PRECALCULUS Print Reference - Pearson Education · 2019-02-20 · Precalculus is a concise and focused presentation of material necessary for a student to be successful in calculus

PRECALCULUS Print ReferenceSecond Edition

ERIC SCHULZ JULIANNE CONNELL SACHSWalla Walla Community College Walla Walla Community College

WILLIAM BRIGGSUniversity of Colorado, Denver

PrefaceAs instructors, we find that students rely heavily on geometric intuition and visualization to construct their knowledge of mathematical concepts. This second edition of Precalculus provides a rich visual base on which analytic skills are built. The foundation provided in Precalculus is not presented as a set of static illustrations, but rather as a visual and intellectual experience in which the student is actively engaged. This eText invites students to explore, be curious, and participate while providing the guidance and support essential for true learning.

Precalculus seamlessly integrates exposition, interactive figures, instructive screencast videos, static illustrations, and progress checks to immerse students in a rich learning environment. The interplay between the narrative and interactive figures in Precalculus is a unique feature of the eText and can have a profound impact on student learning. The eText initially appears on screen with the familiar features of a printed text, with exposition, beautifully typeset mathematics, examples, and so forth, but quickly comes alive with interactive visualizations strategically placed in the exposition and exercises. The interactive figures have been designed to be storytellers on equal footing with the narrative. The narrative is written to complement the interactive figures, which in turn enhance the narrative; that is, the interactive figures illuminate the meaning of the words, while the words support the interactive figures.

Content

Precalculus is a concise and focused presentation of material necessary for a student to be successful in calculus. Our presentation of a new concept seeks to first build a student’s geometric intuition of the idea, and then continue with its analytic development. We believe in the value of multiple representations and weave numeric (tabular), graphical, symbolic, and descriptive presentations of functions and concepts throughout Precalculus. Exercise sets are written to access understand-ing in multiple ways, and include exercises that require technology.

We have found that inclusion of technology specifics, such as graphing calculator screens, keystrokes, and screen captures of software output, included in a textbook become quickly outdated and a distraction from the mathematical con-cepts and ideas central to the discussion. Mathematical knowledge transcends the technology of today, can be applied using a variety of technological devices, and lasts far longer than the device in fashion. Technology-specific manuals for graphing calculators and computer algebra systems are available as supplements in MyMathLab.

The contents of Precalculus naturally consists of topics we believe to be essential for success in calculus. To accommo-date curricula that include a study of vectors, systems of equations, and matrices, we have added two new chapters to the second edition. The ten chapters in Precalculus are as follows:

1 Functions2 Polynomials and Rational Functions3 Exponentials and Logarithms4 Unit Circle Trigonometry5 Triangle Trigonometry6 Trigonometric Identities7 Parametric and Polar Graphs

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Copyright © 2019 Pearson Education, Inc.

8 Conic Sections9 Vectors

10 Systems of Equations and Matrices

The material on trigonometry in Chapters 4 and 5 has been written so the two chapters can be covered in either order, with the exception of Section 5.2, where connections are established between unit circle trigonometry and triangle trigonometry. Two appendices composed of material from Calculus, by Briggs, Cochran, Gillett, and Schulz, are titled Introduction to Limits and Introduction to Sequences and Series. and provide introductions to these topics.

Pedagogical Features

Figures

Given the importance of visualizations to student understanding, we devoted considerable time and thought to the figures in this eText. When there is an opportunity to enhance student learning and understanding, we have created interactive figures to communicate essential mathematical ideas. There are several hundred interactive figures in the eText. Instructors can use these interactive figures in the classroom to illustrate important concepts, and students can learn from them as they use the eText.

Material in the narrative of the eText has been written specifically to support the interactive figures, giving directions, asking questions, and providing explanations of the concepts that the figures visualize. The narrative and the interactive figures have been designed to actively work together. An Interactive Figure Index at the end of the eText shows thumbnail images of all interactive figures listed by section.

A number of the interactive figures are accompanied by a screencast video in which we (the authors) interact with the figure while talking about the important mathematical concepts being illustrated. Clicking the video icon located near the upper right corner of an interactive figure allows the reader to view its associated screencast video.

In addition, Explorations & Notes (described below) are available to help guide students toward deeper understanding as they work with interactive figures.

Annotated Step-by-Step Work

We have carefully annotated steps in our work throughout the eText. For example,

g(x) = x2 + 6 x + 2 Notice 6 is the coefficient of x.= x2 + 6 x +32 - 32

= 0+ 2 Add and subtract the square of one-half the coefficient of x.

= x2 + 6 x + 32

perfect square trinomial

- 32 + 2 Regroup.

