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This book covers calculus with an emphasis on cross-discipline principles and practices. Designed to be both reader-friendly and accessible, it develops a thorough, functional understanding of mathematical concepts in preparation for their application in other areas. Each chapter concentrates on developing concepts and ideas followed immediately by developing computational skills and problem solving. Two-part coverage presents a library of elementary functions and calculus. For individuals looking for a view of mathematical ideas and processes, and an illustration of the relevance of mathematics to the real world.

Emphasizing computational skills, ideas, and problem solving rather than theory, this new edition of a undergraduate level text uses problem examples from other fields to illustrate concepts regarding derivatives, integration, multivariable calculus, and trigonometric functions. Annotation c. by Book News, Inc., Portland, Or.

Raymond A. Barnett, a native of California, received his B.A. in mathematical statistics from the University of California at Berkeley and his M.A. in mathematics from the University of Southern California. He has been a member of the Merritt College Mathematics Department, and was chairman of the department for four years. Raymond Barnett has authored or co-authored eighteen textbooks in mathematics, most of which are still in use. In addition to international English editions, a number of books have been translated into Spanish. Co-authors include Michael Ziegler, Marquette University; Thomas Kearns, Northern University; Charles Burke, City College of San Francisco; John Fuji, Merritt College; and Karl Byleen, Marquette University.

Michael R. Ziegler (late) received his B.S. from Shippensburg State College and his M.S. and Ph.D. from the University of Delaware. After completing post doctoral work at the University of Kentucky, he was appointed to the faculty of Marquette University where he held the rank of Professor in the Department of Mathematics, Statistics, and Computer Science. Dr. Ziegler published over a dozen research articles in complex analysis and co-authored eleven undergraduate mathematics textbooks with Raymond A. Barnett, and more recently, Karl E. Byleen.

Karl E. Byleen received his B.S., M.A. and Ph.D. degrees in mathematics from the University of Nebraska. He is currently an Associate Professor in the Department of Mathematics, Statistics and Computer Science of Marquette University. He has published a dozen research articles on the algebraic theory of semigroups.

The ninth edition of Calculus for Business, Economics, Life Sciences, and Social Sciences is designed for a one-term course in calculus and for students who have had 1 - 2 years of high school algebra or the equivalent. The choice and independence of topics make the text readily adaptable to a variety of courses (see the Chapter Dependency Chart on page xi). It is one of five books in the authors' college mathematics series.

Improvements in this edition evolved out of the generous response from a large number of users of the last and previous editions as well as survey results from instructors, mathematics departments, course outlines, and college catalogs. Fundamental to a book's growth and effectiveness is classroom use and feedback. Now in its ninth edition, Calculus for Business, Economics, Life Sciences, and Social Sciences has had the benefit of having a substantial amount of both.

Emphasis and Style

The text is written for student comprehension. Great care has been taken to write a book that is mathematically correct and accessible to students. Emphasis is on computational skills, ideas, and problem solving rather than mathematical theory. Most derivations and proofs are omitted except where their inclusion adds significant insight into a particular concept. General concepts and results are usually presented only after particular cases have been discussed.

Examples and Matched Problems

Over 290 completely worked examples are used to introduce concepts and to demonstrate problem-solving techniques. Many examples have multiple parts, significantly increasing the total number of worked examples. Eachexample is followed by a similar matched problem for the student to work while reading the material. This actively involves the student in the learning process. The answers to these matched problems are included at the end of each section for easy reference.

Exploration and Discussion

Every section contains Explore - Discuss problems interspersed at appropriate places to encourage the student to think about a relationship or process before a result is stated, or to investigate additional consequences of a development in the text. Verbalization of mathematical concepts, results, and processes is encouraged in these Explore - Discuss problems, as well as in some matched problems, and in some problems in almost every exercise set. The Explore - Discuss material also can be used as in-class or out-of-class group activities. In addition, at the end of every chapter, we have included two special chapter group activities that involve several of the concepts discussed in the chapter. Problems in the exercise sets that require verbalization are indicated by color problem numbers.

Exercise Sets

The book contains over 4,100 problems. Many problems have multiple parts, significantly increasing the total number of problems. Each exercise set is designed so that an average or below-average student will experience success and a very capable student will be challenged. Exercise sets are mostly divided into A (routine, easy mechanics), B (more difficult mechanics), and C (difficult mechanics and some theory) levels.

Applications

A major objective of this book is to give the student substantial experience in modeling and solving real-world problems. Enough applications are included to convince even the most skeptical student that mathematics is really useful (see the Applications Index inside the back cover). Worked examples involving applications are identified by or . Almost every exercise set contains application problems, usually divided into business and economics, life science, and social science groupings. An instructor with students from all three disciplines can let them choose applications from their own field of interest; if most students are from one of the three areas, then special emphasis can be placed there. Most of the applications are simplified versions of actual real-world problems taken from professional journals and books. No specialized experience is required to solve any of the applications.

Internet Connections

The Internet provides a wealth of material that can be related to this book, from sources for the data in application problems to interactive exercises that provide additional insight into the various mathematical processes. Every section of the book contains Internet connections identified by . Links to the related web sites can be found at the PH Companion Website discussed later in this preface

Technology

The generic term graphing utility is used to refer to any of the various graphing calculators or computer software packages that might be available to a student using this book. (See the description of the software accompanying this book later in this Preface.) Although access to a graphing utility is not assumed, it is likely that many students will want to make use of one of these devices. To assist these students, optional graphing utility activities are included in appropriate places in the book. These include brief discussions in the text, examples or portions of examples solved on a graphing utility, problems for the student to solve, and a group activity that involves the use of technology at the end of each chapter. Beginning with the group activity at the end of Chapter 1, and continuing throughout the text, linear regression on a graphing utility is used at appropriate points to illustrate mathematical modeling with real data. All the optional graphing utility material is clearly identified by either or and can be omitted without loss of continuity, if desired.

Graphs

All graphs are computer-generated to ensure mathematical accuracy. Graphing utility screens displayed in the text are actual output from a graphing calculator.

Additional Pedagogical Features

Annotation of examples and developments, in small color type, is found throughout the text to help students through critical stages (see Sections 1-1 and 3-2). Think boxes (dashed boxes) are used to enclose steps that are usually performed mentally (see Sections 1-1 and 3-4). Boxes are used to highlight important definitions, results, and step-by-step processes (see Sections 1-1 and 3-2). Caution statements appear throughout the text where student errors often occur (see Sections 3-2 and 4-1). Functional use of color improves the clarity of many illustrations, graphs, and developments, and guides students through certain critical steps (see Sections 1-1 and 3-2). Boldface type is used to introduce new terms and highlight important comments. Chapter review sections include a review of all important terms and symbols and a comprehensive review exercise. Answers to most review exercises, keyed to appropriate sections, are included in the back of the book. Answers to all other odd-numbered problems are also in the back of the book.

Content

The text begins with the development of a library of elementary functions in Chapters 1 and 2, including their properties and uses. We encourage students to investigate mathematical ideas and processes graphically and numerically, as well as algebraically. This development lays a firm foundation for studying mathematics both in this book and in future endeavors. Depending on the syllabus for the course and the background of the students, some or all of this material can be covered at the beginning of a course, or selected portions can be referred to as needed later in the course.

The material in Part Two (Calculus) consists of differential calculus (Chapters 3 - 5), integral calculus (Chapters 6 - 7), multivariable calculus (Chapter 8), and a brief discussion of differentiation and integration of trigonometric functions (Chapter 9). In general, Chapters 3 - 6 must be covered in sequence; however, certain sections can be omitted or given brief treatments, as pointed out in the discussion that follows (see the Chapter Dependency Chart on page xi).

Chapter 3 introduces the derivative, covers the limit properties essential to understanding the definition of the derivative, develops the rules of differentiation (including the chain rule for power forms), and introduces applications of derivatives in business and economics. The interplay between graphical, numerical, and algebraic concepts is emphasized here and throughout the text.

