Calculus An Integrated Approach to Functions and Their Rates of Change PRELIMINARY EDITION Calculus An Integrated Approach to Functions and Their Rates of Change PRELIMINARY EDITION ROBIN J. GOTTLIEB HARVARD UNIVERSITY Sponsoring Editor: Laurie Rosatone Managing Editor: Karen Guardino Project Editor: Ellen Keohane Marketing Manager: Michael Boezi Manufacturing Buyer: Evelyn Beaton Associate Production Supervisor: Julie LaChance Cover Design: Night and Day Design Cover Art: The Japanese Bridge by Claude Monet; Suzuki Collection, Tokyo/Superstock Interior Design: Sandra Rigney Senior Designer: Barbara Atkinson Composition: Windfall Software Library of Congress Cataloging-in-Publication Data Gottlieb, Robin (Robin Joan) Calculus: an integrated approach to functions and their rates of change / by Robin Gottlieb.—Preliminary ed. p. cm. ISBN 0-201-70929-5 (alk. paper) 1. Calculus. I. Title. QA303 .G685 2001 00-061855 515—dc21 Copyright © 2002 by Addison-Wesley Reprinted with corrections. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United States of America. 2345678910—CRS—04 03 02 01 To my family, especially my grandmother, Sonia Gottlieb. Preface The concepts of calculus are intriguing and powerful. Yet for a learner not fluent in the language of functions and their graphs, the learner arriving at the study of calculus poorly equipped, calculus may become a daunting hurdle rather than a fascinating exploration. The impetus to develop a course integrating calculus with material traditionally labeled “precalculus” emerged from years of working with a sequential college system. Few students regard the prospect of taking a precalculus course as inspiring. In the eyes of many students it lacks the glamour and prestige of calculus. For some students, taking precalculus means retaking material “forgotten” from high school, and, bringing the same learning skills to the subject matter, such a student may easily “forget” again. At many colleges, students who successfully complete a precalculus course subsequently enroll in a calculus course that compresses into one semester what their better prepared fellow students have studied back in high school over the course of a full year. Yet any lack of success in such a course is bemoaned by teachers and students alike. The idea behind an integrated course is to give ample time to the concepts of calculus, while also developing the students’ notion of a function, increasing the students’ facility in working with different types of functions, facilitating the accumulation of a robust set of problem-solving skills, and strengthening the students as learners of mathematics and science. An integrated course offers freedom, new possibilities, and an invigorating freshness of outlook. Freshness in particular is valuable for the student who has taken some precalculus (or even calculus) but come away without an understanding of its conceptual underpinnings. This text grew out of an integrated calculus and precalculus course. Three general principles informed the creation of both the course and the text. Developing mathematical reasoning and problem-solving skills must not be made subservient to developing the subject matter. Making connections between mathematical ideas and representations and making con- nections between functions and the world around us are important to fostering a con- ceptual framework that will be both sturdy and portable. Generating intellectual excitement and a sense of the usefulness of the subject matter is important for both the students’ short-term investment in learning and their long-term benefits. vii viii Preface Developing Mathematical Reasoning Mathematical reasoning skills are developed by learning to make conjectures and convinc- ing mathematical arguments. Mathematics, like any other language, must be spoken before being spoken well. Students initially need to learn to make mathematical arguments at their own level, whatever that level may be. The instructor can model logically convincing ar- guments, but if students are not given the opportunity to engage in discussion themselves, they are not likely to acquire discussion skills. The structure of this text is intended to facilitate learning deductive and inductive reasoning, learning to use examples and counterexamples, and learning to understand the usefulness of a variety of perspectives in devising an argument. Students are encouraged to seek patterns and connections, to make conjectures and construct hypotheses. The reflective thinking this fosters helps students develop judgment and confidence. Making Connections Ideas are presented and discussed graphically, analytically, and numerically, as well as in words, with an emphasis throughout on the connections between different representations. There is an emphasis on visual representations. Topics are introduced through examples and, often, via applications and modeling, in order to build connections between mathematics, the students’ experience outside mathematics, and problems in other disciplines, such as economics, biology, and physics. Generating Enthusiasm When the storylines of mathematics get buried under technicalities and carefully polished definitions, both those storylines and the