Mastering Skills and Concepts

As always, this text continues to maintain a strong skill-building emphasis. Throughout this edition, the authors have included examples and discussions encouraging students to think visually, analytically, and numerically. Almost every exercise set contains problems requiring students to generate and interpret graphs as a tool for understanding mathematical or real-world relationships. Many sections also contain problems to extend the range of applications, mathematical ideas, and rigor.

Students are asked to explore and explain a variety of calculus concepts and applications in writing exercises placed throughout the text. In addition, each chapter ends with a list of questions to help students review and summarize what they have learned. Many of these review questions make great writing assignments.

Problem-Solving Strategies

The authors believe students learn best when procedural techniques are laid out as clearly and simply as possible. To this end, stepwise problem-solving summaries are included as appropriate, especially for the more difficult or complicated procedures. As always, the authors are especially careful that examples in the text illustrate the steps outlined by the summaries.

Exercises

Exercise sets have been carefully reviewed and revised in this new edition. They are grouped by topic, with special sections for computer explorations. These sections contain computer algebra system (CAS) explorations and projects. Within exercise sets, there are practice and applied problems, critical thinking and challenging exercises (in subsections marked "Applications and Theory"), as well as exercises requiring students to write about important calculus concepts. Writing exercises appear throughout exercise sets. Exercises generally follow the order of presentation in the text and those requiring a graphing utility (such as a graphing calculator) are identified throughout the text by the icon T.

Chapter End Support Material

At the end of each chapter there are three features summarizing the chapter contents.

Questions to Guide Your Review ask students to think about key chapter concepts and then verbalize their understanding of them and include illustrative examples. These are suitable for writing exercises.

Practice Exercises provide a review of techniques, computational and numerical skills, and key applications.

Additional Exercises---Theory, Examples, Applications provide students with more theoretical or challenging applications and problems to further deepen their understanding of the mathematical ideas.

Applications and Examples

A hallmark of this book has been the application of calculus to science and engineering. These applied problems have been updated, improved and extended continually over the last several editions. With this edition, the authors include more problems based on real data requiring graphical and numerical techniques for their solution. Throughout the text, the authors cite sources for the data and/or articles from which these applications are drawn, helping students understand that calculus is a current, dynamic field requiring a multiplicity of different techniques and approaches. Most of these applications are directed toward the physical sciences and engineering, but there are many from biology and the social sciences as well.

Technology: Graphing Utility and Computer Explorations

Virtually every section of the text contains exercises to explore numerical patterns, or graphing utility exercises that ask students to generate and interpret graphs as a tool to understanding mathematical and real-world relationships. Many of the graphing utility exercises are suitable for classroom demonstration or for group work by students in or out of class. These exercises are identified throughout the text by the icon T or the heading "Computer Explorations".

Computer Explorations

Numbering more than 200, these computer exercises have been solved using both Mathematica and Maple. In addition, Mathematica and Maple modules are available on the Web site and CD-ROM. These modules have been carefully designed to help students develop a geometric intuition and deeper understanding and appreciation of calculus concepts, methodologies, and applications. CD/Web site icons mark the locations in the text where material related to these modules is covered.

Notes also appear throughout the text encouraging students to explore with graphing utilities, and helping students to assess when the use of technology is helpful, and when it may be misleading.

Expanded History and Biographies Any student is enriched by seeing the human side of mathematics through its historical development. In previous editions we featured history boxes describing the origins of ideas, conflicts concerning ownership of these ideas, and interesting sidelights into modern topics such as fractals and chaos. For the Tenth Edition, Col. A. Chris Arney and Joe B. Albree have expanded and written more biographies and historical essays. These essays are now available on the CD-ROM and Web site, referenced by icons throughout the text, leaving more room in the margins for student notes, observations, and annotations.

The Many Faces of This Book

Mathematics Is a Formal and Beautiful Language

Calculus is one of the most powerful of human intellectual achievements. One of the goals of this book is to give students an appreciation of the beauty of calculus. As in previous editions, the authors have been careful to say only what is true and mathematically sound. Every definition, theorem, corollary, and proof has been reviewed for clarity and mathematical correctness.

Whether calculus is taught in a traditional lecture format or entirely in labs with individual and group learning focusing on numerical and graphical experimentation, its ideas and techniques need to be articulated clearly and accurately.