**Unformatted text preview: **PRECALCULUS
Mathematics for Calculus
FIFTH EDITION This page intentionally left blank PRECALCULUS
Mathematics for Calculus
FIFTH EDITION James Stewart
McMaster University Lothar Redlin
The Pennsylvania State University Saleem Watson
California State University, Long Beach Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States Precalculus: Mathematics for Calculus, Fifth
Edition, Enhanced WebAssign Edition
James Stewart, Lothar Redlin, Saleem Watson
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1 2 3 4 5 6 7 12 11 10 09 08 07 To our students,
from whom we have learned so much. About the Cover
The art on the cover was created by Bill Ralph, a mathematician who uses modern mathematics to produce visual representations of “dynamical systems.” Examples of
dynamical systems in nature include the weather, blood
pressure, the motions of the planets, and other phenomena that involve continual change. Such systems, which
tend to be unpredictable and even chaotic at times, are
modeled mathematically using the concepts of composition and iteration of functions (see Section 2.7 and the
Discovery Project on pages 223–224). The basic idea is to
start with a particular function and evaluate it at some
point in its domain, yielding a new number. The function
is then evaluated at the new number. Repeating this process produces a sequence of numbers called iterates of
the function. The original domain is “painted” by assigning a color to each starting point; the color is determined
by certain properties of its sequence of iterates and the
mathematical concept of “dimension.” The result is a
picture that reveals the complex patterns of the dynamical system. In a sense, these pictures allow us to look,
through the lens of mathematics, at exotic little universes that have never been seen before.
Professor Ralph teaches at Brock University in Canada.
He can be contacted by e-mail at
[email protected] About the Authors
James Stewart was educated at the
University of Toronto and Stanford
University, did research at the
University of London, and now
teaches at McMaster University. His
research ﬁeld is harmonic analysis. He is the author of a best-selling
calculus textbook series published by
Brooks/Cole, including Calculus, 5th
Ed., Calculus: Early Transcendentals,
5th Ed., and Calculus: Concepts and
Contexts, 3rd Ed., as well as a series
of high-school mathematics
textbooks. Lothar Redlin grew up on Vancouver
Island, received a Bachelor of
Science degree from the University
of Victoria, and a Ph.D. from
McMaster University in 1978. He
subsequently did research and
taught at the University of
Washington, the University of
Waterloo, and California State
University, Long Beach.
He is currently Professor of
Mathematics at The Pennsylvania
State University, Abington College.
His research ﬁeld is topology. Saleem Watson received his
Bachelor of Science degree from
Andrews University in Michigan. He
did graduate studies at Dalhousie
University and McMaster University,
where he received his Ph.D. in 1978.
He subsequently did research at the
Mathematics Institute of the
University of Warsaw in Poland. He
also taught at The Pennsylvania
State University.
He is currently Professor of
Mathematics at California State
University, Long Beach. His research
ﬁeld is functional analysis. The authors have also published College Algebra, Fourth Edition (Brooks/Cole, 2004), Algebra and Trigonometry, Second Edition (Brooks/Cole, 2007), and Trigonometry (Brooks/Cole, 2003). Contents Preface
xiii
To the Student
xxi
Calculators and Calculations 1 Fundamentals
■ 1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
1.10
1.11 2 2.1
2.2 1 Chapter Overview
1
Real Numbers
2
Exponents and Radicals
12
Algebraic Expressions
24
● DISCOVERY PROJECT Visualizing a Formula
34
Rational Expressions
35
Equations
44
Modeling with Equations
58
● DISCOVERY PROJECT Equations through the Ages
75
Inequalities
76
Coordinate Geometry
87
Graphing Calculators; Solving Equations and Inequalities
Graphically
101
Lines
111
Modeling Variation
123
Chapter 1 Review
130
Chapter 1 Test
135
■ FOCUS ON PROBLEM SOLVING General Principles
138 Functions
■ xxii 146 Chapter Overview
147
What is a Function?
