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Table of Contents
1. Graphs.
Rectangular Coordinates; Graphing Utilities. Introduction to Graphing Equations. Symmetry; Graphing Key Equations; Circles. Solving Equations Using a Graphing Utility. Solving Inequalities. Lines. Chapter Projects. Cumulative Review.
2. Functions and Their Graphs.
Functions. Linear Functions and Models. Properties of Functions. Library of Functions; PiecewiseDefined Functions. Graphing Techniques: Transformations. Operations on Functions; Composite Functions. Mathematical Models: Constructing Functions. Chapter Projects. Cumulative Review.
3. Polynomial and Rational Functions.
Quadratic Functions and Models. Power Functions and Models. Polynomial Functions and Models. Rational Functions I. Rational Functions II: Analyzing Graphs. Polynomial and Rational Inequalities. The Real Zeros of a Polynomial Function. Complex Zeros; Fundamental Theorem of Algebra. Chapter Projects. Cumulative Review.
4. Exponential and Logarithmic Functions.
OnetoOne Functions; Inverse Functions. Exponential Functions. Logarithmic Functions. Properties of Logarithms. Logarithmic and Exponential Equations. Compound Interest. Growth and Decay. Exponential, Logarithmic, and Logistic Curve Fitting. Chapter Projects. Cumulative Review.
5. Trigonometric Functions.
Angles and Their Measure. Trigonometric Functions: Unit Circle Approach. Properties of the Trigonometric Functions. Graphs of the Sine and Cosine Functions. Graphs of the Tangent, Cotangent, Secant, and Cosecant Functions. Phase Shifts; Sinusoidal Curve Fitting. Chapter Projects. Cumulative Review.
6. Analytic Trigonometry.
The InverseSine, Cosine, and Tangent Functions. The Inverse Trigonometric Functions (continued) Trigonometric Identities. Sum and Difference Formulas. Doubleangle and Halfangle Formulas. ProducttoSum and SumtoProduct Formulas. Trigonometric Equations II. Trigonometric Equations II Chapter Review. Chapter Projects. Cumulative Review.
7. Applications of Trigonometric Functions.
Right Triangle Trigonometry. Law of Sines. Law of Cosines. Area of a Triangle. Simple Harmonic Motion; Damped Motion; Combining Waves. Chapter Projects. Cumulative Review.
8. Polar Coordinates; Vectors.
Polar Coordinates. Polar Equations and Graphs. The Complex Plane; DeMoivre's Theorem. Vectors. The Dot Product. Vectors in Space. The Cross Product. Chapter Projects. Cumulative Review.
9. Analytic Geometry.
Conics. The Parabola. The Ellipse. The Hyperbola. Rotation of Axes; General Form of a Conic. Polar Equations of Conics. Plane Curves and Parametric Equations. Chapter Projects. Cumulative Review.
10. Systems of Equations and Inequalities.
Systems of Linear Equations: Two Equations Containing Two Variables. Systems of Linear Equations: Three Equations Containing Three Variables. Systems of Linear Equations: Matrices. Systems of Linear Equations: Determinants. Matrix Algebra. Partial Fraction Decomposition. Systems of Nonlinear Equations. Systems of Inequalities. Linear Programming. Chapter Projects. Cumulative Review.
11. Sequences, Induction, the Binomial Theorem.
Sequences. Arithmetic Sequences. Geometric Sequences; Geometric Series. Mathematical Induction. The Binomial Theorem. Chapter Projects. Cumulative Review.
12. Counting and Probability.
Sets and Counting. Permutations and Combinations. Probability of Equally Likely Outcomes. Obtaining Probabilities from Data. Chapter Projects. Cumulative Review.
13. A Preview of Calculus: The Limit, Derivative, and Integral of a Function.
Finding Limits Using Tables and Graphs. Algebra Techniques for Finding Limits. Onesided Limits; Continuous Functions. The Tangent Problem; the Derivative. The Area Problem; the Integral. Chapter Projects. Cumulative Review.
Appendix Review.
Algebra Review. Geometry Review. Polynomials. Rational Expressions. Polynomial Division; Synthetic Division. Equations. Complex Numbers; Quadratic Equations with a Negative Discriminant. Setting up Equations; Applications. nth Roots; Rational Exponents.
Answers.
Index.
Introduction
TO THE INSTRUCTOR
As professors at both an urban public university and a community college, Michael Sullivan and Michael Sullivan, III are aware of the varied needs of Precalculus students. As a teacher, and as an author of precalculus, engineering calculus, finite mathematics, and business calculus texts, Michael understands what students must know if they are to be focused and successful in upper level mathematics courses. As a father of four, including the coauthor, he also understands the realities of college life. His coauthor and son, Michael III, believes passionately in the value of technology as a tool for 'learning that enhances understanding without sacrificing important skills.
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 inPrecalculus to further insure their success in calculus.
Together, Michael and Michael, III have taken great pains to ensure that this text contains solid, studentfriendly examples and exercises, as well as a clear, seamless writing style. Please share with them your experiences teaching from this text.
The Third Edition
The Third Edition builds upon a strong 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. One important benefit of authoring a successful series is the broadbased feedback upon which improvements and additions are ultimately based. Virtually every change to this edition is the result of thoughtful comments and suggestions from colleagues and students who used previous editions. We are sincerely grateful for this feedback and have tried to make changes that improve the usefulness of the text for both instructors and students.
