Transcription

College Algebra and Trigonometrya.k.a. PrecalculusbyCarl Stitz, Ph.D.Lakeland Community CollegeJeff Zeager, Ph.D.Lorain County Community CollegeAugust 30, 2010

iiAcknowledgementsThe authors are indebted to the many people who support this project. From Lakeland CommunityCollege, we wish to thank the following people: Bill Previts, who not only class tested the bookbut added an extraordinary amount of exercises to it; Rich Basich and Ivana Gorgievska, whoclass tested and promoted the book; Don Anthan and Ken White, who designed the electric circuitapplications used in the text; Gwen Sevits, Assistant Bookstore Manager, for her patience andher efforts to get the book to the students in an efficient and economical fashion; Jessica Novak,Marketing and Communication Specialist, for her efforts to promote the book; Corrie Bergeron,Instructional Designer, for his enthusiasm and support of the text and accompanying YouTubevideos; Dr. Fred Law, Provost, and the Board of Trustees of Lakeland Community College for theirstrong support and deep commitment to the project. From Lorain County Community College, wewish to thank: Irina Lomonosov for class testing the book and generating accompanying PowerPointslides; Jorge Gerszonowicz, Kathryn Arocho, Heather Bubnick, and Florin Muscutariu for theirunwaivering support of the project; Drs. Wendy Marley and Marcia Ballinger, Lorain CCC, forthe Lorain CCC enrollment data used in the text. We would also like to extend a special thanksto Chancellor Eric Fingerhut and the Ohio Board of Regents for their support and promotion ofthe project. Last, but certainly not least, we wish to thank Dimitri Moonen, our dear friend fromacross the Atlantic, who took the time each week to e-mail us typos and other corrections.

Table of ContentsPreface1 Relations and Functions1.1The Cartesian Coordinate Plane .1.1.1 Distance in the Plane . . .1.1.2 Exercises . . . . . . . . . .1.1.3 Answers . . . . . . . . . . .1.2Relations . . . . . . . . . . . . . .1.2.1 Exercises . . . . . . . . . .1.2.2 Answers . . . . . . . . . . .1.3Graphs of Equations . . . . . . . .1.3.1 Exercises . . . . . . . . . .1.3.2 Answers . . . . . . . . . . .1.4Introduction to Functions . . . . .1.4.1 Exercises . . . . . . . . . .1.4.2 Answers . . . . . . . . . . .1.5Function Notation . . . . . . . . .1.5.1 Exercises . . . . . . . . . .1.5.2 Answers . . . . . . . . . . .1.6Function Arithmetic . . . . . . . .1.6.1 Exercises . . . . . . . . . .1.6.2 Answers . . . . . . . . . . .1.7Graphs of Functions . . . . . . . .1.7.1 General Function Behavior1.7.2 Exercises . . . . . . . . . .1.7.3 Answers . . . . . . . . . . .1.8Transformations . . . . . . . . . . .1.8.1 Exercises . . . . . . . . . .1.8.2 Answers . . . . . . . . . . 4104107

iv2 Linear and Quadratic Functions2.1Linear Functions . . . . . . .2.1.1 Exercises . . . . . . .2.1.2 Answers . . . . . . . .2.2Absolute Value Functions . .2.2.1 Exercises . . . . . . .2.2.2 Answers . . . . . . . .2.3Quadratic Functions . . . . .2.3.1 Exercises . . . . . . .2.3.2 Answers . . . . . . . .2.4Inequalities . . . . . . . . . .2.4.1 Exercises . . . . . . .2.4.2 Answers . . . . . . . .2.5Regression . . . . . . . . . . .2.5.1 Exercises . . . . . . .2.5.2 Answers . . . . . . . .Table of 01751783 Polynomial Functions3.1Graphs of Polynomials . . . . . . . . . . . . . . . . . . . . . . .3.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . .3.1.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . .3.2The Factor Theorem and The Remainder Theorem . . . . . . .3.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . .3.2.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . .3.3Real Zeros of Polynomials . . . . . . . . . . . . . . . . . . . . .3.3.1 For Those Wishing to use a Graphing Calculator . . . .3.3.2 For Those Wishing NOT to use a Graphing Calculator3.3.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . .3.3.4 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . .3.4Complex Zeros and the Fundamental Theorem of Algebra . . .3.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . .3.4.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . 231. 242. 244. 246. 259. 261. 267. 272. 276.4 Rational Functions4.1Introduction to Rational Functions . .4.1.1 Exercises . . . . . . . . . . . .4.1.2 Answers . . . . . . . . . . . . .4.2Graphs of Rational Functions . . . . .4.2.1 Exercises . . . . . . . . . . . .4.2.2 Answers . . . . . . . . . . . . .4.3Rational Inequalities and Applications4.3.1 Variation . . . . . . . . . . . .4.3.2 Exercises . . . . . . . . . . . .

