Precalculus: A Problems-Oriented Approach, Sixth Edition

1222 Pages·2009·10.51 MB·English

by  David Cohen

Precalculus
A Problems-Oriented Approach
Sixth Edition
David Cohen
Late of University of California
Los Angeles
With
Theodore Lee
City College of San Francisco
David Sklar
San Francisco State University
Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States
Precalculus: A Problems-Oriented © 2010Brooks/Cole, Cengage Learning
Approach, Sixth Edition
ALL RIGHTS RESERVED. No part of this work covered by the copyright herein may
David Cohen, Theodore Lee, and be reproduced, transmitted, stored, or used in any form or by anymeans graphic,
David Sklar electronic, or mechanical, including but not limited to photocopying, recording,
scanning, digitizing, taping, Web distribution, information networks, or informa-
Acquisitions Editor: John-Paul Ramin
tion storage and retrieval systems, except as permitted under Section 107or 108
Assistant Editor: Katherine Brayton of the 1976United States Copyright Act, without the prior written permission of
Editorial Assistant: Darlene Amidon-Brent the publisher.
Technology Project Manager: Rachael
Sturgeon
Senior Marketing Manager: Karen Sandberg For product information and technology assistance, contact us at
Marketing Assistant: Erin Mitchell Cengage Learning Customer & Sales Support, 1-800-354-9706
Managing Marketing Communication For permission to use material from this text or product,
submit all requests online www.cengage.com/permissions
Manager: Bryan Vann
Further permissions questions can be emailed to
Senior Project Manager, Editorial Production: [email protected]
Janet Hill
Senior Art Director: Vernon Boes
Senior Print/Media Buyer: Karen Hunt Library of Congress Control Number: 2009929179
Permissions Editor: Kiely Sexton ISBN-13: 978-1-4390-4460-5
Production Service: Martha Emry
ISBN-10: 1-4390-4460-0
Text Designer: Rokusek Design
Art Editor: Martha Emry
Brooks/Cole
Photo Researcher: Sue Howard
10Davis Drive
Copy Editor: Barbara Willette Belmont, CA 94002
Illustrator: Jade Myers USA
Cover Designer: Cheryl Carrington
Cover Image: Yva Momatiuk/John Eastcott
Cengage Learning is a leading provider of customized learning solutions with
Compositor: G&S Typesetters, Inc. office locations around the globe, including Singapore, the United Kingdom,
Australia, Mexico, Brazil, and Japan. Locate your local office at
www.cengage.com/global
Cengage Learning products are represented in Canada by
Nelson Education, Ltd.
To learn more about Brooks/Cole, visit www.cengage.com/brookscole
Purchase any of our products at your local college store or at our preferred online
store www.ichapters.com
Printed in the United States of America
1 2 3 4 5 6 7 11 10 09
To our parents:
Ruth and Ernest Cohen
Lorraine and Robert Lee
Helen and Rubin Sklar
Tribute to David Cohen
Because he loved the part in “The Myth of Sisyphus,” where Camus re-envisions
Sisyphus eternally condemned to the task of pushing a giant boulder to the summit
of a mountain only to watch it roll back down again—descending the hill and smil-
ing, I like to remember my dad in similar moments of inbetweenness. When the un-
known and hope would combine to make anything seem attainable. When becom-
ing conscious of a situation did not necessarily signify limitation by it, but instead,
liberation from it.
And so I imagine my dad, after hours of working at the computer, between a
thoughtful sip of coffee and suddenly realizing the clearest way to word a problem.
After years of early morning commutes to avoid 405 traffic, solitary interludes that
began with waking to a still-dark sky and still-dreaming family, and ended with
walking, transformed into a professor, into a UCLA lecture hall. After a lifetime
of doing his best for those he cared about, intervals of good and poor health sim-
ply challenged him to give a little more. My dad endured his task bravely, from his
decision 15 years ago to fight leukemia until last May, when the illness reminded us
that even fathers and teachers are mortal, regardless of scheduled office hours or
books left unwritten.
A couple months ago, my dad told me the five most important things in life to
him;teachingandwritingthesemathbookswerebothonthatlist.Iwasnevera“reg-
istered” student of his, but to anyone using this book, especially students, I pass on
what I know he would have told me: Your best is always good enough. Enjoy.
Emily Cohen
• • •
My dad was gifted at teaching. He had an innate ability to explain things simply and
relevantly. He excelled at combining his explanations with a patient ear and his self-
defined nerdy sense of humor.
