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 and performing certain matrix operations. (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 a two-term course in finite mathematics and calculus that emphasizes 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 350 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 4,700 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 Perform basic matrix operations Most popular graphing calculators support all of these operations, as do Explorations in Finite Mathematics and Visual Calculus, the software supplements 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 1-1 and 4-2). Think boxes (dashed boxes) are used to enclose steps that are usually performed mentally (see Sections 1-2 and 4-1). 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 4-3 and 4-5). 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 4-2). 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 (Finite Mathematics) can be thought of as four units: mathematics of finance (Chapter 3); linear algebra, including matrices, linear systems, and linear programming (Chapters 4 and 5); probability and statistics (Chapter 6); and applications of linear algebra and probability to Markov chains (Chapter 7). The first three units are independent of each other, while the last chapter is dependent on some of the earlier chapters (see the Chapter Dependencies chart preceding this Preface). Chapter 3 presents a thorough treatment of simple and compound interest and present and future value of ordinary annuities. Chapter 4 covers linear systems and matrices with an emphasis on using Gauss–Jordan elimination to solve systems. The row operations discussed in Sections 4-2 and 4-3 are required for the simplex method in Chapter 5. Matrix multiplication, matrix inverses, and systems of equations are required for Markov chains in Chapter 7. Chapter 5 provides broad and flexible coverage of linear programming. The first two sections cover two-variable graphing techniques. Instructors who wish to emphasize techniques can cover the basic simplex method in Sections 5-3 and 5-4 and then discuss any or all of the following: the dual method (Section 5-5), the big M method (Section 5-6), or the two-phase simplex method (Chapter 5 Group Activity). Those who want to emphasize modeling can discuss the formation of the mathematical model for any of the application examples in Sections 5-4, 5-5, and 5-6, and either omit the solution or use software to find the solution (see the description of the software that accompanies this text later in this Preface). To facilitate this approach, all the answers in the back of the book to application problems in Exercises 5-4, 5-5, 5-6, and the Chapter 5 Review Exercise contain both the mathematical model and the numeric solution. Chapter 6 covers counting techniques and basic probability, including Bayes' formula and random variables. Some of the topics discussed in Chapter 6 are required for Chapter 7. Chapter 7 ties together concepts developed in earlier chapters and applies them to Markov chains. This provides an excellent unifying conclusion to the finite mathematics portion of the text. The material in Part Three (Calculus) consists of differential calculus (Chapters 8–10), integral calculus (Chapters 11 and 12), multivariable calculus (Chapter 13), and differential equations (Chapter 14). In general, Chapters 8–11 must be covered in sequence; however, certain sections can be omitted or given brief treatments, as pointed out in the discussion that follows (see also the Chapter Dependencies chart preceding this Preface). Chapter 8 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 9 focuses on graphical analysis 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 9-4. 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 9-5, including examples and problems involving end-point solutions. The first three sections of Chapter 10 extend the derivative concepts discussed in Chapters 8 and 9 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 10-4 and applied to related rate problems in Section 10-5. These topics are not referred to elsewhere in the text and can be omitted without loss of continuity. Chapter 11 introduces integration. The first two sections cover antidifferentiation techniques essential to the remainder of the text. Section 11-3 discusses some applications involving differential equations that can be omitted. Sections 11-4 and 11-5 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 12 covers additional integration topics and is organized to provide maximum flexibility for the instructor. The first section extends the area concepts introduced in Chapter 11 to the area between two curves and related applications. Section 12-2 covers three more applications of integration, and Sections 12-3 and 12-4 deal with additional techniques of integration. Any or all of the topics in Chapter 12 can be omitted. The first five sections of Chapter 13 deal with differential multivariable calculus and can be covered any time after Section 10-3 has been completed. Section 13-5 applies multivariable optimization techniques to linear regression, providing a foundation for the regression techniques introduced in Section 2-4. Section 13-6 requires the integration concepts discussed in Chapter 11. After introducing the basic concepts and terminology used in the study of differential equations in Section 14-1, separable differential equations (Section 14-2) and first-order linear differential equations (Section 14-3) and related applications are covered thoroughly. All the growth laws introduced in Section 11-3 are covered again, this time with more emphasis on student recognition of the relevant growth laws. 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 is not defined at x = - 2 and x = 2, but has vertical asymptotes at these points. Armed with this knowledge, you will be able to look at the graphing utility graph in Figure 1A, visualize the more accurate graph in Figure 1B, and sketch a good representation of the graph by hand. To assist you in the development of visualization and sketching skills, the text contains hundreds of both graphing utility graphs and more accurate graphs, and also contains many exercises that ask you to sketch graphs by hand.

Problems or Suggestions? Order Information [ Legal Statement ] © 1998 Prentice-Hall, Inc. A Simon & Schuster Company Upper Saddle River, New Jersey 07458