Preface

To the Instructor

Philosophy

Some years ago we published a short text called Basic Mathematics for Calculus. In the preface of the first edition we stated

An instructor is always faced with the dilemma of too much material and too little time. In the vast precalculus market, one can find texts that cover everything from topics in elementary algebra to topics in matrix algebra…. We feel there is a great need for a text that quickly gets to the heart of the matter, a text the presents only those topics that will be of direct and immediate use in most calculus courses.

After three editions, Basic Mathematics for Calculus faded into obscurity as the original publishing company was merged into a large company and then the large company was taken over by an even larger one. In the intervening years we have not changed our opinion. If anything, we feel more strongly that many precalculus texts fall short in concentrating on the specific mathematics that are needed in the study of calculus. Because modern texts tend to be almost encyclopedic in length, the instructor can either move through many chapters at a very fast pace or proceed through a reasonable number of chapters leaving much of a very expensive book unused. This is why we decided to reprise our Basic Mathematics for Calculus. The present text, now renamed as Essentials of Precalculus with Calculus Previews, Fifth Edition represents a thorough revision of this earlier work.

The list that follows reflects some of our pedagogical philosophy that underpins this fifth edition.

• Throughout the revision we held firmly to our belief in a “no nonsense” approach to precalculus. We have deliberately kept the coverage to a reasonable number of topics. The six chapters that comprise this text can easily be covered in a one-term course. Our style is informal, intuitive, and straightforward—we have avoided the theorem–proof format. We try to talk directly to the student.

• We emphasize the basics, especially algebra. Through the many examples and numerous and varied exercises, we provide opportunities for students to practice operations such as factoring, expanding a power of a binomial, completing the square, synthetic and long division, rationalization, and solving inequalities and equations in situations similar to those they will encounter in calculus. Throughout we stress the importance of being familiar with key formulas from algebra, the laws of exponents, the laws of logarithms, and fundamental trigonometric definitions and identities.

• The topics presented in this version of our two precalculus texts are only those that we feel are essential for success in calculus courses. This should allow time for instructors to work with their students to strengthen their algebraic, logarithmic, and trigonometric skills. Instructors wishing a wider variety of topics such as systems of equations, sequences, series, mathematical induction, and convergence of sequences and series are referred to our Precalculus with Calculus Previews, Fourth Edition, Expanded Volume.

• We pace the introduction of new topics. Students can be overwhelmed when a text presents various types of functions and related functional concepts all in one or two sections. Hence our pace, especially in Chapter 2, Functions, is more deliberate. For example, we hold off introducing word problems until after essential function concepts are presented. Similarly, the important topic of piecewise-defined functions is delayed and is given its own section.

• Throughout the text we envision the course as a bridge to calculus. In particular, we use some of the terminology of calculus in an informal way to acclimatize the student to these terms. For example, we use the words “continuous function” when describing graphs of polynomial and exponential functions and “discontinuous functions” in the context of rational and piecewise-defined functions. When the concept of secant lines for the graph of a function is introduced we use the words “difference quotient” to describe their slopes.

• In calculus it is extremely important to be able to sketch graphs of basic functions and equations quickly and accurately. Therefore we have placed a great emphasis on honing the student’s understanding of how concepts such as intercepts, symmetry, rigid and nonrigid transformations, asymptotes, and end behaviors, in conjunction with recognition of the type of function or equation, are valuable aids in sketching its graph by hand. The use of technology is limited to problems of the sort where this mathematical analysis fails. These problems are placed near the end of an exercise set and are clearly marked.

• We firmly believe that a figure should be used to illuminate an example, a discussion, or a problem in the exercises whenever possible. So there are numerous figures in this text; approximately 1600.

• As in previous editions, we feel that the proper approach to trigonometry for calculus preparation is through the unit circle.

An Emphasis on the Algebra of Calculus Many times we have seen students in a calculus class perform an operation such as differentiation flawlessly, but fail to complete the problem because they had difficulty simplifying the resulting expression or solving a related equation. So as mentioned above, in this edition we continue to make an effort to reinforce algebraic skills. Marginal side notes and in-text annotations fill in the details of solutions of examples and convey additional information to the reader.

An Emphasis on the Terminology of Calculus Words such as “continuous function” and “limit” are used as matter of course. The idea is to give students a good intuitive sense of what these words mean prior to their exposure to their formal definitions in calculus.