enthusiasm new learners often bring to their studies may well be lost. When all the technical details and theory are laid out in full at the start, students may become lost and, not understanding the subtleties involved, simply suspend judgment and substitute rote memorization. The intrepid learner has more potential than the timid, self-doubting learner. For these reasons, answers to questions students are unready to ask are often omitted. Definitions may be given informally before they are provided formally; likewise, proofs may be given informally or not given in the body of the text but placed in an appendix. In this text, the presentation is not always linear, not all knots are tied immediately, and some loose ends are picked up later. The goal is to have students learn material and to have them keep concepts solidly in their minds, as opposed to setting the material out on paper in a neat and exhaustive form. About the Problems Problems are the heart of any mathematics text. They are the vehicles through which the learner engages with the material. Certainly, they consume the bulk of students’ time and energy. A lot of class time can be constructively spent discussing problems as well. To do mathematics requires reflection, and discussion both encourages and enriches reflection. The first 16 chapters offer “Exploratory Problems.” These are integral to the text, and some are referred to in later sections. Exploratory problems can be incorporated into the course in many ways, but the bottom line remains that they need to be worked and discussed by students. Exploratory problems can be done as in-class group exercises, given as group homework problems, or given as homework to be discussed by the class during the following class meeting. Many of these problems combine or encourage different viewpoints and require the student to move between representations. Some exploratory problems call for Preface ix conjectures that will subsequently be proven; some call for experimentation. The problems attempt to exercise and stretch mathematical reasoning and the process of discussing them is meant to constitute a common core experience of the class. This preliminary edition includes many problems that require basic analytic manipula- tion. In the sense that mathematics is a language, these problems are analogousto vocabulary drills. They are exercises designed to support the less routine problems. But doing only these warm-up exercises would mean missing the spirit of the text and circumventing the goals laid out in this preface. Because problem solving involves determining which tools to use in a given situation, sometimes a few problems at the end of a section may best be solved by using tools from a previous section. The text assumes that students have access to either a graphing calculator or a computer. Technology may be incorporated to a greater or lesser extent depending upon the philosophy and goals of the instructor. Structuring the Content The text covers the equivalent of a precalculus course plus one year of one-variable calculus. Parts I through VII meld precalculus and first-semester calculus. A yearlong course might cover Parts I through VIII and sections of Part IX, although the composition of the syllabus is, of course, at the instructor’s discretion. Part I provides an introduction to functions and their representations with an emphasis on the relationship between meaning and symbolic and graphic representations. From the outset the study of functions and the study of calculus are intertwined. For example, although the first set of exploratory problems requires no particular mathematical knowledge, the ensuing discussion inevitably involves the notion of relative rates of change. Similarly, in extracting information about velocity from a graph of position versus time, or extracting information about relative position from a graph of velocity versus time, students explore the relationship between a function and its derivative without being formally introduced to the derivative. Part II focuses on rates of change and modeling using linear and quadratic functions. Linearity and interpretation of slope precede the derivative and its interpretation. Knowing about lines and the relationship between a function and its derivative provides a new window into quadratics. A chapter devoted to quadratics allows students to work through issues of sign and the relationship between a function and its graph as well as tackle optimization problems both with and without using calculus. Traditionally applied optimization problems appear in a course after all of the for- mal symbolic derivative manipulations have been mastered. Taking on these problems incrementally permits the topic to be revisited multiple times. The most difficult aspect of optimization involves translating the problem into mathematics and expressing the quan- tity to be optimized as a function of a single variable. Part I, Chapter 1 and Part II, Chapters 4 and 6 address these skills. Once students are able to appreciate the usefulness of computing derivatives, the notions of limits and continuity can be addressed more thoroughly in Chapter 7. Chapter 8 builds on that basis, revisiting the idea of local