Precalculus Textbook.pdf - Precalculus OpenStax College Rice University 6100 Main Street MS-375 Houston Texas 77005 To learn more about OpenStax College

Precalculus Textbook.pdf - Precalculus OpenStax College...

This preview shows page 1 out of 1483 pages.

You've reached the end of your free preview.

Want to read all 1483 pages?

Unformatted text preview: Precalculus OpenStax College Rice University 6100 Main Street MS-375 Houston, Texas 77005 To learn more about OpenStax College, visit . Individual print copies and bulk orders can be purchased through our website. © 2014 Rice University. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution 4.0 International License. Under this license, any user of this textbook or the textbook contents herein must provide proper attribution as follows: - - - If you redistribute this textbook in a digital format (including but not limited to EPUB, PDF, and HTML), then you must retain on every page the following attribution: “Download for free at .” If you redistribute this textbook in a print format, then you must include on every physical page the following attribution: “Download for free at .” If you redistribute part of this textbook, then you must retain in every digital format page view (including but not limited to EPUB, PDF, and HTML) and on every physical printed page the following attribution: “Download for free at .” If you use this textbook as a bibliographic reference, then you should cite it as follows: OpenStax College, Precalculus. OpenStax College. 23 October 2014. < ;. For questions regarding this licensing, please contact [email protected]penstaxcollege.org. Trademarks The OpenStax College name, OpenStax College logo, OpenStax College book covers, Connexions name, and Connexions logo are not subject to the license and may not be reproduced without the prior and express written consent of Rice University. ISBN-10 1938168348 ISBN-13 978-1-938168-34-5 Revision PR-1-000-RS OpenStax College OpenStax College is a non-profit organization committed to improving student access to quality learning materials. Our free textbooks are developed and peer-reviewed by educators to ensure they are readable, accurate, and meet the scope and sequence requirements of modern college courses. Through our partnerships with companies and foundations committed to reducing costs for students, OpenStax College is working to improve access to higher education for all. OpenStax CNX The technology platform supporting OpenStax College is OpenStax CNX ( ), one of the world’s first and largest openeducation projects. OpenStax CNX provides students with free online and low-cost print editions of the OpenStax College library and provides instructors with tools to customize the content so that they can have the perfect book for their course. Rice University OpenStax College and OpenStax CNX are initiatives of Rice University. As a leading research university with a distinctive commitment to undergraduate education, Rice University aspires to path-breaking research, unsurpassed teaching, and contributions to the betterment of our world. It seeks to fulfill this mission by cultivating a diverse community of learning and discovery that produces leaders across the spectrum of human endeavor. Foundation Support OpenStax College is grateful for the tremendous support of our sponsors. Without their strong engagement, the goal of free access to high-quality textbooks would remain just a dream. Laura and John Arnold Foundation (LJAF) actively seeks opportunities to invest in organizations and thought leaders that have a sincere interest in implementing fundamental changes that not only yield immediate gains, but also repair broken systems for future generations. LJAF currently focuses its strategic investments on education, criminal justice, research integrity, and public accountability. The William and Flora Hewlett Foundation has been making grants since 1967 to help solve social and environmental problems at home and around the world. The Foundation concentrates its resources on activities in education, the environment, global development and population, performing arts, and philanthropy, and makes grants to support disadvantaged communities in the San Francisco Bay Area. Guided by the belief that every life has equal value, the Bill & Melinda Gates Foundation works to help all people lead healthy, productive lives. In developing countries, it focuses on improving people’s health with vaccines and other life-saving tools and giving them the chance to lift themselves out of hunger and extreme poverty. In the United States, it seeks to significantly improve education so that all young people have the opportunity to reach their full potential. Based in Seattle, Washington, the foundation is led by CEO Jeff Raikes and Co-chair William H. Gates Sr., under the direction of Bill and Melinda Gates and Warren Buffett. The Maxfield Foundation supports projects with potential for high impact in science, education, sustainability, and other areas of social importance. Our mission at the Twenty Million Minds Foundation is to grow access and success by eliminating unnecessary hurdles to affordability. We support the creation, sharing, and proliferation of more effective, more affordable educational content by leveraging disruptive technologies, open educational resources, and new models for collaboration between for-profit, nonprofit, and public entities. Table of Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 1: Functions . . . . . . . . . . . . . . . . . . . . . 1.1 Functions and Function Notation . . . . . . . . . . . 1.2 Domain and Range . . . . . . . . . . . . . . . . . 