Chapter 4 focuses on graphing and optimization. The first three sections cover continuity and first-derivative and second-derivative graph properties, while emphasizing polynomial graphing. Rational function graphing is covered in Section 4-4. In a course that does not include graphing rational functions, this section can be omitted or given a brief treatment. Optimization is covered in Section 4-5, including examples and problems involving end-point solutions.

The first three sections of Chapter 5 extend the derivative concepts discussed in Chapters 3 and 4 to exponential and logarithmic functions (including the general form of the chain rule). This material is required for all the remaining chapters. Implicit differentiation is introduced in Section 5-4 and applied to related rate problems in Section 5-5. These topics are not referred to elsewhere in the text and can be omitted.

Chapter 6 introduces integration. The first two sections cover antidifferentiation techniques essential to the remainder of the text. Section 6-3 discusses some applications involving differential equations that can be omitted. Sections 6-4 and 6-5 discuss the definite integral in terms of Riemann sums, including approximations with various types of sums and some simple error estimation. As before, the interplay between the graphical, numeric, and algebraic properties is emphasized. These two sections also are required for the remaining chapters in the text.

Chapter 7 covers additional integration topics and is organized to provide maximum flexibility for the instructor. The first section extends the area concepts introduced in Chapter 6 to the area between two curves and related applications. Section 7-2 covers three more applications of integration, and Sections 7-3 and 7-4 deal with additional techniques of integration. Any or all of the topics in Chapter 7 can be omitted.

The first five sections of Chapter 8 deal with differential multivariable calculus and can be covered any time after Section 5-3 has been completed. Section 8-6 requires the integration concepts discussed in Chapter 6.

Chapter 9 provides brief coverage of trigonometric functions that can be incorporated into the course, if desired. Section 9-1 provides a review of basic trigonometric concepts. Section 9-2 can be covered any time after Section 5-3 has been completed. Section 9-3 requires the material in Chapter 6.

Appendix A contains a self-test and a concise review of basic algebra that also may be covered as part of the course or referred to as needed. As mentioned above, Appendix B contains additional topics that can be covered in conjunction with certain sections in the text, if desired.

Supplements for the Student

1. A Student Solutions Manual and Visual Calculus by Garret J. Etgen and David Schneider is available through your book store. The manual includes detailed solutions to all odd-numbered problems and all review exercises. Visual Calculus by David Schneider contains over twenty routines that provide additional insight into the topics discussed in the text. Although this software has much of the computing power of standard calculus software packages, it is primarily a teaching tool that focuses on understanding mathematical concepts, rather than on computing. These routines incorporate graphics whenever possible to illustrate topics such as secant lines; tangent lines; velocity; optimization; the relationship between the graphs of f, , ; and the various approaches to approximating definite integrals. All the routines in this software package are menu-driven and very easy to use. The software will run on DOS or Windows platforms.

2. The PH Companion Website, designed to complement and expand upon the text, offers a variety of teaching and learning tools, including links to related websites, practice work for students, and the ability for instructors to monitor and evaluate students' work on the website. For more information, contact your local Prentice Hall representative

3. CourseCompass/Blackboard/WebCT offers Course compatible content including Excel Projects, Quizzes, Chapter Destinations, Lecture Notes, and Graphing Calculator Help. CourseCompass is the perfect course management solution that combines quality Pearson Education content with state-of-the-art Blackboard technology! It is a dynamic, interactive online course management tool powered by Blackboard. This exciting product allows you to teach with market-leading Pearson Education content in an easy-to-use customizable format. Blackboard 5SM is a comprehensive and flexible e-Learning software platform that delivers a course management system, customizable institution-wide portals, online communities, and an advanced architecture that allows for Web-based integration of multiple administrative systems. WebCT is one of the most popular Web course platforms in higher education today. It is the first destination site for the higher education marketplace to offer both teaching and learning resources and a community of peers across course and institutional boundaries.

Supplements for the Instructor

For a summary of all available supplementary materials and detailed information regarding examination copy requests and orders, see page xix.

1. PH Custom Test, a menu-driven random test system for either Windows or Macintosh is available to instructors. The test system has been greatly expanded and now offers on-line testing. Carefully constructed algorithms use random-number generators to produce different, yet equivalent, versions of each of these problems. In addition, the system incorporates a unique editing function that allows the instructor to create additional problems, or alter any of the existing problems in the test, using a full set of mathematical notation. The test system offers free-response, multiple-choice, and mixed exams. An almost unlimited number of quizzes, review exercises, chapter tests, midterms, and final examinations, each different from the other, can be generated quickly and easily. At the same time, the system will produce answer keys, student worksheets, and a gradebook for the instructor, if desired.

2. A Test Item File, prepared by Laurel Technical Services, provides a hard copy of the test items available in PH Custom Test.

3. An Instructor's Solutions Manual provides detailed solutions to the problems not solved in the Student Solutions Manual. This manual is available to instructors without charge.

4. A Student Solutions Manual and Visual Calculus by Garret J. Etgen and David Schneider (see Supplements for the Student) is available to instructors.

5. The PH Companion Website, designed to complement and expand upon the text, offers a variety of interactive teaching and learning tools, including links to related websites, practice work for students, and the ability for instructors to monitor and evaluate students' work on the website. For more information, contact your local Prentice Hall representative

6. CourseCompass/Blackboard/WebCT offers Course compatible content including Excel Projects, Quizzes, Chapter Destinations, Lecture Notes, and Graphing Calculator Help. CourseCompass is the perfect course management solution that combines quality Pearson Education content with state-of-the-art Blackboard technology! It is a dynamic, interactive online course management tool powered by Blackboard. This exciting product allows you to teach with market-leading Pearson Education content in an easy-to-use customizable format. Blackboard 5SM is a comprehensive and flexible e-Learning software platform that delivers a course management system, customizable institution-wide portals, online communities, and an advanced architecture that allows for Web-based integration of multiple administrative systems. WebCT is one of the most popular Web course platforms in higher education today. It is the first destination site for the higher education marketplace to offer both teaching and learning resources and a community of peers across course and institutional boundaries.

Error Check

Acknowledgments

In addition to the authors, many others are involved in the successful publication of a book. We wish to thank the following reviewers of the eighth edition: Ann Pellegrino, The College of Charleston; Yesmine Akl, Widener University; Lorenzo Pitts, Jr., DeKalb Technical Institute; and, Martha Goshaw, Piedmont Virginia Community College.

We also wish to thank our colleagues who have provided input on previous editions: Chris Boldt, Bob Bradshaw, Celeste Carter, Bruce Chaffee, Robert Chaney, Dianne Clark, Charles E. Cleaver, Barbara Cohen, Richard L. Conlon, Catherine Cron, Madhu Deshpande, John Dickerson, Kenneth A. Dodaro, Michael W. Ecker, Jerry R. Ehman, Lucina Gallagher, Joel Haack, Martha M. Harvey, Sue Henderson, Lloyd R. Hicks, Louis F. Hoelzle, Paul Hutchins, K. Wayne James, Robert H. Johnston, Robert Krystock, James T. Loats, Frank Lopez, Roy H. Luke, Mel Mitchell, Michael Montano, Ronald Persky, Shala Peterman, Kenneth A. Peters, Jr., Tom Plavchak, Bob Prielipp, Stephen Rodi, Arthur Rosenthal, Sheldon Rothman, Elaine Russell, Daniel E. Scanlon, George R. Schriro, Arnold L. Schroeder, Hari Shanker, Larry Small, Joan Smith, Steven Terry, Delores A. Williams, Caroline Woods, Charles W. Zimmerman, and Pat Zrolka.

