com Calculus Applications and Technology THIRD EDITION www.com This page intentionally left blank www.com Calculus Applications and Technology THIRD EDITION Edmond C. Tomastik University of Connecticut With Interactive Illustrations by Hu Hohn, Massachusetts School of Art Jean Marie McDill, California Polytechnic State University, San Luis Obispo Agnes Rash, St. Joseph’s University Australia • Canada • Mexico • Singapore • Spain United Kingdom • United States www.com Publisher: Curt Hinrichs Text Designer: John Edeen Development Editor: Cheryll Linthicum Art Editor: Ann Seitz Assistant Editor: Ann Day Photo Researcher: Gretchen Miller Editorial Assistant: Katherine Brayton Copy Editor: Barbara Willette Technology Project Manager: Earl Perry Illustrator: Hearthside Publishing Services/Jade Myers Marketing Manager: Tom Ziolkowski Cover Designer: Ron Stanton Marketing Assistant: Jessica Bothwell Cover Image: Gary Holscher Advertising Project Manager: Nathaniel Bergson-Michelson Interior Printer: Quebecor/Taunton Senior Project Manager, Editorial Production: Janet Hill Cover Printer: Phoenix Color Corp Print/Media Buyer: Barbara Britton Compositor: Techsetters, Inc. Production Service: Hearthside Publication Services/Anne Seitz COPYRIGHT © 2005 Brooks/Cole, a division of Thomson Thomson Brooks/Cole Learning, Inc.
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Student Edition: ISBN 0-534-46496-3 Mexico Spain/Portugal Instructor’s Edition: ISBN 0-534-46498-X Paraninfo Calle Magallanes, 25 28015 Madrid, Spain www.com An Overview of Third Edition Changes 1. In this new edition we have followed a general philosophy of dividing the material into smaller, more manageable sections. This has resulted in an increase in the number of sections. We think this makes it easier for the instructor and the student, gives more flexibility, and creates a better flow of material.
To add to the flexibility, many sections now have enrichment subsections. Ma- terial in such enrichment subsections is not needed in the subsequent text (except possibly in later enrichment subsections). Now instructors can easily tailor the material in the text to teach a course at different levels. The third edition has even more referenced real-life examples.
It is important to realize that the mathematical models presented in these referenced examples are models created by the experts in their fields and published in refereed journals. So not only is the data in these referenced examples real data, but the mathematical models based on this real data have been created by experts in their fields (and not by us). Mathematical modeling is stressed in this edition. Mathematical modeling is an attempt to describe some part of the real world in mathematical terms.
Already at the beginning of Section 1.2 we describe the three steps in mathematical mod- eling: formulation, mathematical manipulation, and evaluation. We return to this theme often. For example, in Section 5.6 on optimization and modeling we give a six-step procedure for mathematical modeling specifically useful in opti- mization. Essentially every section has examples and exercises in mathematical modeling.
This edition also includes many more opportunities to model by curve fitting. In this kind of modeling we have a set of data connecting two variables, x and y, and graphed in the xy-plane. We then try to find a function y = f (x) whose graph comes as close as possible to the data. This material is found in a new Chapter 2 and can be skipped without any loss of continuity in the remainder of the text.
Curve-fitting exercises are clearly marked as such. The text is now technology independent. Graphing calculators or computers work just as well with the text. A disk with interactive illustrations is now included with each text.
These in- teractive illustrations provide the student and instructor with wonderful demon- strations of many of the important ideas in the calculus. They appear in every chapter. These demonstrations and explorations are highlighted in the text at appropriate times. They provide an extraordinary means of obtaining deep and clear insights into the important concepts.
We are extremely excited to present these in this format.com vi An Overview of Third Edition Changes Chapter 1. This chapter now contains five sections: 1.4, Combinations of Functions; and 1. The material that covered modeling with least squares has all been moved to a new Chapter 2. Most of the material in the sections on quadratics and special functions has been moved to the Review Appendix.
A geometric definition of continuity now appears in the first section. Modeling with Least Squares. This is a new chapter and places all the material on least squares that was originally in Chapter 1 into this new chapter. Instructors who wish can ignore the material in this chapter.
Limits and the Derivative. This chapter has been substantially revised. The material on the limit definition of continuity is now an “enrichment” subsection of the first section on limits and is not needed in the remainder of the text. The material on limits at infinity has been moved to a later chapter.
