Brief Calculus & Its Applications / Edition 12

by Larry J. Goldstein, David I. Schneider, David C. Lay, Nakhle H. Asmar

Overview

This extremely readable, highly regarded, and widely adopted text present innovative ways for applying calculus to real-world situations in the business, economics, life science, and social science disciplines. The text's straightforward, engaging approach fosters the growth of both mathematical maturity and an appreciation for the usefulness of mathematics. The authors' tried and true formula — pairing substantial amounts of graphical analysis and informal geometric proofs with an abundance of hands-on exercises — has proven to be tremendously successful. Functions, derivatives, applications of the derivative, techniques of differentiations, exponential and natural logarithm functions, definite integral, variables and trigonometric functions. For individuals interested in a brief introduction to calculus applications.

Product Details

ISBN-13:	9780321568564
Publisher:	Pearson
Publication date:	03/17/2009
Edition description:	Older Edition
Pages:	512
Product dimensions:	7.90(w) x 9.80(h) x 1.20(d)

About the Author

Larry Goldstein has received several distinguished teaching awards, given more than fifty Conference and Colloquium talks & addresses, and written more than fifty books in math and computer programming. He received his PhD at Princeton and his BA and MA at the University of Pennsylvania. He also teaches part time at Drexel University.

David Schneider, who is known widely for his tutorial software, holds a BA degree from Oberlin College and a PhD from MIT. He is currently an associate professor of mathematics at the University of Maryland. He has authored eight widely used math texts, fourteen highly acclaimed computer books, and three widely used mathematics software packages. He has also produced instructional videotapes at both the University of Maryland and the BBC.

Martha Siegel holds a BA from Russell Sage College, attended Rensselear Polytechnic Institute as a special student, and received his PhD at the University of Rochester. From 1966 until 1971 she taught at Goucher University in Baltimore. Since 1971 she has been a professor at Towson State University, also in Maryland. Professor Siegel has been on the writing team of this book since the fifth edition and is also the co-author of a precalculus reform book.

Read an Excerpt

PREFACE:

Preface

We have been very pleased with the enthusiastic response to the first seven editions of Brief Calculus and Its Applications by teachers and students alike. The present work incorporates many of the suggestions they have put forward.

Although there are many changes, we have preserved the approach and the flavor. Our goals remain the same: to begin the calculus as soon as possible; to present calculus in an intuitive yet intellectually satisfying way; and to illustrate the many applications of calculus to the biological, social, and management sciences.

The distinctive order of topics has proven over the years to be successful—easier for students to learn, and more interesting because students see significant applications early. For instance, the derivative is explained geometrically before the analytic material on limits is presented. This approach gives the students an understanding of the derivative at least as strong as that obtained from the traditional approach. To reach the applications in Chapter 2 quickly, we present only the differentiation rules and the curve sketching needed for those applications. Advanced topics come later when they are needed. Other aspects of this student-oriented approach follow below.

Applications

We provide realistic applications that illustrate the uses of calculus in other disciplines. See the Index of Applications on the inside cover. Wherever possible, we have attempted to use applications to motivate the mathematics.

Examples

The text includes many more worked examples than is customary. Furthermore, we have included computational details toenhance readability by students whose basic skills are weak.

Exercises

The exercises comprise about one-quarter of the text-the most important part of the text in our opinion. The exercises at the ends of the sections are usually arranged in the order in which the text proceeds, so that the homework assignments may easily be made after only part of a section is discussed. Interesting applications and more challenging problems tend to be located near the ends of the exercise sets. Supplementary exercises at the end of each chapter expand the other exercise sets and include problems that require skills from earlier chapters.

Practice Problems

The practice problems have proven to be a popular and useful feature. Practice Problems are carefully selected questions located at the end of each section, just before the exercise set. Complete solutions are given following the exercise set. The practice problems often focus on points that are potentially confusing or are likely to be overlooked. We recommend that the reader seriously attempt the practice problems and study their solutions before moving on to the exercises. In effect, the practice problems constitute a built-in workbook.

Minimal Prerequisites

In Chapter 0, we review those concepts that the reader needs to study calculus. Some important topics, such as the laws of exponents, are reviewed again when they are used in a later chapter. Section 0.6 prepares students for applied problems that appear throughout the text. A reader familiar with the content of Chapter 0 should begin with Chapter 1 and use Chapter 0 as a reference, whenever needed.

