Thomas’ Calculus: Early Transcendentals 13th Edition

Book Preface

Thomas’ Calculus: Early Transcendentals, Thirteenth Edition, provides a modern introduction to calculus that focuses on conceptual understanding in developing the essential  elements of a traditional course. This material supports a three-semester or four-quarter  calculus sequence typically taken by students in mathematics, engineering, and the natural  sciences. Precise explanations, thoughtfully chosen examples, superior figures, and timetested exercise sets are the foundation of this text. We continue to improve this text in  keeping with shifts in both the preparation and the ambitions of today’s students, and the applications of calculus to a changing world.

Many of today’s students have been exposed to the terminology and computational  methods of calculus in high school. Despite this familiarity, their acquired algebra and  trigonometry skills sometimes limit their ability to master calculus at the college level. In  this text, we seek to balance students’ prior experience in calculus with the algebraic skill  development they may still need, without slowing their progress through calculus itself. We  have taken care to provide enough review material (in the text and appendices), detailed solutions, and variety of examples and exercises, to support a complete understanding of  calculus for students at varying levels. We present the material in a way to encourage student thinking, going beyond memorizing formulas and routine procedures, and we show  students how to generalize key concepts once they are introduced. References are made  throughout which tie a new concept to a related one that was studied earlier, or to a generalization they will see later on.

After studying calculus from Thomas, students will have developed problem solving and reasoning abilities that will serve them well in many important aspects of their lives.

Mastering this beautiful and creative subject, with its many  practical applications across so many fields of endeavor, is its own reward. But the real gift  of studying calculus is acquiring the ability to think logically and factually, and learning  how to generalize conceptually. We intend this book to encourage and support those goals.

New to this Edition

In this new edition we further blend conceptual thinking with the overall logic and structure of single and multivariable calculus. We continue to improve clarity and precision,  taking into account helpful suggestions from readers and users of our previous texts. While  keeping a careful eye on length, we have created additional examples throughout the text.  Numerous new exercises have been added at all levels of difficulty, but the focus in this  revision has been on the mid-level exercises. A number of figures have been reworked and  new ones added to improve visualization. We have written a new section on probability, which provides an important application of integration to the life sciences.

We have maintained the basic structure of the Table of Contents, and retained improvements from the twelfth edition. In keeping with this process, we have added more  improvements throughout, which we detail here • Functions In discussing the use of software for graphing purposes, we added a brief  subsection on least squares curve fitting, which allows students to take advantage of  this widely used and available application. Prerequisite material continues to be reviewed in Appendices 1–3.
• Continuity We clarified the continuity definitions by confining the term “endpoints” to  intervals instead of more general domains, and we moved the subsection on continuous  extension of a function to the end of the continuity section.
• Derivatives We included a brief geometric insight justifying l’Hôpital’s Rule. We also  enhanced and clarified the meaning of differentiability for functions of several variables, and added a result on the Chain Rule for functions defined along a path.
• Integrals We wrote a new section reviewing basic integration formulas and the Substitution Rule, using them in combination with algebraic and trigonometric identities,  before presenting other techniques of integration.
• Probability We created a new section applying improper integrals to some commonly  used probability distributions, including the exponential and normal distributions. Many examples and exercises apply to the life sciences.
• Series We now present the idea of absolute convergence before giving the Ratio and  Root Tests, and then state these tests in their stronger form. Conditional convergence is introduced later on with the Alternating Series Test.
• Multivariable and Vector Calculus We give more geometric insight into the idea of  multiple integrals, and we enhance the meaning of the Jacobian in using substitutions  to evaluate them. The idea of surface integrals of vector fields now parallels the notion for line integrals of vector fields. We have improved our discussion of the divergence  and curl of a vector field.
• Exercises and Examples Strong exercise sets are traditional with Thomas’ Calculus,  and we continue to strengthen them with each new edition. Here, we have updated,  changed, and added many new exercises and examples, with particular attention to including more applications to the life science areas and to contemporary problems. For  instance, we updated an exercise on the growth of the U.S. GNP and added new exercises addressing drug concentrations and dosages, estimating the spill rate of a ruptured oil pipeline, and predicting rising costs for college tuition.

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  6 we discuss the reliance of the validity 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 for students to review and summarize what they  have learned. Many of these exercises make good writing assignments.

END-OF-CHAPTER REVIEWS AND PROJECTS

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 serving to include  more challenging or synthesizing problems. Most chapters also include descriptions of  several Technology Application Projects that can be worked by individual students or  groups of students over a longer period of time. These projects require the use of a com puter running Mathematica or Maple and additional material that is available over the Internet at www.pearsonhighered.com/thomas and in MyMathLab.

