Proofs to Accompany Chapter 30, Series 1131 lim n→∞ S n ≤ lim n→∞ a 1 + n 1 f(x)dx =a 1 +A The sequence of partial sums is bounded and increasing. Therefore, by the Bounded Increasing Partial Sums Theorem, ∞ k=1 a k converges. Suppose ∞ 1 f(x)dx =∞.Refer to Equation (H.1) to obtain n−1 1 f(x)dx ≤a 1 +a 2 +···+a n =S n Taking the limit as n →∞gives ∞≤ lim n→∞ S n . So S n grows without bound and ∞ k=1 a k diverges. Now we show that the behavior of the integral can be determined by that of the series. Because f(x)>0,decreasing, and continuous on [1, ∞), lim b→∞ b 1 f(x)dx is either finite or grows without bound. Therefore, if we can find an upper bound, the integral converges. If it has no upper bound, it diverges. Suppose ∞ k=1 a k converges. Denote its sum by S. From Equation (H.1) we know n 1 f(x)dx ≤a 1 +a 2 +···+a n−1 lim n→∞ n 1 f(x)dx ≤ lim n→∞ n−1 k=1 a k = S. If lim n→∞ n 1 f(x)dx is bounded, so too is lim b→∞ b 1 f(x)dx (given the hypothe- ses). Suppose ∞ k=1 a k diverges. Because the terms are all positive, we know lim n→∞ n k=1 a k =∞.From Equation (H.1) we know a 2 +···+a n ≤ n 1 f(x)dx lim n→∞ n k=2 a k ≤ lim n→∞ n 1 f(x)dx. Weconclude that lim b→∞ b 1 f(x)dx =∞;the improper integral diverges. Index Abel, Niels, 382, 1078n Absolute convergence conditional, 953 explanation of, 952–953 implies convergence, 1128–1129 Absolute maximum point, 347 Absolute maximum value, 252, 347 Absolute minimum point, 347 Absolute minimum value, 347, 352 Absolute value function explanation of, 61–62 maximum and minimum and, 350 Absolute values analytic principle for working with, 66 elements of, 65 functions and, 129 geometric principle for working with, 66–68 Acceleration due to force of gravity, 150n, 406–407 explanation of, 76 Accruement, 746 Accumulation, 746 Addition of functions, 101–103 principles of, 1059–1061 Addition formulas, 668–669, 709 Additive integrand property, 738 Algebra equations and, 1053 explanation of, 1051–1052 exponential, 247, 309–312 exponents and, 1053–1054 expressions and, 1052–1053, 1056–1070 order of operations and, 1054 solving equations using, 1071–1083 (See also Equations) square roots and, 1054–1055 Alternating series error estimate, 954–955 explanation of, 953 Alternating series test, 953–956, 977 Amount added, 746 Amplitude definition of, 603 modifications to, 604, 606 Ancient Egyptians, 95, 97n, 627, 854 Angles complementary, 630 of depression, 631 of elevation, 631 initial side of, 619 measurement of, 619–622 right, 620 terminal side of, 619–620 trigonometric functions of, 622–623 vertex of, 619 Antiderivatives definition of, 761, 762 for integrand, 805–806 list of basic, 783–786 table of, 789–790 use of, 763, 764, 766, 770, 771 Antidifferentiate, 805 Approximations constant, 920 of definite integrals, 805–816, 820–825 Euler’s method and, 1022 examples of, 828–829 higher degree, 923–924 linear, 159, 163 local linearity and, 279–282 net change, 715–718 Newton’s method, 1121–1125 polynomial, 693–694, 919–931 second degree, 921–922 successive, 170, 176, 208, 715–718 tangent line, 282–284, 920–921, 937 Taylor polynomial, 924–931 third degree, 922–923 Arbitrarily close, 249 Arbitrarily small, 249, 250 Arccosine, 646, 647 Archimedes, 97n, 245, 1079n Arc length definite integrals and, 865–867 definition of, 594, 708 explanation of, 621–622 Arcsine, 646, 647 Arctangent, 646 Area of circle, 17 of oblique angles, 662–663 of oblique triangles, 662–664 slicing to find, 843–845 Area function amount added, accumulation, accruement and, 745 characteristics of, 747–755 definition of, 745 explanation of, 743–744 general principles of, 744–745 Astronomy, 1107 Asymptotes horizontal, 64–65, 407–409, 411 overview of, 64–65 vertical, 64, 407, 408, 617 Autonomous differential equations explanation of, 997 qualitative analysis of solutions to, 1002–1014 Average rate of change calculation of, 170 definition of, 75 explanation of, 73–76, 176 Average value definition of, 777 of functions, 775–780 Rates of Change and Behavior of Graphs Rates of Change and Behavior of Graphs By: OpenStaxCollege Gasoline costs have experienced some wild fluctuations over the last several decades [link] http://www.eia.gov/totalenergy/data/annual/showtext.cfm?t=ptb0524 