Precalculus / Edition 7

by Michael Sullivan

Overview

This book motivates through realistic applications to people of diverse mathematical backgrounds. Develops the trigonometric functions using a unit circle approach and progresses to the right triangle approach. Graphing techniques are emphasized, including a thorough discussion of polar coordinates, parametric equations, and conics using polar coordinates.

Product Details

ISBN-13:	9780131913998
Publisher:	Pearson
Publication date:	03/12/2004
Edition description:	New Edition
Pages:	1001
Product dimensions:	6.50(w) x 1.50(h) x 9.50(d)

About the Author

Mike Sullivan recently retired as Professor of Mathematics at Chicago State University, having taught there for more than 30 years. He received his PhD in mathematics from Illinois Institute of Technology. He is a native of Chicago’s South Side and currently resides in Oak Lawn, Illinois. Mike has 4 children; the 2 oldest have degrees in mathematics and assisted in proofing, checking examples and exercises, and writing solutions manuals for this project. His son Mike Sullivan, III co-authored the Sullivan Graphing with Data Analysis series as well as this series. Mike has authored or co-authored more than 10 books. He owns a travel agency, and splits his time between a condo in Naples, Florida and a home in Oak Lawn, where he enjoys gardening.

Mike Sullivan, III is a professor of mathematics at Joliet Junior College. He holds graduate degrees from DePaul University in both mathematics and economics. Mike is an author or co-author on more than 20 books, including a statistics book and a developmental mathematics series. Mike is the father of 3 children and an avid golfer who tries to spend as much of his limited free time as possible on the golf course.

Mike Sullivan recently retired as Professor of Mathematics at Chicago State University, having taught there for more than 30 years. He received his PhD in mathematics from Illinois Institute of Technology. He is a native of Chicago’s South Side and currently resides in Oak Lawn, Illinois. Mike has 4 children; the 2 oldest have degrees in mathematics and assisted in proofing, checking examples and exercises, and writing solutions manuals for this project. His son Mike Sullivan, III co-authored the Sullivan Graphing with Data Analysis series as well as this series. Mike has authored or co-authored more than 10 books. He owns a travel agency, and splits his time between a condo in Naples, Florida and a home in Oak Lawn, where he enjoys gardening.

Mike Sullivan, III is a professor of mathematics at Joliet Junior College. He holds graduate degrees from DePaul University in both mathematics and economics. Mike is an author or co-author on more than 20 books, including a statistics book and a developmental mathematics series. Mike is the father of 3 children and an avid golfer who tries to spend as much of his limited free time as possible on the golf course.