We also express our thanks to:

Hossein Hamedani, Carolyn Meitler, Stephen Merrill, Robert Mullins, and Caroline Woods for providing a careful and thorough check of all the mathematical calculations in the book, and to Priscilla Gathoni for checking the Student Solutions Manual, and the Instructor's Solutions Manual (a tedious but extremely important job).

Garret Etgen, Hossein Hamedani, Carolyn Meitler, and David Schneider for developing the supplemental manuals that are so important to the success of a text.

Jeanne Wallace for accurately and efficiently producing most of the manuals that supplement the text.

George Morris and his staff at Scientific Illustrators for their effective illustrations and accurate graphs.

All the people at Prentice Hall who contributed their efforts to the production of this book, especially Quincy McDonald, our acquisitions editor, and Lynn Savino Wendel, our Production Editor.

Producing this new edition with the help of all these extremely competent people has been a most satisfying experience.

The ninth edition of Calculus for Business, Economics, Life Sciences, and Social Sciences is designed for a one-term course in calculus and for students who have had 1 - 2 years of high school algebra or the equivalent. The choice and independence of topics make the text readily adaptable to a variety of courses (see the Chapter Dependency Chart on page xi). It is one of five books in the authors' college mathematics series.

Improvements in this edition evolved out of the generous response from a large number of users of the last and previous editions as well as survey results from instructors, mathematics departments, course outlines, and college catalogs. Fundamental to a book's growth and effectiveness is classroom use and feedback. Now in its ninth edition, Calculus for Business, Economics, Life Sciences, and Social Sciences has had the benefit of having a substantial amount of both.

Emphasis and Style

The text is written for student comprehension. Great care has been taken to write a book that is mathematically correct and accessible to students. Emphasis is on computational skills, ideas, and problem solving rather than mathematical theory. Most derivations and proofs are omitted except where their inclusion adds significant insight into a particular concept. General concepts and results are usually presented only after particular cases have been discussed.

Examples and Matched Problems

Over 290 completely worked examples are used to introduce concepts and to demonstrate problem-solving techniques. Many examples have multiple parts, significantly increasing the total number of worked examples. Eachexample is followed by a similar matched problem for the student to work while reading the material. This actively involves the student in the learning process. The answers to these matched problems are included at the end of each section for easy reference.

Exploration and Discussion

Every section contains Explore - Discuss problems interspersed at appropriate places to encourage the student to think about a relationship or process before a result is stated, or to investigate additional consequences of a development in the text. Verbalization of mathematical concepts, results, and processes is encouraged in these Explore - Discuss problems, as well as in some matched problems, and in some problems in almost every exercise set. The Explore - Discuss material also can be used as in-class or out-of-class group activities. In addition, at the end of every chapter, we have included two special chapter group activities that involve several of the concepts discussed in the chapter. Problems in the exercise sets that require verbalization are indicated by color problem numbers.

Exercise Sets

The book contains over 4,100 problems. Many problems have multiple parts, significantly increasing the total number of problems. Each exercise set is designed so that an average or below-average student will experience success and a very capable student will be challenged. Exercise sets are mostly divided into A (routine, easy mechanics), B (more difficult mechanics), and C (difficult mechanics and some theory) levels.

Applications

A major objective of this book is to give the student substantial experience in modeling and solving real-world problems. Enough applications are included to convince even the most skeptical student that mathematics is really useful (see the Applications Index inside the back cover). Worked examples involving applications are identified by or . Almost every exercise set contains application problems, usually divided into business and economics, life science, and social science groupings. An instructor with students from all three disciplines can let them choose applications from their own field of interest; if most students are from one of the three areas, then special emphasis can be placed there. Most of the applications are simplified versions of actual real-world problems taken from professional journals and books. No specialized experience is required to solve any of the applications.

Internet Connections

The Internet provides a wealth of material that can be related to this book, from sources for the data in application problems to interactive exercises that provide additional insight into the various mathematical processes. Every section of the book contains Internet connections identified by . Links to the related web sites can be found at the PH Companion Website discussed later in this preface: www.prenhall.com/barnett

Technology

The generic term graphing utility is used to refer to any of the various graphing calculators or computer software packages that might be available to a student using this book. (See the description of the software accompanying this book later in this Preface.) Although access to a graphing utility is not assumed, it is likely that many students will want to make use of one of these devices. To assist these students, optional graphing utility activities are included in appropriate places in the book. These include brief discussions in the text, examples or portions of examples solved on a graphing utility, problems for the student to solve, and a group activity that involves the use of technology at the end of each chapter. Beginning with the group activity at the end of Chapter 1, and continuing throughout the text, linear regression on a graphing utility is used at appropriate points to illustrate mathematical modeling with real data. All the optional graphing utility material is clearly identified by either or and can be omitted without loss of continuity, if desired.

Graphs

All graphs are computer-generated to ensure mathematical accuracy. Graphing utility screens displayed in the text are actual output from a graphing calculator.

Additional Pedagogical Features

Annotation of examples and developments, in small color type, is found throughout the text to help students through critical stages (see Sections 1-1 and 3-2). Think boxes (dashed boxes) are used to enclose steps that are usually performed mentally (see Sections 1-1 and 3-4). Boxes are used to highlight important definitions, results, and step-by-step processes (see Sections 1-1 and 3-2). Caution statements appear throughout the text where student errors often occur (see Sections 3-2 and 4-1). Functional use of color improves the clarity of many illustrations, graphs, and developments, and guides students through certain critical steps (see Sections 1-1 and 3-2). Boldface type is used to introduce new terms and highlight important comments. Chapter review sections include a review of all important terms and symbols and a comprehensive review exercise. Answers to most review exercises, keyed to appropriate sections, are included in the back of the book. Answers to all other odd-numbered problems are also in the back of the book.

Content

The text begins with the development of a library of elementary functions in Chapters 1 and 2, including their properties and uses. We encourage students to investigate mathematical ideas and processes graphically and numerically, as well as algebraically. This development lays a firm foundation for studying mathematics both in this book and in future endeavors. Depending on the syllabus for the course and the background of the students, some or all of this material can be covered at the beginning of a course, or selected portions can be referred to as needed later in the course.

The material in Part Two (Calculus) consists of differential calculus (Chapters 3 - 5), integral calculus (Chapters 6 - 7), multivariable calculus (Chapter 8), and a brief discussion of differentiation and integration of trigonometric functions (Chapter 9). In general, Chapters 3 - 6 must be covered in sequence; however, certain sections can be omitted or given brief treatments, as pointed out in the discussion that follows (see the Chapter Dependency Chart on page xi).

Chapter 3 introduces the derivative, covers the limit properties essential to understanding the definition of the derivative, develops the rules of differentiation (including the chain rule for power forms), and introduces applications of derivatives in business and economics. The interplay between graphical, numerical, and algebraic concepts is emphasized here and throughout the text.

Chapter 4 focuses on graphing and optimization. The first three sections cover continuity and first-derivative and second-derivative graph properties, while emphasizing polynomial graphing. Rational function graphing is covered in Section 4-4. In a course that does not include graphing rational functions, this section can be omitted or given a brief treatment. Optimization is covered in Section 4-5, including examples and problems involving end-point solutions.

The first three sections of Chapter 5 extend the derivative concepts discussed in Chapters 3 and 4 to exponential and logarithmic functions (including the general form of the chain rule). This material is required for all the remaining chapters. Implicit differentiation is introduced in Section 5-4 and applied to related rate problems in Section 5-5. These topics are not referred to elsewhere in the text and can be omitted.

Chapter 6 introduces integration. The first two sections cover antidifferentiation techniques essential to the remainder of the text. Section 6-3 discusses some applications involving differential equations that can be omitted. Sections 6-4 and 6-5 discuss the definite integral in terms of Riemann sums, including approximations with various types of sums and some simple error estimation. As before, the interplay between the graphical, numeric, and algebraic properties is emphasized. These two sections also are required for the remaining chapters in the text.