Accessed 3/5/ 2014 lists the average cost, in dollars, of a gallon of gasoline for the years 2005–2012 The cost of gasoline can be considered as a function of year y 2005 2006 2007 2008 2009 2010 2011 2012 C(y) 2.31 2.62 2.84 3.30 2.41 2.84 3.58 3.68 If we were interested only in how the gasoline prices changed between 2005 and 2012, we could compute that the cost per gallon had increased from $2.31 to $3.68, an increase of $1.37 While this is interesting, it might be more useful to look at how much the price changed per year In this section, we will investigate changes such as these Finding the Average Rate of Change of a Function The price change per year is a rate of change because it describes how an output quantity changes relative to the change in the input quantity We can see that the price of gasoline in [link] did not change by the same amount each year, so the rate of change was not constant If we use only the beginning and ending data, we would be finding the average rate of change over the specified period of time To find the average rate of change, we divide the change in the output value by the change in the input value 1/30 Rates of Change and Behavior of Graphs Average rate of change = Change in output Change in input = Δy Δx = y2 − y1 x2 − x1 = f(x2) − f(x1) x2 − x1 The Greek letterΔ (delta) signifies the change in a quantity; we read the ratio as “delta-y over delta-x” or “the change in y divided by the change in x.” Occasionally we write Δf instead of Δy, which still represents the change in the function’s output value resulting from a change to its input value It does not mean we are changing the function into some other function In our example, the gasoline price increased by $1.37 from 2005 to 2012 Over years, the average rate of change was Δy $1.37 = ≈ 0.196 dollars per year Δx years On average, the price of gas increased by about 19.6¢ each year Other examples of rates of change include: • A population of rats increasing by 40 rats per week • A car traveling 68 miles per hour (distance traveled changes by 68 miles each hour as time passes) • A car driving 27 miles per gallon (distance traveled changes by 27 miles for each gallon) • The current through an electrical circuit increasing by 0.125 amperes for every volt of increased voltage • The amount of money in a college account decreasing by $4,000 per quarter A General Note Rate of Change A rate of change describes how an output quantity changes relative to the change in the input quantity The units on a rate of change are “output units per input units.” The average rate of change between two input values is the total change of the function values (output values) divided by the change in the input values 2/30 Rates of Change and Behavior of Graphs Δy Δx = ( ) ( ) f x2 − f x1 x2 − x1 How To Given the value of a function at different points, calculate the average rate of change of a function for the interval between two values x1 and x2 Calculate the difference y2 − y1 = Δy Calculate the difference x2 − x1 = Δx Δy Find the ratio Δx Computing an Average Rate of Change Using the data in [link], find the average rate of change of the price of gasoline between 2007 and 2009 In 2007, the price of gasoline was $2.84 In 2009, the cost was $2.41 The average rate of change is Δy y2 − y1 = Δx x2 − x1 = $2.41 − $2.84 2009 − 2007 = − $0.43 years = − $0.22 per year Analysis Note that a decrease is expressed by a negative change or “negative increase.” A rate of change is negative when the output decreases as the input increases or when the output increases as the input decreases Try It Using the data in [link], find the average rate of change between 2005 and 2010 $2.84 − $2.31 years = $0.53 years = $0.106 per year Computing Average Rate of Change from a Graph Given the function g(t) shown in [link], find the average rate of change on the interval [ − 1, 2] 3/30 Rates of Change and Behavior of Graphs At t = − 1, [link] shows g(−1) = At t = 2, the graph shows g(2) = The horizontal change Δt = is shown by the red arrow, and the vertical change Δg(t) = − is shown by the turquoise arrow The output changes by –3 while the input changes by 3, giving an average rate of change of 1−4 −3 = = −1 − ( − 1) Analysis 4/30 Rates of Change and Behavior of Graphs Note that the order we choose is very important If, for example, we use y2 − y1 x1 − x2 , we will not get the correct answer Decide which point will be and which point will be 2, and keep the coordinates fixed as (x1, y1) and (x2, y2) Computing Average Rate of Change from a Table After picking up a friend who lives 10 miles away, Anna records her distance from home over time The values are shown in [link] Find her average speed over the first hours t ...