= (x + 3)2 - 7. Factor the perfect square trinomial.

Quick Check, Notes, and Quick Quizzes

The narrative of the eText is interspersed with Quick Check questions that encourage students to actively work through the content. These questions resemble the kinds of questions instructors pose in class. Answers to the Quick Check questions follow each question and are revealed with a single click. Notes offer reminders, provide insight, and clarify technical points. Preceding every section exercise set is an interactive Quick Quiz composed of multiple-choice questions intended to quickly test students’ understanding of the basic concepts of the section. These can also be used as clicker questions in the classroom.

NEW! Algebra Checks

We have added a feature at the beginning of each section to provide a brief check of prerequisite algebra skills needed for the section. The Algebra Checks are designed to help guide students in reviewing necessary skills and preparing for new content. Algebra Checks are assignable in MyMathLab.

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Exercises

The exercises at the end of each section are a strong feature of the eText. They are graded, varied, and original. In addition, they are labeled and carefully organized into categories.

! Each exercise set begins with Getting Started exercises that check students’ conceptual understanding of the essential ideas from the section and lead students toward subsequent problem solving.

! Practice exercises are intended to build skills and strengthen conceptual understanding. These exercises offer many opportunities to apply mathematics in context.

! Explorations and Challenges challenge students to think creatively and to generalize. They include proofs of results cited in the narrative and explorations of new ideas.

Each chapter concludes with a comprehensive Summary and a set of Review Exercises. Answers to odd exercises are included at the end of every section. All of the answers in the eText, and the corresponding solutions in the Instructor’s Solutions Manual and Student’s Solutions Manual (available in MyMathLab), have been written by the authors to assure consistency between the presentation of material in the eText and techniques used to solve the exercises.

Graphing Technology

We believe that a precalculus text should help students strengthen their analytical skills and demonstrate how technology can extend (not replace) those skills. The exercises and examples in this eText emphasize this balance. Calculators and graphing utilities are additional tools in the kit, and students must learn when and when not to use them.

We assume that students have access to graphing technology, and the decision as to whether the technology is a graphing calculator, computer algebra system, mobile app, or a web site is left to the discretion of the reader. Throughout the eText, exercises marked with T indicate that the use of technology will likely be necessary. Technology manuals giving detailed instructions for Texas Instruments graphing calculators, Mathematica, and Maple are provided in MyMathLab.

Explorations & Notes (Updated for Second Edition)

To guide students as they actively work through the eText, we have created printable Explorations & Notes for each section. The Explorations & Notes include the objectives, Algebra Check questions, interactive figure questions, definitions, theo-rems, important statements, examples followed by dedicated space to take notes on their solutions, and the exercise set. The Explorations & Notes will also be available as a separate print supplement.

Interactive figures are powerful learning tools for students, but only when they are used, investigated, and understood. Generalization of mathematical concepts visualized in the interactive figures is encouraged throughout the eText and Explo-rations & Notes.

eText Features

An orientation video (for both instructors and students) and Instructor Resource videos will help make using this eText as user-friendly as possible. The Instructor Resource videos explain the authors’ philosophy and provide some teaching resources. The orientation video outlines the functionality and main features of the eText; a sampling of what is included in this video follows.

! Each eText section contains a navigation button that provides quick navigation to any item (example, figure, definition, etc.) in the section. Interactive figures are denoted with a ( symbol in the navigation list.

! The eText can be magnified; text will automatically be adjusted to fit the screen width when the magnification changes—no horizontal scrolling is required.

! The eText is searchable using the CDF Player’s Find feature. An index of terms and an index of applications are provided at the end of the eText.

! The design of the eText has been optimized for the screen; however, the material can be printed section-by-section if desired. We designed Precalculus primarily as an eText, so figures will appear in print larger than they would if we had designed for print.

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Resources for SuccessMyLab Math Online Course (access code required)

MyLab Math empowers you to reach every student. By partnering trusted authors’ content with digital tools and a flexible platform, MyLab Math personalizes the experience for each student and improves results.

Video Program

NEW! Guided Solution Videos walk students step-by-step through the problem-solving process for key examples.

Guided Figure Videos appear in the eText beside key Interactive Figures. Authors Eric and Julianne develop the mathemati-cal story through co-teaching moments, supplying students with insightful explanations and additional context for the interactive figures. These can also be assigned in MyLab Math with accompanying assessment questions.

Video Assessment exercises are tied to the Guided Figure Videos, giving students a more interactive experience with the videos and presenting another opportunity to engage with the interactive figures.

