The widespread availability of graphing calculators and computers is revolutionizing mathematics education. Each year, more and more students enter college mathematics courses with a wide variety of technological experiences and an expectation of using and building on these experiences as they continue their mathematical education. Because of this trend, we have written this book under the assumption that each student has access to an electronic device capable of displaying graphs. (We will refer to any such device as a graphing utility—see the sections titled "Technology" and "To the Student" later in this Preface.) However, we do not assume any previous experience with a graphing utility. Instead, we carefully demonstrate through discussions and examples how a graphing utility often gives additional insight into mathematical concepts and processes. (Specific instructions for a number of popular graphing calculators are included in a supplement—see the section titled "Supplements for the Student" later in this Preface.) While we assume that the student will attack appropriate problems with a graphing utility at hand and we present problem-solving techniques that will take advantage of this technology, we also include many activities that promote understanding of concepts independent of the use of any technology. This book is designed for either one- or two-term calculus courses that emphasize the topics most useful to students in business, economics, life sciences, and social sciences, and for students who have had 1 1/2–2 years of high school algebra, or the equivalent.
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. General results are usually presented only after particular cases have been discussed.
EXAMPLES AND MATCHED PROBLEMS
Over 240 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. Each example 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 boxes 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 boxes, 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 (before the chapter review), we have included a special chapter group activity that involves several of the concepts discussed in the chapter. All these special activities are highlighted with color shading to emphasize their importance.
EXERCISE SETS
The book contains over 3,400 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 divided into A (routine, easy mechanics), B (more difficult mechanics), and C (difficult mechanics and some theory) levels. Most exercise sets contain some problems that require a graphing utility for their solution and others that could be solved either with or without a graphing utility. Since a student must learn when to use a graphing utility, as well as how to use one, the text generally does not identify problems that must be solved with the aid of a graphing utility.
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 
. 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, and many involve real data. No specialized experience is required to solve any of the applications.
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. We assume that each student has access to a graphing utility that can perform the following operations:
- Simultaneously display several graphs in a user-selected viewing window
- Explore graphs using zoom and trace
- Approximate roots
- Approximate intersection points
- Approximate values of the derivative of a function
- Approximate maxima and minima
- Approximate definite integrals
- Plot data sets and find associated regression equations
Most popular graphing calculators support all of these operations, as doesVisual Calculus, the software supplement for this text described later in this Preface.
GRAPHS
All graphs are computer-generated to ensure mathematical accuracy. Graphing utility screens displayed in the text are actual output from a graphing calculator—in most cases, the Texas Instruments TI-82 graphics calculator.
STUDENT AIDS
Annotation of examples and developments, in small color type, is found throughout the text to help students through critical stages (see Sections 1-2 and 3-2). Think boxes (dashed boxes) are used to enclose steps that are usually performed mentally (see Sections 1-2 and 3-4). Boxes are used to highlight important definitions, theorems, results, and step-by-step processes (see Sections 1-2 and 1-4). 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 1-2 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 an introduction to graphing utilities and the development of a library of elementary functions in Chapters 1 and 2. The development focuses on a discussion of the properties of these functions, related applications, and graphing utility techniques, including zoom and trace, root approximation, intersection points, and regression techniques for data analysis. Students are encouraged to investigate mathematical ideas and processes graphically and numerically, as well as algebraically. This development substantially enriches the study of mathematics. 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 and 7), multivariable calculus (Chapter 8), and differential equations (Chapter 9), and a brief discussion of differentiation and integration of trigonometric functions (Chapter 10). 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 Dependencies chart on page ix).
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), discusses numerical differentiation on a graphing utility, 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 functions. Analysis of graphs of rational functions is covered in Section 4-4. In a course that does not emphasize graphical analysis of 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 without loss of continuity.
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, simple error estimation, and numerical integration on a graphing utility. As before, the interplay between the graphical, numerical, and algebraic properties is emphasized. These two sections 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 8-5 applies multivariable optimization techniques to linear regression, providing a foundation for the regression techniques introduced in Section 2-4. Section 8-6 requires the integration concepts discussed in Chapter 6.
After introducing the basic concepts and terminology used in the study of differential equations in Section 9-1, separable differential equations (Section 9-2) and first-order linear differential equations (Section 9-3) and related applications are covered thoroughly. All the growth laws introduced in Section 6-3 are covered again, this time with more emphasis on student recognition of the relevant growth laws.
Chapter 10 provides brief coverage of trigonometric functions that can be incorporated into the course, if desired. Section 10-1 provides a brief review of basic trigonometric concepts. Section 10-2 can be covered any time after Section 5-3 has been completed. Section 10-3 requires the material in Chapter 6.
TO THE STUDENT
A graphing utility is a powerful tool for analyzing graphs, but like any good tool, it must be used properly. The display on most graphing utilities produces only rough approximations to graphs. One of the skills you must develop as you study the material in this book is the ability to visualize the correct appearance of a graph, based on the rough graph produced by a graphing utility and your understanding of the mathematics under discussion. For example, consider Figures 1A and 1B (taken from page 95).
FIGURE 1
The graphing utility graph in Figure 1A seems to indicate that the graph is one continuous curve. However, the more accurate graph in Figure 1B clearly shows that the graph consists of three separate curves. When you study this function in the text, you will learn that f
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