1.3 Rates of Change and Behavior of Graphs . . . . . . 1.4 Composition of Functions . . . . . . . . . . . . . . 1.5 Transformation of Functions . . . . . . . . . . . . . 1.6 Absolute Value Functions . . . . . . . . . . . . . . 1.7 Inverse Functions . . . . . . . . . . . . . . . . . . Chapter 2: Linear Functions . . . . . . . . . . . . . . . . . 2.1 Linear Functions . . . . . . . . . . . . . . . . . . . 2.2 Graphs of Linear Functions . . . . . . . . . . . . . 2.3 Modeling with Linear Functions . . . . . . . . . . . 2.4 Fitting Linear Models to Data . . . . . . . . . . . . Chapter 3: Polynomial and Rational Functions . . . . . . . 3.1 Complex Numbers . . . . . . . . . . . . . . . . . . 3.2 Quadratic Functions . . . . . . . . . . . . . . . . . 3.3 Power Functions and Polynomial Functions . . . . . 3.4 Graphs of Polynomial Functions . . . . . . . . . . . 3.5 Dividing Polynomials . . . . . . . . . . . . . . . . . 3.6 Zeros of Polynomial Functions . . . . . . . . . . . . 3.7 Rational Functions . . . . . . . . . . . . . . . . . . 3.8 Inverses and Radical Functions . . . . . . . . . . . 3.9 Modeling Using Variation . . . . . . . . . . . . . . . Chapter 4: Exponential and Logarithmic Functions . . . . 4.1 Exponential Functions . . . . . . . . . . . . . . . . 4.2 Graphs of Exponential Functions . . . . . . . . . . 4.3 Logarithmic Functions . . . . . . . . . . . . . . . . 4.4 Graphs of Logarithmic Functions . . . . . . . . . . 4.5 Logarithmic Properties . . . . . . . . . . . . . . . . 4.6 Exponential and Logarithmic Equations . . . . . . . 4.7 Exponential and Logarithmic Models . . . . . . . . 4.8 Fitting Exponential Models to Data . . . . . . . . . . Chapter 5: Trigonometric Functions . . . . . . . . . . . . 5.1 Angles . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Unit Circle: Sine and Cosine Functions . . . . . . . 5.3 The Other Trigonometric Functions . . . . . . . . . 5.4 Right Triangle Trigonometry . . . . . . . . . . . . . Chapter 6: Periodic Functions . . . . . . . . . . . . . . . . 6.1 Graphs of the Sine and Cosine Functions . . . . . . 6.2 Graphs of the Other Trigonometric Functions . . . . 6.3 Inverse Trigonometric Functions . . . . . . . . . . . Chapter 7: Trigonometric Identities and Equations . . . . 7.1 Solving Trigonometric Equations with Identities . . . 7.2 Sum and Difference Identities . . . . . . . . . . . . 7.3 Double-Angle, Half-Angle, and Reduction Formulas . 7.4 Sum-to-Product and Product-to-Sum Formulas . . . 7.5 Solving Trigonometric Equations . . . . . . . . . . . 7.6 Modeling with Trigonometric Equations . . . . . . . Chapter 8: Further Applications of Trigonometry . . . . . 8.1 Non-right Triangles: Law of Sines . . . . . . . . . . 8.2 Non-right Triangles: Law of Cosines . . . . . . . . . 8.3 Polar Coordinates . . . . . . . . . . . . . . . . . . 8.4 Polar Coordinates: Graphs . . . . . . . . . . . . . . 8.5 Polar Form of Complex Numbers . . . . . . . . . . 8.6 Parametric Equations . . . . . . . . . . . . . . . . 8.7 Parametric Equations: Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . 9 . 10 . 41 . 67 . 84 . 101 . 137 . 151 . 181 . 182 . 205 . 233 . 247 . 273 . 274 . 286 . 311 . 335 . 362 . 375 . 392 . 421 . 438 . 459 . 460 . 482 . 501 . 513 . 538 . 553 . 569 . 590 . 625 . 626 . 651 . 674 . 693 . 715 . 716 . 739 . 764 . 789 . 790 . 803 . 821 . 835 . 844 . 862 . 903 . 904 . 923 . 939 . 955 . 978 . 992 1008 8.8 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 9: Systems of Equations and Inequalities . . . . . . . . . . . 9.1 Systems of Linear Equations: Two Variables . . . . . . . . . . 9.2 Systems of Linear Equations: Three Variables . . . . . . . . . . 9.3 Systems of Nonlinear Equations and Inequalities: Two Variables 9.4 Partial Fractions . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Matrices and Matrix Operations . . . . . . . . . . . . . . . . . 9.6 Solving Systems with Gaussian Elimination . . . . . . . . . . . 9.7 Solving Systems with Inverses . . . . . . . . . . . . . . . . . . 9.8 Solving Systems with Cramer's Rule . . . . . . . . . . . . . . . Chapter 10: Analytic Geometry . . . . . . . . . . . . . . . . . . . . . 10.1 The Ellipse . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 The Hyperbola . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 The Parabola . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Rotation of Axes . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Conic Sections in Polar Coordinates . . . . . . . . . . . . . . Chapter 11: Sequences, Probability and Counting Theory . . . . . . 11.1 Sequences and Their Notations . . . . . . . . . . . . . . . . 11.2 Arithmetic Sequences . . . . . . . . . . . . . . . . . . . . . . 11.3 Geometric Sequences . . . . . . . . . . . . . . . . . . . . . 11.4 Series and Their Notations . . . . . . . . . . . . . . . . . . . 11.5 Counting Principles . . . . . . . . . . . . . . . . . . . . . . . 11.6 Binomial Theorem . . . . . . . . . . . . . . . . . . . . . . . . 11.7 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 12: Introduction to Calculus . . . . . . . . . . . . . . . . . . 12.1 Finding Limits: Numerical and Graphical Approaches . . . . . 12.2 Finding Limits: Properties of Limits . . . . . . . . . . . . . . . 12.3 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . A Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . This content is available for free at . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1022 1055 1056 1077 1090 1104 1116 1130 1145 1161 1185 1186 1208 1230 1248 1266 1289 1290 1307 1320 1331 1346 1358 1366 1387 1388 1404 1416 1432 1467 1473 Preface 1 PREFACE Welcome to Precalculus, an OpenStax College resource. This textbook has been created with several goals in mind: accessibility, customization, and student engagement—all while encouraging students toward high levels of academic scholarship. Instructors and students alike will find that this textbook offers a strong foundation in precalculus in an accessible format. About OpenStax College OpenStax College is a non-profit organization committed to improving student access to quality learning materials. Our free textbooks go through a rigorous editorial publishing process. Our texts are developed and peer-reviewed by educators to ensure they are readable, accurate, and meet the scope and sequence requirements of today’s college courses. Unlike traditional textbooks, OpenStax College resources live online and are owned by the community of educators using them. Through our partnerships with companies and foundations committed to reducing costs for students, OpenStax College is working to improve access to higher education for all. OpenStax College is an initiative of Rice University and is made possible through the generous support of several philanthropic foundations. OpenStax College textbooks are used at many colleges and universities around the world. Please go to to see our rapidly expanding number of adoptions. About OpenStax College’s Resources OpenStax College resources provide quality academic instruction. Three key features set our materials apart from others: they can be customized by instructors for each class, they are a "living" resource that grows online through contributions from educators, and they are available free or for minimal cost. Customization OpenStax College learning resources are designed to be customized for each course. Our textbooks provide a solid foundation on which instructors can build, and our resources are conceived and written with flexibility in mind. Instructors can select the sections most relevant to their curricula and create a textbook that speaks directly to the needs of their classes and student body. Teachers are encouraged to expand on existing examples by adding unique context via geographically localized applications and topical connections. Precalculus can be easily customized using our online platform ( ). Simply select the content most relevant to your current semester and create a textbook that speaks directly to the needs of your class. Precalculus is organized as a collection of sections that can be rearranged, modified, and enhanced through localized examples or to incorporate a specific theme to your course. This customization feature will ensure that your textbook truly reflects the goals of your course. Curation To broaden access and encourage community curation, Precalculus is “open source” licensed under a Creative Commons Attribution (CC-BY) license. The mathematics community is invited to submit feedback to enhance and strengthen the material and keep it current and relevant for today’s students. Submit your suggestions to [email protected], and check in on edition status, alternate versions, errata, and news on the StaxDash at . Cost Our textbooks are available for free online, and in low-cost print and e-book editions. About Precalculus Precalculus is intended for college-level precalculus students. Since precalculus courses vary from one institution to the next, we have attempted to meet the needs of as broad an audience as possible, including all of the content that might be covered in any particular course. The result is a comprehensive book that covers more ground than an instructor could likely cover in a typical one- or two-semester course; but instructors should find, almost without fail, that the topics they wish to include in their syllabus are covered in the text. Many chapters of Openstax College Precalculus are suitable for other freshman and sophomore math courses such as College Algebra and Trigonometry; however, instructors of those courses might need to supplement or adjust the material. 2 Preface Openstax will also be releasing College Algebra and Algebra and Trigonometry titles tailored to the particular scope, sequence, and pedagogy of those courses. Coverage and Scope Precalculus contains twelve chapters, roughly divided into three groups. Chapters 1-4 discuss various types of functions, providing a foundation for the remainder of the course. Chapter 1: Functions Chapter 2: Linear Functions Chapter 3: Polynomial and Rational Functions Chapter 4: Exponential and Logarithmic Functions Chapters 5-8 focus on Trigonometry. In Precalculus, we approach trigonometry by first introducing angles and the unit circle, as opposed to the right triangle approach more commonly used in College Algebra and Trigonometry courses. Chapter 5: Trigonometric Functions Chapter 6: Periodic Functions Chapter 7: Trigonometric Identities and Equations Chapter 8: Further Applications of Trigonometry Chapters 9-12 present some advanced Precalculus topics that build on topics introduced in chapters 1-8. Most Precalculus syllabi include some of the topics in these chapters, but few include all. Instructors can select material as needed from this group of chapters, since they are not cumulative. Chapter 9: Systems of Equations and Inequalities Chapter 10: Analytic Geometry Chapter 11: Sequences, Probability and Counting Theory Chapter 12: Introduction to Calculus All chapters are broken down into multiple sections, the titles of which can be viewed in the Table of Contents. Development Overview Openstax Precalculus is the product of a collaborative effort by a group of dedicated authors, editors, and instructors whose collec...
View Full Document

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture