Precalculus (5th Edition) Blitzer.pdf,Description
Download Free PDF. Precalculus fifth edition Mathematics for Calculus Download Precalculus Mathematics For Calculus 5th Edition PDF. Found 49 PDF Ebooks MATH , Section 8 Precalculus Course Syllabus Fall Semester View PDF MB. Create Date. May 1, Last Updated. May 1, Download. File. Calculus 5th Edition - James blogger.com Download View and download Precalculus (5th Edition) blogger.com on DocDroid 3/07/ · Precalculus fifth edition mathematics for calculus Pdf. July 3, by adminpeso. Are you looking for a ebook precalculus fifth edition mathematics for calculus pdf? We have the ... read more
Algebra's importance lies in the student's future. as essential preparation for the serious study of science, engineering, economics, or for more advanced types of mathematics. The primary importance of trigonometry is not in its applications to surveying and navigation, or in making computations about triangles, but rather in the mathematical description of vibrations, rotations, and periodic phenomena of all kinds, including light, sound, alternating currents, and the orbits of the planets around the sun. Ó In this brief, clearly written book, the essentials of geometry, algebra, and trigonometry are pulled together into three complementary and convenient small packages, providing an excellent preview and review for anyone who wishes to prepare to master calculus with a minimum of misunderstanding and wasted time and effort. Students and other readers will find here all they need to pull them through. Problem solving and mathematical modeling are reinforced throughout. This comprehensive, evenly paced book provides complete coverage of the function concept and integrates substantial graphing calculator materials that help you develop insight into mathematical ideas.
Online resources available with the text give you the practice you need to improve your grade in the course. This best selling author team explains concepts simply and clearly, without glossing over difficult points. Problem solving and mathematical modeling are introduced early and reinforced throughout, providing students with a solid foundation in the principles of mathematical thinking. Comprehensive and evenly paced, the book provides complete coverage of the function concept, and integrates a significant amount of graphing calculator material to help students develop insight into mathematical ideas.
The authors' attention to detail and clarity, the same as found in James Stewart's market-leading Calculus text, is what makes this text the market leader. Important Notice: Media content referenced within the product description or the product text may not be available in the ebook version. Problem solving and mathematical modeling are introduced early and reinforced throughout, so that when students finish the course, they have a solid foundation in the principles of mathematical thinking. This comprehensive, evenly paced book provides complete coverage of the function concept and integrates substantial graphing calculator materials that help students develop insight into mathematical ideas. Precalculus is adaptable and designed to fit the needs of a variety of precalculus courses. It is a comprehensive text that covers more ground than a typical one- or two-semester college-level precalculus course.
All Rights Reserved. May not be copied, scanned, or duplicated, in whole o. Expand your learning experience with the Tools for Enriching Calculus CD-ROM The Tools for Enriching Calculus CD-ROM is. FIFTH EDITION Precalculus A Graphing Approach Ron Larson The Pennsylvania State University The Behrend College Robert. No part of this work covered by the copyright herein may be reproduced, transmitted, stored or used in any form or by any means graphic, electronic, or mechanical, including but not limited to photocopying, recording, scanning, digitizing, taping, Web distribution, information networks, or information storage and retrieval systems, except as permitted under Sec- Editorial Assistant: Rebecca Dashiell Technology Project Manager: Sam Subity ten permission of the publisher.
Marketing Manager: Joe Rogove For product information and technology assistance, contact us at Marketing Assistant: Ashley Pickering Marketing Communications Manager: Darlene Amidon-Brent For permission to use material from this text or product, submit all requests online at cengage. Neuhaus Cover Image: Bill Ralph Compositor: Newgen—India Cengage Learning is a leading provider of customized learning solutions with office locations around the globe, including Singapore, the United Kingdom, Australia, Mexico, Brazil, and Japan. Locate your local office at international. Cengage Learning products are represented in Canada by Nelson Education, Ltd. For your course and learning solutions, visit academic. Purchase any of our products at your local college store or at our preferred online store www.
Printed in the United States of America 1 2 3 4 5 6 7 12 11 10 09 08 07 To our students, from whom we have learned so much. Such systems, which tend to be unpredictable and even chaotic at times, are modeled mathematically using the concepts of composition and iteration of functions see Section 2. The basic idea is to start with a particular function and evaluate it at some point in its domain, yielding a new number. The function is then evaluated at the new number. Repeating this process produces a sequence of numbers called iterates of the function. In a sense, these pictures allow us to look, through the lens of mathematics, at exotic little universes that have never been seen before. Professor Ralph teaches at Brock University in Canada. He can be contacted by e-mail at [email protected] About the Authors James Stewart was educated at the University of Toronto and Stanford University, did research at the University of London, and now teaches at McMaster University.
