Essentials of precalculus

668 Pages·2006·16.405 MB·English

by  Richard N. Aufmann, Richard Nation

E
SSENTIALS OF
P
RECALCULUS
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E
SSENTIALS OF
P
RECALCULUS
Richard N.Aufmann
Richard D.Nation
Palomar College
Houghton Mifflin Company
Boston New York
Publisher: Jack Shira
Senior Sponsoring Editor: Lynn Cox
Associate Editor: Jennifer King
Assistant Editor: Melissa Parkin
Senior Project Editor: Tamela Ambush
Editorial Assistant: Sage Anderson
Manufacturing Manager: Karen Banks
Senior Marketing Manager: Danielle Potvin
Marketing Coordinator: Nicole Mollica
Cover photograph: PunchStock
PHOTO CREDITS:
Chapter 1: p. 1Charles O’Rear / CORBIS; p. 8The Granger Collection; p. 77CORBIS. Chapter 2: p. 115
Sonda Dawes / The Image Works, Inc.; p. 116David Young-Wolff / PhotoEdit, Inc.; p. 136Syndicated
Features Limited / The Image Works, Inc.; p. 157The Granger Collection; p. 167Bettmann/CORBIS;
p. 169Bettman / CORBIS; p. 190Richard T. Nowitz / CORBIS; Chapter3:p. 201Chris McLaughlin /
CORBIS; p. 221Bettman / CORBIS; p. 227Charles O’Rear / CORBIS; p. 227David James / Getty
Images; p. 231Bettman / CORBIS. Chapter 4:p. 304Courtesy of NASAand STScI; p. 318Reuters /
CORBIS. Chapter 5:p. 357Massimo Listri / CORBIS; p. 373Art; p. 373Tony Craddock / Getty
Images; p. 402Courtesy of Richard Nation. Chapter 6:p. 461Stephen Johnson / Getty Images; p. 461
Stephen Johnson / Getty Images.
Copyright © 2006 Houghton Mifflin Company. All rights reserved.
No part of this work may be reproduced or transmitted in any form or by any means, electronic or
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permitted by federal copyright law. Address inquiries to College Permissions, Houghton Mifflin
Company, 222 Berkeley Street, Boston, MA02116-3764.
Printed in the U.S.A.
Library of Congress Control Number: 2003110130
ISBNs:
Student's Edition: 0-618-44702-4
Instructor's Annotated Edition: 0-618-44703-2
1 2 3 4 5 6 7 8 9-VH-09 08 07 06 05
C
ONTENTS
Preface ix
1
FUNCTIONS AND GRAPHS 1
1.1 NSEEANQOESINITUADIL quations and Inequalities 3
1.2 ATwo-Dimensional Coordinate System and Graphs 16
1.3 Introduction to Functions 30
1.4 Linear Functions 51
1.5 Quadratic Functions 67
1.6 Properties of Graphs 81
1.7 The Algebra of Functions 96
EXPLORINGCONCEPTSWITHTECHNOLOGY: Graphing Piecewise Functions
with a Graphing Calculator 108
Chapter 1 Summary 109
Chapter 1 True/False Exercises 111
Chapter 1 Review Exercises 111
Chapter 1 Test 114
2
POLYNOMIAL AND RATIONAL FUNCTIONS 115
2.1 Complex Numbers 117
2.2 The Remainder Theorem and the Factor Theorem 127
2.3 Polynomial Functions of Higher Degree 138
2.4 Zeros of Polynomial Functions 154
2.5 The Fundamental Theorem of Algebra 168
2.6 Graphs of Rational Functions and Their Applications 177
EXPLORINGCONCEPTSWITHTECHNOLOGY: Finding Zeros of a Polynomial
Using Mathematica 193
Chapter 2 Summary 194
Chapter 2 True/False Exercises 195
Chapter 2 Review Exercises 196
Chapter 2 Test 198
Cumulative Review Exercises 199
v
vi Contents
3
EXPONENTIAL AND LOGARITHMIC FUNCTIONS 201
3.1 Inverse Functions 203
3.2 Exponential Functions and Their Applications 215
3.3 Logarithmic Functions and Their Applications 230
3.4 Logarithms and Logarithmic Scales 243
3.5 Exponential and Logarithmic Equations 256
3.6 Exponential Growth and Decay 268
EXPLORINGCONCEPTSWITHTECHNOLOGY: Using a Semilog Graph to
Model Exponential Decay 281
Chapter 3 Summary 282
Chapter 3 True/False Exercises 284
Chapter 3 Review Exercises 284
Chapter 3 Test 286
Cumulative Review Exercises 287
4
TRIGONOMETRIC FUNCTIONS 289
4.1 Angles and Arcs 291
4.2 Trigonometric Functions of Real Numbers 307
4.3 Graphs of the Sine and Cosine Functions 322
4.4 Graphs of the Other Trigonometric Functions 331
4.5 Graphing Techniques 340
EXPLORINGCONCEPTSWITHTECHNOLOGY: Sinusoidal Families 350
Chapter 4 Summary 351
Chapter 4 True/False Exercises 352
Chapter 4 Review Exercises 352
Chapter 4 Test 354
Cumulative Review Exercises 354
5
APPLICATIONS OF TRIGONOMETRY AND