New in this Edition

Among the many changes in this edition, the following are the most significant:

1. Delta Notation We introduce delta notation in Chapter 0 and use it in our discussion of the derivative. As in previous editions, we have tried to minimize the use of complicated notation, preferring instead verbal descriptions. However, in the case of the delta notation, we feel that the clarity achieved is worth the extra notation.
2. Derivative as a Rate of Change We preview the derivative as a rate of change at the beginning of Chapter 1, anticipating the more detailed discussion in Section 1.8. Since students have difficulty interpreting the derivative as a rate of change, we felt it prudent to allow them to practice repeatedly with the concept.
3. Analysis of Data We added a broad theme that might best be described as "calculus for functions defined by data." Throughout the book, we include discussions about real-life applications whose underlying functions are defined by tables of data.More on Regression (optional) We added the optional Section 7.6 on multiple and nonlinear regression analysis. The goal in this section is to provide a taste of what a business student will encounter in a course in regression analysis. Our emphasis is on using technology, especially spreadsheets, to do the computations for various flavors of regression (multiple-linear, quadratic, exponential, etc.).
4. Additional Technology (optional) The new technology appendix to Chapter 0 includes the graphing calculator material previously found within the chapter, a discussion of calculus and spreadsheets, and a new exercise set testing student technology skills.
5. Real-Life Data We have collected spreadsheets containing real-life statistical data and made them available to students and faculty on the Web site www.prenhall.com/goldstein.
6. Projects Each chapter now includes a project, designed to provide more open-ended problem solving, critical thinking, verbal expression, and integration of mathematical techniques, both manual and technological.
7. Other Changes We made improvements throughout the text based on suggestions from students, teachers, reviewers, and editors. Our thanks to all who assisted us with their valuable suggestions.

This edition contains more material than can be covered in most one-semester courses. Optional sections are starred in the table of contents. In addition, the level of theoretical material may be adjusted to the needs of the students. For instance, only the first two pages of Section 1.4 are required in order to introduce the limit notation.

A Study Guide for students containing detailed explanations and solutions for every sixth exercise is available. The Study Guide also includes helpful hints and strategies for studying that will help students improve their performance in the course. In addition, the Study Guide contains a copy of Visual Calculus, the popular, easy-to-use software for IBM compatible computers. Visual Calculus contains over 20 routines that provide additional insights into the topics discussed in the text. Also, instructors find the software valuable for constructing graphs for exams.

An Instructor's Solutions Manual contains worked solutions to every exercise.

TestGen EQ provides nearly 1000 suggested test questions, keyed to chapter and section. TestGen EQ is a text-specific testing program networkable for administering tests and capturing grades online. Edit and add your own questions, or use the new "Function Plotter" to create a nearly unlimited number of tests and drill worksheets.

Designed to complement and expand upon the text, the text Web site offers a variety of interactive teaching and learning tools. Since many of the text projects use real-life data, we made the data easier to use by making it available in Excel spreadsheets on the Web site. The Web site also includes links to related Web sites, quizzes, Syllabus Builder, and more. For more information, visit www.prenhall.com/goldstein or contact your local Prentice Hall representative.

Table of Contents

0. Functions.

Functions and Their Graphs. Some Important Functions. The Algebra of Functions. Zeros of Functions — The Quadratic Formula and Factoring. Exponents and Power Functions. Functions and Graphs in Applications. Appendix: Graphing Functions Using Technology.

1. The Derivative.

The Slope of a Straight Line. The Slope of a Curve at a Point. The Derivative. Limits and the Derivative. Differentiability and Continuity. Some Rules for Differentiation. More About Derivatives. The Derivative as a Rate of Change.

2. Applications of the Derivative.

Describing Graphs of Functions. The First and Second Derivative Rules. Curve Sketching (Introduction.) Curve Sketching (Conclusion.) Optimization Problems. Further Optimization Problems. Applications of Derivatives to Business and Economics.

3. Techniques of Differentiation.

The Product and Quotient Rules. The Chain Rule and the General Power Rule. Implicit Differentiation and Related Rates.

4. The Exponential and Natural Logarithm Functions.

Exponential Functions. The Exponential Function e^x. Differentiation of Exponential Functions. The Natural Logarithm Function. The Derivative of ln x. Properties of the Natural Logarithm Function.

5. Applications of the Exponential and Natural Logarithm Functions.

Exponential Growth and Decay. Compound Interest.Applications of the Natural Logarithm Function to Economics. Further Exponential Models.

6. The Definite Integral.

Antidifferentiation. Areas and Reimann Sums. Definite Integrals and the Fundamental Theorem. Areas in the xy-Plane. Applications of the Definite Integral.

7. Functions of Several Variables.

Examples of Functions of Several Variables. Partial Derivatives. Maxima and Minima of Functions of Several Variables. Lagrange Multipliers and Constrained Optimization. The Method of Least Squares. Nonlinear Regression. Double Integrals.

8. The Trigonometric Functions.

Radian Measure of Angles. The Sine and the Cosine. Differentiation of sin t and cos t. The Tangent and Other Trigonometric Functions.

Preface

Preface

We have been very pleased with the enthusiastic response to the first seven editions of Brief Calculus and Its Applications by teachers and students alike. The present work incorporates many of the suggestions they have put forward.

Although there are many changes, we have preserved the approach and the flavor. Our goals remain the same: to begin the calculus as soon as possible; to present calculus in an intuitive yet intellectually satisfying way; and to illustrate the many applications of calculus to the biological, social, and management sciences.

