Welcome to Calculus with Analytic Geometry, Sixth Edition! We are excited about the Sixth Edition and hope you will be too after you hear about why we wrote it, what's new about it, and how it will carry you and your calculus students into the twenty-first century!

Reform in Mathematics Education

As you know, the current reform movement in math education started about 15 years ago and has involved all levels of mathematics education from kindergarten through college. There may be some who say reform was not needed. However, the vast majority of math educators agree that reform was essential. In fact, the new math of the 1960s, 1970s, and early 1980s was a national disaster. It was far too abstract and far too removed from the real-life applications that were the actual foundation of mathematics. The result was evident to everyone -- math phobia, falling test scores, high drop-out rates, and a general sense that students were not learning to be creative problem solvers.

So what were the proposed solutions? There were many-more real-life connections, more incorporation of technology, curriculum revisions, and the development of alternative forms of teaching, assessment, and learning.

Where This Text Stands with Respect to Reform

Where does the Sixth Edition stand with respect to reform? This is one of the most common questions we are asked. Our answer: The text has definitely benefited from reform. More than that, over the years our calculus text has actually led the way in developing many innovative learning techniques. From its first edition, the text stressed the importance of graphical learning -- much more than other texts in use in the late 1970s and early 1980s. This text was one of the first to incorporate computer-generated art -- both two-dimensional and three-dimensional -- to aid in the visualization of complex mathematical concepts.

We have always paid careful attention to the presentation -- using precise mathematical language, innovative full-color designs for emphasis and clarity, and a level of exposition that appeals to students -- to create an effective teaching and learning tool. Although difficult to quantify, this feature has been praised by thousands of students and their instructors over the past 20 years. With each edition, we have continued to incorporate the best strategies for teaching calculus, using the pedagogy we have developed as a result of our teaching experiences, as well as many suggestions from thoughtful users.

This Sixth Edition might best be described as fitting midway between texts that define themselves as traditional and those that are considered reform texts. Our approach is like that of a traditional text in that we firmly believe in the importance of carefully developed theory, correct statements of theorems, inclusion of proofs, and mastery of traditional calculus skills. We have found no evidence that it is somehow possible to apply calculus in real-life situations without first being able to understand and "do" calculus.

On the other hand, we wholeheartedly embrace many of the features of calculus reform. For instance, this edition features additional opportunities to use technology, an increased emphasis on real-life applications and modeling data, new motivational features, many more conceptual exercises and multipart exercises, new explorations in many sections, and a myriad of student and teacher aids.

We believe in giving teachers options to teach calculus the way they want. Because of this, you will find a wide variety of approaches and features in the text. By choosing the options that best fit you as a teacher, you can customize the book in dozens of different ways: lecture or discovery approaches, pencil and paper skills or technology, formal or informal, theoretical or intuitive, analytic or graphical, mathematics-centered or applied, classroom presentation or distance learning -- they are all here as options for you.

What We Changed in the Sixth Edition

In the Sixth Edition, we continue to lead the way in incorporating the best aspects of reform in a meaningful yet easy-to-use manner. Here are some of the most significant new features.

New Explorations. For instructors and students who benefit from a frequent or occasional "discovery" mode, we have incorporated this option into the Sixth Edition. By discovering concepts that are new to them, students get a taste of what it is like to be a real mathematician. Also, some students find it easier to remember concepts they have "discovered." Truly effective explorations are difficult to create. The challenge is finding a good balance between what is given and what is expected to be discovered. Our Explorations represent a combination of suggestions by users and the results of our own classroom experience.

New Motivating the Chapter. Each chapter now begins with a full-page chapter motivator. As all calculus teachers know, one of the difficulties in teaching calculus is that we are often answering questions that students have not yet asked. The motivators address this dilemma by presenting real-life situations with exploratory questions. As students attempt to use the techniques of their current skill set to answer the questions, they will learn to appreciate the power and efficiency of the new calculus techniques presented in the chapter. We spent a lot of time researching effective settings and developing meaningful questions, and we found that the ones selected for this text sparked interest among the students in our classes.

