3 + 3x − 4 4. h(z) = z3 + z−3

5. g(t) = 1 t + 2 6. f (x) =

1

x2 + 4

7. G(u) = 1 u2 − 4 8. f (x) =

√ x

x2 − 9

9. f (x) = x−4 + (x − 1)−3 10. F(s) = sin (

s

s + 1

)

11. g(y) = 10 √

y+y−1 12. f (x) = x + x −1

(x − 3)(x + 4) In Exercises 13–24, identify each of the following functions as polyno- mial, rational, algebraic, or transcendental.

13. f (x) = 4x3 + 9x2 − 8 14. f (x) = x−4

15. f (x) = √x 16. f (x) = √

1 − x2

S E C T I O N 1.4 Trigonometric Functions 23

17. f (x) = x 2

x + sin x 18. f (x) = 2 x

19. f (x) = 2x 3 + 3x

9 − 7x2 20. f (x) = 3x − 9x−1/2

9 − 7x2

21. f (x) = sin(x2) 22. f (x) = x√ x + 1

23. f (x) = x2 + 3x−1 24. f (x) = sin(3x)

25. Is f (x) = 2x2 a transcendental function?

26. Show that f (x) = x2 + 3x−1 and g(x) = 3x3 − 9x + x−2 are ra- tional functions—that is, quotients of polynomials.

In Exercises 27–34, calculate the composite functions f ◦ g and g ◦ f , and determine their domains.

27. f (x) = √x, g(x) = x + 1

28. f (x) = 1 x

, g(x) = x−4

29. f (x) = 2x , g(x) = x2

30. f (x) = |x|, g(θ) = sin θ

31. f (θ) = cos θ , g(x) = x3 + x2

32. f (x) = 1 x2 + 1 , g(x) = x

−2

33. f (t) = 1√ t

, g(t) = −t2

34. f (t) = √t , g(t) = 1 − t3

In Exercises 35–38, draw the graphs of each of the piecewise-defined functions.

35.

f (x) = {

3 when x < 0 x2 + 3 when x ≥ 0

36.

f (x) = { x + 1 when x < 0 1 − x when x ≥ 0

37.

f (x) = { x2 when x < 0 −x2 when x ≥ 0

38.

f (x) = {

2x − 2 when x < 0 x when x ≥ 0

39. The population (in millions) of a country as a function of time t (years) is P(t) = 30 · 20.1t . Show that the population doubles every 10 years. Show more generally that for any positive constants a and k, the function g(t) = a2kt doubles after 1/k years.

40. Find all values of c such that f (x) = x + 1 x2 + 2cx + 4 has domain R.

Further Insights and Challenges In Exercises 41–47, we define the first difference δf of a function f by δf (x) = f (x + 1) − f (x).

41. Show that if f (x) = x2, then δf (x) = 2x + 1. Calculate δf for f (x) = x and f (x) = x3.

42. Show that δ(10x) = 9 · 10x and, more generally, that δ(bx) = (b − 1)bx .

43. Show that for any two functions f and g, δ(f + g) = δf + δg and δ(cf ) = cδ(f ), where c is any constant.

44. Suppose we can find a function P such that δP(x) = (x + 1)k and P(0) = 0. Prove that P(1) = 1k , P(2) = 1k + 2k , and, more gener- ally, for every whole number n,

P(n) = 1k + 2k + · · · + nk 1

45. First show that

P(x) = x(x + 1) 2

satisfies δP = (x + 1). Then apply Exercise 44 to conclude that

1 + 2 + 3 + · · · + n = n(n + 1) 2

46. Calculate δ(x3), δ(x2), and δ(x). Then find a polynomial P of degree 3 such that δP = (x + 1)2 and P(0) = 0. Conclude that P(n) = 12 + 22 + · · · + n2.

47. This exercise combined with Exercise 44 shows that for all whole numbers k, there exists a polynomial P satisfying Eq. (1). The solu- tion requires the Binomial Theorem and proof by induction (see Ap- pendix C). (a) Show that δ(xk+1) = (k + 1) xk + · · · , where the dots indicate terms involving smaller powers of x. (b) Show by induction that there exists a polynomial of degree k + 1 with leading coefficient 1/(k + 1):

P(x) = 1 k + 1x

k+1 + · · ·

such that δP = (x + 1)k and P(0) = 0.

1.4 Trigonometric Functions We begin our trigonometric review by recalling the two systems of angle measurement: radians and degrees. They are best described using the relationship between angles and rotation. As is customary, we often use the lowercase Greek letter θ (“theta”) to denote angles and rotations.

24 C H A P T E R 1 PRECALCULUS REVIEW

1

(A) (B) (C) (D)

O P

Q

P

Q

θ

1O P = Q

θ = 2π θ =

1O

θ = −

P

Q

1 O

π 4

π 2

FIGURE 1 The radian measure θ of a counterclockwise rotation is the length along the unit circle of the arc traversed by P as it rotates into Q.

Figure 1(A) shows a unit circle with radius OP rotating counterclockwise into radius

rO θ

θr

FIGURE 2 On a circle of radius r , the arc traversed by a counterclockwise rotation of θ radians has length θr .