Chapter 7 covers additional integration topics and is organized to provide maximum flexibility for the instructor. The first section extends the area concepts introduced in Chapter 6 to the area between two curves and related applications. Section 7-2 covers three more applications of integration, and Sections 7-3 and 7-4 deal with additional techniques of integration. Any or all of the topics in Chapter 7 can be omitted.

The first five sections of Chapter 8 deal with differential multivariable calculus and can be covered any time after Section 5-3 has been completed. Section 8-6 requires the integration concepts discussed in Chapter 6.

Chapter 9 provides brief coverage of trigonometric functions that can be incorporated into the course, if desired. Section 9-1 provides a review of basic trigonometric concepts. Section 9-2 can be covered any time after Section 5-3 has been completed. Section 9-3 requires the material in Chapter 6.

Appendix A contains a self-test and a concise review of basic algebra that also may be covered as part of the course or referred to as needed. As mentioned above, Appendix B contains additional topics that can be covered in conjunction with certain sections in the text, if desired.

Supplements for the Student

1. A Student Solutions Manual and Visual Calculus by Garret J. Etgen and David Schneider is available through your book store. The manual includes detailed solutions to all odd-numbered problems and all review exercises. Visual Calculus by David Schneider contains over twenty routines that provide additional insight into the topics discussed in the text. Although this software has much of the computing power of standard calculus software packages, it is primarily a teaching tool that focuses on understanding mathematical concepts, rather than on computing. These routines incorporate graphics whenever possible to illustrate topics such as secant lines; tangent lines; velocity; optimization; the relationship between the graphs of f, , ; and the various approaches to approximating definite integrals. All the routines in this software package are menu-driven and very easy to use. The software will run on DOS or Windows platforms.

2. The PH Companion Website, designed to complement and expand upon the text, offers a variety of teaching and learning tools, including links to related websites, practice work for students, and the ability for instructors to monitor and evaluate students' work on the website. For more information, contact your local Prentice Hall representative: www.prenhall.com/barnett

3. CourseCompass/Blackboard/WebCT offers Course compatible content including Excel Projects, Quizzes, Chapter Destinations, Lecture Notes, and Graphing Calculator Help. CourseCompass is the perfect course management solution that combines quality Pearson Education content with state-of-the-art Blackboard technology! It is a dynamic, interactive online course management tool powered by Blackboard. This exciting product allows you to teach with market-leading Pearson Education content in an easy-to-use customizable format. Blackboard 5SM is a comprehensive and flexible e-Learning software platform that delivers a course management system, customizable institution-wide portals, online communities, and an advanced architecture that allows for Web-based integration of multiple administrative systems. WebCT is one of the most popular Web course platforms in higher education today. It is the first destination site for the higher education marketplace to offer both teaching and learning resources and a community of peers across course and institutional boundaries.

Supplements for the Instructor

For a summary of all available supplementary materials and detailed information regarding examination copy requests and orders, see page xix.

1. PH Custom Test, a menu-driven random test system for either Windows or Macintosh is available to instructors. The test system has been greatly expanded and now offers on-line testing. Carefully constructed algorithms use random-number generators to produce different, yet equivalent, versions of each of these problems. In addition, the system incorporates a unique editing function that allows the instructor to create additional problems, or alter any of the existing problems in the test, using a full set of mathematical notation. The test system offers free-response, multiple-choice, and mixed exams. An almost unlimited number of quizzes, review exercises, chapter tests, midterms, and final examinations, each different from the other, can be generated quickly and easily. At the same time, the system will produce answer keys, student worksheets, and a gradebook for the instructor, if desired.

2. A Test Item File, prepared by Laurel Technical Services, provides a hard copy of the test items available in PH Custom Test.

3. An Instructor's Solutions Manual provides detailed solutions to the problems not solved in the Student Solutions Manual. This manual is available to instructors without charge.

4. A Student Solutions Manual and Visual Calculus by Garret J. Etgen and David Schneider (see Supplements for the Student) is available to instructors.

5. The PH Companion Website, designed to complement and expand upon the text, offers a variety of interactive teaching and learning tools, including links to related websites, practice work for students, and the ability for instructors to monitor and evaluate students' work on the website. For more information, contact your local Prentice Hall representative: www.prenhall.com/barnett

6. CourseCompass/Blackboard/WebCT offers Course compatible content including Excel Projects, Quizzes, Chapter Destinations, Lecture Notes, and Graphing Calculator Help. CourseCompass is the perfect course management solution that combines quality Pearson Education content with state-of-the-art Blackboard technology! It is a dynamic, interactive online course management tool powered by Blackboard. This exciting product allows you to teach with market-leading Pearson Education content in an easy-to-use customizable format. Blackboard 5SM is a comprehensive and flexible e-Learning software platform that delivers a course management system, customizable institution-wide portals, online communities, and an advanced architecture that allows for Web-based integration of multiple administrative systems. WebCT is one of the most popular Web course platforms in higher education today. It is the first destination site for the higher education marketplace to offer both teaching and learning resources and a community of peers across course and institutional boundaries.

Error Check

Because of the careful checking and proofing by a number of mathematics instructors (acting independently), the authors and publisher believe this book to be substantially error-free. For any errors remaining, the authors would be grateful if they were sent to: Karl E. Byleen, 9322 W. Garden Court, Hales Corners, WI 53130; or, by e-mail, to: byleen@execpc.com

Acknowledgments

In addition to the authors, many others are involved in the successful publication of a book. We wish to thank the following reviewers of the eighth edition: Ann Pellegrino, The College of Charleston; Yesmine Akl, Widener University; Lorenzo Pitts, Jr., DeKalb Technical Institute; and, Martha Goshaw, Piedmont Virginia Community College.

We also wish to thank our colleagues who have provided input on previous editions: Chris Boldt, Bob Bradshaw, Celeste Carter, Bruce Chaffee, Robert Chaney, Dianne Clark, Charles E. Cleaver, Barbara Cohen, Richard L. Conlon, Catherine Cron, Madhu Deshpande, John Dickerson, Kenneth A. Dodaro, Michael W. Ecker, Jerry R. Ehman, Lucina Gallagher, Joel Haack, Martha M. Harvey, Sue Henderson, Lloyd R. Hicks, Louis F. Hoelzle, Paul Hutchins, K. Wayne James, Robert H. Johnston, Robert Krystock, James T. Loats, Frank Lopez, Roy H. Luke, Mel Mitchell, Michael Montano, Ronald Persky, Shala Peterman, Kenneth A. Peters, Jr., Tom Plavchak, Bob Prielipp, Stephen Rodi, Arthur Rosenthal, Sheldon Rothman, Elaine Russell, Daniel E. Scanlon, George R. Schriro, Arnold L. Schroeder, Hari Shanker, Larry Small, Joan Smith, Steven Terry, Delores A. Williams, Caroline Woods, Charles W. Zimmerman, and Pat Zrolka.

We also express our thanks to:

Hossein Hamedani, Carolyn Meitler, Stephen Merrill, Robert Mullins, and Caroline Woods for providing a careful and thorough check of all the mathematical calculations in the book, and to Priscilla Gathoni for checking the Student Solutions Manual, and the Instructor's Solutions Manual (a tedious but extremely important job).

Garret Etgen, Hossein Hamedani, Carolyn Meitler, and David Schneider for developing the supplemental manuals that are so important to the success of a text.

Jeanne Wallace for accurately and efficiently producing most of the manuals that supplement the text.

George Morris and his staff at Scientific Illustrators for their effective illustrations and accurate graphs.

All the people at Prentice Hall who contributed their efforts to the production of this book, especially Quincy McDonald, our acquisitions editor, and Lynn Savino Wendel, our Production Editor.

Producing this new edition with the help of all these extremely competent people has been a most satisfying experience.

