University Calculus, Early Transcendentals (2nd Edition)

Book Preface

We have significantly revised this edition of University Calculus, Early Transcendentals to meet the changing needs of today’s instructors and students. The result is a book with more examples, more mid-level exercises, more figures, better conceptual flow, and increased clarity and precision. As with the previous edition, this new edition provides a briefer, modern introduction to calculus that supports conceptual understanding but retains the essential elements of a traditional course. These enhancements are closely tied to an expanded version of MyMathLab® for this text (discussed further on), providing additional support for students and flexibility for instructors.

In this second edition, we introduce the basic transcendental functions in Chapter 1. After reviewing the basic trigonometric functions, we present the family of exponential functions using an algebraic and graphical approach, with the natural exponential described as a particular member of this family. Logarithms are then defined as the inverse functions of the exponentials, and we also discuss briefly the inverse trigonometric functions. We fully incorporate these functions throughout our developments of limits, derivatives, and integrals in the next five chapters of the book, including the examples and exercises. This approach gives students the opportunity to work early with exponential and logarithmic functions in combinations with polynomials, rational and algebraic functions, and trigonometric functions as they learn the concepts, operations, and applications of single-variable calculus. Later, in Chapter 7, we revisit the definition of transcendental functions, now giving a more rigorous presentation. Here we define the natural logarithm function as an integral with the natural exponential as its inverse.

Today, an increasing number of students become familiar with the terminology and operational methods of calculus in high school. However, their conceptual understanding of calculus is often quite limited when they enter college. We have acknowledged this reality by concentrating on concepts and their applications throughout.

We encourage students to think beyond memorizing formulas and to generalize concepts as they are introduced. Our hope is that after taking calculus, students will be confident in their problem-solving and reasoning abilities. Mastering a beautiful subject with practical applications to the world is its own reward, but the real gift is the ability to think and generalize. We intend this book to provide support and encouragement for both.