Calculus: Early Transcendentals – First Edition

Book Preface

On Teaching Mathematics

As a young instructor, I enjoyed teaching but didn’t appreciate how difficult it is to communicate mathematics effectively. Early in my teaching career, I was confronted with a student rebellion when my efforts to explain epsilon-delta proofs were not greeted with the enthusiasm I anticipated. Experiences of this type taught me two basic principles:

1. We should try to teach students as much as possible, but not more so.
2. As math teachers, how we say it is as important as what we say.

When a concept is wrapped in dry mathematical formalism, the majority of students cannot assimilate it. The formal language of mathematics is intimidating to the uninitiated. By presenting the same concept in everyday language, which may be more casual but no less precise, we open the way for students to understand the underlying idea and integrate it into their way of thinking. Students are then in a better position to appreciate the need for formal definitions and proofs and to grasp their logic.

My most valuable resource in writing this text has been my classroom experience of the past 25 years. I have learned how to teach from my students, and I hope my text reflects a true sensitivity to student needs and capabilities. I also hope it helps students experience the joy of understanding and mastering the beautiful ideas of the subject.

On Writing a New Calculus Text

Calculus has a deservedly central role in higher education. It is not only the key to the full range of scientific and engineering disciplines; it is also a crucial component in a student’s intellectual development. I hope my text will help to open up this multifaceted world of ideas for the student. Like a great symphony, the ideas in calculus never grow stale. There is always something new to appreciate and delight in, and I enjoy the challenge of finding the best way to communicate these ideas to students.

My text builds on the tradition of several generations of calculus authors. There is no perfect text, given the choices and inevitable compromises that must be made, butmany of the existing textbooks reflect years of careful thought, meticulous craft, and accumulated wisdom, all of which cannot and should not be ignored in the writing of a new text.

I have made a sustained effort to communicate the underlying concepts, ideas, and “reasons why things work” in language that is accessible to students. Throughout the text, student difficulties are anticipated and addressed. Problem-solving skills are systematically developed in the examples and problem sets. I have also included a wide range of applications, both innovative and traditional, which provide additional insight into the mathematics and communicate the important message that calculus plays a vital role in the modern world.

My textbook follows a largely traditional organization, with a few exceptions. Two such exceptions are the structure of Chapter 2 on limits and the placement of Taylor polynomials in Chapter 8