148
Graphs of Functions
158
● DISCOVERY PROJECT Relations and Functions 171 vii viii Contents 2.3
2.4
2.5
2.6
2.7
2.8 3 Polynomial and Rational Functions
■ 3.1
3.2
3.3
3.4
3.5
3.6 4 Increasing and Decreasing Functions; Average Rate of
Change
173
Transformations of Functions
182
Quadratic Functions; Maxima and Minima
193
Modeling with Functions
203
Combining Functions
214
● DISCOVERY PROJECT Iteration and Chaos
223
One-to-One Functions and Their Inverses
225
Chapter 2 Review
233
Chapter 2 Test
237
■ FOCUS ON MODELING Fitting Lines to Data
239 248 Chapter Overview
249
Polynomial Functions and Their Graphs
250
Dividing Polynomials
265
Real Zeros of Polynomials
272
● DISCOVERY PROJECT Zeroing in on a Zero
283
Complex Numbers
285
Complex Zeros and the Fundamental Theorem of
Algebra
291
Rational Functions
299
Chapter 3 Review
316
Chapter 3 Test
319
■ FOCUS ON MODELING Fitting Polynomial Curves to
Data
320 Exponential and Logarithmic
326
Functions
■ 4.1
4.2
4.3
4.4
4.5 Chapter Overview
327
Exponential Functions
328
● DISCOVERY PROJECT Exponential Explosion
341
Logarithmic Functions
342
Laws of Logarithms
352
Exponential and Logarithmic Equations
358
Modeling with Exponential and Logarithmic
Functions
369
Chapter 4 Review
382
Chapter 4 Test
385
■ FOCUS ON MODELING Fitting Exponential and Power
Curves to Data
386 Contents 5 Trigonometric Functions of Real
398
Numbers
■ 5.1
5.2
5.3
5.4
5.5 6 Trigonometric Functions of Angles
■ 6.1
6.2
6.3
6.4
6.5 7 Chapter Overview
399
The Unit Circle
400
Trigonometric Functions of Real Numbers
408
Trigonometric Graphs
418
● DISCOVERY PROJECT Predator/Prey Models
432
More Trigonometric Graphs
434
Modeling Harmonic Motion
442
Chapter 5 Review
454
Chapter 5 Test
458
■ FOCUS ON MODELING Fitting Sinusoidal Curves to
Data
459 Chapter Overview
467
Angle Measure
468
Trigonometry of Right Triangles
478
Trigonometric Functions of Angles
488
● DISCOVERY PROJECT Similarity
499
The Law of Sines
501
The Law of Cosines
508
Chapter 6 Review
516
Chapter 6 Test
520
■ FOCUS ON MODELING Surveying
522 Analytic Trigonometry
■ 7.1
7.2
7.3
7.4
7.5 466 526 Chapter Overview
527
Trigonometric Identities
528
Addition and Subtraction Formulas
535
Double-Angle, Half-Angle, and Sum-Product
Formulas
541
Inverse Trigonometric Functions
550
● DISCOVERY PROJECT Where to Sit at the Movies
Trigonometric Equations
561
Chapter 7 Review
571
Chapter 7 Test
574
■ FOCUS ON MODELING Traveling and Standing
Waves
575 560 ix x Contents 8 Polar Coordinates and Vectors
■ 8.1
8.2
8.3 8.4
8.5 9 580 Chapter Overview
581
Polar Coordinates
582
Graphs of Polar Equations
587
Polar Form of Complex Numbers; DeMoivre’s
Theorem
596
● DISCOVERY PROJECT Fractals
605
Vectors
607
The Dot Product
617
● DISCOVERY PROJECT Sailing Against the Wind
626
Chapter 8 Review
627
Chapter 8 Test
629
■ FOCUS ON MODELING Mapping the World
630 Systems of Equations and
634
Inequalities
■ 9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
9.9 Chapter Overview
635
Systems of Equations
636
Systems of Linear Equations in Two Variables
644
Systems of Linear Equations in Several Variables
651
● DISCOVERY PROJECT Best Fit versus Exact Fit
660
Systems of Linear Equations: Matrices
662
The Algebra of Matrices
675
● DISCOVERY PROJECT Will the Species Survive?