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Getting Ready

Skill review quizzes are assignable throughout the course, testing students on prerequisite knowledge. From these quizzes, each student receives a personalized, just-in-time review assignment, which serves to refresh forgotten concepts.

Study Plan

The Study Plan acts as a tutor, providing personalized recommendations for each of your students based on his or her ability to master the learning objectives in your course. This allows students to focus their study time by pinpointing the precise areas they need to review, and allowing them to use customized practice and learning aids—such as videos, eText, tutorials, and more—to get them back on track. Using the report available in the Gradebook, you can then tailor course lectures to prioritize the content where students need the most support—offering you better insight into classroom and individual performance.

With Companion Study Plan Assignments, you can now assign the Study Plan as a prerequisite to a test or quiz, guiding students through the concepts they need to master.

Learning Catalytics

Foster student engagement and peer-to-peer learningGenerate class discussion, guide your lecture, and promote peer-to-peer learning with real-time analytics. MyLab Math and MyLab Statistics now provide Learning Catalytics™—an interactive student response tool that uses students’ smartphones, tablets, or laptops to engage them in more sophisticated tasks and thinking.

Instructors can:

! Pose a variety of open-ended questions that help students develop critical thinking skills.! Monitor responses to find out where students are struggling.! Use real-time data to adjust instructional strategy and try other ways of engaging students during class.! Manage student interactions by automatically grouping students for discussion, teamwork, and peer-to-peer learning.

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Instructor ResourcesOrientation VideoThe orientation video helps both instructors and students navigate through the eText. The video outlines the functionality and main features of the eText, including how to quickly “jump” to different parts of the section and how to easily find interactive figures within the eText.

Implementation VideosThe authors have created implementation videos to make using this eText as easy as possible. This video series explains the authors’ philosophy underlying the interactive eText and offers best practices and tips for teaching a course with these materials. These videos are available for viewing in MyLab Math.

Enhanced Sample Assignments by authors Eric Schulz and Julianne Connell Sachs make course set-up easier by giving instructors a starting point for each section. Each assignment, handpicked by the authors to align with this text, includes a thoughtful mix of question types (conceptual, skills, etc.) specific to that topic.

Magnification Functionality Every page of the eText has a pulldown menu that allows the user to increase the size of the on-screen material. This allows instructors to enlarge the interactive figures so students can clearly see the image, even from the back of the room.

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Instructor’s Solutions ManualThe Instructor’s Solutions Manual includes worked-out solutions, written by the authors, to all the exercises in the eText. The solutions are available as downloadable CDF files in MyLab Math.

Explorations & NotesTo ensure that students stay focused and curious while working within the all-digital environment of the eText and MyLab Math, the authors have written an accompanying note-taking guide:

! “Explore” questions to drive students back to the rich Interactive Figures! “Algebra Check” for reviewing prerequisite material! Key definitions and concepts! Examples, with space to work out solutions! Full set of section exercises

Explorations & Notes are designed for students to use independently and during lectures. In this second edition, instructors have the ability to edit this resource to align it with their syllabus. Explorations & Notes can be packaged with the MyLab Math access code, or it can be accessed electronically within the MyLab Math course as CDFs, PDFs, or Word files.

Student ResourcesGuided Figure VideosThroughout the eText, the authors created annotated videos to accompany key interactive figures, providing additional support for difficult math topics. In these videos, Eric Schulz and Julianne Connell Sachs develop the mathematical story, giving students insightful explanations and additional context. The videos are assignable in MyLab Math with assessment exercises developed by the authors.

Explorations & NotesThis notebook parallels the eText, creating a hybrid learning experience. It is designed to be used during lecture and outside of class when doing assignments.

Explorations & Notes includes:

! “Explore” questions, to dive deeper into the Interactive Figures

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! “Algebra Check” questions to make sure you're ready for the content coming up! Key definitions and concepts for efficient review! All Examples from the eText, with space to work out your solutions! The full set of section exercises so you can work on paper, if you prefer

Explorations & Notes may come packaged with your MyLab Math access code, or you can access it electronically within the MyLab Math course as CDFs, PDFs, or Word files.

Orientation Video SeriesTo help you quickly get familiar with the digital course layout,a series of orientation videos made especially for students is available in the MyLab Math menu bar.