Building Functions from Words As teachers we know that the related rate and applied max-min, or optimization, problems can be a discouraging experience for some students of calculus. Typically, correctly interpreting the words of such a problem in order to set up an equation or a function presents the greatest challenge for many students. It follows then that it is appropriate to emphasize such material in a precalculus course. In Section 2.8, entitled Building a Function from Words, we begin by illustrating how to translate a verbal description into a symbolic representation of a function. We then present actual problems taken from Calculus: Early Transcendentals, Fourth Edition by Dennis G. Zill and Warren S. Wright (Jones and Bartlett Publishers, 2011) and demonstrate how to decode the statement of the problem and transform those words into an objective function. We discuss the importance of drawing pictures, using variables to describe pertinent quantities, identifying a constraint between the variables, using the constraint to eliminate an extra variable, and observing that the domain of the objective function may not be the same as its implicit domain. To ensure that the focus is squarely on the process of fashioning a symbolic function from the words, we have chosen not to discuss how such optimization problems are actually solved.

Notes from the Classroom Selected sections of this text conclude with remarks called Notes from the Classroom. These remarks are aimed directly at the student and address a variety of student/textbook/classroom/calculus issues such as alternative terminology, reinforcement of important concepts, what material is or is not recommended for memorization, misinterpretations, common errors, solution procedures, calculators, and advice on the importance of neatness and organization.

Calculus Previews The chapters in this text conclude with a section subtitled Calculus Previews. Each of these sections is devoted to a single calculus concept:

• Chapter 1, Section 1.5: Algebra and Limits

• Chapter 2, Section 2.9: The Tangent Line Problem

• Chapter 3, Section 3.7: The Area Problem

• Chapter 4, Section 4.11: The Limit Concept Revisited

• Chapter 5, Section 5.4: The Hyperbolic Functions

• Chapter 6, Section 6.8: 3-Space

In these sections the discussion is kept at a level easily within the reach of a precalculus student. The emphasis is not on the calculus; the calculus topic provides a framework and motivation for the precalculus mathematics we discuss. The focus in these sections is on the algebraic, logarithmic, and trigonometric manipulations that are necessary for the successful completion of typical calculus problems related to the Calculus Preview topic. Consequently, the Calculus Previews are intended to be taught as part of a regular course in precalculus mathematics. In Algebra and Limits we examine the analytical calculation of a limit as x → a, where a represents a real number. The material is presented in such a manner that the instructor has a choice: he/she can choose either to review the important algebra of simplifying fractional expressions using binomial expansions, factoring, common denominators, and rationalizations, or to go the extra step and actually calculate a limit. Similarly, we discuss The Tangent Line Problem and then examine the four-step calculation of the limit of the difference quotient as h → 0. Once again the stress is on the non-calculus steps, but the instructor can choose to extend the discussion to include the concept of a derivative of a function. In Sections 1.5 and 2.9 we do not delve into the theoretical aspects of the existence or nonexistence of limits. All limits given in the exercises or illustrated in examples exist. In The Area Problem, we discuss the geometry and algebra required to use a limiting process to find the area under a curve. In The Limit Concept Revisited we examine the evaluation of some trigonometric limits and the calculation of the difference quotient when f is either the sine or cosine function. In The Hyperbolic Functions we first use the difference quotient to show the student why the number e is the most natural base for the exponential function in a calculus setting and then conclude the section with a discussion of the hyperbolic sine and hyperbolic cosine. Section 6.8, 3-Space is new to this edition. In this section we introduce the rudiments of the rectangular coordinate system in three dimensions and consider the equations of spheres and planes.

Final Examination Following the six chapters of the text we present a list of 70 questions called the Final Examination. This “test” is mostly fill-in-the-blank and true/false questions. It was not our intention to emulate an actual final examination in precalculus, rather our thought was to offer a vehicle for an informal wrap-up of the entire course. We suggest that a part of a class period be devoted to a discussion of these questions to help students prepare for their actual final examination and their subsequent transition to calculus. To facilitate the students’ review, the answers of the Final Examination are given both in the Student Resource Manual as well as in the instructor’s Complete Solutions Manual. Of course, the instructor is free to utilize this material in whatever manner he/she chooses (including ignoring it completely).

Student Resource Manual We feel that the Student Resource Manual (SRM) that accompanies this text can be of significant help to a student’s success in this course as well as in calculus. Unlike the traditional student solutions manual, where a selected subset of the problems are worked out, the SRM is divided into five parts:

• ALGEBRA TOPICS • USE OF A CALCULATOR • BASIC SKILLS

• SELECTED SOLUTIONS • ANSWERS TO THE FINAL EXAM

In ALGEBRA TOPICS, selected topics from algebra (such as multiplication of an inequality by an unknown, implicit conditions in a word problem, Pascal’s triangle, factoring techniques, the binomial theorem, rationalization of a numerator or a denominator, adding symbolic fractions, complex numbers and their properties, long division of polynomials, synthetic division, and factorial notation) are reviewed because of their relevance to calculus. Since we do not discuss how to use technology within the text proper, we have devoted the section USE OF A CALCULATOR to the review of graphing calculator essentials. In SELECTED SOLUTIONS, a detailed solution of every third problem in the exercise sets is given. ANSWERS TO THE FINAL EXAM is a list of answers for all the questions in the Final Examination.