Table of Contents

Preface to the Instructor     xiii
List of Applications     xix
Photo Credits     xxv
To the Student     xxvi
Graphs     1
The Distance and Midpoint Formulas     2
Graphs of Equations in Two Variables; Intercepts; Symmetry     9
Lines     19
Circles     35
Chapter Review     42
Chapter Test     44
Chapter Projects     45
Functions and Their Graphs     47
Functions     48
The Graph of a Function     62
Properties of Functions     71
Library of Functions; Piecewise-defined Functions     82
Graphing Techniques: Transformations     92
Mathematical Models: Building Functions     104
Chapter Review     109
Chapter Test     114
Cumulative Review     115
Chapter Projects     116
Linear and Quadratic Functions     117
Linear Functions and Their Properties     118
Building Linear Functions from Data     127
Quadratic Functions and Their Properties     133
Quadratic Models; Building Quadratic Functions from Data     145
Inequalities Involving Quadratic Functions     154
Chapter Review     158
Chapter Test     160
Cumulative Review     161
Chapter Projects     162
Polynomial and Rational Functions     163
Polynomial Functions and Models     164
Properties of Rational Functions     184
The Graph of a Rational Function     195
Polynomial and Rational Inequalities     209
The Real Zeros of a Polynomial Function     215
Complex Zeros; Fundamental Theorem of Algebra     229
Chapter Review     234
Chapter Test     238
Cumulative Review     238
Chapter Projects     239
Exponential and Logarithmic Functions     241
Composite Functions     242
One-to-One Functions; Inverse Functions     249
Exponential Functions     263
Logarithmic Functions     277
Properties of Logarithms     290
Logarithmic and Exponential Equations     299
Compound Interest     305
Exponential Growth and Decay Models; Newton's Law; Logistic Growth and Decay Models     315
Building Exponential, Logarithmic, and Logistic Models from Data      327
Chapter Review     334
Chapter Test     340
Cumulative Review     341
Chapter Projects     342
Trigonometric Functions     343
Angles and Their Measure     344
Trigonometric Functions: Unit Circle Approach     357
Properties of the Trigonometric Functions     373
Graphs of the Sine and Cosine Functions     386
Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions     401
Phase Shift; Sinusoidal Curve Fitting     408
Chapter Review     419
Chapter Test     424
Cumulative Review     425
Chapter Projects     426
Analytic Trigonometry     427
The Inverse Sine, Cosine, and Tangent Functions     428
The Inverse Trigonometric Functions (Continued)     440
Trigonometric Identities     446
Sum and Difference Formulas     453
Double-angle and Half-angle Formulas     463
Product-to-Sum and Sum-to-Product Formulas     472
Trigonometric Equations (I)     475
Trigonometric Equations (II)     482
Chapter Review     489
Chapter Test     492
Cumulative Review      493
Chapter Projects     493
Applications of Trigonometric Functions     495
Applications Involving Right Triangles     496
The Law of Sines     508
The Law of Cosines     519
Area of a Triangle     525
Simple Harmonic Motion; Damped Motion; Combining Waves     531
Chapter Review     541
Chapter Test     545
Cumulative Review     546
Chapter Projects     546
Polar Coordinates; Vectors     549
Polar Coordinates     550
Polar Equations and Graphs     558
The Complex Plane; DeMoivre's Theorem     574
Vectors     582
The Dot Product     593
Vectors in Space     601
The Cross Product     610
Chapter Review     616
Chapter Test     620
Cumulative Review     620
Chapter Projects     621
Analytic Geometry     623
Conics     624
The Parabola     625
The Ellipse     634
The Hyperbola     644
Rotation of Axes; General Form of a Conic     656
Polar Equations of Conics      664
Plane Curves and Parametric Equations     670
Chapter Review     682
Chapter Test     685
Cumulative Review     685
Chapter Projects     686
Systems of Equations and Inequalities     687
Systems of Linear Equations: Substitution and Elimination     688
Systems of Linear Equations: Matrices     702
Systems of Linear Equations: Determinants     717
Matrix Algebra     727
Partial Fraction Decomposition     743
Systems of Nonlinear Equations     750
Systems of Inequalities     759
Linear Programming     767
Chapter Review     773
Chapter Test     777
Cumulative Review     778
Chapter Projects     779
Sequences; Induction; the Binomial Theorem     781
Sequences     782
Arithmetic Sequences     791
Geometric Sequences; Geometric Series     797
Mathematical Induction     809
The Binomial Theorem     813
Chapter Review     819
Chapter Test     822
Cumulative Review     822
Chapter Projects     823
Counting and Probability      825
Counting     826
Permutations and Combinations     831
Probability     840
Chapter Review     850
Chapter Test     852
Cumulative Review     853
Chapter Projects     853
A Preview of Calculus: The Limit, Derivative, and Integral of a Function     855
Finding Limits Using Tables and Graphs     856
Algebra Techniques for Finding Limits     861
One-sided Limits; Continuous Functions     868
The Tangent Problem; The Derivative     875
The Area Problem; The Integral     883
Chapter Review     889
Chapter Test     893
Chapter Projects     894
Review     A1
Algebra Essentials     A1
Geometry Essentials     A14
Polynomials     A22
Synthetic Division     A31
Rational Expressions     A35
Solving Equations     A43
Complex Numbers; Quadratic Equations in the Complex Number System     A54
Problem Solving: Interest, Mixture, Uniform Motion, Constant Rate Job Applications     A62
Interval Notation; Solving Inequalities     A72
nth Roots; Rational Exponents      A81
Graphing Utilities     B1
The Viewing Rectangle     B1
Using a Graphing Utility to Graph Equations     B3
Using a Graphing Utility to Locate Intercepts and Check for Symmetry     B6
Using a Graphing Utility to Solve Equations     B8
Square Screens     B10
Using a Graphing Utility to Graph Inequalities     B11
Using a Graphing Utility to Solve Systems of Linear Equations     B12
Using a Graphing Utility to Graph a Polar Equation     B13
Using a Graphing Utility to Graph Parametric Equations     B14
Answers     AN1
Index     I1