Videos include:

! Using and navigating the eText! How to get the most out of working/studying with Interactive Figures! How to get the most out of the Explorations & Notes

Student’s Solutions ManualThe Student’s Solution Manual includes worked-out solutions, written by the authors, to all of the odd exercises in the eText. The solutions are available as downloadable CDF files in MyLab Math

AcknowledgmentsThis project would not have happened without the encouragement and support of a sensational team at Pearson:

Anne Kelly, Director, Portfolio ManagerChelsea Kharakozova, Courseware Portfolio ManagerAshley Gordon, Courseware Portfolio Management AssistantKathy Manley, Content ProducerKaren Wernholm, Managing ProducerMarielle Guiney, ProducerKristina Evans, Manager, Content DevelopmentStacey Sveum, Product Marketing ManagerShannon McCormack, Product Marketing AssistantPeggy Lucas, Field Marketing ManagerAdrianna Valencia, Field Marketing AssistantBarbara Atkinson, Senior Design LeadJoe Vetere, Senior Author Support/Technology Specialist

We would like to extend a special thank you to our development and copy editors, Leslie Lahr and Jane Hoover, for their detailed, insightful, and spot-on feedback throughout the development of the second edition of Precalculus. We would also like to thank Sarah Sponholz for her dedication to the project in her accuracy checking and copy editing of the second edition manuscript and solutions.

We are thankful for feedback and comments from many colleagues who participated in focus groups and reviewed the manuscript. Precalculus is better because of contributions from:

Marwan Abu-Sawwa, Florida State College at JacksonvilleJeremy Aikin, Middle Georgia State CollegeBenjamin Allgeier, Marian UniversityK. M. Aminul, College of Southern NevadaKenneth R. Anderson, Chemeketa Community CollegeDarry Andrews, Ohio State UniversityHolly Ashton, Pikes Peak Community CollegePeter Woo Banwarth, Oregon State UniversityWayne G. Barber, Chemeketa Community CollegeScott E. Barnett, Henry Ford Community CollegeTerry (Tee) Barron, Georgia Gwinnett CollegeLinda Estes Barton, Ball State UniversityRobert Bates, Honolulu Community CollegeColleen R. Beaudoin, University of TampaMohamed Benbourenane, California University of PennsylvaniaAnnette Benbow, Tarrant County College Jeffrey V. Berg, Arapahoe Community CollegeNorma Bisulca, University of Maine-AugustaDavid Britz, Raritan Valley Community CollegeJodie Broussard, Florida State College at JacksonvilleTammy Calhoun, Hill CollegePaul Canepa, Delaware County Community CollegeFlorence Chambers, Southern Maine Community CollegeJerry Chen, Suffolk County Community CollegeMarcela Chiorescu, Georgia CollegeIvette Chuca, El Paso Community CollegeScott E. Clark, University of ArizonaBeth Clickner, Hillsborough Community CollegeShawn Clift, Eastern Kentucky UniversityJane R. Cushman, Buffalo State College-SUNYRichard Danner, California State University-ChicoKumer Pial Das, Lamar UniversityNasser Dastrange, Buena Vista UniversityNelson De La Rosa, Miami Dade College-KendallJoyati Debnath, Winona State UniversityLucy Dechéne, Fitchburg State UniversityDonna B. Densmore, Bossier Parish Community CollegeRobert Diaz, Fullerton CollegeElena Dilai, Monroe Community CollegeSusan Dimick, Spokane Community CollegeNancy D. Eschen, Florida State College at JacksonvilleA. S. Elkhader, Northern State UniversityRobert Everest, Citrus CollegeDan Fahringer, Harrisburg Area Community CollegeJeff Feuer, Dowling CollegeBrian Fisher, Pepperdine UniversityThomas L. Fitzkee, Francis Marion UniversityNicole Francis, Linn-Benton Community CollegeGreg Fry, El Camino CollegeDoug Gardner, Rogue Community CollegeCornell Grant, Georgia Piedmont Technical CollegeJack Green, Mt. Hood Community CollegeLisa Greenberg, Florida Atlantic UniversityJames Hamblin, Shippensburg UniversityDonna Harbin, University of Hawaii-Maui Community CollegeMako E. Haruta, University of HartfordCarla Hill, Marist CollegeDiane R. Hollister, Reading Area Community CollegeKevin W. Hopkins, Southwest Baptist UniversityMiles Hubbard, St. Cloud State UniversityEric Hutchinson, College of Southern NevadaElizabeth Hylton, Clatsop Community CollegeGlenn Jablonski, Triton CollegeChristy Jacobs, Washington State UniversityDavid M. James, Howard UniversityDeborah W. Jeter, College of CharlestonKiandra Johnson, Morehouse CollegeMike Kawai, University of Colorado-DenverMike Kirby, Tidewater Community College-Virginia BeachElena Kravchuk, University of Alabama-BirminghamDr. Carole King Krueger, The University of Texas ArlingtonNamyong Lee, Minnesota State UniversityMary Margarita Legner, Riverside City CollegeLucas Lembrick, Klamath Community CollegeRita Lindsay, PhD, Indian River State CollegeElsa Veronica Lopez, El Paso Community CollegeYun Lu, Kutztown UniversityJames Martin, Wake Technical Community CollegeRuby R. Martinez, San Antonio CollegeCynthia Martinez, Temple CollegeChad Mathews, West Georgia Technical College Eric Matsuoka, Leeward Community CollegeLobna Mazzawi, Everett Community CollegeMichael McAfee, Mt. Hood Community CollegeRinav C. Mehta, Central Piedmont Community CollegeGabriel Mendoza, El Paso Community CollegeMary Merchant, Cedar Valley CollegeVal Mohanakumar, Hillsborough Community CollegeRam N. Mohapatra, University of Central FloridaSharon L. Morrison, St. Petersburg College Amal Mostafa, New Mexico State UniversityCarrie Muir, University of Colorado-BoulderDr. Stacey O. Nicholls, Anne Arundel Community CollegeDr. Kate S. Owens, College of CharlestonC. Altay Ozgener, State College of FloridaDonnabelle B. Pascual, Leeward Community CollegeStan Perrine, Georgia Gwinnett CollegeJeffrey L. Poet, Missouri Western State UniversitySandra Poinsett, College of Southern MarylandMichael Price, University of OregonStephen Prioetti, Northern Essex Community CollegeBrooke P. Quinlan, Hillsborough Community CollegeDr. Denise Race, Eastfield CollegeHarriet Markos Roadman, New River Community CollegeDouglas Scheib, Saint Mary-of-the-Woods CollegeKeith Schloeman, Oregon State UniversityHoward Schwesky, Pierce CollegeNicholas Sedlock, Framingham State UniversityJiajia Seffrood, Leeward Community CollegePavel Sikorskii, Michigan State UniversityAbraham D. Smith, Fordham UniversityAlin Stancu, Columbus State UniversityBrad D. Stetson, Schoolcraft CollegeKristin Stoley, Blinn College-Bryan Bob Strozak, Old Dominion UniversityPanyada Sullivan, Yakima Valley Community CollegeJennie Thompson, Leeward Community CollegeSusan E. Thompson, Otterbein UniversityHansun To, Worcester State UniversityEllen Turnell, Lone Star College-North HarrisJulian Viera Jr., The University of Texas at El PasoBeverly Vredevelt, Spokane Falls Community CollegeThomas P. Wakefield, Youngstown State UniversityDr. Charles D. Watson, University of Central ArkansasJudith B. Wood, PhD, College of Central Florida Teu-Yi Alan Yang, Columbus State Community CollegeCathleen Zucco-Teveloff, Rider University