Exercises All of the exercise sets have been updated and many new problems have been added. Most of the exercise sets conclude with conceptual problems that are labeled For Discussion. We hope that instructors will utilize these problems, which are primarily conceptual in nature, and their expertise to engage in a classroom exchange of ideas with the students on how these problems can be solved. These problems could also be the basis for assigned writing projects. To encourage original thought we purposely have not included answers to these problems.

New to the Fifth Edition Here are some items that are new to this fifth edition.

• A new section on simple harmonic motion has been added to Chapter 4.

• A separate section on parametric equations has been added to Chapter 6.

• A new calculus preview of 3-space has been added to Chapter 6.

• Rotation of polar graphs is now discussed (because of its similarity to shifted rectangular graphs) in Section 6.6.

• The discussion of the hyperbolic functions in Section 5.4 has been expanded.

• Many new problems have been added throughout the text.

• The final exam at the end of the text has been expanded.

Supplements

For the Instructor

The following materials are available online, at http://www.jblearning.com/catalog/9781449614973/

• Complete Solutions Manual (CSM) prepared by Warren S. Wright and Carol D. Wright.

• Computerized Testing System for both Windows^® and Mac OS^® operating systems. This system allows instructors to create customized tests and quizzes. The questions and answers are sorted by chapter and can be easily installed on a computer. Publisher-supplied .rtf files can also be uploaded to the instructor’s Learning Management System.

• PowerPoint^® slides feature all labeled figures as they appear in the text. This useful tool allows instructors to easily display and discuss figures and problems found within the textbook.

• WebAssign™ developed by instructors for instructors, is a premier independent online teaching and learning environment, guiding several million students through their academic careers since 1997. With WebAssign, instructors can create and distribute algorithmic assignments using selected questions specific to this textbook. Instructors can also grade, record, and analyze student responses and performance instantly; offer more practice exercises, quizzes, and homework; and upload additional resources to share and communicate with your students seamlessly, such as the PowerPoint slides and the test items supplied by Jones & Bartlett Learning Computerized Testing System.

• eBook format—As an added convenience this complete textbook is now available in eBook format for purchase by the student through WebAssign.

• CourseSmart is a new way for instructors and students to access this textbook in digital format, anytime from anywhere. Jones & Bartlett Learning has partnered with CourseSmart to make available many of our leading mathematics textbooks in the CourseSmart eTextbook store.

For more information on CourseSmart Editions please visit www.jblearning.com/elearning/econtent/coursesmart/

Please contact your Jones & Bartlett Learning Account Specialist for information on, access to, and online demonstrations of the supplements and services described above.

For the Student

• Student Resource Manual (SRM) prepared by Warren S. Wright and Carol D. Wright. This manual continues to be popular with students using any one of the Zill series of mathematics textbooks. A complete description of the content specific to this text can be found on page xiv of this preface. Available in both print and online formats this student manual can be purchased separately or ordered bundled with the textbook at substantial savings.

• Student Companion Website is available at www.jblearning.com/catalog/9781449614973/. This online tutorial learning center can be accessed at any time during the term. The resources are tied directly to the text and include: Practice Quizzes, an Online Glossary of Key Terms, and Animated Flashcards.

• Graphing Calculator Manual by Jeffery M. Gervasi, EdD, of Porterville College may be ordered through the bookstore or online directly from Jones & Bartlett Learning.

• WebAssign Access card can be bundled with this text or purchased separately by the student online at www.webassign.net.

• eBook with course access card can also be purchased separately by the student online at www.webassign.net.

• CourseSmart is a new way for students to access college textbooks in digital format, anytime from anywhere. Jones & Bartlett Learning has partnered with CourseSmart to make this textbook available in the CourseSmart eTextbook store.

For students, this CourseSmart Edition has many features designed to make studying more efficient such as highlighting, online search, note-taking, and print capabilities.

For more information on purchasing this CourseSmart Edition please visit www.jblearning.com/elearning/econtent/coursesmart/.