Preface

To the Instructor

As a professor at an urban public university for over 30 years, I am aware of the varied needs of precalculus students. As the author of precalculus, engineering calculus, finite mathematics and business calculus texts, and, as a teacher, I understand what students must know if they are to be focused and successful in upper level mathematics courses. However, as a father of four college graduates, I also understand the realities of college life.

Precalculus texts too often are simply condensed versions of algebra and trigonometry texts. College algebra and algebra and trigonometry students are different from precalculus students and their texts should reflect this difference. For example, Chapter 13 A Preview of Calculus; the Limit, Derivative, and Integral of a Function, not only demonstrates to students how the material of Precalculus applies to calculus, but also moves the student into calculus. Throughout this text there are references to calculus, shown by a calculus icon ~ to further motivate and remind the student that this mathematics will be used later. There are other, more subtle, aspects of this text that prepare the student for calculus. For example, many applications that are traditional to calculus have been inserted as algebra and trigonometry problems. These examples and exercises are designed to emphasize the role of algebra and trigonometry in calculus and to encourage and motivate students in Precalculus to further insure their success in calculus.

I have taken great pains to insure that the text contains solid, student-friendly examples and problems, as well as a clear, seamless, writing style. I encourage you toshare with me your experiences teaching from this text.

THE SIXTH EDITION

The Sixth Edition builds upon a solid foundation by integrating new features and techniques that further enhance student interest and involvement. The elements of previous editions that have proved successful remain, while many changes, some obvious, others subtle, have been made. A huge benefit of authoring a successful series is the broad-based feedback upon which improvements and additions are ultimately based. Virtually every change to this edition is the result of thoughtful comments and suggestions made from colleagues and students who have used previous editions. I am sincerely grateful for this feedback and have tried to make changes that improve the flow and usability of the text.

NEW TO THE SIXTH EDITION

Real Mathematics at Motorola
Each chapter begins with Field Trip to Motorola, a brief description of a current situation at Motorola, followed by Interview at Motorola, a biographical sketch of a Motorola employee. At the end of each chapter is Project at Motorola, written by the Motorola employee, that contains a description, with exercises, of a problem at Motorola that relates to the mathematics found in the chapter. It doesn't get more REAL than this.

Preparing for This Section
Most sections now open with a referenced list (by section and page number) of key items to review in preparation for the section ahead. This provides a just-in-time review for students.

Appendix A Review
The content here consists of the first half of the old Chapter 1, Synthetic Division, and Complex Numbers; Quadratic Equations with a Negative Discriminant. Although it could be used as the first part of a course in Precalculus, its real value lies in its use as a just-in-time review of material. Specific references to Appendix A occur throughout the text to assist in the review process. Appropriate use of this appendix will allow students to review when they need to and will allow the instructor more time to cover the course content.

Content:

• Appendix B, Graphing Utilities, has been updated and expanded to include the latest features of the graphing calculator. While the graphing calculator remains an option, identified by a graphing icon, references to Appendix B occur at appropriate places in the text for those inclined to use the graphing calculator features of the text.
• The Cross Product is a new section in Chapter 8 on Vectors.
• The Area Problem; the Integral is a new section in Chapter 13 A Preview of Calculus
• Area of a Sector is a new sub-section in Chapter 5, Section 5.1, Angles and Their Measure
• Combining Waves is a new sub-section in Chapter 7 Applications of Trigonometry, Section 7.5.