688
Inverses of Matrices and Matrix Equations
689
● DISCOVERY PROJECT Computer Graphics I
700
Determinants and Cramer’s Rule
704
Partial Fractions
715
Systems of Inequalities
721
Chapter 9 Review
728
Chapter 9 Test
733
■ FOCUS ON MODELING Linear Programming
735 10 Analytic Geometry
■ 10.1
10.2
10.3 Chapter Overview
Parabolas
744
Ellipses
753
Hyperbolas
762 743 742 Contents ● DISCOVERY PROJECT 10.4
10.5
10.6
10.7 Conics in Architecture
771
Shifted Conics
775
Rotation of Axes
783
● DISCOVERY PROJECT Computer Graphics II
792
Polar Equations of Conics
795
Plane Curves and Parametric Equations
801
Chapter 10 Review
810
Chapter 10 Test
814
■ FOCUS ON MODELING The Path of a Projectile
816 11 Sequences and Series
■ 11.1
11.2
11.3
11.4
11.5
11.6 820 Chapter Overview
821
Sequences and Summation Notation
822
Arithmetic Sequences
833
Geometric Sequences
838
● DISCOVERY PROJECT Finding Patterns
847
Mathematics of Finance
848
Mathematical Induction
854
The Binomial Theorem
860
Chapter 11 Review
870
Chapter 11 Test
873
■ FOCUS ON MODELING Modeling with Recursive
Sequences
874 12 Limits: A Preview of Calculus
■ 12.1
12.2
12.3
12.4
12.5 880 Chapter Overview
881
Finding Limits Numerically and Graphically
882
Finding Limits Algebraically
890
Tangent Lines and Derivatives
898
● DISCOVERY PROJECT Designing a Roller Coaster
908
Limits at Inﬁnity: Limits of Sequences
908
Areas
916
Chapter 12 Review
925
Chapter 12 Test
928
■ FOCUS ON MODELING Interpretations of Area
929 Cumulative Review
Answers
A1
Index
I1
Photo Credits
C1 CR1 xi This page intentionally left blank Preface The art of teaching is the art of assisting discovery.
MARK VAN DOREN What do students really need to know to be prepared for calculus? What tools do instructors really need to assist their students in preparing for calculus? These two
questions have motivated the writing of this book.
To be prepared for calculus a student needs not only technical skill but also a clear
understanding of concepts. Indeed, conceptual understanding and technical skill go
hand in hand, each reinforcing the other. A student also needs to gain an appreciation
for the power and utility of mathematics in modeling the real-world. Every feature of
this textbook is devoted to fostering these goals.
We are keenly aware that good teaching comes in many different forms, and that
each instructor brings unique strengths and imagination to the classroom. Some instructors use technology to help students become active learners; others use the rule
of four, “topics should be presented geometrically, numerically, algebraically, and
verbally,” to promote conceptual reasoning; some use an expanded emphasis on applications to promote an appreciation for mathematics in everyday life; still others
use group learning, extended projects, or writing exercises as a way of encouraging
students to explore their own understanding of a given concept; and all present mathematics as a problem-solving endeavor. In this book we have included all these methods of teaching precalculus as enhancements to a central core of fundamental skills.
These methods are tools to be utilized by instructors and their students to navigate
their own course of action in preparing for calculus.
In writing this ﬁfth edition our purpose was to further enhance the utility of the
book as an instructional tool. The main change in this edition is an expanded emphasis on modeling and applications: In each section the applications exercises have been
expanded and are grouped together under the heading Applications, and each chapter (except Chapter 1) now ends with a Focus on Modeling section. We have also
made some organizational changes, including dividing the chapter on analytic
trigonometry into two chapters, each of more manageable size. There are numerous
other smaller changes—as we worked through the book we sometimes realized that
an additional example was needed, or an explanation could be clariﬁed, or a section
could beneﬁt from different types of exercises. Throughout these changes, however,
we have retained the overall structure and the main features that have contributed to
the success of this book. xiii xiv Preface Many of the changes in this edition have been drawn from our own experience in
teaching, but, more importantly, we have listened carefully to the users of the current
edition, including many of our closest colleagues. We are also grateful to the many
letters and e-mails we have received from users of this book, instructors as well as
students, recommending changes and suggesting additions. Many of these have
helped tremendously in making this edition even more user-friendly. Special Features
The most important way to foster conceptual understanding and
hone technical skill is through the problems that the instructor assigns. To that end we
have provided a wide selection of exercises. EXERCISE SETS ■ ■ ■ Exercises Each exercise set is carefully graded, progressing from basic conceptual exercises and skill-development problems to more challenging problems
requiring synthesis of previously learned material with new concepts.
Applications Exercises We have included substantial applied problems that
we believe will capture the interest of students. These are integrated throughout
the text in both examples and exercises. In the exercise sets, applied problems
are grouped together under the heading, Applications. (See, for example, pages
127, 156, 314, and 451.)