OQ. The radian measure of this rotation is the length θ of the circular arc traversed by P as it rotates into Q. On a circle of radius r , the arc traversed by a counterclockwise rotation of θ radians has length θr (Figure 2).

The unit circle has circumference 2π . Therefore, a rotation through a full circle has radian measure θ = 2π [Figure 1(B)]. The radian measure of a rotation through one- quarter of a circle is θ = 2π/4 = π/2 [Figure 1(C)] and, in general, the rotation through one-nth of a circle has radian measure 2π/n (Table 1). A negative rotation (with θ < 0) is a rotation in the clockwise direction [Figure 1(D)].

The radian measure of an angle such as ̸ POQ in Figure 1(A) is defined as the radian

TABLE 1

Rotation through Radian measure

Two full circles 4π Full circle 2π Half circle π Quarter circle 2π/4 = π/2 One-sixth circle 2π/6 = π/3

measure of a rotation that carries OP to OQ. Notice, however, that the radian measure of an angle is not unique. The rotations through θ and θ + 2π both carry OP to OQ. Therefore, θ and θ + 2π represent the same angle even though the rotation through θ + 2π takes an extra trip around the circle. In general, two radian measures represent the same an- gle if the corresponding rotations differ by an integer multiple of 2π . For example, π/4, 9π/4, and −15π/4 all represent the same angle because they differ by multiples of 2π :

π

4 = 9π

4 − 2π = −15π

4 + 4π

Every angle has a unique radian measure satisfying 0 ≤ θ < 2π . With this choice, the angle θ subtends an arc of length θr on a circle of radius r (Figure 2).

Degrees are defined by dividing the circle (not necessarily the unit circle) into 360 equal parts. A degree is 1360 of a circle. A rotation through θ degrees (denoted θ

◦) is a rotation through the fraction θ/360 of the complete circle. For example, a rotation through 90◦ is a rotation through the fraction 90360 , or

1 4 , of a circle.

As with radians, the degree measure of an angle is not unique. Two degree measures represent that same angle if they differ by an integer multiple of 360. For example, the angles −45◦ and 675◦ coincide because 675 = −45 + 2(360). Every angle has a unique degree measure θ with 0 ≤ θ < 360.

To convert between radians and degrees, remember that 2π radians is equal to 360◦.

Radians Degrees

0 0◦ π

6 30◦

π

4 45◦

π

3 60◦

π

2 90◦ Therefore, 1 radian equals 360/2π or 180/π degrees.

• To convert from radians to degrees, multiply by 180/π . • To convert from degrees to radians, multiply by π/180.

EXAMPLE 1 Convert (a) 55◦ to radians and (b) 0.5 radians to degrees.

Solution

(a) 55◦ × π 180◦

≈ 0.9599 radians (b) 0.5 radians × 180 ◦

π ≈ 28.648◦

Convention Unless otherwise stated, we always measure angles in radians.

Radian measurement is usually the better choice for mathematical purposes, but there are good practical reasons for using degrees. The number 360 has many divisors (360 = 8 · 9 · 5), and consequently, many fractional parts of the circle can be expressed as an integer number of degrees. For example, one-fifth of the circle is 72◦, two-ninths is 80◦, three-eighths is 135◦, etc.

S E C T I O N 1.4 Trigonometric Functions 25

The trigonometric functions sine and cosine can be defined in terms of right triangles. Let θ be an acute angle in a right triangle, and let us label the sides as in Figure 3. Then

a

b c

Hypotenuse

Adjacent

Opposite

θ

FIGURE 3

sin θ = b c

= opposite hypotenuse

, cos θ = a c

= adjacent hypotenuse

A disadvantage of this definition is that it makes sense only if θ lies between 0 and π/2 (because an angle in a right triangle cannot exceed π/2). However, sine and cosine can be defined for all angles in terms of the unit circle. Let P = (x, y) be the point on the unit circle corresponding to the angle θ as in Figures 4(A) and (B), and define

cos θ = x-coordinate of P , sin θ = y-coordinate of P This agrees with the right-triangle definition when 0 < θ < π2 . On the circle of radius r (centered at the origin), the point corresponding to the angle θ has coordinates

(r cos θ, r sin θ)

Furthermore, we see from Figure 4(C) that f (x) = sin θ is an odd function and f (x) = cos θ is an even function:

sin(−θ) = − sin θ, cos(−θ) = cos θ

P = (cos θ, sin θ)

x

1 y

θ

(A)

P = (cos θ, sin θ)

x y

θ

(B) (C)

(x, y)

(x, −y)

θ −θ

FIGURE 4 The unit circle definition of sine and cosine is valid for all angles θ .

Although we use a calculator to evaluate sine and cosine for general angles, the standard values listed in Figure 5 and Table 2 appear often and should be memorized.

(0, 1)

π/6 π/2π/3π/4

( ) , .32 12 ( ) , .22 .22 ( ) ,

.3 2

1 2

FIGURE 5 Four standard angles: The x- and y-coordinates of the points are cos θ and sin θ .