Organization:

• The discussion on Rational Functions now appears in two sections, Rational Functions I and Rational Functions II: Analyzing Graphs. This division should allow the sections to be covered in one teaching period each.
• The discussion of Polynomial and Rational Inequalities now appears after Polynomial and Rational Functions. This allows us to use information obtained about the graphs to solve the inequalities. Students and instructors will appreciate how easy this usually tough concept is now handled.
• The chapter on Trigonometric Functions now has a single section devoted to the graphs of the sine and cosine functions, including a discussion of sinusoidal graphs. Separate sections follow on Graphs of the remaining Trigonometric Functions and Phase Shift; Sinusoidal Curve Fitting. These changes will allow the material of each section to be taught in a single period and provide flexibility in choice of content.
• The chapter on Analytic Trigonometry now begins with two sections that discuss the inverse trigonometric functions. The chapter concludes with two sections devoted to Trigonometric Equations. These changes will allow each section to be taught in a single period.
• Separate chapters on Sequences; Induction; the Binomial Theorem and Counting and Probability also provide more flexibility in coverage.

FEATURES IN THE 6TH EDITION

• Section OBJECTIVES appear in a numbered list to begin each section.
• Now Work Problem XX appears after a concept has been introduced. This directs the student to a problem in the exercises that tests the concept, insuring that the concept has been mastered before moving on. The Now Work problems are identified in the exercises using yellow numbers and a pencil icon.
• Optional Comments, Explorations, Seeing the Concept, Examples, and Exercises that utilize the graphing calculator are clearly marked with a calculator icon. Calculator exercises are also identified by the calculator icon and green numbers.
• Discussion, Writing, and Research problems appear in each exercise set, identified by an icon and red numbers. These provide the basis for class discussion, writing projects, and collaborative learning experiences. References to Calculus are identified by a calculus icon.
• Historical Perspectives, sometimes with exercises, are presented in context and provide interesting anecdotal information.
• Varied applications are abundant both in Examples and in Exercises. Many contain sourced data.
• An extensive Chapter Review provides a list of important formulas, definitions, theorems, and objectives, as well as a complete set of Review Exercises, with sample test questions identified by blue numbers.

USING THE 6TH EDITION EFFECTIVELY AND EFFICIENTLY WITH YOUR SYLLABUS

To meet the varied needs of diverse syllabi, this book contains more content than expected in a precalculus course. The illustration shows the dependencies of chapters on each other.

As the chart indicates, this book has been organized with flexibility of use in mind. Even within a given chapter, certain sections can be skipped without fear of future problems.

Chapter 1 Graphs
This chapter is the last half of the old Chapter 1. A quick coverage of this short chapter, which is mainly review material, will enable you to get to Chapter 2 Functions and their Graphs earlier. If curve fitting is not part of your syllabus, Section 1.4 may be omitted with any adverse effects.

Chapter 2 Functions and Their Graphs
Perhaps the most important chapter. Section 2.6 can be skipped without adverse effects.

Chapter 3 Polynomial and, Rational Functions
Topic selection is dependent on your syllabus.

Chapter 4 Exponential and Logarithmic Functions
Sections 4.1-4.5 follow in sequence; Sections 4.6, 4.7, and 4.8 each require Section 4.3.

Chapter 5 Trigonometric Functions
The sections follow in sequence.

Chapter 6 Analytic Trigonometry
The sections follow in sequence. Sections 6.2, 6.6, and 6.8 may be skipped in a brief course.

Chapter 7 Applications of Trigonometric Functions
The sections follow in sequence. Sections 7.4 and 7.5 may be skipped in a brief course.

Chapter 8 Polar Coordinates; Vectors
Sections 8.1-8.3 and Sections 8.4-8.7 are independent and may be covered separately.

Chapter 9 Analytic Geometry
Sections 9.1-9.4 follow in sequence. Sections 9.5, 9.6, and 9.7 are independent of each other, but do depend on Sections 9.1-9.4.