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## More About This Textbook

## Overview

## Editorial Reviews

## Booknews

Emphasizing computational skills, ideas, and problem solving rather than theory, this new edition of a undergraduate level text uses problem examples from other fields to illustrate concepts regarding derivatives, integration, multivariable calculus, and trigonometric functions. Annotation c. by Book News, Inc., Portland, Or.## Product Details

## Related Subjects

## Meet the Author

Normal 0 false false false

Raymond A. Barnett, a native of California, received his B.A. in mathematical statistics from the University of California at Berkeley and his M.A. in mathematics from the University of Southern California. He has been a member of the Merritt College Mathematics Department, and was chairman of the department for four years. Raymond Barnett has authored or co-authored eighteen textbooks in mathematics, most of which are still in use. In addition to international English editions, a number of books have been translated into Spanish. Co-authors include Michael Ziegler, Marquette University; Thomas Kearns, Northern University; Charles Burke, City College of San Francisco; John Fuji, Merritt College; and Karl Byleen, Marquette University.

Michael R. Ziegler (late) received his B.S. from Shippensburg State College and his M.S. and Ph.D. from the University of Delaware. After completing post doctoral work at the University of Kentucky, he was appointed to the faculty of Marquette University where he held the rank of Professor in the Department of Mathematics, Statistics, and Computer Science. Dr. Ziegler published over a dozen research articles in complex analysis and co-authored eleven undergraduate mathematics textbooks with Raymond A. Barnett, and more recently, Karl E. Byleen.

Karl E. Byleen received his B.S., M.A. and Ph.D. degrees in mathematics from the University of Nebraska. He is currently an Associate Professor in the Department of Mathematics, Statistics and Computer Science of Marquette University. He has published a dozen research articles on the algebraic theory of semigroups.

## Read an Excerpt

## Preface

The ninth edition of Calculus for Business, Economics, Life Sciences, and Social Sciences is designed for a one-term course in calculus and for students who have had 1 - 2 years of high school algebra or the equivalent. The choice and independence of topics make the text readily adaptable to a variety of courses (see the Chapter Dependency Chart on page xi). It is one of five books in the authors' college mathematics series.

Improvements in this edition evolved out of the generous response from a large number of users of the last and previous editions as well as survey results from instructors, mathematics departments, course outlines, and college catalogs. Fundamental to a book's growth and effectiveness is classroom use and feedback. Now in its ninth edition, Calculus for Business, Economics, Life Sciences, and Social Sciences has had the benefit of having a substantial amount of both.

## Emphasis and Style

The text is written for student comprehension. Great care has been taken to write a book that is mathematically correct and accessible to students. Emphasis is on computational skills, ideas, and problem solving rather than mathematical theory. Most derivations and proofs are omitted except where their inclusion adds significant insight into a particular concept. General concepts and results are usually presented only after particular cases have been discussed.

## Examples and Matched Problems

Over 290 completely worked examples are used to introduce concepts and to demonstrate problem-solving techniques. Many examples have multiple parts, significantly increasing the total number of worked examples. Eachexample is followed by a similar matched problem for the student to work while reading the material. This actively involves the student in the learning process. The answers to these matched problems are included at the end of each section for easy reference.

## Exploration and Discussion

Every section contains Explore - Discuss problems interspersed at appropriate places to encourage the student to think about a relationship or process before a result is stated, or to investigate additional consequences of a development in the text. Verbalization of mathematical concepts, results, and processes is encouraged in these Explore - Discuss problems, as well as in some matched problems, and in some problems in almost every exercise set. The Explore - Discuss material also can be used as in-class or out-of-class group activities. In addition, at the end of every chapter, we have included two special chapter group activities that involve several of the concepts discussed in the chapter. Problems in the exercise sets that require verbalization are indicated by color problem numbers.

## Exercise Sets

The book contains over 4,100 problems. Many problems have multiple parts, significantly increasing the total number of problems. Each exercise set is designed so that an average or below-average student will experience success and a very capable student will be challenged. Exercise sets are mostly divided into A (routine, easy mechanics), B (more difficult mechanics), and C (difficult mechanics and some theory) levels.

## Applications

A major objective of this book is to give the student substantial experience in modeling and solving real-world problems. Enough applications are included to convince even the most skeptical student that mathematics is really useful (see the Applications Index inside the back cover). Worked examples involving applications are identified by or . Almost every exercise set contains application problems, usually divided into business and economics, life science, and social science groupings. An instructor with students from all three disciplines can let them choose applications from their own field of interest; if most students are from one of the three areas, then special emphasis can be placed there. Most of the applications are simplified versions of actual real-world problems taken from professional journals and books. No specialized experience is required to solve any of the applications.

## Internet Connections

The Internet provides a wealth of material that can be related to this book, from sources for the data in application problems to interactive exercises that provide additional insight into the various mathematical processes. Every section of the book contains Internet connections identified by . Links to the related web sites can be found at the PH Companion Website discussed later in this preface

## Technology

The generic term graphing utility is used to refer to any of the various graphing calculators or computer software packages that might be available to a student using this book. (See the description of the software accompanying this book later in this Preface.) Although access to a graphing utility is not assumed, it is likely that many students will want to make use of one of these devices. To assist these students, optional graphing utility activities are included in appropriate places in the book. These include brief discussions in the text, examples or portions of examples solved on a graphing utility, problems for the student to solve, and a group activity that involves the use of technology at the end of each chapter. Beginning with the group activity at the end of Chapter 1, and continuing throughout the text, linear regression on a graphing utility is used at appropriate points to illustrate mathematical modeling with real data. All the optional graphing utility material is clearly identified by either or and can be omitted without loss of continuity, if desired.

## Graphs

All graphs are computer-generated to ensure mathematical accuracy. Graphing utility screens displayed in the text are actual output from a graphing calculator.

## Additional Pedagogical Features

Annotation of examples and developments, in small color type, is found throughout the text to help students through critical stages (see Sections 1-1 and 3-2). Think boxes (dashed boxes) are used to enclose steps that are usually performed mentally (see Sections 1-1 and 3-4). Boxes are used to highlight important definitions, results, and step-by-step processes (see Sections 1-1 and 3-2). Caution statements appear throughout the text where student errors often occur (see Sections 3-2 and 4-1). Functional use of color improves the clarity of many illustrations, graphs, and developments, and guides students through certain critical steps (see Sections 1-1 and 3-2). Boldface type is used to introduce new terms and highlight important comments. Chapter review sections include a review of all important terms and symbols and a comprehensive review exercise. Answers to most review exercises, keyed to appropriate sections, are included in the back of the book. Answers to all other odd-numbered problems are also in the back of the book.

## Content

The text begins with the development of a library of elementary functions in Chapters 1 and 2, including their properties and uses. We encourage students to investigate mathematical ideas and processes graphically and numerically, as well as algebraically. This development lays a firm foundation for studying mathematics both in this book and in future endeavors. Depending on the syllabus for the course and the background of the students, some or all of this material can be covered at the beginning of a course, or selected portions can be referred to as needed later in the course.

The material in Part Two (Calculus) consists of differential calculus (Chapters 3 - 5), integral calculus (Chapters 6 - 7), multivariable calculus (Chapter 8), and a brief discussion of differentiation and integration of trigonometric functions (Chapter 9). In general, Chapters 3 - 6 must be covered in sequence; however, certain sections can be omitted or given brief treatments, as pointed out in the discussion that follows (see the Chapter Dependency Chart on page xi).

Chapter 3 introduces the derivative, covers the limit properties essential to understanding the definition of the derivative, develops the rules of differentiation (including the chain rule for power forms), and introduces applications of derivatives in business and economics. The interplay between graphical, numerical, and algebraic concepts is emphasized here and throughout the text.