His research field is harmonic analysis. Lothar Redlin grew up on Vancouver Island, received a Bachelor of Science degree from the University of Victoria, and a Ph. from McMaster University in He subsequently did research and taught at the University of Washington, the University of Waterloo, and California State University, Long Beach. He is currently Professor of Mathematics at The Pennsylvania State University, Abington College. His research field is topology. Saleem Watson received his Bachelor of Science degree from Andrews University in Michigan. He did graduate studies at Dalhousie University and McMaster University, where he received his Ph. in He subsequently did research at the Mathematics Institute of the University of Warsaw in Poland.
He also taught at The Pennsylvania State University. He is currently Professor of Mathematics at California State University, Long Beach. His research field is functional analysis. MARK VAN DOREN What do students really need to know to be prepared for calculus? What tools do instructors really need to assist their students in preparing for calculus? These two questions have motivated the writing of this book. To be prepared for calculus a student needs not only technical skill but also a clear understanding of concepts.
Indeed, conceptual understanding and technical skill go hand in hand, each reinforcing the other. A student also needs to gain an appreciation for the power and utility of mathematics in modeling the real-world. Every feature of this textbook is devoted to fostering these goals. We are keenly aware that good teaching comes in many different forms, and that each instructor brings unique strengths and imagination to the classroom. In this book we have included all these methods of teaching precalculus as enhancements to a central core of fundamental skills. These methods are tools to be utilized by instructors and their students to navigate their own course of action in preparing for calculus. In writing this fifth edition our purpose was to further enhance the utility of the book as an instructional tool.
The main change in this edition is an expanded emphasis on modeling and applications: In each section the applications exercises have been expanded and are grouped together under the heading Applications, and each chapter except Chapter 1 now ends with a Focus on Modeling section. We have also made some organizational changes, including dividing the chapter on analytic trigonometry into two chapters, each of more manageable size. There are numerous other smaller changes—as we worked through the book we sometimes realized that an additional example was needed, or an explanation could be clarified, or a section could benefit from different types of exercises.
Throughout these changes, however, we have retained the overall structure and the main features that have contributed to the success of this book. xiii xiv Preface Many of the changes in this edition have been drawn from our own experience in teaching, but, more importantly, we have listened carefully to the users of the current edition, including many of our closest colleagues. We are also grateful to the many letters and e-mails we have received from users of this book, instructors as well as students, recommending changes and suggesting additions. Many of these have helped tremendously in making this edition even more user-friendly. Special Features The most important way to foster conceptual understanding and hone technical skill is through the problems that the instructor assigns.
To that end we have provided a wide selection of exercises. Applications Exercises We have included substantial applied problems that we believe will capture the interest of students. These are integrated throughout the text in both examples and exercises. In the exercise sets, applied problems are grouped together under the heading, Applications. See, for example, pages , , , and A COMPLETE REVIEW CHAPTER We have included an extensive review chapter primarily as a handy reference for the student to revisit basic concepts in algebra and analytic geometry. As much or as little of this chapter can be covered in class as needed, depending on the background of the students. Chapter 1 Test The test at the end of Chapter 1 is intended as a diagnostic instrument for determining what parts of this review chapter need to be taught.
It also serves to help students gauge exactly what topics they need to review. The trigonometry chapters of this text have been written so that either the right triangle approach or the unit circle approach may be taught first. Putting these two approaches in different chapters, each with its relevant applications, helps clarify the purpose of each approach. This approach emphasizes that the trigonometric functions are functions of real numbers, just like the polynomial and exponential functions with which students are already familiar. Chapter 6: Trigonometric Functions of Angles This chapter introduces trigonometry through the right triangle approach. This approach builds on the foundation of a conventional high-school course in trigonometry.
Preface xv Another way to teach trigonometry is to intertwine the two approaches. Some instructors teach this material in the following order: Sections 5. Our organization makes it easy to do this without obscuring the fact that the two approaches involve distinct representations of the same functions. The availability of graphing calculators makes it not less important, but far more important to understand the concepts that underlie what the calculator produces. Accordingly, all our calculator-oriented subsections are preceded by sections in which students must graph or calculate by hand, so that they can understand precisely what the calculator is doing when they later use it to simplify the routine, mechanical part of their work.
The graphing calculator sections, subsections, examples, and exercises, all marked with the special symbol , are optional and may be omitted without loss of continuity. Simple Programs We exploit the programming capabilities of a graphing calculator to simulate real-life situations, to sum series, or to compute the terms of a recursive sequence. See, for instance, pages , , and We have made a special effort, in these modeling sections and subsections, to clarify the essential process of translating problems from English into the language of mathematics. See pages or But the material on modeling, where students are required to construct mathematical models for themselves, has been organized into clearly defined sections and subsections see, for example, pages , , , and Focus on Modeling Each chapter concludes with a Focus on Modeling section.
The first such section, after Chapter 2, introduces the basic idea of modeling a real-life situation by fitting lines to data linear regression. Other sections present ways in which polynomial, exponential, logarithmic, and trigonometric functions, and systems of inequalities can all be used to model familiar phenomena from the sciences and from everyday life see, for example, pages , , or Chapter 1 concludes with a section entitled Focus on Problem Solving. One way to engage students and make them active learners is to have them work perhaps in groups on extended projects that give a feeling of substantial accomplishment when completed.