The distinctive order of topics has proven over the years to be successful—easier for students to learn, and more interesting because students see significant applications early. For instance, the derivative is explained geometrically before the analytic material on limits is presented. This approach gives the students an understanding of the derivative at least as strong as that obtained from the traditional approach. To reach the applications in Chapter 2 quickly, we present only the differentiation rules and the curve sketching needed for those applications. Advanced topics come later when they are needed. Other aspects of this student-oriented approach follow below.

Applications

We provide realistic applications that illustrate the uses of calculus in other disciplines. See the Index of Applications on the inside cover. Wherever possible, we have attempted to use applications to motivate the mathematics.

Examples

The text includes many more worked examples than is customary. Furthermore, we have included computational details to enhancereadability by students whose basic skills are weak.

Exercises

The exercises comprise about one-quarter of the text-the most important part of the text in our opinion. The exercises at the ends of the sections are usually arranged in the order in which the text proceeds, so that the homework assignments may easily be made after only part of a section is discussed. Interesting applications and more challenging problems tend to be located near the ends of the exercise sets. Supplementary exercises at the end of each chapter expand the other exercise sets and include problems that require skills from earlier chapters.

Practice Problems

The practice problems have proven to be a popular and useful feature. Practice Problems are carefully selected questions located at the end of each section, just before the exercise set. Complete solutions are given following the exercise set. The practice problems often focus on points that are potentially confusing or are likely to be overlooked. We recommend that the reader seriously attempt the practice problems and study their solutions before moving on to the exercises. In effect, the practice problems constitute a built-in workbook.

Minimal Prerequisites

In Chapter 0, we review those concepts that the reader needs to study calculus. Some important topics, such as the laws of exponents, are reviewed again when they are used in a later chapter. Section 0.6 prepares students for applied problems that appear throughout the text. A reader familiar with the content of Chapter 0 should begin with Chapter 1 and use Chapter 0 as a reference, whenever needed.

New in this Edition

Among the many changes in this edition, the following are the most significant:

1. Delta Notation We introduce delta notation in Chapter 0 and use it in our discussion of the derivative. As in previous editions, we have tried to minimize the use of complicated notation, preferring instead verbal descriptions. However, in the case of the delta notation, we feel that the clarity achieved is worth the extra notation.
2. Derivative as a Rate of Change We preview the derivative as a rate of change at the beginning of Chapter 1, anticipating the more detailed discussion in Section 1.8. Since students have difficulty interpreting the derivative as a rate of change, we felt it prudent to allow them to practice repeatedly with the concept.
3. Analysis of Data We added a broad theme that might best be described as "calculus for functions defined by data." Throughout the book, we include discussions about real-life applications whose underlying functions are defined by tables of data.More on Regression (optional) We added the optional Section 7.6 on multiple and nonlinear regression analysis. The goal in this section is to provide a taste of what a business student will encounter in a course in regression analysis. Our emphasis is on using technology, especially spreadsheets, to do the computations for various flavors of regression (multiple-linear, quadratic, exponential, etc.).
4. Additional Technology (optional) The new technology appendix to Chapter 0 includes the graphing calculator material previously found within the chapter, a discussion of calculus and spreadsheets, and a new exercise set testing student technology skills.
5. Real-Life Data We have collected spreadsheets containing real-life statistical data and made them available to students and faculty on the Web site www.prenhall.com/goldstein.
6. Projects Each chapter now includes a project, designed to provide more open-ended problem solving, critical thinking, verbal expression, and integration of mathematical techniques, both manual and technological.
7. Other Changes We made improvements throughout the text based on suggestions from students, teachers, reviewers, and editors. Our thanks to all who assisted us with their valuable suggestions.

This edition contains more material than can be covered in most one-semester courses. Optional sections are starred in the table of contents. In addition, the level of theoretical material may be adjusted to the needs of the students. For instance, only the first two pages of Section 1.4 are required in order to introduce the limit notation.

A Study Guide for students containing detailed explanations and solutions for every sixth exercise is available. The Study Guide also includes helpful hints and strategies for studying that will help students improve their performance in the course. In addition, the Study Guide contains a copy of Visual Calculus, the popular, easy-to-use software for IBM compatible computers. Visual Calculus contains over 20 routines that provide additional insights into the topics discussed in the text. Also, instructors find the software valuable for constructing graphs for exams.

An Instructor's Solutions Manual contains worked solutions to every exercise.

TestGen EQ provides nearly 1000 suggested test questions, keyed to chapter and section. TestGen EQ is a text-specific testing program networkable for administering tests and capturing grades online. Edit and add your own questions, or use the new "Function Plotter" to create a nearly unlimited number of tests and drill worksheets.

Designed to complement and expand upon the text, the text Web site offers a variety of interactive teaching and learning tools. Since many of the text projects use real-life data, we made the data easier to use by making it available in Excel spreadsheets on the Web site. The Web site also includes links to related Web sites, quizzes, Syllabus Builder, and more. For more information, visit www.prenhall.com/goldstein or contact your local Prentice Hall representative.