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-to-understand  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 last several editions.

TECHNOLOGY

In a course using the text, technology can be incorporated according to  the taste 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.

Additional Resources

INSTRUCTOR’S SOLUTIONS MANUAL
Single Variable Calculus (Chapters 1–11), ISBN 0-321-88408-6 | 978-0-321-88408-4  Multivariable Calculus (Chapters 10–16), ISBN 0-321-87901-5 | 978-0-321-87901-1  The Instructor’s Solutions Manual contains complete worked-out solutions to all of the  exercises in Thomas’ Calculus: Early Transcendentals.

STUDENT’S SOLUTIONS MANUAL
Single Variable Calculus (Chapters 1–11), ISBN 0-321-88410-8 | 978-0-321-88410-7
Multivariable Calculus (Chapters 10–16), ISBN 0-321-87897-3 | 978-0-321-87897-7
The Student’s Solutions Manual is designed for the student and contains carefully  worked-out solutions to all the odd numbered exercises in Thomas’ Calculus: Early Transcendentals.

JUST-IN-TIME ALGEBRA AND TRIGONOMETRY FOR  EARLY TRANSCENDENTALS CALCULUS, Fourth Edition
ISBN 0-321-67103-1 | 978-0-321-67103-5
Sharp algebra and trigonometry skills are critical to mastering calculus, and Just-in-Time  Algebra and Trigonometry for Early Transcendentals Calculus by Guntram Mueller and  Ronald I. Brent is designed to bolster these skills while students study calculus. As students make their way through calculus, this text is with them every step of the way, showing them the necessary algebra or trigonometry topics and pointing out potential problem  spots. The easy-to-use table of contents has algebra and trigonometry topics arranged in  the order in which students will need them as they study calculus.

Technology Resource Manuals
Maple Manual by Marie Vanisko, Carroll College Mathematica Manual by Marie Vanisko, Carroll College TI-Graphing Calculator Manual by Elaine McDonald-Newman, Sonoma State University  These manuals cover Maple 17, Mathematica 8, and the TI-83 Plus/TI-84 Plus and TI-89,  respectively. Each manual provides detailed guidance for integrating a specific software  package or graphing calculator throughout the course, including syntax and commands.  These manuals are available to qualified instructors through the Thomas’ Calculus: Early  Transcendentals Web site, www.pearsonhighered.com/thomas, and MyMathLab.

WEB SITE www.pearsonhighered.com/thomas
The Thomas’ Calculus: Early Transcendentals Web site contains the chapter on SecondOrder Differential Equations, including odd-numbered answers, and provides the expanded historical biographies and essays referenced in the text. The Technology Resource
Manuals and the Technology Application Projects, which can be used as projects by individual students or groups of students, are also available.
MyMathLab® Online Course (access code required)
MyMathLab from Pearson is the world’s leading online resource in mathematics, integrating interactive homework, assessment, and media in a flexible, easy-to-use format. MyMathLab delivers proven results in helping individual students succeed.
• MyMathLab has a consistently positive impact on the quality of learning in higher  education math instruction. MyMathLab can be successfully implemented in any  environment—lab-based, hybrid, fully online, traditional—and demonstrates the quantifiable difference that integrated usage makes in regard to student retention, subsequent success, and overall achievement.
• MyMathLab’s comprehensive online gradebook automatically tracks your students’ results on tests, quizzes, homework, and in the study plan. You can use the gradebook to  quickly intervene if your students have trouble, or to provide positive feedback on a job  well done. The data within MyMathLab are easily exported to a variety of spreadsheet  programs, such as Microsoft Excel. You can determine which points of data you want  to export, and then analyze the results to determine success. MyMathLab provides engaging experiences that personalize, stimulate, and measure  learning for each student.
• “Getting Ready” chapter includes hundreds of exercises that address prerequisite  skills in algebra and trigonometry. Each student can receive remediation for just those  skills he or she needs help with.
• Exercises: The homework and practice exercises in MyMathLab are correlated to the  exercises in the textbook, and they regenerate algorithmically to give students unlimited opportunity for practice and mastery. The software offers immediate, helpful feedback when students enter incorrect answers.
• Multimedia Learning Aids: Exercises include guided solutions, sample problems,  animations, Java™ applets, videos, and eText access for extra help at point-of-use.
• Expert Tutoring: Although many students describe the whole of MyMathLab as “like  having your own personal tutor,” students using MyMathLab do have access to live  tutoring from Pearson, from qualified math and statistics instructors