Each chapter contains one or more Discovery Projects see the table of contents ; these provide a challenging but accessible set of activities that enable students to explore in greater depth an interesting aspect of the topic they have just learned. See, for instance, pages , , or These serve to enliven the material and show that mathematics is an important, vital activity, and that even at this elementary level it is fundamental to everyday life. See, for instance, the vignettes on Viète, page 49; coordinates as addresses, page 88; and radiocarbon dating, page See pages , , and , for example. The Check Your Answer feature is used wherever possible to emphasize the importance of looking back to check whether an answer is reasonable. See, for instance, page CHECK YOUR ANSWER REVIEW MATERIAL The review material in this edition covers individual chapters as well as groups of chapters.
This material is an important tool for helping students see the unity of the different precalculus topics. The questions and exercises in each review section combine the topics from an entire chapter or from groups of chapters. The review material is organized as follows. These can be used as writing exercises, in a classroom discussion setting, or for personal study. Review Exercises The Concept Checks are followed by review exercises designed to provide additional practice for working with the chapter material. Answers to odd-numbered review exercises are given in the back of the book. Chapter Test Each chapter ends with a Chapter Test designed to help the students assess their ability to work with the chapter material as a whole.
Answers to both even and odd test questions are given in the back of the book. Cumulative Review The Cumulative Reviews at the end of the text cover the material of several related chapters, very much like midterm exams. Each such review begins with a checklist of the topics the students should have mastered after completing the respective chapters. This is followed by a Cumulative Review Test. As with the Chapter Tests, answers to all cumulative test questions are given in the back of the book. New exercises have been chosen to provide more practice with basic concepts, as well as to explore ideas that we do not have space to cover in the discussion and examples in the text itself. Many new applied exercises have been added. Each chapter now begins with a Chapter Overview that introduces the main themes of the chapter and explains why the material is important.
Six new Focus on Modeling sections have been added, with topics ranging from Mapping the World Chapter 8 to Traveling and Standing Waves Chapter 7. A few more mathematical vignettes have been added see for example the vignette on splines, page , and the one on Maria Agnesi, page We have moved the section on variation from Chapter 2 to Chapter 1, thus focusing Chapter 2 more clearly on the essential concept of a function. In Chapter 5, Trigonometric Functions of Real Numbers, we have incorporated the material on harmonic motion as a new section.
The Focus on Modeling section is now about fitting sinusoidal curves to data. In Chapter 7, Analytic Trigonometry, we now include only the material on trigonometric identities and equations. This change was done at the request of users. Chapter 8, Polar Coordinates and Vectors, is a new chapter, incorporating material that was previously in other chapters. The topics in this chapter, which also include the polar representation of complex numbers, are united by the theme of using the trigonometric functions to locate the coordinates of a point or describe the components of a vector. In Chapter 9, Systems of Equations and Inequalities, we have put the section on graphing of inequalities as the last section, so it now immediately precedes the material on linear programming in the Focus on Modeling section. Chapter 10, Analytic Geometry, now includes only the conic sections and parametric equations. The material on polar coordinates is in the new Chapter 8. In Chapter 11, Sequence and Series, we have expanded the material on recursive sequences by adding a Focus on Modeling section on the use of such sequences in modeling real-world phenomena.
Acknowledgments We thank the following reviewers for their thoughtful and constructive comments. Michelle Benedict, Augusta State University; Linda Crawford, Augusta State University; Vivian G. Kostyk, Inver Hills Community College; and Heather C. McGilvray, Seattle University. REVIEWERS FOR THE FOURTH EDITION REVIEWERS FOR THE FIFTH EDITION Kenneth Berg, University of Maryland; Elizabeth Bowman, University of Alabama at Huntsville; William Cherry, University of North Texas; Barbara Cortzen, DePaul University; Gerry Fitch, Louisiana State University; Lana Grishchenko, Cal Poly State University, San Luis Obispo; Bryce Jenkins, Cal Poly State University, San Luis Obispo; Margaret Mary Jones, Rutgers University; Victoria Kauffman, University of New Mexico; Sharon Keener, Georgia Perimeter College; YongHee Kim-Park, California State University Long Beach; Mangala Kothari, Rutgers University; Andre Mathurin, Bellarmine College Prep; Donald Robertson, Olympic College; Jude Socrates, Pasadena City College; Enefiok Umana, Georgia Perimeter College; Michele Wallace, Washington State University; and Linda Waymire, Daytona Beach Community College.