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We dedicate Precalculus to our students: those we have learned from in the past, those who learn with us now, and those we are able to serve in the future. We are grateful to the teachers and colleagues that have guided, challenged, and encouraged each of us along our paths.

About the AuthorsFrom the authors who brought you the most successful calculus text published in the last two decades, Pearson is excited to present to you the second edition of Precalculus, by Eric Schulz, Julianne Connell Sachs, and William Briggs.

Eric Schulz has been on the mathematics faculty at Walla Walla Community College in Walla Walla, Washington, since 1989. He has a BS degree in mathematics from Seattle Pacific University and an MS degree in mathematics from the University of Washington. He has been working with technology throughout his career and is especially interested in innovative and effective uses of technology in mathematics. Eric is an author on multiple textbooks, including Calculus titles with Briggs, Cochran, and Gillett and Precalculus with Sachs and Briggs, for which he writes, codes, and builds eTexts that combine narrative and interactive figures using Mathematica and CDF technology. He is the developer of the Basic Math Assistant, Classroom Assistant, and Writing Assistant palettes included in Mathematica by Wolfram Research, Inc.

Julianne Connell Sachs has been on the mathematics faculty at Walla Walla Community College in Walla Walla, Washing-ton, since 2001. She has a BS degree in mathematics, a BA degree in Spanish, and a MS in mathematics from Western Washington University. She is particularly interested in pedagogies and technologies that engage students and guide learners toward a deeper understanding of mathematics.

William Briggs was on the mathematics faculty at the University of Colorado at Denver for 23 years. He received his BA in mathematics from the University of Colorado and his MS and PhD in applied mathematics from Harvard University. He taught undergraduate and graduate courses throughout the mathematics curriculum with a special interest in mathematical modeling and differential equations as they apply to problems in the biosciences.

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PRECALCULUS Print Reference - Pearson Education · 2019-02-20 · Precalculus is a concise and focused presentation of material necessary for a student to be successful in calculus