Chapter 4 focuses on graphing and optimization. The first three sections cover continuity and first-derivative and second-derivative graph properties, while emphasizing polynomial graphing. Rational function graphing is covered in Section 4-4. In a course that does not include graphing rational functions, this section can be omitted or given a brief treatment. Optimization is covered in Section 4-5, including examples and problems involving end-point solutions.

The first three sections of Chapter 5 extend the derivative concepts discussed in Chapters 3 and 4 to exponential and logarithmic functions (including the general form of the chain rule). This material is required for all the remaining chapters. Implicit differentiation is introduced in Section 5-4 and applied to related rate problems in Section 5-5. These topics are not referred to elsewhere in the text and can be omitted.

Chapter 6 introduces integration. The first two sections cover antidifferentiation techniques essential to the remainder of the text. Section 6-3 discusses some applications involving differential equations that can be omitted. Sections 6-4 and 6-5 discuss the definite integral in terms of Riemann sums, including approximations with various types of sums and some simple error estimation. As before, the interplay between the graphical, numeric, and algebraic properties is emphasized. These two sections also are required for the remaining chapters in the text.

Chapter 7 covers additional integration topics and is organized to provide maximum flexibility for the instructor. The first section extends the area concepts introduced in Chapter 6 to the area between two curves and related applications. Section 7-2 covers three more applications of integration, and Sections 7-3 and 7-4 deal with additional techniques of integration. Any or all of the topics in Chapter 7 can be omitted.

The first five sections of Chapter 8 deal with differential multivariable calculus and can be covered any time after Section 5-3 has been completed. Section 8-6 requires the integration concepts discussed in Chapter 6.

Chapter 9 provides brief coverage of trigonometric functions that can be incorporated into the course, if desired. Section 9-1 provides a review of basic trigonometric concepts. Section 9-2 can be covered any time after Section 5-3 has been completed. Section 9-3 requires the material in Chapter 6.

Appendix A contains a self-test and a concise review of basic algebra that also may be covered as part of the course or referred to as needed. As mentioned above, Appendix B contains additional topics that can be covered in conjunction with certain sections in the text, if desired.

## Supplements for the Student

1. A Student Solutions Manual and Visual Calculus by Garret J. Etgen and David Schneider is available through your book store. The manual includes detailed solutions to all odd-numbered problems and all review exercises. Visual Calculus by David Schneider contains over twenty routines that provide additional insight into the topics discussed in the text. Although this software has much of the computing power of standard calculus software packages, it is primarily a teaching tool that focuses on understanding mathematical concepts, rather than on computing. These routines incorporate graphics whenever possible to illustrate topics such as secant lines; tangent lines; velocity; optimization; the relationship between the graphs of f, , ; and the various approaches to approximating definite integrals. All the routines in this software package are menu-driven and very easy to use. The software will run on DOS or Windows platforms.

2. The PH Companion Website, designed to complement and expand upon the text, offers a variety of teaching and learning tools, including links to related websites, practice work for students, and the ability for instructors to monitor and evaluate students' work on the website. For more information, contact your local Prentice Hall representative

3. CourseCompass/Blackboard/WebCT offers Course compatible content including Excel Projects, Quizzes, Chapter Destinations, Lecture Notes, and Graphing Calculator Help. CourseCompass is the perfect course management solution that combines quality Pearson Education content with state-of-the-art Blackboard technology! It is a dynamic, interactive online course management tool powered by Blackboard. This exciting product allows you to teach with market-leading Pearson Education content in an easy-to-use customizable format. Blackboard 5SM is a comprehensive and flexible e-Learning software platform that delivers a course management system, customizable institution-wide portals, online communities, and an advanced architecture that allows for Web-based integration of multiple administrative systems. WebCT is one of the most popular Web course platforms in higher education today. It is the first destination site for the higher education marketplace to offer both teaching and learning resources and a community of peers across course and institutional boundaries.

## Supplements for the Instructor

For a summary of all available supplementary materials and detailed information regarding examination copy requests and orders, see page xix.

1. PH Custom Test, a menu-driven random test system for either Windows or Macintosh is available to instructors. The test system has been greatly expanded and now offers on-line testing. Carefully constructed algorithms use random-number generators to produce different, yet equivalent, versions of each of these problems. In addition, the system incorporates a unique editing function that allows the instructor to create additional problems, or alter any of the existing problems in the test, using a full set of mathematical notation. The test system offers free-response, multiple-choice, and mixed exams. An almost unlimited number of quizzes, review exercises, chapter tests, midterms, and final examinations, each different from the other, can be generated quickly and easily. At the same time, the system will produce answer keys, student worksheets, and a gradebook for the instructor, if desired.

2. A Test Item File, prepared by Laurel Technical Services, provides a hard copy of the test items available in PH Custom Test.

3. An Instructor's Solutions Manual provides detailed solutions to the problems not solved in the Student Solutions Manual. This manual is available to instructors without charge.

4. A Student Solutions Manual and Visual Calculus by Garret J. Etgen and David Schneider (see Supplements for the Student) is available to instructors.

5. The PH Companion Website, designed to complement and expand upon the text, offers a variety of interactive teaching and learning tools, including links to related websites, practice work for students, and the ability for instructors to monitor and evaluate students' work on the website. For more information, contact your local Prentice Hall representative

6. CourseCompass/Blackboard/WebCT offers Course compatible content including Excel Projects, Quizzes, Chapter Destinations, Lecture Notes, and Graphing Calculator Help. CourseCompass is the perfect course management solution that combines quality Pearson Education content with state-of-the-art Blackboard technology! It is a dynamic, interactive online course management tool powered by Blackboard. This exciting product allows you to teach with market-leading Pearson Education content in an easy-to-use customizable format. Blackboard 5SM is a comprehensive and flexible e-Learning software platform that delivers a course management system, customizable institution-wide portals, online communities, and an advanced architecture that allows for Web-based integration of multiple administrative systems. WebCT is one of the most popular Web course platforms in higher education today. It is the first destination site for the higher education marketplace to offer both teaching and learning resources and a community of peers across course and institutional boundaries.

## Error Check

## Acknowledgments

In addition to the authors, many others are involved in the successful publication of a book. We wish to thank the following reviewers of the eighth edition: Ann Pellegrino, The College of Charleston; Yesmine Akl, Widener University; Lorenzo Pitts, Jr., DeKalb Technical Institute; and, Martha Goshaw, Piedmont Virginia Community College.

We also wish to thank our colleagues who have provided input on previous editions: Chris Boldt, Bob Bradshaw, Celeste Carter, Bruce Chaffee, Robert Chaney, Dianne Clark, Charles E. Cleaver, Barbara Cohen, Richard L. Conlon, Catherine Cron, Madhu Deshpande, John Dickerson, Kenneth A. Dodaro, Michael W. Ecker, Jerry R. Ehman, Lucina Gallagher, Joel Haack, Martha M. Harvey, Sue Henderson, Lloyd R. Hicks, Louis F. Hoelzle, Paul Hutchins, K. Wayne James, Robert H. Johnston, Robert Krystock, James T. Loats, Frank Lopez, Roy H. Luke, Mel Mitchell, Michael Montano, Ronald Persky, Shala Peterman, Kenneth A. Peters, Jr., Tom Plavchak, Bob Prielipp, Stephen Rodi, Arthur Rosenthal, Sheldon Rothman, Elaine Russell, Daniel E. Scanlon, George R. Schriro, Arnold L. Schroeder, Hari Shanker, Larry Small, Joan Smith, Steven Terry, Delores A. Williams, Caroline Woods, Charles W. Zimmerman, and Pat Zrolka.

We also express our thanks to:

Hossein Hamedani, Carolyn Meitler, Stephen Merrill, Robert Mullins, and Caroline Woods for providing a careful and thorough check of all the mathematical calculations in the book, and to Priscilla Gathoni for checking the Student Solutions Manual, and the Instructor's Solutions Manual (a tedious but extremely important job).