xviii Preface We have benefited greatly from the suggestions and comments of our colleagues who have used our books in previous editions. We extend special thanks in this regard to Linda Byun, Bruce Chaderjian, David Gau, Daniel Hernandez, YongHee Kim-Park, Daniel Martinez, David McKay, Robert Mena, Kent Merryfield, Florence Newberger, Viet Ngo, Marilyn Oba, Alan Safer, Angelo Segalla, Robert Valentini, and Derming Wang, from California State University, Long Beach; to Karen Gold, Betsy Gensamer, Cecilia McVoy, Mike McVoy, Samir Ouzomgi, and Ralph Rush, of the Pennsylvania State University, Abington College; to Gloria Dion, of Educational Testing Service, Princeton, New Jersey; to Mark Ashbaugh and Nakhlé Asmar of the University of Missouri, Columbia; to Fred Safier, of the City College of San Francisco; and Steve Edwards, of Southern Polytechnic State University in Marietta, Georgia.
We have also received much invaluable advice from our students, especially Devaki Shah and Ellen Newman. We especially thank Martha Emry, manager of our production service, for her excellent work and her tireless attention to quality and detail. Her energy, devotion, experience, and intelligence were essential components in the creation of this book. We are grateful to Luana Richards, our copy editor, who over the years has expertly shaped the language and style of all of our books. At Matrix Art Services we thank Jade Myers for his elegant graphics. Our special thanks to Phyllis Panman-Watson for her dedication and care in creating the answer section. We are particularly grateful to our publisher Bob Pirtle for guiding this book through every stage of writing and production.
His support and editorial insight when crucial decisions had to be made were invaluable. Complete Solutions Manual This manual provides worked-out solutions to all of the problems in the text. Solution Builder This is an electronic version of the complete solutions manual. It provides instructors with an efficient method for creating solution sets to homework or exams that can then be printed or posted. Each chapter of the text is broken down into to 20minute problem-solving lessons that cover each section of the chapter. Easy to use, it offers you complete control when creating assessments; you can draw from the wealth of exercises provided or create your own questions. A real timesaver, iLrn Assessment offers automatic grading of text-specific homework, quizzes, and tests with results flowing directly into the gradebook. The auto-enrollment feature also saves time with course set up as students selfenroll into the course gradebook. A wide range of problem types provides greater variety and more diverse challenges in your tests.
iLrn Assessment provides seamless integration with Blackboard® and WebCT®. Test Bank The Test Bank consists of two parts. Part 1 includes six tests per chapter including 3 final exams. Part 2 of the Test Bank contains test questions broken down by section. Question types are free response and multiple choice. For the Student HOMEWORK TOOLS Student Solutions Manual The student solutions manual provides worked out solutions to the odd-numbered problems in the text. Enhanced WebAssign Enhanced WebAssign is designed for you to do your homework online. Automatically graded homework allows you to focus on your learning and get interactive study assistance outside of class. Enhanced WebAssign is compatible with recent operating systems, works with most web browsers, and requires no proprietary plug-ins.
LEARNING TOOLS Interactive Video Skillbuilder CD ROM The Interactive Video Skillbuilder CD-ROM contains hours of video instruction. To help students evaluate their progress, each section contains a question web quiz and each chapter contains a chapter test, with answers to each problem on each test. Also includes MathCue Tutorial, dualplatform software that presents and scores problems and tutors students by displaying annotated, step-by-step solutions. Problem sets may be customized as desired. Study Guide Contains detailed explanations, worked-out examples, practice problems, and key ideas to master. Each section of the main text has a corresponding section in the Study Guide. com This outstanding site features chapter-by-chapter online tutorial quizzes, a sample final exam, chapter outlines, chapter review, chapter-by-chapter web links, flashcards, and more! ExamView® Computerized Testing Create, deliver, and customize tests both print and online in minutes with this easy-to-use assessment system.
ExamView is a registered trademark of FSCreations, Inc. Used herein under license. xix This page intentionally left blank To the Student This textbook was written for you to use as a guide to mastering precalculus mathematics. Here are some suggestions to help you get the most out of your course. First of all, you should read the appropriate section of text before you attempt your homework problems. Reading a mathematics text is quite different from reading a novel, a newspaper, or even another textbook. You may find that you have to reread a passage several times before you understand it. Pay special attention to the examples, and work them out yourself with pencil and paper as you read. With this kind of preparation you will be able to do your homework much more quickly and with more understanding.
Mathematics is a problem-solving art, not just a collection of facts. To master the subject you must solve problems—lots of problems. Do as many of the exercises as you can. Be sure to write your solutions in a logical, step-by-step fashion. Try to understand the problem more clearly—reread it thoughtfully and relate it to what you have learned from your teacher and from the examples in the text. Struggle with it until you solve it. Once you have done this a few times you will begin to understand what mathematics is really all about. Answers to the odd-numbered exercises, as well as all the answers to each chapter test, appear at the back of the book.
There may be a calculation that connects the two answers and makes both correct. The symbol is used to warn against committing an error. We have placed this symbol in the margin to point out situations where we have found that many of our students make the same mistake. xxi Calculators and Calculations Calculators are essential in most mathematics and science subjects. They free us from performing routine tasks, so we can focus more clearly on the concepts we are studying. Calculators are powerful tools but their results need to be interpreted with care.