Irememberwhenmydadtaughtmetoread.Ihadbecomeupsetonenightwhile
we were out to dinner because I couldn’t read the menu. Frustrated, I expressed my
dismay from my booster chair. My dad responded with the first of my reading les-
sons. He began by pointing out something I already knew, the letter “a.” He care-
fully explained that this letter was actually a word ... every day after that he would
write new words on one of his ubiquitous lined yellow notepads and teach them to
me. Eventually we began to form simple sentences. He always kept my attention
because he made up sentences that made me laugh. The frogs go jog, the cat loves
the dog, so on and so forth. My dad’s sense of humor was quite captivating.
I was lucky in high school to have someone to assist me with all of my mathe-
matical questions. My math teachers seemed to have a special aptitude for making
formulas, theories, and problems both complicated and boring. My dad was good
at simplifying these matters for me. He was so good at explaining math to me that
I would often remark afterward, with the hindsight of an enlightened one, “Oh,
that’s all? Why didn’t they just say that in the first place?” I took comfort in know-
ing that math didn’t have to be complicated when it was explained well.
iv
Tribute to David Cohen v
Hemingway said that the key to immortality was to write a book. I feel lucky
because my dad left behind several. Try reading one—you might find that you ac-
tually like it!
Jennifer Cohen
• • •
I first met Dave Cohen in spring 1981, when I signed up for an algebra course he
was teaching as part of my graduate studies at UCLA. I quickly took a liking to his
conversational style of teaching and the clear explanations he would use to illus-
trate an idea. Dave and I began an informal, weekly meeting to discuss solutions to
various problems he would pose. Usually he would buy me a cup of coffee and pre-
sent me with some new trigonometric identity he had discovered, or some archaic
conic section property he had come across in an old 1886 algebra book. We would
discuss the problem (“Did it have a solution? Was it really an identity? Could we
prove it?”), then go about our business.
In summer 1981, I had the opportunity to TA a course called Precalculus, which
used Dave’s notes (sometimes written by hand), rather than a traditional textbook.
I had no idea what the term “precalculus” meant, and soon learned that these notes
Dave had prepared were beginning to define, or at the least greatly expand, the sub-
ject. It was the beginning of a textbook.
Two years later, after graduating UCLA, I was pleased to receive a copy of Pre-
calculus,by David Cohen. Dave had scribbled a note to me saying: “Thanks for the
great coffee breaks.” I was pleased to see that some of our weekly problems ap-
peared in that book (others showed up years later). More importantly, that book,
and every other text Dave has ever written, talked to you. Even now, when I read
this book, I can hear Dave talking, and sometimes even listening. Lecturing was
never a part of Dave Cohen’s vocabulary.
Since that time Dave has written books entitled College Algebra, Trigonometry,
and Algebra and Trigonometry.Most have gone on to multiple editions, including
this one. In each edition Dave has sought the better explanation, the “cooler” prob-
lem, the more interesting data set. I’ve had the honor of working with him on all
ofthese books, and we had developed a mathematical friendship. Perhaps some of
you reading this have a great study partner; one who intuitively knows what the
other is thinking. That was the relationship Dave and I had.
I last saw Dave in November 2001, when we met at UCLA to (naturally) have a
cup of coffee and discuss this textbook. He was excited about the quality of prob-
lems and interesting data sets he had found to use in this book. He felt this was go-
ing to be the best book he had ever written.
In May 2002, just after completing much of this manuscript, Dave Cohen passed
away from complications caused by his leukemia. His legacy of fine textbooks (and
this is, perhaps, his finest) and great teaching has influenced a generation of stu-
dents and teachers. Everyone who had contact with Dave feels a bit richer from the
experience. His total lack of ego, curiosity about the world, and respect for what
others think, made him one of the finest human beings I have ever met.
Ross Rueger
About the Co-Authors
David Sklar was a longtime friend of David Cohen, whom he met while they were
both attending graduate school at San Francisco State University. Because of his re-
lationship with the author, he has followed the progress of this book since work
started on the first edition. He brings a unique blend of teaching and professional
experience to the table, having taught at San Francisco State University, Sonoma
State University,Menlo College, and City College of San Francisco. At the same
time, David is a researcher and consultant in the field of optics—you’ll notice that
interest in some of the new group projects in the text. David is an active member of
the American Mathematical Society, Mathematical Association of America, is past
chairman of the Northern California section of the MAA, and serves on the math-
ematics department advisory board at San Francisco State University.