Garret Etgen, Hossein Hamedani, Carolyn Meitler, and David Schneider for developing the supplemental manuals that are so important to the success of a text.

Jeanne Wallace for accurately and efficiently producing most of the manuals that supplement the text.

George Morris and his staff at Scientific Illustrators for their effective illustrations and accurate graphs.

All the people at Prentice Hall who contributed their efforts to the production of this book, especially Quincy McDonald, our acquisitions editor, and Lynn Savino Wendel, our Production Editor.

Producing this new edition with the help of all these extremely competent people has been a most satisfying experience.

R. A. Barnett

M. R. Ziegler

K. E. Byleen

## Table of Contents

## Preface

## Preface

The ninth edition of Calculus for Business, Economics, Life Sciences, and Social Sciences is designed for a one-term course in calculus and for students who have had 1 - 2 years of high school algebra or the equivalent. The choice and independence of topics make the text readily adaptable to a variety of courses (see the Chapter Dependency Chart on page xi). It is one of five books in the authors' college mathematics series.

Improvements in this edition evolved out of the generous response from a large number of users of the last and previous editions as well as survey results from instructors, mathematics departments, course outlines, and college catalogs. Fundamental to a book's growth and effectiveness is classroom use and feedback. Now in its ninth edition, Calculus for Business, Economics, Life Sciences, and Social Sciences has had the benefit of having a substantial amount of both.

## Emphasis and Style

The text is written for student comprehension. Great care has been taken to write a book that is mathematically correct and accessible to students. Emphasis is on computational skills, ideas, and problem solving rather than mathematical theory. Most derivations and proofs are omitted except where their inclusion adds significant insight into a particular concept. General concepts and results are usually presented only after particular cases have been discussed.

## Examples and Matched Problems

Over 290 completely worked examples are used to introduce concepts and to demonstrate problem-solving techniques. Many examples have multiple parts, significantly increasing the total number of worked examples. Eachexample is followed by a similar matched problem for the student to work while reading the material. This actively involves the student in the learning process. The answers to these matched problems are included at the end of each section for easy reference.

## Exploration and Discussion

Every section contains Explore - Discuss problems interspersed at appropriate places to encourage the student to think about a relationship or process before a result is stated, or to investigate additional consequences of a development in the text. Verbalization of mathematical concepts, results, and processes is encouraged in these Explore - Discuss problems, as well as in some matched problems, and in some problems in almost every exercise set. The Explore - Discuss material also can be used as in-class or out-of-class group activities. In addition, at the end of every chapter, we have included two special chapter group activities that involve several of the concepts discussed in the chapter. Problems in the exercise sets that require verbalization are indicated by color problem numbers.

## Exercise Sets

The book contains over 4,100 problems. Many problems have multiple parts, significantly increasing the total number of problems. Each exercise set is designed so that an average or below-average student will experience success and a very capable student will be challenged. Exercise sets are mostly divided into A (routine, easy mechanics), B (more difficult mechanics), and C (difficult mechanics and some theory) levels.

## Applications

A major objective of this book is to give the student substantial experience in modeling and solving real-world problems. Enough applications are included to convince even the most skeptical student that mathematics is really useful (see the Applications Index inside the back cover). Worked examples involving applications are identified by or . Almost every exercise set contains application problems, usually divided into business and economics, life science, and social science groupings. An instructor with students from all three disciplines can let them choose applications from their own field of interest; if most students are from one of the three areas, then special emphasis can be placed there. Most of the applications are simplified versions of actual real-world problems taken from professional journals and books. No specialized experience is required to solve any of the applications.

## Internet Connections

The Internet provides a wealth of material that can be related to this book, from sources for the data in application problems to interactive exercises that provide additional insight into the various mathematical processes. Every section of the book contains Internet connections identified by . Links to the related web sites can be found at the PH Companion Website discussed later in this preface: www.prenhall.com/barnett

## Technology

The generic term graphing utility is used to refer to any of the various graphing calculators or computer software packages that might be available to a student using this book. (See the description of the software accompanying this book later in this Preface.) Although access to a graphing utility is not assumed, it is likely that many students will want to make use of one of these devices. To assist these students, optional graphing utility activities are included in appropriate places in the book. These include brief discussions in the text, examples or portions of examples solved on a graphing utility, problems for the student to solve, and a group activity that involves the use of technology at the end of each chapter. Beginning with the group activity at the end of Chapter 1, and continuing throughout the text, linear regression on a graphing utility is used at appropriate points to illustrate mathematical modeling with real data. All the optional graphing utility material is clearly identified by either or and can be omitted without loss of continuity, if desired.

## Graphs

All graphs are computer-generated to ensure mathematical accuracy. Graphing utility screens displayed in the text are actual output from a graphing calculator.

## Additional Pedagogical Features

Annotation of examples and developments, in small color type, is found throughout the text to help students through critical stages (see Sections 1-1 and 3-2). Think boxes (dashed boxes) are used to enclose steps that are usually performed mentally (see Sections 1-1 and 3-4). Boxes are used to highlight important definitions, results, and step-by-step processes (see Sections 1-1 and 3-2). Caution statements appear throughout the text where student errors often occur (see Sections 3-2 and 4-1). Functional use of color improves the clarity of many illustrations, graphs, and developments, and guides students through certain critical steps (see Sections 1-1 and 3-2). Boldface type is used to introduce new terms and highlight important comments. Chapter review sections include a review of all important terms and symbols and a comprehensive review exercise. Answers to most review exercises, keyed to appropriate sections, are included in the back of the book. Answers to all other odd-numbered problems are also in the back of the book.

## Content

The text begins with the development of a library of elementary functions in Chapters 1 and 2, including their properties and uses. We encourage students to investigate mathematical ideas and processes graphically and numerically, as well as algebraically. This development lays a firm foundation for studying mathematics both in this book and in future endeavors. Depending on the syllabus for the course and the background of the students, some or all of this material can be covered at the beginning of a course, or selected portions can be referred to as needed later in the course.

The material in Part Two (Calculus) consists of differential calculus (Chapters 3 - 5), integral calculus (Chapters 6 - 7), multivariable calculus (Chapter 8), and a brief discussion of differentiation and integration of trigonometric functions (Chapter 9). In general, Chapters 3 - 6 must be covered in sequence; however, certain sections can be omitted or given brief treatments, as pointed out in the discussion that follows (see the Chapter Dependency Chart on page xi).

Chapter 3 introduces the derivative, covers the limit properties essential to understanding the definition of the derivative, develops the rules of differentiation (including the chain rule for power forms), and introduces applications of derivatives in business and economics. The interplay between graphical, numerical, and algebraic concepts is emphasized here and throughout the text.

Chapter 4 focuses on graphing and optimization. The first three sections cover continuity and first-derivative and second-derivative graph properties, while emphasizing polynomial graphing. Rational function graphing is covered in Section 4-4. In a course that does not include graphing rational functions, this section can be omitted or given a brief treatment. Optimization is covered in Section 4-5, including examples and problems involving end-point solutions.

The first three sections of Chapter 5 extend the derivative concepts discussed in Chapters 3 and 4 to exponential and logarithmic functions (including the general form of the chain rule). This material is required for all the remaining chapters. Implicit differentiation is introduced in Section 5-4 and applied to related rate problems in Section 5-5. These topics are not referred to elsewhere in the text and can be omitted.

Chapter 6 introduces integration. The first two sections cover antidifferentiation techniques essential to the remainder of the text. Section 6-3 discusses some applications involving differential equations that can be omitted. Sections 6-4 and 6-5 discuss the definite integral in terms of Riemann sums, including approximations with various types of sums and some simple error estimation. As before, the interplay between the graphical, numeric, and algebraic properties is emphasized. These two sections also are required for the remaining chapters in the text.