TRIGONOMETRIC IDENTITIES 357
5.1 Trigonometric Functions of Angles 359
5.2 Verification of Trigonometric Identities 375
5.3 More on Trigonometric Identities 388
5.4 Inverse Trigonometric Functions 402
Contents vii
5.5 Trigonometric Equations 414
5.6 The Law of Sines and the Law of Cosines 425
5.7 Vectors 438
EXPLORINGCONCEPTSWITHTECHNOLOGY: Approximate an Inverse Trigonometric
Function with Polynomials 453
Chapter 5 Summary 454
Chapter 5 True/False Exercises 455
Chapter 5 Review Exercises 456
Chapter 5 Test 458
Cumulative Review Exercises 459
6
ADDITIONAL TOPICS IN MATHEMATICS 461
6.1 Conic Sections 463
6.2 Polar Coordinates 484
6.3 Parametric Equations 499
6.4 Sequences, Series, and Summation Notation 508
6.5 The Binomial Theorem 525
EXPLORINGCONCEPTSWITHTECHNOLOGY: Using a Graphing Calculator to
Find the nth Roots of z 531
Chapter 6 Summary 532
Chapter 6 True/False Exercises 533
Chapter 6 Review Exercises 534
Chapter 6 Test 535
Cumulative Review Exercises 536
ALGEBRA REVIEW APPENDIX
A.1 Integer and Rational Number Exponents 537
A.2 Polynomials 550
A.3 Factoring 556
SOLUTIONSTOTHETRYEXERCISES S1
ANSWERSTOSELECTEDEXERCISES A1
INDEX I1
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P
REFACE
Essentials of Precalculus provides students with material that focuses on
selected key concepts of precalculus and how those concepts can be applied to a
variety of problems. To help students master these concepts, we have tried to
maintain a balance among theory, application, modeling, and drill. Carefully
developed mathematics is complemented by applications that are both con-
temporary and representative of a wide range of disciplines. Many application
exercises are accompanied by a diagram that helps the student visualize the
mathematics of the application.
Technology is introduced naturally to support and advance better under-
standing of a concept. The optional Integrating Technologyboxes and graphing cal-
culator exercises are designed to promote an appreciation of both the power and
the limitations of technology.
Features
Interactive Presentation Essentials of Precalculus is written in a style that
encourages the student to interact with the textbook. At various places through-
out the text, we pose a question to the student about the material being presented.
This question encourages the reader to pause and think about the current discus-
sion and to answer the question. To ensure that the student does not miss impor-
tant information, the answer to the question is provided as a footnote on the
same page.
Each section contains a variety of worked examples. Each example is given a
title so that the student can see at a glance the type of problem being illustrated.
Most examples are accompanied by annotations that assist the student in moving
from step to step. Following the worked example is a suggested exercise for
the student to work. The complete solution to that exercise can be found in an ap-
pendix in the text. This feature allows students to self-assess their progress and to
get immediate feedback by means of not just an answer, but a complete solution.
Focus on Problem Solving Each chapter begins with a Focus on Problem Solving
that demonstrates various strategies that are used by successful problem solvers.
At the completion of the Focus on Problem Solving, the student is directed to an
exercise in the text that can be solved using the problem-solving strategy that was
just discussed.
Mathematics and Technology Technology is introduced in the text to illustrate
or enhance a concept. We attempt to foster the idea that technology, combined
with analytical thinking, can lead to deeper understanding of a concept. The
optional Integrating Technology boxes and graphing calculator exercises are de-
signed to develop an awareness of technology’s capabilities and limitations.
Topics for Discussionare found at the end of each section of the text. These topics
can form the basis for a group discussion or serve as writing assignments.
ix