David is joined by Theodore Lee, professor of mathematics at City College
ofSan Francisco. Ted is a highly respected teacher who brings nearly 30 years of
teaching experience to this project and provides a valuable perspective on teaching
precalculus mathematics. On three separate occasions he has been honored with
distinguished teaching awards from his colleagues at CCSF. Ted has also been hon-
ored by Alpha Sigma Gamma, CCSF’s student honor society, as a favorite teacher.
A fourth-generation Californian, Ted received his bachelor’s degree from the Uni-
versity of California, Berkeley, and his master’s degree from the University of Cali-
fornia, Los Angeles.
Both David and Ted understand the factors that make the book so special to the
people who use it—the clear writing, the conversational style, the variety of prob-
lems(includingmanychallengingones),andthethoughtfuluseoftechnology.They
havemadeeveryefforttomaintainthestandardandqualityofDavidCohen’swork.
vi
Contents
4.5 Maximum and Minimum
1
Fundamentals 1 Problems 272
4.6 Polynomial Functions 287
1.1 Sets of Real Numbers 1
4.7 Rational Functions 304
1.2 Absolute Value 6
1.3 Solving Equations (Review and
Preview) 10 5
1.4 Rectangular Coordinates. Visualizing Exponential and Logarithmic
Data 18 Functions 325
1.5 Graphs and Graphing Utilities 32
5.1 Exponential Functions 327
1.6 Equations of Lines 43 5.2 The Exponential Function y(cid:2)ex 336
1.7 Symmetry and Graphs. Circles 57
5.3 Logarithmic Functions 347
5.4 Properties of Logarithms 361
5.5 Equations and Inequalities with Logs
2
Equations and Inequalities 80 and Exponents 371
5.6 Compound Interest 382
2.1 Quadratic Equations: Theory and
5.7 Exponential Growth and Decay 394
Examples 80
2.2 Other Types of Equations 91
2.3 Inequalities 103 6
2.4 More on Inequalities 113 Trigonometric Functions of
Angles 423
6.1 Trigonometric Functions of Acute
3
Functions 129 Angles 423
6.2 Algebra and the Trigonometric
3.1 The Definition of a Function 129
Functions 437
3.2 The Graph of a Function 145
6.3 Right-Triangle Applications 445
3.3 Shapes of Graphs. Average Rate of
6.4 Trigonometric Functions of
Change 159
Angles 459
3.4 Techniques in Graphing 171
6.5 Trigonometric Identities 472
3.5 Methods of Combining Functions.
Iteration 182
3.6 Inverse Functions 194 7
Trigonometric Functions
of Real Numbers 487
4
Polynomial and Rational 7.1 Radian Measure 487
Functions. Applications to 7.2 Radian Measure and Geometry 496
Optimization 213 7.3 Trigonometric Functions of Real
Numbers 506
4.1 Linear Functions 213
7.4 Graphs of the Sine and Cosine
4.2 Quadratic Functions 229
Functions 521
4.3 Using Iteration to Model Population 7.5 Graphs of y(cid:2)Asin(Bx(cid:3)C) and
Growth (Optional Section) 246 y(cid:2)Acos(Bx(cid:3)C) 538
4.4 Setting Up Equations That Define
7.6 Simple Harmonic Motion 554
Functions 260
vii
viii Contents
7.7 Graphs of the Tangent and the Reciprocal 11.7 The Conics in Polar Coordinates 867
Functions 560 11.8 Rotation of Axes 872
8 12
Analytical Trigonometry 577 Roots of Polynomial
Equations 890
8.1 The Addition Formulas 577
8.2 The Double-Angle Formulas 589 12.1 The Complex Number System 891
8.3 The Product-to-Sum and Sum-to-Product 12.2 Division of Polynomials 898
Formulas 599 12.3 The Remainder Theorem and the Factor
8.4 Trigonometric Equations 607 Theorem 905
8.5 The Inverse Trigonometric 12.4 The Fundamental Theorem of
Functions 619 Algebra 914
12.5 Rational and Irrational Roots 926
12.6 Conjugate Roots and Descartes’s Rule
9 of Signs 934
Additional Topics in
12.7 Introduction to Partial Fractions 941
Trigonometry 645
12.8 More About Partial Fractions 948
9.1 The Law of Sines and the Law of
Cosines 645
