The wide-ranging debate brought about by the calculus reform movement has had a significant impact on the calculus textbook market. In response to many of the questions and concerns surrounding this debate, we have written a modern calculus textbook, intended for students majoring in mathematics, physics, chemistry, engineering and related fields.

While following a relatively traditional table of contents, we have attempted to address many of the concerns of calculus reformers. We have written this book for the average student, that is, one who does not already know the subject, whose background is somewhat weak in spots and who requires significant motivation to study the subject. Our intention is that students should be able to read our book, rather than merely use it as an encyclopedia filled with the facts of calculus. We have written the text in a casual, almost conversational style, one that our students have found easy to read. Given the widespread availability of graphing calculators and powerful computer algebra systems, we assume that students using this text will have one of these systems at their disposal.

In an effort to ensure that this textbook successfully addresses our concerns about the effective teaching of calculus, as well as others' concerns, we continually asked instructors around North America for their opinions on the calculus curriculum, the strengths and weaknesses of current textbooks and our manuscript and ideas. In preparing this finished product, we had the benefit of countless insightful comments from a panel of reviewers that were carefully selected to help us with this project. Their detailed reviews of our materials and their opinions about the teaching of calculus were invaluable to us during our development of this textbook. We are deeply indebted to them for their time and effort.

Philosophy

We agree with many of the ideas that have come out of the calculus reform movement. In particular, we believe in the Rule of Three: that concepts should be presented graphically, numerically, and algebraically, whenever these are appropriate. In fact, we would add verbally and physically to this list, because the communication of mathematical ideas and the modeling of physical problems are important skills that students need to develop. We also believe that, while the calculus curriculum has been in need of reform, we should not throw out those things that already work well. Our book thus represents a fresh approach to the traditional topics of calculus. We follow an essentially traditional order of presentation, while integrating technology and more thought-provoking exercises throughout.

One of the thrusts of the calculus reform movement has been to place greater emphasis on problem solving and to present students with more realistic applications, as well as open-ended problems. We have incorporated meaningful writing exercises and extended, open-ended problems into every problem set. You will also find a much wider range of applications than in most traditional texts. We frequently make use of applications from within students' experience to both motivate the development of new topics, as well as to illustrate concepts we have already presented. In particular, we have included numerous examples from the physics of sports to give students a familiar context in which to think of various concepts.

We believe that a conceptual development of the calculus must drive the book. Although we have integrated technology throughout, we have not allowed the technology to drive the book. We have also not given in to the temptation to show off what technology can do, except where this has a direct bearing on learning the calculus. Our goal is to use the available technology to help students reach a conceptual understanding of the calculus as it is used today.

Throughout the text, we have assumed that students have access to calculator- or computer-generated graphs. This allows us to routinely use graphs as the first step in solving a problem or as a check on the reasonableness of an answer. Being able to visualize a problem is an invaluable aid to students, and we try to take full advantage of this. One benefit of readily available graphics is the ability to solve more realistic application problems. Functions associated with realistic problems are often not mathematically simple, but we can approximate zeros or extrema graphically and numerically. Further, concepts such as the convergence of Taylor series are more meaningful when graphs are used to show this convergence. This same graphical approach benefits our presentation of Fourier series, which is an important tool for understanding much of our digitally enhanced world.

Throughout the text, we have made use of the ability of calculators and computers to solve equations and quickly complete lengthy calculations. Limits, derivatives, definite integrals and infinite series are all investigated using tables of computations, where students can clearly see what convergence or divergence represents. To avoid having students blindly using technology, we include a section on loss-of-significance errors. Here, we give a highly accessible introduction to common computing errors and how to avoid them. Numerical analysis also plays a large role in our development of Euler's method for approximating solutions of differential equations. Euler's method is closely related to direction fields and plots of vector fields, two places where we rely on computer graphics. Such connections are vital to what we try to accomplish using technology: not new topics added on for completeness, but new ways to visualize and understand important concepts of calculus.

Perhaps the most difficult task when preparing a new calculus text is the actual writing of it. We have endeavored to write this text in a manner that combines an appropriate level of informality with an honest discussion regarding the difficulties that students commonly face in their study of calculus. In addition to the concepts and applications of calculus, we have also included many frank discussions about what is practical and impractical, and what is difficult and not so difficult to students of calculus. We have attempted to provide clarity of presentation in the creation of every example, application and exercise.