New Lab Manuals. The real-life application that is introduced in each Motivating the Chapter forms the basis for the extended technology lab projects in the supplementary Lab Manuals. In each Motivating the Chapter, students are asked to solve a problem using the techniques of their current skill set. As the chapter progresses, students can be assigned projects from the Lab Manuals that ask them to take a new look at these problems using concepts learned in that chapter as well as technology. The Lab Manuals come in five versions (each with data disks) designed for use with Maple, Mathematica, Derive, Mathcad, and the TI-92 graphing calculator.

New Art Program. Visualization is a problem-solving skill that is critical for the understanding of complex calculus theory and concepts. To help students develop this skill, the Sixth Edition features a completely new art program that was created using state-of-the-art computer technology for accuracy, clarity, and realism. The effect of the new three-dimensional art is particularly striking. Using computer graphics software, we adjusted the color and transparency, view, light sources, and shadows on the solids and surfaces until we found the optimal combination of features to show true perspective. The result, we believe, is the most accurate representation of three-dimensional calculus surfaces ever! For instance, the image on the book's cover was produced with this computer graphics system and the illusion of three-dimensionality is amazing.

Revised Exercise Sets. All of the exercise sets of the previous edition were considered for revision, and many new exercises were added to the Sixth Edition. We focused on adding problems that were technology-oriented, thought-provoking, conceptual, creative, real, and engaging. There are many more opportunities for writing, for individual and group projects, and for solving problems with graphical, numerical, and analytical approaches. We expanded on the wide variety of applications, which were already distinctive for their relevance and originality. Also new to the Sixth Edition are many modeling data exercises that ask students to find and interpret mathematical models from the real-life data that are given. To the many who helped us in our search for such exercises, we offer our thanks.

Revised Interactive Calculus. The Fifth Edition of this text was available in an interactive CD-ROM format that was innovative and well received. Now, four years later, we have continued to push the limits of what technology can do as a medium for teaching calculus. Among the new and enhanced features of the CD-ROM version of the Sixth Edition are: all of the content of the revised text, more active mathematics, editable two-dimensional graphs, dazzling three-dimensional graphs, additional explorations and simulations, and a syllabus builder for instructors.

New Internet Calculus. This ground-breaking on-line subscription service offers the first complete calculus package on the web. It contains the entire content of Calculus, Sixth Edition, plus all the interactive features of the CD-ROM, Interactive Calculus version 2.0. Furthermore, it also provides features available exclusively through Internet subscription such as chat rooms, mailing lists, and newsgroups.

Revised Table of Contents. Some of the text was revised to enhance the flexibility of the presentation and to reflect our perspective on reform. We rewrote Chapter P, moving much of the precalculus review to Appendix A. What remains of precalculus in Chapter P was restructured to introduce students to new ways of thinking about math, including graphical, numerical, and analytical approaches; modeling; problem solving; and data analysis. Instructors who skipped this chapter in previous editions might want to reconsider that decision.

Chapter 1 now begins with a new section, "A Preview of Calculus," which introduces students to the distinctions between precalculus and calculus, emphasizing the necessity of calculus in our dynamic, everyday surroundings. One section on differential equations was moved from Chapter 15 to Chapter 5 to allow students to make more use of this material, and to better prepare them for their courses in other disciplines such as physics and chemistry. In Chapter 6, we repositioned the section on moments, centers of mass, and centroids so that it precedes the section on fluid pressure and fluid force, to address changing teaching styles. We reduced the coverage of conic sections in Chapter 9, because this material is mostly a review for students at this level. This edition discusses conics in the same chapter that covers parametric equations and the polar coordinate system. Finally, the section on rotation and the general second-degree equation was moved to Appendix E, leaving it optional for those instructors who wish to cover it.

What We Didn't Change

Although we carefully and thoroughly revised the text by enhancing the usefulness of some features and topics and adding others, we didn't change many of the things that our colleagues and the 1,500,000 students who have used this book have told us worked for them. We still offer comprehensive coverage of the material required by students in a three-semester calculus course, including carefully stated theory and proofs. Additionally, as we do with all our books, we painstakingly formatted every page of the Sixth Edition to achieve a clear presentation of the material. Finally, the text was carefully written in a style that is mathematically precise, as well as engaging, direct, and readable.

We hope you will enjoy the Sixth Edition. We are proud to have it as our calculus entry for the next century!

Roland E. Larson

Robert P. Hostetler

Bruce H. Edwards

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