Table of Contents4.3.3vAnswers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2785 Further Topics in Functions5.1Function Composition . .5.1.1 Exercises . . . . .5.1.2 Answers . . . . . .5.2Inverse Functions . . . . .5.2.1 Exercises . . . . .5.2.2 Answers . . . . . .5.3Other Algebraic Functions5.3.1 Exercises . . . . .5.3.2 Answers . . . . . .2792792892912933093103113213246 Exponential and Logarithmic Functions6.1Introduction to Exponential and Logarithmic Functions6.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . .6.1.2 Answers . . . . . . . . . . . . . . . . . . . . . . .6.2Properties of Logarithms . . . . . . . . . . . . . . . . . .6.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . .6.2.2 Answers . . . . . . . . . . . . . . . . . . . . . . .6.3Exponential Equations and Inequalities . . . . . . . . . .6.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . .6.3.2 Answers . . . . . . . . . . . . . . . . . . . . . . .6.4Logarithmic Equations and Inequalities . . . . . . . . .6.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . .6.4.2 Answers . . . . . . . . . . . . . . . . . . . . . . .6.5Applications of Exponential and Logarithmic Functions6.5.1 Applications of Exponential Functions . . . . . .6.5.2 Applications of Logarithms . . . . . . . . . . . .6.5.3 Exercises . . . . . . . . . . . . . . . . . . . . . .6.5.4 Answers . . . . . . . . . . . . . . . . . . . . . . 913957 Hooked on Conics7.1Introduction to Conics7.2Circles . . . . . . . . .7.2.1 Exercises . . .7.2.2 Answers . . . .7.3Parabolas . . . . . . .7.3.1 Exercises . . .7.3.2 Answers . . . .7.4Ellipses . . . . . . . .7.4.1 Exercises . . .7.4.2 Answers . . . .397397400404405407415416419428430.

viTable of Contents7.5Hyperbolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4337.5.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4447.5.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4468 Systems of Equations and Matrices8.1Systems of Linear Equations: Gaussian Elimination .8.1.1 Exercises . . . . . . . . . . . . . . . . . . . .8.1.2 Answers . . . . . . . . . . . . . . . . . . . . .8.2Systems of Linear Equations: Augmented Matrices .8.2.1 Exercises . . . . . . . . . . . . . . . . . . . .8.2.2 Answers . . . . . . . . . . . . . . . . . . . . .8.3Matrix Arithmetic . . . . . . . . . . . . . . . . . . .8.3.1 Exercises . . . . . . . . . . . . . . . . . . . .8.3.2 Answers . . . . . . . . . . . . . . . . . . . . .8.4Systems of Linear Equations: Matrix Inverses . . . .8.4.1 Exercises . . . . . . . . . . . . . . . . . . . .8.4.2 Answers . . . . . . . . . . . . . . . . . . . . .8.5Determinants and Cramer’s Rule . . . . . . . . . . .8.5.1 Definition and Properties of the Determinant8.5.2 Cramer’s Rule and Matrix Adjoints . . . . .8.5.3 Exercises . . . . . . . . . . . . . . . . . . . .8.5.4 Answers . . . . . . . . . . . . . . . . . . . . .8.6Partial Fraction Decomposition . . . . . . . . . . . .8.6.1 Exercises . . . . . . . . . . . . . . . . . . . .8.6.2 Answers . . . . . . . . . . . . . . . . . . . . .8.7Systems of Non-Linear Equations and Inequalities . .8.7.1 Exercises . . . . . . . . . . . . . . . . . . . .8.7.2 Answers . . . . . . . . . . . . . . . . . . . . .9 Sequences and the Binomial Theorem9.1Sequences . . . . . . . . . . . . . . .9.1.1 Exercises . . . . . . . . . . .9.1.2 Answers . . . . . . . . . . . .9.2Summation Notation . . . . . . . . .9.2.1 Exercises . . . . . . . . . . .9.2.2 Answers . . . . . . . . . . . .9.3Mathematical Induction . . . . . . .9.3.1 Exercises . . . . . . . . . . .9.3.2 Selected Answers . . . . . . .9.4The Binomial Theorem . . . . . . . .9.4.1 Exercises . . . . . . . . . . .9.4.2 Answers . . . . . . . . . . . 17521522530531532544547.551. 551. 559. 561. 562. 571. 572. 573. 578. 579. 581. 590. 591

Table of Contentsvii10 Foundations of Trigonometry10.1 Angles and their Measure . . . . . . . . . . . . . . . . . . . . . . . . . . .10.1.1 Applications of Radian Measure: Circular Motion . . . . . . . . .10.1.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10.1.3 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10.2 The Unit Circle: Cosine and Sine . . . . . . . . . . . . . . . . . . . . . . .10.2.1 Beyond the Unit Circle . . . . . . . . . . . . . . . . . . . . . . . .10.2.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10.2.3 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10.3 The Six Circular Functions and Fundamental Identities . . . . . . . . . . .10.3.1 Beyond the Unit Circle . . . . . . . . . . . . . . . . . . . . . . . .10.3.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10.3.3 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10.4 Trigonometric Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . .10.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10.4.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10.5 Graphs of the Trigonometric Functions . . . . . . . . . . . . . . . . . . . .10.5.1 Graphs of the Cosine and Sine Functions . . . . . . . . . . . . . .10.5.2 Graphs of the Secant and Cosecant Functions . . . . . . . . . . .10.5.3 Graphs of the Tangent and Cotangent Functions . . . . . . . . . .10.5.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10.5.5 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10.6 The Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . .10.6.1 Inverses of Secant and Cosecant: Trigonometry Friendly Approach10.6.2 Inverses of Secant and Cosecant: Calculus Friendly Approach . . .10.6.3 Using a Calculator to Approximate Inverse Function Values. . . .10.6.4 Solving Equations Using the Inverse Trigonometric Functions. . .10.6.5 Exercises . . . . . . . . . . . .