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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.
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