New to the Third Edition
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 justintime review for students.
Concepts and Vocabulary
At the end of every section, there is a short list of FillintheBlank and True/False items that test concepts and vocabulary in a short answer format. Several quickanswer questions are also included.
Cumulative Reviews
At the end of Chapters 212, exercises are provided that require skills learned in the earlier chapters. These cumulative reviews serve to continually rein force the important concepts of Precalculus. They also make it easier for the student to prepare for a comprehensive final examination.
Content
Â• The formula for the area of a sector and related exercises are now part of Section 5.1 Angles and Their Measure.
Â• Combining Waves is a new subsection in Section 7.5 Simple Harmonic Motion; Damped Motion; Combining Waves.
Â• The Cross Product is a new section in Chapter 8 Polar Coordinates; Vectors.
Â• The Area Problem; the Integral is a new section in Chapter 13 A Preview of Calculus: the Limit, Derivative, and Integral of a Function
Â• New Chapter Projects have been added that discuss topics of current interest.
Organization
Â• Scatter diagrams, formerly part of Section 1.1, now appear in Section 2.2 Linear Functions and Models. This change positions the content where it is being used.
Â• Setting up Equations; Applications, formerly Section 1.4, now is part of the Appendix. This change allows those who wish to finish Chapter 1 more quickly, the opportunity to do so. If you wish, it can be covered anytime after Section 1.4.
Â• Circles, formerly Section 1.7, now is part of Section 1.3.
Â• The discussion on Rational Functions, formerly Section 3.7, now appears earlier in Chapter 3 as two sections. This division makes it possible to teach the sections in one period each.
Â• Complex Numbers; Quadratic Equations with a Negative Discriminant, formerly Section 3.5, now is part of the Appendix. This change now makes the material of Chapter 3 more consistent in level of difficulty. For those who wish to cover quadratic equations with a negative discriminant early, this section may be covered anytime after Section 1.4. Finally, this material can be used as a justintime review of complex numbers before Section 3.8 Complex Zeros; Fundamental Theorem of Algebra and before Section 8.3 The Complex Plane; DeMoivre's Theorem.
Â• Section 4.3, Exponential Functions, now contains a subsection on exponential equations; Section 4.4, Logarithmic Functions, now contains a subsection on logarithmic equations.
Â• The discussion on Right Triangle Trigonometry, formerly Section 5.4, has been relocated to Chapter 7, Applications of Trigonometry. This change makes the material of Chapter 5 more consistent in level of difficulty. For those who wish to cover this material earlier, this section can be covered anytime after Section 5.3.
Â• Graphs of the Trigonometric Functions and Sinusoidal Graphs; Sinusoidal Curve Fitting, formerly Sections 5.5 and 5.6, now is covered in three sees dons, 5.4, 5.5, and 5.6. This change makes it possible to teach the section in one period each.
Â• The Inverse Trigonometric Functions, formerly Section 6.5, now is covered in two sections at the beginning of the chapter. This change makes it possible to teach the sections in one period each. It also places the content closer to the discussion of the trigonometric functions and their graphs.
Features in the 3^{rd} Edition
Using the 3^{rd} 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 are optional and can be skipped without fear of future problems.
Chapter 1. Graphs
This chapter is a briefer version of the former Chapter 1. A quick coverage of this chapter, which is mainly review material, will enable you to get to Chapter 2 Functions and their Graphs earlier.
Chapter 2. Functions and Their Graphs
Perhaps the most important chapter. Section 2.7 is optional.
Chapter 3. Polynomial and Rational Functions
Topic selection is dependent on your syllabus.
Chapter 4. Exponential and Logarithmic Functions
Sections 4.14.5 follow in sequence. Sections 4.6, 4.7, and 4.8 are optional; each of these requires Section 4.3.
Chapter 5. Trigonometric Functions
The sections follow in sequence. Section 5.6 is optional.
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 Trigonometry
The sections follow in sequence. Sections 7.4 and 7.5 are optional.
Chapter 8. Polar Coordinates; Vectors
Sections 8.18.3 and Sections 8.48.7 are independent and may be covered separately.
Chapter 9. Analytic Geometry
Sections 9.19.4 follow in sequence. Sections 9.5,9.6, and 9.7 are independent of each other, but do depend on sections 9.19.4.
Chapter 10. Systems of Equations and Inequalities
Sections 10.110.2 follow in sequence; Sections 10.310.8 require Sections 10.110.2, but may be covered in any order. Section 10.7 also requires Sections 9.19.4. Section 10.9 requires Section 10.8.
Chapter 11. Sequences; Induction; The Binomial Theorem
There are three independent parts: Sections 11.111.3; Section 11.4; and Section 11.5.
Chapter 12. Counting and Probability
Sections 12.112.3 follow in sequence; Section 12.4 is optional.
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 headstart in calculus. The sections follow in sequence.
Appendix Review
Sections A.1A.6 and Section A.9 consist of review material, which can be used as the first part of a course in Precalculus or as a justintime review. Specific references to this material occur throughout the text to assist in the review process. Coverage of Sections A.7 and A.8 depends on your syllabus.