9.2 Vectors in the Plane: A Geometric 13
Additional Topics in
Approach 661
Algebra 965
9.3 Vectors in the Plane: An Algebraic
Approach 671 13.1 Mathematical Induction 965
9.4 Parametric Equations 684 13.2 The Binomial Theorem 971
9.5 Introduction to Polar Coordinates 694 13.3 Introduction to Sequences and
9.6 Curves in Polar Coordinates 705 Series 981
13.4 Arithmetic Sequences and Series 991
13.5 Geometric Sequences and
10 Series 1002
Systems of Equations 721
13.6 DeMoivre’s Theorem 1007
10.1 Systems of Two Linear Equations in Two
Unknowns 721
10.2 Gaussian Elimination 735 A
Appendix A-1
10.3 Matrices 749
10.4 The Inverse of a Square Matrix 763 A.1 Significant Digits A-1
10.5 Determinants and Cramer’s Rule 776 A.2 Properties of the Real Numbers A-4
10.6 Nonlinear Systems of Equations 789 A.3 12is Irrational A-7
10.7 Systems of Inequalities 797
B
11 Appendix A-8
The Conic Sections 809
B.1 Review of Integer Exponents A-8
11.1 The Basic Equations 809
B.2 Review of nth Roots A-14
11.2 The Parabola 817
B.3 Review of Rational Exponents A-21
11.3 Tangents to Parabolas (Optional
B.4 Review of Factoring A-25
Section) 831
B.5 Review of Fractional Expressions A-31
11.4 The Ellipse 833
11.5 The Hyperbola 848
Answers A-37
11.6 The Focus–Directrix Property of
Conics 859 Index I-1
I
LIST OF PROJECTS
Section Title and Page Number Section Title and Page Number
1.4 Discuss, Compute, Reassess 31 6.2 Constructing a Regular Pentagon 444
1.5 Drawing Conclusions from Visual Evidence 43 6.3 Snell’s Law and an Ancient Experiment 455
1.6 Thinking About Slope 57 6.5 Identities and Graphs 479
1.7 Thinking About Symmetry 69 7.3 A Linear Approximation for the Sine
2.1 Put the Quadratic Equation in Its Place? 90 Function 518
2.2 Flying the Flag 102 7.4 Making Waves 536
2.2 Specific or General? Whatever Works! 102 7.5 Fourier Series 550
2.3 An Inequality for the Garden 112 7.6 The Motion of a Piston 559
2.4 Wind Power 121 8.1 The Design of a Fresnel Lens 586
3.1 A Prime Function 143 8.3 Superposition 605
3.2 Implicit Functions: Batteries Required! 157 8.4 Astigmatism and Eyeglass Lenses 618
3.4 Correcting a Graphing Utility Display 181 8.5 Inverse Secant Functions 633
3.5 A Graphical Approach to Composition 9.2 Vector Algebra Using Vector Geometry 669
of Functions 194 9.3 Lines, Circles, and Ray Tracing with
3.6 A Frequently Asked Question About Vectors 679
Inverses 206 9.4 Parameterizations for Lines and Circles 690
4.1 Who Are Better Runners, Men or 10.1 Geometry Workbooks on the Euler Line and
Women? 228 the Nine-Point Circle 732
4.2 How Do You Know that the Graph of a 10.2 The Leontief Input-Output Model 746
Quadratic Function Is Always Symmetric about 10.3 Communications and Matrices 761
a Vertical Line? 243 10.4 The Leontief Model Revisited 774
4.2 What’s Left in the Tank? 244 11.2 A Bridge with a Parabolic Arch 830
4.4 Group Work on Functions of Time 271 11.2 Constructing a Parabola 830
4.5 The Least-Squares Line 284 11.4 The Circumference of an Ellipse 847
4.6 Finding Some Maximum Values Without Using 11.5 Using Hyperbolas to Determine a
Calculus 302 Location 858
4.7 Finding Some Minimum Values Without Using 12.1 A Geometric Interpretation of Complex
Calculus 317 Roots 898
5.1 Using Differences to Compare Exponential and 12.4 Two Methods for Solving Certain Cubic
Polynomial Growth 336 Equations 924
5.2 Coffee Temperature 346 12.7 Checking a Partial Fraction
5.3 More Coffee 361 Decomposition 947
5.6 Loan Payments 392 12.8 An Unusual Partial Fractions Problem 957
5.7 A Variable Growth Constant? 417 13.3 Perspective and Alternate Scenarios for
6.1 Transits of Venus and the Scale of the Solar Example 5 991
System 433 13.4 More on Sums 997
ix