To the Student

After teaching collegiate mathematics for many years, we have seen almost every type of student, from a budding genius who invented his own calculus, to students who struggled to master the most rudimentary mechanics of the subject. Frequently the source of difficulty in calculus can be traced to weak algebra skills and an inadequate background in trigonometry. Calculus builds immediately on your prior knowledge and skills and there is much new ground to be covered. Consequently, there is very little time to review precalculus mathematics in the calculus classroom. So those who teach calculus must assume that you can factor, simplify and solve equations, solve inequalities, handle absolute values, use a calculator, apply the laws of exponents, find equations of lines, plot points, sketch basic graphs, and apply important logarithmic and trigonometric identities. The ability to do algebra and trigonometry, work with exponentials and logarithms, and sketch by hand basic graphs quickly and accurately are keys to success in a calculus course. This book focuses on the specific mathematical topics and skills we consider essential for calculus.

In this text we have tried to give you as much help as possible within the confines of the printed page using such features as marginal annotations, annotations within examples, notes of caution, Notes from the Classroom, and the Final Examination. The many marginal and in-text annotations provide additional information or further explanation of the steps in the solution of an example. The Student Resource Manual (described previously) was written just for you. It contains review material not found in the text, extra examples, information on calculators, solutions of problems, and answers to the Final Examination.

Those of us who teach and write mathematics texts strive to communicate clearly how to do mathematics. This text reflects our philosophy that a mathematics text for the beginning college/university level should be readable, straightforward, and loaded with motivation. The principal reason for studying precalculus is to become well-prepared for calculus. To show you how the material covered in this text is essential for success in calculus, we end each chapter with a section called Calculus Preview. In each of these previews a calculus topic provides a framework and motivation for precalculus mathematics and shows you how this mathematics is a vital part of the calculus problem.

Finally, we caution you that learning mathematics is not like learning how to ride a bicycle—that once learned, the ability sticks for a lifetime. Mathematics is more like learning another language or learning to play a musical instrument: it requires time and effort to memorize basic formulas and to understand when to apply them, and most importantly, it requires a lot of practice to develop and maintain proficiency. Even experienced musicians still practice the fundamental scales. So ultimately, you the student can learn mathematics (that is, make it stick) only through the hard work of doing mathematics.

In conclusion, we sincerely wish you the best of luck in this preparatory course and in your subsequent course in calculus.

Acknowledgments

Our warm gratitude goes out to all the good people at Jones & Bartlett Learning who worked on this text. Because of their great number, they perforce will remain nameless. But we do want to single out for special thanks Timothy Anderson, senior acquisitions editor, and Amy Rose, production director, for their hard work, cooperation, and patience in making this fifth edition a reality, and Warren S. Wright for his careful reading of the first proof pages of this text.

Lastly, all the mistakes in the text are ours. If you run across any errors, or if you have any suggestions for improvement of the text, we would greatly appreciate it if you would bring them to our attention through our editor at: [email protected].

Dennis G. Zill

Jacqueline M. Dewar

About the Cover

Over three thousand light years away, the fascinating and complex Cat’s Eye Nebula (NGC 6543) is the only planetary nebula in the constellation Draco (meaning “dragon”), which wraps around the well-known and easily spotted Little Dipper and Big Dipper. Seen clearly in this book’s front cover image, the cosmic eye measures over half a light year across, or 3 trillion miles. At a fairly low 30–40x magnification, you can just begin to distinguish the nebula from the surrounding stars; it looks slightly fuzzy with perhaps a greenish glow. But in this detailed image from NASA’s Hubble Space Telescope, an astonishing bull’s-eye pattern of eleven or more concentric, gaseous rings, or shells, can be seen around the dying central star. These layers of gases that manifest as bright rings are actually spherical bubbles projected onto the sky. Observations suggest that this simple outer pattern of dust shells was produced when the central Sun-like star cast off its mass in a series of explosive convulsions that took place nearly 1500 years apart. Then, approximately 1000 years ago, the pattern of mass loss suddenly changed and the planetary nebula itself started forming inside the dust shells.

Another perspective of the Cat’s Eye Nebula can be seen on the back cover of this book. Although the glowing concentric shells are not visible in this image, you can clearly see the complex inner bubble, or core, of the nebula that surrounds the bright central star in helix-like formations. This intricate inner structure consists of tightly formed knots and glowing filaments and is much hotter than the nebula’s outer shells. It emits a large amount of ultraviolet radiation that dramatically illuminates the glowing outer gases. Quite noticeable in this image, the high-velocity stellar winds have thinned out the inner bubble of the nebula and appear to have burst the outer filamentary portion at both ends (where the glowing green arcs are visible).

In astronomical terms, planetary nebulae are an extremely short-lived phenomenon. After an aging star has shed its outer layers, the gaseous rings will dissipate and the remaining hot core will cool and fade away as a white dwarf for billions of years. Scientists believe that our Sun is destined to enter its own planetary nebula phase similar to the Cat’s Eye, but not for another 5 billion years or so.