The section on rates of change now has more examples of average rates of change. More emphasis is put on interpretations of rates of change and on units. The old section on derivatives has been made into two sections, the first on derivatives and the second on local linearity. The new section on derivatives has more emphasis on graphing the derivative given the function and also on interpretations.
The section on local linearity now includes marginal analysis and the economic interpretation of the derivatives of cost, revenue, and profits. This latter material was formerly in a later chapter. Rules for Derivatives. This chapter now includes more “intuitive,” that is, geometrical and numerical, sketches of a number of proofs, the formal proofs being given in enrichment subsections.
Thus, a geometrical sketch of the proof for the derivative of a constant times a function is given, and numerical evidence for the proof for the derivative of the sum of two functions is given. The formal proofs of these, together with the proof of the derivative of the product, are in optional subsections. More geometrical insight has been added to the chain rule, and more emphasis is put on determining units. The more difficult proofs in the exponential and logarithm section have been placed in an enrichment subsection.
Elasticity of demand now has it’s own section. The introductory material on elasticity has been rewritten to make the topic more transparent. The last section on applications on renewable resources has been updated with timely new material. Curve Sketching and Optimization.
This chapter has been exten- sively reorganized. The second section on the second derivative now contains only material specific to concavity and the second derivative test and is much shorter and much more manageable. The material on additional curve sketching that was previ- ously in this section has been given its own section, Section 5. Limits at infinity are now discussed in Section 3, having been moved from an earlier chapter.
It is in this chapter that this material is actually used, so it seems appropriate that it be located here. The old section on optimization has been split into two sections, the first on absolute extrema and the second on optimization and mathematical modeling. A new section on the logistic curve has been created from material found scattered in various sections. With its own section, new material has been added to give this important model its proper due (although instructors can omit this material without effecting the flow of the text).
The section on substitution has been refocused to have a more intuitive as opposed to formal approach and is now more easily accessible. To the third section, on distance traveled, more examples of Riemann sums have been added, and taking the limit as n → ∞ is postponed until the next section. The section on the definite integral now contains some properties of integrals that were not found in the last edition. The section on the fundamental theorem of calculus has www.com An Overview of Third Edition Changes vii been extensively rewritten, with a different proof of the fundamental theorem given.
x We first show that the derivative of f (t) dt is f (x) using a geometric argument a using the new properties of integrals that were included in the previous section and b then proceed to prove f (t) dt = F (b) − F (a), where F is an antiderivative. The a more formal proof is given in an enrichment subsection. Finally, a new Section 6.7 has been created to include the various applications of the integral that had been scattered in previous sections. Additional Topics in Integration.
The interactive illustrations in the numerical integration section yield considerable insight into the subject. Students can move from one method to another and choose any n and see the graphs and the numerical answers immediately. Functions of Several Variables. Graphing in several variables and visualizing the geometric interpretation of partial derivatives is always difficult.
There are several interactive illustrations in this chapter that are extremely helpful in this regard. The Trigonometric Functions. This chapter covers an introduction to the trigonometric functions, including differentiation and integration. Taylor Polynomials and Infinite Series.
This chapter covers Taylor polynomials and infinite series.7 constitute a subchapter on Taylor polynomials.7 is written so that the reader can go from Section 10.2 directly to Section 10. Probability and Calculus. This chapter is on probability.1 is a brief review of discrete probability.2 considers continuous prob- ability density functions and Section 11.3 presents the expected value and variance of these functions.4 covers the normal distribution. This chapter is a brief introduction to differential equations and includes the technique of separation of variables, approx- imate solutions using Euler’s method, some qualitative analysis, and mathematical problems involving the harvesting of a renewable natural resource.com This page intentionally left blank www.com Preface Calculus: Applications and Technology is designed to be used in a one- or two- semester calculus course aimed at students majoring in business, management, eco- nomics, or the life or social sciences.
The text is written for a student with two years of high school algebra. A wide range of topics is included, giving the instructor considerable flexibility in designing a course. Since the text uses technology as a major tool, the reader is required to use a computer or a graphing calculator. The Student’s Suite CD with the text, gives all the details, in user friendly terms, needed to use the technology in conjunction with the text.
This text, together with the accompanying Student’s Suite CD, constitutes a completely organized, self-contained, user-friendly set of material, even for students without any knowledge of computers or graphing calculators. Philosophy The writing of this text has been guided by four basic principles, all of which are consistent with the call by national mathematics organizations for reform in calculus teaching and learning. The Rule of Four: Where appropriate, every topic should be presented graph- ically, numerically, algebraically, and verbally. Technology: Incorporate technology into the calculus instruction.