In what follows, we describe the features that a calculator suitable for a precalculus course should have, and we give guidelines for interpreting the results of its calculations. In addition, a memory and at least some degree of programmability will be useful. Your instructor may recommend or require that you purchase a graphing calculator. This book has optional subsections and exercises that require the use of a graphing calculator or a computer with graphing software. These special subsections and exercises are indicated by the symbol. Besides graphing functions, graphing calculators can also be used to find functions that model real-life data, solve equations, perform matrix calculations which are studied in Chapter 9 , and help you perform other mathematical operations.
All these uses are discussed in this book. It is important to realize that, because of limited resolution, a graphing calculator gives only an approximation to the graph of a function. It plots only a finite number of points and then connects them to form a representation of the graph. In Section 1. Calculations and Significant Figures Most of the applied examples and exercises in this book involve approximate values. For example, one exercise states that the moon has a radius of miles. One simple method for specifying the accuracy of a number is to state how many significant digits it has. The significant digits in a number are the ones from the first nonzero digit to the last nonzero digit reading from left to right. Thus, has four significant digits, has three, has two, and has one significant digit.
This rule may sometimes lead to ambiguities. For example, if a distance is km to xxii Calculators and Calculations xxiii the nearest kilometer, then the number really has three significant digits, not just one. This ambiguity is avoided if we use scientific notation—that is, if we express the number as a multiple of a power of 2. The final result can be no more accurate than the measurements given in the problem. For example, suppose we are told that the two shorter sides of a right triangle are measured to be 1. By the Pythagorean Theorem, we find, using a calculator, that the hypotenuse has length But since the given lengths were expressed to three significant digits, the answer cannot be any more accurate. We can therefore say only that the hypotenuse is 2. long, rounding to the nearest hundredth. In general, the final answer should be expressed with the same accuracy as the least-accurate measurement given in the statement of the problem.
The following rules make this principle more precise. Rules for Working with Approximate Data 1. When multiplying or dividing, round off the final result so that it has as many significant digits as the given value with the fewest number of significant digits. When adding or subtracting, round off the final result so that it has its last significant digit in the decimal place in which the least-accurate given value has its last significant digit. When taking powers or roots, round off the final result so that it has the same number of significant digits as the given value. As an example, suppose that a rectangular table top is measured to be by Tenths digit Note that in the formula for the perimeter, the value 2 is an exact value, not an approximate measurement. It therefore does not affect the accuracy of the final result.
In general, if a problem involves only exact values, we may express the final answer with as many significant digits as we wish. Note also that to make the final result as accurate as possible, you should wait until the last step to round off your answer. If necessary, use the memory feature of your calculator to retain the results of intermediate calculations. Abbreviations cm dB F ft g gal h H Hz in. You are probably already familiar with these concepts, but it is helpful to get a fresh look at how these ideas work together to solve problems and model or describe realworld situations. We are interested in how much money you make. To describe your pay we use real numbers. In fact, we use real numbers every day—to describe how tall we are, how much money we have, how cold or warm it is, and so on. In algebra, we express properties of the real numbers by using letters to stand for numbers.
This and other properties of the real numbers constitute the rules for working with numbers, or the rules of algebra. We can also model your pay for any number of hours by a formula. An equation is a sentence written in the language of algebra that expresses a fact about an unknown quantity x. For example, how many hours would you need to work to get paid 60 dollars? The coordinate plane allows us to sketch a graph of an equation in two variables. In this chapter we will see many examples of how the real numbers, equations, and the coordinate plane all work together to help us solve real-life problems. We start with the natural numbers: 1, 2, 3, 4,. The different types of real numbers were invented to meet specific needs. The integers consist of the natural numbers together with their negatives and We construct the rational numbers by taking ratios of integers. There are also real numbers, such as 12, that cannot be expressed as a ratio of integers and are therefore called irrational numbers.
Rational numbers Irrational numbers —21 , -—37 , 3 œ3 , œ5 , œ2 , π , — 2 Integers 3 46, 0. is a rational number. The idea is to multiply x by appropriate powers of 10, and then subtract to eliminate the repeating part. Figure 1 The real number system Every real number has a decimal representation. If the number is rational, then its corresponding decimal is repeating. SECTION 1. From our experience with numbers we know that the properties in the following box are also valid. When we multiply a number by a sum of two numbers, we get the same result as multiplying the number by each of the terms and then adding the results.
The Distributive Property applies whenever we multiply a number by a sum. Figure 2 explains why this property works for the case in which all the numbers are positive integers, but the property is true for any real numbers a, b, and c. The Distributive Property is crucial because it describes the way addition and multiplication interact with each other. Subtraction is the operation that undoes addition; to subtract a number from another, we simply add the negative of that number. Properties of Negatives Property Example 1. Division is the operation that undoes multiplication; to divide by a number, we multiply by the inverse of that number.
