College Algebra And Trigonometry - Stitz Zeager
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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10.6.6 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10.7 Trigonometric Equations and Inequalities . . . . . . . . . . . . . . . . . .10.7.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10.7.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11 Applications of Trigonometry11.1 Applications of Sinusoids . .11.1.1 Harmonic Motion .11.1.2 Exercises . . . . . .11.1.3 Answers . . . . . . .11.2 The Law of Sines . . . . . .11.2.1 Exercises . . . . . .11.2.2 Answers . . . . . . .11.3 The Law of Cosines . . . . 7. 747. 751. 757. 759. 761. 769. 772. 773
viii11.3.1 Exercises . . . . . . . . .11.3.2 Answers . . . . . . . . . .11.4 Polar Coordinates . . . . . . . . .11.4.1 Exercises . . . . . . . . .11.4.2 Answers . . . . . . . . . .11.5 Graphs of Polar Equations . . . .11.5.1 Exercises . . . . . . . . .11.5.2 Answers . . . . . . . . . .11.6 Hooked on Conics Again . . . . .11.6.1 Rotation of Axes . . . . .11.6.2 The Polar Form of Conics11.6.3 Exercises . . . . . . . . .11.6.4 Answers . . . . . . . . . .11.7 Polar Form of Complex Numbers11.7.1 Exercises . . . . . . . . .11.7.2 Answers . . . . . . . . . .11.8 Vectors . . . . . . . . . . . . . . .11.8.1 Exercises . . . . . . . . .11.8.2 Answers . . . . . . . . . .11.9 The Dot Product and Projection11.9.1 Exercises . . . . . . . . .11.9.2 Answers . . . . . . . . . .11.10 Parametric Equations . . . . . .11.10.1 Exercises . . . . . . . . .11.10.2 Answers . . . . . . . . . .IndexTable of 2855857859872874875883884885896899901
PrefaceThank you for your interest in our book, but more importantly, thank you for taking the time toread the Preface. I always read the Prefaces of the textbooks which I use in my classes becauseI believe it is in the Preface where I begin to understand the authors - who they are, what theirmotivation for writing the book was, and what they hope the reader will get out of reading thetext. Pedagogical issues such as content organization and how professors and students should bestuse a book can usually be gleaned out of its Table of Contents, but the reasons behind the choicesauthors make should be shared in the Preface. Also, I feel that the Preface of a textbook shoulddemonstrate the authors’ love of their discipline and passion for teaching, so that I come awaybelieving that they really want to help students and not just make money. Thus, I thank my fellowPreface-readers again for giving me the opportunity to share with you the need and vision whichguided the creation of this book and passion which both Carl and I hold for Mathematics and theteaching of it.Carl and I are natives of Northeast Ohio. We met in graduate school at Kent State Universityin 1997. I finished my Ph.D in Pure Mathematics in August 1998 and started teaching at LorainCounty Community College in Elyria, Ohio just two days after graduation. Carl earned his Ph.D inPure Mathematics in August 2000 and started teaching at Lakeland Community College in Kirtland,Ohio that same month. Our schools are fairly similar in size and mission and each serves a similarpopulation of students. The students range in age from about 16 (Ohio has a Post-SecondaryEnrollment Option program which allows high school students to take college courses for free whilestill in high school.) to over 65. Many of the “non-traditional” students are returning to school inorder to change careers. A majority of the students at both schools receive some sort of financialaid, be it scholarships from the schools’ foundations, state-funded grants or federal financial aidlike student loans, and many of them have lives busied by family and job demands. Some willbe taking their Associate degrees and entering (or re-entering) the workforce while others will becontinuing on to a four-year college or university. Despite their many differences, our studentsshare one common attribute: they do not want to spend 200 on a College Algebra book.The challenge of reducing the cost of textbooks is one that many states, including Ohio, are takingquite seriously. Indeed, state-level leaders have started to work with faculty from several of thecolleges and universities in Ohio and with the major publishers as well. That process will takeconsiderable time so Carl and I came up with a plan of our own. We decided that the bestway to help our students right now was to write our own College Algebra book and give it awayelectronically for free. We were granted sabbaticals from our respective institutions for the Spring
xPrefacesemester of 2009 and actually began writing the textbook on December 16, 2008. Using an opensource text editor called TexNicCenter and an open-source distrib
College Algebra and Trigonometry a.k.a. Precalculus by Carl Stitz, Ph.D. Jeff Zeager, Ph.D. . Enrollment Option program which allows high school students to take college courses for free while still in high school.) to over 65. Many of the \non-traditional" students are returning to school in . they do not want to spend 200 on a College .
College Trigonometry Version bˇc Corrected Edition by Carl Stitz, Ph.D. Je Zeager, Ph.D. Lakeland Community College Lorain County Community College July 4, 2013. ii Acknowledgements While the cover of this textbook lists only two names, the book as it stands today would simply
Robert Gerver, Ph.D. North Shore High School 450 Glen Cove Avenue Glen Head, NY 11545 gerverr@northshoreschools.org Rob has been teaching at . Algebra 1 Financial Algebra Geometry Algebra 2 Algebra 1 Geometry Financial Algebra Algebra 2 Algebra 1 Geometry Algebra 2 Financial Algebra ! Concurrently with Geometry, Algebra 2, or Precalculus
College Algebra Version bˇc Corrected Edition by Carl Stitz, Ph.D. Je Zeager, Ph.D. Lakeland Community College Lorain County Community College July 4, 2013
College Algebra and Trigonometry by Alexander Rozenblyum is licensed under a . Creative Commons Attribution-Noncommercial-Share Alike 4.0 License. Preface . This textbook is based on the course “College Algebra and Trigonometry” taught at New York City College of Tec
Trigonometry Trigonometry is a three-unit course at my college. I used the following open-source text-book: Smith, P. A. (2015). College Trigonometry With Extensive Use of the Tau Transcen-dental. [Derivative work based upon Stitz, C. & Zeager, J. (2013). Precalculus (Cor-
I used the book in three sections of College Algebra at Lorain County Community College in the Fall of 2009 and Carl's colleague, Dr. Bill Previts, taught a section of College Algebra at Lakeland with the book that semester as well. Students had the option of downloading the book as a .pdf le from our websitewww.stitz-zeager.comor buying a
College Algebra, Geometry, and Trigonometry Placement Tests . College Algebra Placement Test . Items in the College Algebra Test focu
Agile Software Development with Scrum Jeff Sutherland Gabrielle Benefield. Agenda Introduction Overview of Methodologies Exercise; empirical learning Agile Manifesto Agile Values History of Scrum Exercise: The offsite customer Scrum 101 Scrum Overview Roles and responsibilities Scrum team Product Owner ScrumMaster. Agenda Scrum In-depth The Sprint Sprint Planning Exercise: Sprint Planning .
