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Book Preface
We have significantly revised this edition of Thomas’ Calculus to meet the changing needs of today’s instructors and students. The result is a book with more examples, more midlevel exercises, more figures, better conceptual flow, and increased clarity and precision. As with previous editions, this new edition provides a 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 for this text of MyMathLab® (discussed further on), providing additional support for students and flexibility for instructors.
Many of our students were exposed to the terminology and computational aspects of calculus during high school. Despite this familiarity, students’ algebra and trigonometry skills often hinder their success in the college calculus sequence. With this text, we have sought to balance the students’ prior experience with calculus with the algebraic skill development they may still need, all without undermining or derailing their confidence. We have taken care to provide enough review material, fully stepped-out solutions, and exercises to support complete understanding for students of all levels.
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.
Changes for the Twelfth Edition
CONTENT In preparing this edition we have maintained the basic structure of the Table of Contents from the eleventh edition. Yet we have paid attention to requests by current users and reviewers to postpone the introduction of parametric equations until we present polar coordinates, and to treat 1’H6pital’s Rule after the transcendental functions have been studied. We have made numerous revisions to most of the chapters, detailed as follows.
• Functions We condensed this chapter even more to focus on reviewing function concepts. Prerequisite material covering real numbers, intervals, increments, straight lines, distances, circles, and parabolas is presented in Appendices 1-3.
• Limits To improve the flow of this chapter, we combined the ideas of limits involving infinity and their associations with asymptotes to the graphs of functions, placing them together in the final chapter section.
• Differentiation While we use rates of change and tangents to curves as motivation for studying the limit concept, we now merge the derivative concept into a single chapter. We reorganized and increased the number ofrelated rates examples, and we added new examples and exercises on graphing rational functions.
• Antiderivatives and Integration We maintain the organization of the eleventh edition in placing antiderivatives as the rmal topic of the chapter covering applications of derivatives. Our focus is on ”recovering a function from its derivative” as the solution to the simplest type offirst-order differential equation. Integrals, as “limits ofRiemann sums,” motivated primarily by the problem of rmding the areas ofgeneral regions with curved boundaries, are a new topic fanning the substance ofChapter 5. After carefully developing the integral concept, we turn our attention to its evaluation and connection to antiderivatives captured in the Fundamental Theorem of Calculus. The ensuing applications then define the various geometric ideas ofarea, volume, lengths ofpaths, and centroids all as limits of Riemann sums giving definite integrals, which can be evaluated by finding an antiderivative ofthe integrand. We return later to the topic of solving more complicated first-order differential equations, after we derme and establish the transcendental functions and their properties.
• Differential Equations Some universities prefer that this subject be treated in a course separate from calculus. Although we do cover solutions to separable differential equations when treating exponential growth and decay applications in the chapter on transcendental functions, we organize the bulk of our material into two chapters (which may be omitted for the calculus sequence). We give an introductory treatment of first-order differential equations in Chapter 9, including a new section on systems and phase planes, with applications to the competitive-hunter and predator-prey models. We present an introduction to second-order differential equations in Chapter 17, which is included in MyMathLab as well as the Thomas’ Calculus Web site, www.pearsonhighered.com/thomas.
• Series We retain the organizational structure and content ofthe eleventh edition for the topics of sequences and series. We have added several new figures and exercises to the various sections, and we revised some of the proofs related to convergence ofpower series in order to improve the accessibility of the material for students. The request stated by one of our users as, “anything you can do to make this material easier for students will be welcomed by our faculty;’ drove our thinking for revisions to this chapter.
Caranza's 12th Edition Pdf Solution
• Parametric Equations Several users requested that we move this topic into Chapter II, where we also cover polar coordinates and conic sections. We have done this, realizing that many departments choose to cover these topics at the beginning ofCalculus m, in preparation for their coverage ofvectors and multivariable calculus.
• Vector-Valued Functions We streamlined the topics in this chapter to place more emphasis on the conceptual ideas supporting the later material on partial derivatives, the gradient vector, and line integrals. We condensed the discussions of the Frenet frame and Kepler’s three laws of planetary motion.
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