To combine real numbers using the operation of division, we use the following properties. Properties of Fractions Property Example Description 1. When adding fractions with different denominators, find a common denominator. Then add the numerators. Instead we rewrite the fractions so that they have the smallest possible common denominator often smaller than the product of the denominators , and then we use Property 3. This denominator is the Least Common Denominator LCD described in the next example. The positive direction toward the right is indicated by an arrow. We choose an arbitrary reference point O, called the origin, which corresponds to the real number 0.
The number associated with the point P is called the coordinate of P, and the line is then called a coordinate line, or a real number line, or simply a real line. Often we identify the point with its coordinate and think of a number as being a point on the real line. Geometrically, this means that a lies to the left of b on the number line. Equivalently, we can say that b is greater than a and write b a. If S is a set, the notation a 僆 S means that a is an element of S, and b 僆 S means that b is not an element of S. Some sets can be described by listing their elements within braces. The intersection of S and T is the set S 傽 T consisting of all elements that are in both S and T. In other words, S 傽 T is the common part of S and T. The empty set, denoted by , is the set that contains no element. The closed interval from a to b includes the endpoints and is denoted 3a, b4. Intervals may also include one endpoint but not the other, or they may extend infinitely far in one direction or both.
The following table lists the possible types of intervals. The notation 1a, q 2 , for instance, simply indicates that the interval has no endpoint on the right but extends infinitely far in the positive direction. In the closed interval 3 0, 14 , the smallest number is 0 and the largest is 1, but the open interval 10, 12 contains no smallest or largest number. To see this, note that 0. So we can always find a number in the interval 10, 1 2 closer to zero than any given number. Since 0 itself is not in the interval, the interval contains no smallest number. Similarly, 0. Since 1 itself is not in the interval, the interval has no largest number. Example 5 Graphing Intervals Express each interval in terms of inequalities, and then graph the interval. a 11, 3 2 艚 32, 74 b 11, 3 2 傼 32, 74 Solution a The intersection of two intervals consists of the numbers that are in both intervals.
b The union of two intervals consists of the numbers that are in either one interval or the other or both. Properties of Absolute Value Property Example 1. A number and its negative have the same absolute value. The absolute value of a product is the product of the absolute values. The absolute value of a quotient is the quotient of the absolute values. From Figure 10 we see that the distance is From this observation we make the following definition see Figure This confirms that, as we would expect, the distance from a to b is the same as the distance from b to a.
Figure 12 1. a 27—28 7. See the margin note on page 2. a b Express each set in interval notation. Temperature Variation The bar graph shows the daily high temperatures for Omak, Washington, and Geneseo, New York, during a certain week in June. Let TO represent the temperature in Omak and TG the temperature in Geneseo. Is the product of two irrational numbers necessarily irrational? What about the sum? Omak, WA Geneseo, NY 80 75 70 65 Sun Mon Tue Wed Day Thu Fri Sat Is 12 12 rational or irrational? In general, what can you say about the sum of a rational and an irrational number? What about the product? Limiting Behavior of Reciprocals Complete the tables. As x gets small? wide, 8 in. deep, and 5 ft long? What about a package that measures 2 ft by 2 ft by 4 ft? b What is the greatest acceptable length for a package that has a square base measuring 9 in. by 9 in?
Irrational Numbers and Geometry Using the following figure, explain how to locate the point 12 on a number line. Can you locate 15 by a similar method? What about 16? List some other irrational numbers that can be located this way. Find the sign of each expression. Sums and Products of Rational and Irrational Numbers Explain why the sum, the difference, and the 1. Commutative and Noncommutative Operations We have seen that addition and multiplication are both commutative operations. a Is subtraction commutative? b Is division of nonzero real numbers commutative? To do this, we need to recall some facts about integer exponents, radicals, and nth roots.
Integer Exponents A product of identical numbers is usually written in exponential notation. For example, 5 5 5 is written as In general, we have the following definition. a n factors The number a is called the base and n is called the exponent. a2 1a a. We would like this rule to be true even when m and n are 0 or negative integers. These observations lead to the following definition. In the table the bases a and b are real numbers, and the exponents m and n are integers. Laws of Exponents Law Example 1. Graphs of Functions. Discovery Project: Relations and Functions. Increasing and Decreasing Functions: Average Rate of Change.
Transformations of Functions. Quadratic Functions: Maxima and Minima. Modeling with Functions. Combining Functions. Project: Iteration and Chaos. One-to-One Functions and Their Inverses Review. Focus on Modeling: Fitting Lines to Data. Polynomial Functions and Their Graphs. Dividing Polynomials. Real Zeros of Polynomials. Discovery Project: Zeroing in on a Zero. Complex Numbers. Complex Zeros and the Fundamental Theorem of Algebra. Rational Functions. Focus on Modeling: Fitting Polynomials to Data. Exponential Functions. Discovery Project: Exponential Explosion.