Chapter 10 Systems of Equations and Inequalities
Sections 10.1-10.2 follow in sequence; Sections 10.3-10.8 require Sections 10.1 and 10.2, and may be covered in any order. Section 10.9 depends on Section 10.8.

Chapter 11 Sequences; Introduction; The Binomial Theorem
The are three independent part: Sections 11.1-11.3,11.4, and 11.5.

Chapter 12 Counting and Probability
Sections 12.1-12.3 follow in order.

Chapter 13 A Preview of Calculus: The Limit, Derivative, and Integral of a Function
If time permits, coverage of this chapter will give your students a beneficial head-start in calculus.

To the Student

As you begin your study of Precalculus you may feel overwhelmed by the number of theorems, definitions, procedures, and equations that confront you. You may even wonder whether or not you can learn all of this material in the time allotted. These concerns are normal. Keep in mind that many elements of Precalculus are all around us as we go through our daily routines. Many of the concepts you will learn to express mathematically, you already know intuitively. For many of you, this may be your last math course, while for others, just the first in a series of many. Either way, this text was written with you in mind. I have taught precalculus courses for over thirty years. I am also the father of four college graduates who called home from time to time, frustrated and with questions. I know what you're going through. So I have written a text that doesn't overwhelm, or unnecessarily complicate Precalculus, while at the same time providing you the skills and practice you need to be successful.

This text is designed to help you, the student, master the terminology and basic concepts of Precalculus. These aims have helped to shape every aspect of the book. Many learning aids are built into the format of the text to make your study of the material easier and more rewarding. This book is meant to be a "machine for learning," one that can help you focus your efforts and get the most from the time and energy you invest.

HOW TO USE THIS BOOK EFFECTIVELY AND EFFICIENTLY

First, and most important, this book is meant to be read-so please, begin by reading the material assigned. You will find that the text has additional explanation and examples that will help you. Also, it is best to read the section before the lecture, so you can ask questions right away about anything you didn't understand.

Many sections begin with "Preparing for This Section," a list of concepts that will be used in the section. Take the short amount of time required to refresh your memory. This will make the section easier to understand and will actually save you time and effort.

A list of OBJECTIVES is provided at the beginning of each section. Read them. They will help you recognize the important ideas and skills developed in the section.

After a concept has been introduced and an example given, you will see NOW WORK PROBLEM XX. Go to the exercises at the end of the section, work the problem cited, and check your answer in the back of the book. If you get it right, you can be confident in continuing on in the section. If you don't get it right, go back over the explanations and examples to see what you might have missed. Then rework the problem. Ask for help if you miss it again.

If you follow these practices throughout the section, you will find that you have probably done many of your homework problems. In the exercises, every "Now Work Problem" number is in yellow with a pencil icon. All the odd-numbered problems have answers in the back of the book and worked-out solutions in the Student Solutions Manual supplement. Be sure you have made an honest effort before looking at a worked-out solution.

At the end of each chapter is a Chapter Review. Use it to be sure you are completely familiar with the equations and formulas listed under "Things to Know." If you are unsure of an item here, use the page reference to go back and review it. Go through the Objectives and be sure you can answer "Yes" to the question "I should be able to ...." If you are uncertain, a page reference to the objective is provided.

Spend the few minutes necessary to answer the "Fill-in-the-Blank" items and the "True/False" items. These are quick and valuable questions to answer.

Lastly, do the problems identified with blue numbers in the Review Exercises. These are my suggestions for a Practice Test. Do some of the other problems in the review for more practice to prepare for your exam.

Please do not hesitate to contact me, through Prentice Hall, with any suggestions or comments that would improve this text. I look forward to hearing from you.

Best Wishes!
Michael Sullivan

Editorial Reviews

Provides undergraduate students with skills they will need to be successful in calculus. Chapters cover equations and graphs through sequences and probability, with worked examples, within-chapter exercises, and chapter reviews with quizes, review exercises, and problems. This fifth edition features Internet exercises, a new chapter previewing calculus, and new exercises and examples utilizing real data in table form. Annotation c. by Book News, Inc., Portland, Or.

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