Brief Calculus for Business, Social Sciences, and Life Sciences

The Consortium's new Brief Calculus text draws heavily from the Applied Calculus text, but also incorporates many new features. The Brief Calculus text is obviously shorter and more directly focused toward the needs of this particular audience. On the other hand, a look at the table of contents shows that it is far more than just a shorter version of the two-semester text. In fact, many of the changes in the new book have worked so well that we plan to incorporate them into the next edition of the two-semester Applied Calculus text.

We noticed that students had more trouble with the distinction between change and rate of change than we originally realized. Since this concept is critical to an understanding of calculus, we begin in the very first section by asking students to focus on the distinction between, for example, 40 mg versus 40 mg per day, or a cost of $400 versus a cost of $400 per unit. The idea of an average rate of change is carried throughout the first chapter, so that when students first encounter the instantaneous rate of change in the second chapter, they have a framework on which to build.

The material on understanding families of functions has been split into two chapters, and appears in Chapters 1 and 4. Chapter 1 includes material on linear functions, exponential functions, and power functions. The sections on linear and exponential functions build on the concept of change, since students must understand the distinction between growing at 500 units per year and growing at 5% per year. Chapter 1 ends with a section on regression, enabling students to use their new knowledge about these functions to do some mathematical modeling.

The book begins with three very conceptual chapters. Chapter 1 focuses on change and functions. Chapter 2 presents the derivative as an instantaneous rate of change and as slope. Chapter 3 presents the definite integral as total change and as area. By the end of these three chapters, students should have a solid understanding of the major ideas of calculus and should be comfortable working with functions. At this point, they should be ready to study the more complicated functions presented in Chapter 4. These functions include exponential functions with base e, logarithms, polynomials, periodic functions, the logistic function, and the surge function. The emphasis in the chapter is on understanding families of functions and on working with parameters. Students are asked, for example, to study the graph of C = te^-bt using different positive values for b, to discuss the shape of the graph, and to explain the effect of the parameter b. The functions covered in this chapter were chosen because of their many applications in the user disciplines, and the applications are emphasized throughout the chapter.

Chapter 5 presents the derivative formulas, and is based on Chapter 4 of the Applied Calculus text. The formulas are motivated graphically and numerically; all algebraic proofs of the formulas are delayed until Section 5.6. In response to requests from users, we have added a new section on antiderivatives at the end of this chapter. Chapter 6 includes specific, more extended, applications of calculus to the user disciplines, but the number of applications included is substantially smaller in this book than in the two-semester version. The chapter on differential equations is omitted from this book. Chapter 7 presents functions of several variables with an emphasis on understanding these functions and their contour diagrams as they are used by the user disciplines. Since surfaces are rarely used in business, social sciences, or life sciences to represent functions of two variables, they are not included in the book.

We have tried to design a book that includes the key concepts needed b this audience and that can be realistically covered in one semester. We were helped by the excellent feedback provided to the Consortium by reviewers, members of the client disciplines, and other users of the Applied Calculus text. We appreciate all input and encourage users to continue to give us feedback.