# University Calculus: Early Transcendentals (3rd Edition)

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Book Preface

This third edition of University Calculus provides a streamlined treatment of the material in a standard three-semester or four-quarter course taught at the university level. As the title suggests, the book aims to go beyond what many students may have seen at the high school level. By emphasizing rigor and mathematical precision, supported with examples and exercises, this book encourages students to think more clearly than if they were using rote procedures. Generalization drives the development of calculus and is pervasive in this book. Slopes of lines generalize to slopes of curves, lengths of line segments to lengths of curves, areas and volumes of regular geometric figures to areas and volumes of shapes with curved boundaries, rational exponents to irrational ones, and finite sums to series. Plane analytic geometry generalizes to the geometry of space, and single variable calculus to the calculus of many variables. Generalization weaves together the many threads of calculus into an elegant tapestry that is rich in ideas and their applications.

Mastering this beautiful subject is its own reward, but the real gift of mastery is the ability to think through problems clearly distinguishing between what is known and what is assumed, and using a logical sequence of steps to reach a solution. We intend this book to capture the richness and powerful applicability of calculus, and to support student thinking and understanding for mastery of the material.

New to this Edition

In this new edition, we have followed the basic structure of earlier editions. Taking into account helpful suggestions from readers and users of previous editions, we continued to improve clarity and readability. We also made the following improvements:
• Updated and added numerous exercises throughout, with emphasis on the mid-level and more in the life science areas
• Reworked many figures and added new ones
• Moved the discussion of conditional convergence to follow the Alternating Series Test
• Enhanced the discussion defining differentiability for functions of several variables with more emphasis on linearization
• Showed that the derivative along a path generalizes the single-variable chain rule
• Added more geometric insight into the idea of multiple integrals and the meaning of the Jacobian in substitutions for their evaluations
• Developed surface integrals of vector fields as generalizations of line integrals
• Extended and clarified the discussion of the curl and divergence, and added new figures to help visualize their meanings

Continuing Features

RIGOR The level of rigor is consistent with that of earlier editions. We continue to distinguish between formal and informal discussions and to point out their differences. We think starting with a more intuitive, less formal, approach helps students understand a new or difficult concept so they can then appreciate its full mathematical precision and outcomes. We pay attention to defining ideas carefully and to proving theorems appropriate for calculus students, while mentioning deeper or subtler issues they would study in a more advanced course. Our organization and distinctions between informal and formal discussions give the instructor a degree of flexibility in the amount and depth of coverage of the various topics. For example, while we do not prove the Intermediate Value Theorem or the Extreme Value Theorem for continuous functions on a … x … b, we do state these theorems precisely, illustrate their meanings in numerous examples, and use them to prove other important results. Furthermore, for those instructors who desire greater depth of coverage, in Appendix 7 we discuss the reliance of these theorems on the completeness of the real numbers.

WRITING EXERCISES Writing exercises placed throughout the text ask students to explore and explain a variety of calculus concepts and applications. In addition, the end of each chapter contains a list of questions to help students review and summarize what they have learned. Many of these exercises make good writing assignments. END-OF-CHAPTER REVIEWS In addition to problems appearing after each section, each chapter culminates with review questions, practice exercises covering the entire chapter, and a series of Additional and Advanced Exercises providing more challenging or synthesizing problems.

WRITING AND APPLICATIONS As always, this text continues to be easy to read, conversational, and mathematically rich. Each new topic is motivated by clear, easy-tounderstand examples and is then reinforced by its application to real-world problems of immediate interest to students. 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 past several editions.

TECHNOLOGY In a course using the text, technology can be incorporated at the discretion of the instructor. Each section contains exercises requiring the use of technology; these are marked with a “T” if suitable for calculator or computer use, or they are labeled “Computer Explorations” if a computer algebra system (CAS, such as Maple or Mathematica) is required.