Logarithmic Functions. Laws of Logarithms. Exponential and Logarithmic Equations. Modeling with Exponential and Logarithmic Functions. Focus on Modeling: Fitting Exponential and Power Curves to Data. The Unit Circle. Trigonometric Functions of Real Numbers. Trigonometric Graphs. Discovery Project: Predator-Prey Models. More Trigonometric Graphs. Modeling Harmonic Motion. Focus on Modeling: Fitting Sinusoidal Curves to Data. Angle Measure. Trigonometry of Right Triangles. Discovery Project: Similarity. Trigonometric Functions of Angles. The Law of Sines. The Law of Cosines. Focus on Modeling:Surveying. Trigonometric Identities. Addition and Subtraction Formulas.
Double-Angle, Half-Angle, and Sum-Product Identities. Inverse Trigonometric Functions. Discovery Project: Where to Sit at the Movies. Trigonometric Equations. Focus on Modeling: Traveling and Standing Waves. Polar Coordinates.
Pages Page size x DOWNLOAD FILE. PreCalculus Copyright Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole o. Expand your learning experience with the Tools for Enriching Calculus CD-ROM The Tools for Enriching Calculus CD-ROM is. FIFTH EDITION Precalculus A Graphing Approach Ron Larson The Pennsylvania State University The Behrend College Robert. No part of this work covered by the copyright herein may be reproduced, transmitted, stored or used in any form or by any means graphic, electronic, or mechanical, including but not limited to photocopying, recording, scanning, digitizing, taping, Web distribution, information networks, or information storage and retrieval systems, except as permitted under Sec- Editorial Assistant: Rebecca Dashiell Technology Project Manager: Sam Subity ten permission of the publisher. Marketing Manager: Joe Rogove For product information and technology assistance, contact us at Marketing Assistant: Ashley Pickering Marketing Communications Manager: Darlene Amidon-Brent For permission to use material from this text or product, submit all requests online at cengage.
Neuhaus Cover Image: Bill Ralph Compositor: Newgen—India Cengage Learning is a leading provider of customized learning solutions with office locations around the globe, including Singapore, the United Kingdom, Australia, Mexico, Brazil, and Japan. Locate your local office at international. Cengage Learning products are represented in Canada by Nelson Education, Ltd. For your course and learning solutions, visit academic. Purchase any of our products at your local college store or at our preferred online store www. Printed in the United States of America 1 2 3 4 5 6 7 12 11 10 09 08 07 To our students, from whom we have learned so much. Such systems, which tend to be unpredictable and even chaotic at times, are modeled mathematically using the concepts of composition and iteration of functions see Section 2.
The basic idea is to start with a particular function and evaluate it at some point in its domain, yielding a new number. The function is then evaluated at the new number. Repeating this process produces a sequence of numbers called iterates of the function. In a sense, these pictures allow us to look, through the lens of mathematics, at exotic little universes that have never been seen before. Professor Ralph teaches at Brock University in Canada. He can be contacted by e-mail at [email protected] About the Authors James Stewart was educated at the University of Toronto and Stanford University, did research at the University of London, and now teaches at McMaster University. His research field is harmonic analysis. Lothar Redlin grew up on Vancouver Island, received a Bachelor of Science degree from the University of Victoria, and a Ph. from McMaster University in He subsequently did research and taught at the University of Washington, the University of Waterloo, and California State University, Long Beach.
He is currently Professor of Mathematics at The Pennsylvania State University, Abington College. His research field is topology. Saleem Watson received his Bachelor of Science degree from Andrews University in Michigan. He did graduate studies at Dalhousie University and McMaster University, where he received his Ph. in He subsequently did research at the Mathematics Institute of the University of Warsaw in Poland. He also taught at The Pennsylvania State University. He is currently Professor of Mathematics at California State University, Long Beach. His research field is functional analysis. MARK VAN DOREN What do students really need to know to be prepared for calculus? What tools do instructors really need to assist their students in preparing for calculus? These two questions have motivated the writing of this book. To be prepared for calculus a student needs not only technical skill but also a clear understanding of concepts.
Indeed, conceptual understanding and technical skill go hand in hand, each reinforcing the other. A student also needs to gain an appreciation for the power and utility of mathematics in modeling the real-world. Every feature of this textbook is devoted to fostering these goals. We are keenly aware that good teaching comes in many different forms, and that each instructor brings unique strengths and imagination to the classroom. In this book we have included all these methods of teaching precalculus as enhancements to a central core of fundamental skills. These methods are tools to be utilized by instructors and their students to navigate their own course of action in preparing for calculus. In writing this fifth edition our purpose was to further enhance the utility of the book as an instructional tool.