Chapter 7 covers additional integration topics and is organized to provide maximum flexibility for the instructor. The first section extends the area concepts introduced in Chapter 6 to the area between two curves and related applications. Section 7-2 covers three more applications of integration, and Sections 7-3 and 7-4 deal with additional techniques of integration. Any or all of the topics in Chapter 7 can be omitted.

The first five sections of Chapter 8 deal with differential multivariable calculus and can be covered any time after Section 5-3 has been completed. Section 8-6 requires the integration concepts discussed in Chapter 6.

Chapter 9 provides brief coverage of trigonometric functions that can be incorporated into the course, if desired. Section 9-1 provides a review of basic trigonometric concepts. Section 9-2 can be covered any time after Section 5-3 has been completed. Section 9-3 requires the material in Chapter 6.

Appendix A contains a self-test and a concise review of basic algebra that also may be covered as part of the course or referred to as needed. As mentioned above, Appendix B contains additional topics that can be covered in conjunction with certain sections in the text, if desired.

## Supplements for the Student

1. A Student Solutions Manual and Visual Calculus by Garret J. Etgen and David Schneider is available through your book store. The manual includes detailed solutions to all odd-numbered problems and all review exercises. Visual Calculus by David Schneider contains over twenty routines that provide additional insight into the topics discussed in the text. Although this software has much of the computing power of standard calculus software packages, it is primarily a teaching tool that focuses on understanding mathematical concepts, rather than on computing. These routines incorporate graphics whenever possible to illustrate topics such as secant lines; tangent lines; velocity; optimization; the relationship between the graphs of f, , ; and the various approaches to approximating definite integrals. All the routines in this software package are menu-driven and very easy to use. The software will run on DOS or Windows platforms.

2. The PH Companion Website, designed to complement and expand upon the text, offers a variety of teaching and learning tools, including links to related websites, practice work for students, and the ability for instructors to monitor and evaluate students' work on the website. For more information, contact your local Prentice Hall representative: www.prenhall.com/barnett

3. CourseCompass/Blackboard/WebCT offers Course compatible content including Excel Projects, Quizzes, Chapter Destinations, Lecture Notes, and Graphing Calculator Help. CourseCompass is the perfect course management solution that combines quality Pearson Education content with state-of-the-art Blackboard technology! It is a dynamic, interactive online course management tool powered by Blackboard. This exciting product allows you to teach with market-leading Pearson Education content in an easy-to-use customizable format. Blackboard 5SM is a comprehensive and flexible e-Learning software platform that delivers a course management system, customizable institution-wide portals, online communities, and an advanced architecture that allows for Web-based integration of multiple administrative systems. WebCT is one of the most popular Web course platforms in higher education today. It is the first destination site for the higher education marketplace to offer both teaching and learning resources and a community of peers across course and institutional boundaries.

## Supplements for the Instructor

For a summary of all available supplementary materials and detailed information regarding examination copy requests and orders, see page xix.

1. PH Custom Test, a menu-driven random test system for either Windows or Macintosh is available to instructors. The test system has been greatly expanded and now offers on-line testing. Carefully constructed algorithms use random-number generators to produce different, yet equivalent, versions of each of these problems. In addition, the system incorporates a unique editing function that allows the instructor to create additional problems, or alter any of the existing problems in the test, using a full set of mathematical notation. The test system offers free-response, multiple-choice, and mixed exams. An almost unlimited number of quizzes, review exercises, chapter tests, midterms, and final examinations, each different from the other, can be generated quickly and easily. At the same time, the system will produce answer keys, student worksheets, and a gradebook for the instructor, if desired.

2. A Test Item File, prepared by Laurel Technical Services, provides a hard copy of the test items available in PH Custom Test.

3. An Instructor's Solutions Manual provides detailed solutions to the problems not solved in the Student Solutions Manual. This manual is available to instructors without charge.

4. A Student Solutions Manual and Visual Calculus by Garret J. Etgen and David Schneider (see Supplements for the Student) is available to instructors.

5. The PH Companion Website, designed to complement and expand upon the text, offers a variety of interactive teaching and learning tools, including links to related websites, practice work for students, and the ability for instructors to monitor and evaluate students' work on the website. For more information, contact your local Prentice Hall representative: www.prenhall.com/barnett

6. CourseCompass/Blackboard/WebCT offers Course compatible content including Excel Projects, Quizzes, Chapter Destinations, Lecture Notes, and Graphing Calculator Help. CourseCompass is the perfect course management solution that combines quality Pearson Education content with state-of-the-art Blackboard technology! It is a dynamic, interactive online course management tool powered by Blackboard. This exciting product allows you to teach with market-leading Pearson Education content in an easy-to-use customizable format. Blackboard 5SM is a comprehensive and flexible e-Learning software platform that delivers a course management system, customizable institution-wide portals, online communities, and an advanced architecture that allows for Web-based integration of multiple administrative systems. WebCT is one of the most popular Web course platforms in higher education today. It is the first destination site for the higher education marketplace to offer both teaching and learning resources and a community of peers across course and institutional boundaries.

## Error Check

Because of the careful checking and proofing by a number of mathematics instructors (acting independently), the authors and publisher believe this book to be substantially error-free. For any errors remaining, the authors would be grateful if they were sent to: Karl E. Byleen, 9322 W. Garden Court, Hales Corners, WI 53130; or, by e-mail, to: byleen@execpc.com

## Acknowledgments

In addition to the authors, many others are involved in the successful publication of a book. We wish to thank the following reviewers of the eighth edition: Ann Pellegrino, The College of Charleston; Yesmine Akl, Widener University; Lorenzo Pitts, Jr., DeKalb Technical Institute; and, Martha Goshaw, Piedmont Virginia Community College.

We also wish to thank our colleagues who have provided input on previous editions: Chris Boldt, Bob Bradshaw, Celeste Carter, Bruce Chaffee, Robert Chaney, Dianne Clark, Charles E. Cleaver, Barbara Cohen, Richard L. Conlon, Catherine Cron, Madhu Deshpande, John Dickerson, Kenneth A. Dodaro, Michael W. Ecker, Jerry R. Ehman, Lucina Gallagher, Joel Haack, Martha M. Harvey, Sue Henderson, Lloyd R. Hicks, Louis F. Hoelzle, Paul Hutchins, K. Wayne James, Robert H. Johnston, Robert Krystock, James T. Loats, Frank Lopez, Roy H. Luke, Mel Mitchell, Michael Montano, Ronald Persky, Shala Peterman, Kenneth A. Peters, Jr., Tom Plavchak, Bob Prielipp, Stephen Rodi, Arthur Rosenthal, Sheldon Rothman, Elaine Russell, Daniel E. Scanlon, George R. Schriro, Arnold L. Schroeder, Hari Shanker, Larry Small, Joan Smith, Steven Terry, Delores A. Williams, Caroline Woods, Charles W. Zimmerman, and Pat Zrolka.

We also express our thanks to:

Hossein Hamedani, Carolyn Meitler, Stephen Merrill, Robert Mullins, and Caroline Woods for providing a careful and thorough check of all the mathematical calculations in the book, and to Priscilla Gathoni for checking the Student Solutions Manual, and the Instructor's Solutions Manual (a tedious but extremely important job).

Garret Etgen, Hossein Hamedani, Carolyn Meitler, and David Schneider for developing the supplemental manuals that are so important to the success of a text.

Jeanne Wallace for accurately and efficiently producing most of the manuals that supplement the text.

George Morris and his staff at Scientific Illustrators for their effective illustrations and accurate graphs.

All the people at Prentice Hall who contributed their efforts to the production of this book, especially Quincy McDonald, our acquisitions editor, and Lynn Savino Wendel, our Production Editor.

Producing this new edition with the help of all these extremely competent people has been a most satisfying experience.

R. A. Barnett

M. R. Ziegler

K. E. Byleen