Discovery, Writing, and Group Learning Each exercise set ends with a
block of exercises called Discovery•Discussion. These exercises are designed to
encourage students to experiment, preferably in groups, with the concepts developed in the section, and then to write out what they have learned, rather than
simply look for “the answer.” (See, for example, pages 232 and 369.) A COMPLETE REVIEW CHAPTER We have included an extensive review chapter
primarily as a handy reference for the student to revisit basic concepts in algebra and
analytic geometry.
■ ■ Chapter 1 This is the review chapter; it contains the fundamental concepts a
student needs to begin a precalculus course. As much or as little of this chapter
can be covered in class as needed, depending on the background of the students.
Chapter 1 Test The test at the end of Chapter 1 is intended as a diagnostic
instrument for determining what parts of this review chapter need to be taught.
It also serves to help students gauge exactly what topics they need to review. The trigonometry chapters of this
text have been written so that either the right triangle approach or the unit circle approach may be taught ﬁrst. Putting these two approaches in different chapters, each
with its relevant applications, helps clarify the purpose of each approach. The chapters introducing trigonometry are as follows: FLEXIBLE APPROACH TO TRIGONOMETRY ■ ■ Chapter 5: Trigonometric Functions of Real Numbers This chapter introduces trigonometry through the unit circle approach. This approach emphasizes
that the trigonometric functions are functions of real numbers, just like the polynomial and exponential functions with which students are already familiar.
Chapter 6: Trigonometric Functions of Angles This chapter introduces
trigonometry through the right triangle approach. This approach builds on the
foundation of a conventional high-school course in trigonometry. Preface xv Another way to teach trigonometry is to intertwine the two approaches. Some instructors teach this material in the following order: Sections 5.1, 5.2, 6.1, 6.2, 6.3, 5.3,
5.4, 6.4, 6.5. Our organization makes it easy to do this without obscuring the fact that
the two approaches involve distinct representations of the same functions.
GRAPHING CALCULATORS AND COMPUTERS Calculator and computer technology extends in a powerful way our ability to calculate and visualize mathematics.
The availability of graphing calculators makes it not less important, but far more important to understand the concepts that underlie what the calculator produces. Accordingly, all our calculator-oriented subsections are preceded by sections in which
students must graph or calculate by hand, so that they can understand precisely what
the calculator is doing when they later use it to simplify the routine, mechanical part
of their work. The graphing calculator sections, subsections, examples, and exercises,
all marked with the special symbol
, are optional and may be omitted without loss
of continuity. We use the following capabilities of the calculator:
■ ■ Graphing Calculators The use of the graphing calculator is integrated
throughout the text to graph and analyze functions, families of functions, and sequences, to calculate and graph regression curves, to perform matrix algebra, to
graph linear inequalities, and other powerful uses.
Simple Programs We exploit the programming capabilities of a graphing calculator to simulate real-life situations, to sum series, or to compute the terms of
a recursive sequence. (See, for instance, pages 702, 825, and 829.) FOCUS ON MODELING The “modeling” theme has been used throughout to unify
and clarify the many applications of precalculus. We have made a special effort, in
these modeling sections and subsections, to clarify the essential process of translating problems from English into the language of mathematics. (See pages 204 or 647.)
■ ■ Constructing Models There are numerous applied problems throughout the
book where students are given a model to analyze (see, for instance, page 200).
But the material on modeling, where students are required to construct mathematical models for themselves, has been organized into clearly deﬁned sections
and subsections (see, for example, pages 203, 369, 442, and 848).
Focus on Modeling Each chapter concludes with a Focus on Modeling section. The ﬁrst such section, after Chapter 2, introduces the basic idea of modeling a real-life situation by ﬁtting lines to data (linear regression). Other sections
present ways in which polynomial, exponential, logarithmic, and trigonometric
functions, and systems of inequalities can all be used to model familiar phenomena from the sciences and from everyday life (see, for example, pages 320, 386,
or 459). Chapter 1 concludes with a section entitled Focus on Problem Solving. One way to engage students and make them active learners is to have them work (perhaps in groups) on extended projects that give a feeling
of substantial accomplishment when completed. Each chapter contains one or more
Discovery Projects (see the table of contents); these provide a challenging but accessible set of activities that enable students to ...

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