The main change in this edition is an expanded emphasis on modeling and applications: In each section the applications exercises have been expanded and are grouped together under the heading Applications, and each chapter except Chapter 1 now ends with a Focus on Modeling section. We have also made some organizational changes, including dividing the chapter on analytic trigonometry into two chapters, each of more manageable size. There are numerous other smaller changes—as we worked through the book we sometimes realized that an additional example was needed, or an explanation could be clarified, or a section could benefit from different types of exercises. Throughout these changes, however, we have retained the overall structure and the main features that have contributed to the success of this book. xiii xiv Preface Many of the changes in this edition have been drawn from our own experience in teaching, but, more importantly, we have listened carefully to the users of the current edition, including many of our closest colleagues.
We are also grateful to the many letters and e-mails we have received from users of this book, instructors as well as students, recommending changes and suggesting additions. Many of these have helped tremendously in making this edition even more user-friendly. Special Features The most important way to foster conceptual understanding and hone technical skill is through the problems that the instructor assigns. To that end we have provided a wide selection of exercises. Applications Exercises We have included substantial applied problems that we believe will capture the interest of students. These are integrated throughout the text in both examples and exercises.
In the exercise sets, applied problems are grouped together under the heading, Applications. See, for example, pages , , , and A COMPLETE REVIEW CHAPTER We have included an extensive review chapter primarily as a handy reference for the student to revisit basic concepts in algebra and analytic geometry. As much or as little of this chapter can be covered in class as needed, depending on the background of the students. Chapter 1 Test The test at the end of Chapter 1 is intended as a diagnostic instrument for determining what parts of this review chapter need to be taught. It also serves to help students gauge exactly what topics they need to review. The trigonometry chapters of this text have been written so that either the right triangle approach or the unit circle approach may be taught first.
Putting these two approaches in different chapters, each with its relevant applications, helps clarify the purpose of each approach. This approach emphasizes that the trigonometric functions are functions of real numbers, just like the polynomial and exponential functions with which students are already familiar. Chapter 6: Trigonometric Functions of Angles This chapter introduces trigonometry through the right triangle approach. This approach builds on the foundation of a conventional high-school course in trigonometry. Preface xv Another way to teach trigonometry is to intertwine the two approaches. Some instructors teach this material in the following order: Sections 5. Our organization makes it easy to do this without obscuring the fact that the two approaches involve distinct representations of the same functions. The availability of graphing calculators makes it not less important, but far more important to understand the concepts that underlie what the calculator produces.
Accordingly, all our calculator-oriented subsections are preceded by sections in which students must graph or calculate by hand, so that they can understand precisely what the calculator is doing when they later use it to simplify the routine, mechanical part of their work. The graphing calculator sections, subsections, examples, and exercises, all marked with the special symbol , are optional and may be omitted without loss of continuity. Simple Programs We exploit the programming capabilities of a graphing calculator to simulate real-life situations, to sum series, or to compute the terms of a recursive sequence. See, for instance, pages , , and We have made a special effort, in these modeling sections and subsections, to clarify the essential process of translating problems from English into the language of mathematics.
See pages or But the material on modeling, where students are required to construct mathematical models for themselves, has been organized into clearly defined sections and subsections see, for example, pages , , , and Focus on Modeling Each chapter concludes with a Focus on Modeling section. The first such section, after Chapter 2, introduces the basic idea of modeling a real-life situation by fitting lines to data linear regression. Other sections present ways in which polynomial, exponential, logarithmic, and trigonometric functions, and systems of inequalities can all be used to model familiar phenomena from the sciences and from everyday life see, for example, pages , , or Chapter 1 concludes with a section entitled Focus on Problem Solving.
One way to engage students and make them active learners is to have them work perhaps in groups on extended projects that give a feeling of substantial accomplishment when completed. Each chapter contains one or more Discovery Projects see the table of contents ; these provide a challenging but accessible set of activities that enable students to explore in greater depth an interesting aspect of the topic they have just learned. See, for instance, pages , , or These serve to enliven the material and show that mathematics is an important, vital activity, and that even at this elementary level it is fundamental to everyday life.
See, for instance, the vignettes on Viète, page 49; coordinates as addresses, page 88; and radiocarbon dating, page See pages , , and , for example. The Check Your Answer feature is used wherever possible to emphasize the importance of looking back to check whether an answer is reasonable. See, for instance, page CHECK YOUR ANSWER REVIEW MATERIAL The review material in this edition covers individual chapters as well as groups of chapters. This material is an important tool for helping students see the unity of the different precalculus topics. The questions and exercises in each review section combine the topics from an entire chapter or from groups of chapters. The review material is organized as follows. These can be used as writing exercises, in a classroom discussion setting, or for personal study. Review Exercises The Concept Checks are followed by review exercises designed to provide additional practice for working with the chapter material.
Answers to odd-numbered review exercises are given in the back of the book. Chapter Test Each chapter ends with a Chapter Test designed to help the students assess their ability to work with the chapter material as a whole.
Precalculus: Mathematics for Calculus – James Stewart – 5th Edition,Newest Books
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