linearity and introducing the Product and Quotient Rules. Part III introduces exponential functions through modeling. These functions are treated early on, because students of biology, chemistry, and economics need facility in dealing with them right away. The derivatives of exponentials are therefore discussed twice, first before the discussion of logarithms and then, more completely, after it. This order leaves some x Preface loose ends, but the subsequent resolution after a few weeks is quite satisfying and makes the natural logarithm seem natural. The number e is introduced as the base for which the derivative of b x is b x . Part III also takes up polynomials and optimization. Part IV deals more fully with logarithmic and exponential functions and their deriva- tives. The number e is revisited here. By design, the Chain Rule is delayed until after the differentiation of the exponential and logarithmic functions. Differentiating these functions without the Chain Rule gives students a lot of practice with logarithmic and exponential ma- nipulations. For instance, to differentiate ln(3x 7 ) the student must rewrite the expression as ln 3 + 7lnx .Chapter 15, which introduces differential equations via the exponential function, can be postponed if the instructor prefers. Part V revisits differentiation by addressing the Chain Rule and implicit differentiation. Part VI provides an excursion into geometric sums and geometric series. A mobile chapter, it can easily be postponed to immediately precede Part X, on series. If, however, the class includes students of economics and biology who will not necessarily study Taylor series, then the students will be well served by studying geometric series. Part VI emphasizes modeling, using examples predominantly drawn from pharmacology and finance. Part VII presents the trigonometric functions, inverse trigonometric functions, and their derivatives. From a practical point of view, this order means that trigonometry is pushed to the second semester. The rationale is twofold. First, some traditional precalculus material must be delayed to make room for the bulk of differential calculus in the first semester. Second, delaying trigonometry has the benefit of returning students to the basics of differential calculus. Too often in a standard calculus course students think about what a derivative is only at the beginning of the course, but by mid-term they are thinking of a derivative as a formula. This text looks at the derivative of sin t from graphic, numeric, analytic, and modeling viewpoints. By the time students have reached Part VII, they are more sophisticated and can follow the more complicated analytic derivations, if the instructor chooses to emphasize them. Delaying trigonometry presents the opportunity to revisit applications previously studied. Students should now have enough confidence to understand that the basic properties of trigonometric functions can be easily derived from the unit circle definitions of sine and cosine; they will be capable of retrieving information forgotten or learning it for the first time without being overwhelmed by detail. Part VIII introduces integration and the Fundamental Theorem of Calculus. There is a geometric flavor to this set of chapters, as well as an emphasis on interpreting the definite integral. Part IX discusses applications and computation of the definite integral, with an emphasis on the notion of slicing, approximating, and summing. Part X focuses on polynomial approximations of functions and Taylor series. (Con- vergence issues are first brought up in Part VI, in the context of a discussion of geometric series.) In Part X the discussion of polynomial approximations motivates the subsequent series discussion. Differential equations are the topic of Part XI. Although the emphasis is on modeling and qualitative behavior, students working through the chapter will come out able to solve separable first order differential equations and second order differential equations with constant coefficients, and they will understand the idea behind Euler’s method. Some discussion of systems of differential equations is also included. The first few sections of Chapter 31 may easily be moved up to follow an introduction to integration and thereby be included at the end of a one-year course. Ending the year with these sections reinforces the basic ideas of differential calculus while simultaneously introducing an important new topic. Certain sections have been made into appendices in order to give the instructor freedom to insert them (or omit them) where they see fit, as determined by the particular goals of the . Calculus An Integrated Approach to Functions and Their Rates of Change PRELIMINARY EDITION Calculus An Integrated Approach to Functions and Their Rates of Change PRELIMINARY EDITION ROBIN. Windfall Software Library of Congress Cataloging-in-Publication Data Gottlieb, Robin (Robin Joan) Calculus: an integrated approach to functions and their rates of change / by Robin Gottlieb. Preliminary. function and its derivative without being formally introduced to the derivative. Part II focuses on rates of change and modeling using linear and quadratic functions. Linearity and interpretation of