TABLE 2

θ 0 π

6 π

4 π

3 π

2 2π 3

3π 4

5π 6

π

sin θ 0 1 2

√ 2

2

√ 3

2 1

√ 3

2

√ 2

2 1 2

0

cos θ 1

√ 3

2

√ 2

2 1 2

0 −1 2

− √

2 2

− √

3 2

−1

The graph of y = sin θ is the familiar “sine wave” shown in Figure 6. Observe how the graph is generated by the y-coordinate of the point P = (cos θ, sin θ) moving around the unit circle.

26 C H A P T E R 1 PRECALCULUS REVIEW

1

pq qq

2p

1 y y

x

FIGURE 6 The graph of y = sin θ is generated as the point P = (cos θ, sin θ) moves around the unit circle.

1

y = sin θ

I II III IV

−1

Quadrant of unit circle

π 2π π 2π θ θ

y = cos θ

I II III IV y y

π 4

π 2

π 4

π 2

3π 4

5π 4

7π 4

3π 4

5π 4

3π 2

3π 2

7π 4

FIGURE 7 Graphs of y = sin θ and y = cos θ over one period of length 2π .

The graph of y = cos θ has the same shape but is shifted to the left π/2 units (Figure 7). The signs of sin θ and cos θ vary as P = (cos θ, sin θ) changes quadrant.

A function f is called periodic with period T if f (x + T ) = f (x) (for all x) and T is the smallest positive number with this property. The sine and cosine functions are periodicWe often write sin x and cos x, using x

instead of θ . Depending on the application, we may think of x as an angle or simply as a real number.

with period T = 2π (Figure 8) because the radian measures x and x + 2πk correspond to the same point on the unit circle for any integer k:

sin x = sin(x + 2πk), cos x = cos(x + 2πk)

y = sin x

2π−2π 4π

y = cos x

2π−2π 4π xx

y 1

y

1

FIGURE 8 Sine and cosine have period 2π .

There are four other standard trigonometric functions, each defined in terms of sin x and cos x or as ratios of sides in a right triangle (Figure 9):

a x

b c

Hypotenuse

Adjacent

Opposite

FIGURE 9

Tangent: tan x = sin x cos x

= b a , Cotangent: cot x = cos x

sin x = a

b

Secant: sec x = 1 cos x

= c a , Cosecant: csc x = 1

sin x = c

b

These functions are periodic (Figure 10):y = tan x andy = cot x have periodπ ;y = sec x and y = csc x have period 2π (see Exercise 57).

x

−1

y = csc xy = sec x

1 1

−1 xx

yy

−1

y = tan x

1

π 2π 2π 2π 2ππ−−π −π π π−π −π x

y

y = cot x

1

−1 x

y

π 2

−π 2

−π 2

π 2

−π 2

π 2

π 2

π 2

3π 2

3π 2

5π 2

3π 2

3π 2

5π 2

FIGURE 10 Graphs of the standard trigonometric functions.

S E C T I O N 1.4 Trigonometric Functions 27

EXAMPLE 2 Computing Values of Trigonometric Functions Find the values of the six trigonometric functions at x = 4π/3. Solution The point P on the unit circle corresponding to the angle x = 4π/3 lies opposite the point with angle π/3 (Figure 11). Thus, we see that (refer to Table 2)

P =

1

(− − ) , .3212

( ), .3212 4π 3

π 3

FIGURE 11

sin 4π 3

= − sin π 3

= − √

3 2

, cos 4π 3

= − cos π 3

= −1 2

The remaining values are

tan 4π 3

= sin 4π/3 cos 4π/3

= − √

3/2 −1/2 =

√ 3, cot

4π 3

= cos 4π/3 sin 4π/3

= √

3 3

sec 4π 3

= 1 cos 4π/3

= 1−1/2 = −2, csc 4π 3

= 1 sin 4π/3

= −2 √

3 3

EXAMPLE 3 Find the angles x such that sec x = 2. Solution Because sec x = 1/ cos x, we must solve cos x = 12 . From Figure 12 we see

1 2

1−

π 3 π 3

FIGURE 12 cos x = 12 for x = ±π3 that x = π/3 and x = −π/3 are solutions. We may add any integer multiple of 2π , so the general solution is x = ±π/3 + 2πk for any integer k.

EXAMPLE 4 Trigonometric Equation Solve sin 4x + sin 2x = 0 for x ∈ [0, 2π). Solution We must find the angles x such that sin 4x = − sin 2x. First, let’s determine when angles θ1 and θ2 satisfy sin θ2 = − sin θ1. Figure 13 shows that this occurs if θ2 = −θ1 or θ2 = θ1 + π . Because the sine function is periodic with period 2π ,

sin θ2 = − sin θ1 ⇔ θ2 = −θ1 + 2πk or θ2 = θ1 + π + 2πk

where k is an integer. Taking θ2 = 4x and θ1 = 2x, we see that

sin 4x = − sin 2x ⇔ 4x = −2x + 2πk or 4x = 2x + π + 2πk

The first equation gives 6x = 2πk or x = (π/3)k and the second equation gives 2x = π + 2πk or x = π/2 + πk. We obtain eight solutions in [0, 2π) (Figure 14):