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 Preface xi course. Sections on algebra, the theoretical basis of calculus, including Rolle’s Theorem and the Mean Value Theorem, induction, conics, l’H ˆ opital’s Rule for using derivatives to evaluate limits of an indeterminate form, and Newton’s method of using derivatives to approximate roots constitute Appendices A, C, D, E, F, and G, respectively. Certain appendices can be transported directly into the course. Others can be used as the basis of independent student projects. This book is a preliminary edition and should be viewed as a work in progress. The exposition and choice and sequencing of topics have evolved over the years and will, I expect, continue to evolve. I welcome instructors’ and students’ comments and suggestions on this edition. I can be contacted at the addresses given below. Robin Gottlieb Department of Mathematics 1 Oxford Street Cambridge, MA 02138 gottlieb@math.harvard.edu Acknowledgments A work in progress incurs many debts. I truly appreciate the good humor that participants have shown while working with an evolving course and text. For its progress to this point I’d like to thank all my students and all my fellow instructors and course assistants for their feedback, cooperation, help, and enthusiasm. They include Kevin Oden, Eric Brussel, Eric Towne, Joseph Harris, Andrew Engelward, Esther Silberstein, Ann Ryu, Peter Gilchrist, Tamara Lefcourt, Luke Hunsberger, Otto Bretscher, Matthew Leerberg, Jason Sunderson, Jeanie Yoon, Dakota Pippins, Ambrose Huang, and Barbara Damianic. Special thanks to Eric Towne, without whose help writing course notes in the academic year 1996-1997 this text would not exist. Special thanks also to Eric Brussel whose support for the project has been invaluable, and Peter Gilchrist whose help this past summer was instrumental in getting this preliminary edition ready. Thanks to Matt Leingang and Oliver Knill for technical assistance, to Janine Clookey and Esther Silberstein for start-up assistance, and to everyone in the Harvard Mathematics department for enabling me to work on this book over these past years. I alsowant to acknowledge the type-setting assistance of Paul Anagnostopoulos, Renata D’Arcangelo, Daniel Larson, Eleanor Williams, and numerous others. For the art, I’d like to acknowledge the work of George Nichols, and also of Ben Stephens and Huan Yang. For their work on solutions, thanks go to Peter Gilchrist, Boris Khentov, Dave Marlow, and Sean Owen and coworkers. My thanks to the team at Addison-Wesley for accepting the assortment of materials they were given and carrying out the Herculean task of turning it into a book, especially to Laurie Rosatone for her encouragement and confidence in the project and Ellen Keohane for her assistance and coordination efforts. It has been a special pleasure to work with Julie LaChance in production; I appreciate her effort and support. Thanks also to Joe Vetere, Caroline Fell, Karen Guardino, Sara Anderson, Michael Boezi, Susan Laferriere, and Barbara Atkinson. And thanks to Elka Block and Frank Purcell, for their comments and suggestions. Finally, I want to thank the following people who reviewed this preliminary edition: Dashan Fan, University of Wisconsin, Milwaukee Baxter Johns, Baylor University Michael Moses, George Washington University xii Preface Peter Philliou, Northeastern University Carol S. Schumacher, Kenyon College Eugene Spiegel, The University of Connecticut Robert Stein, California State University, San Bernardino James A. Walsh, Oberlin College To the Student This text has multiple goals. To begin with, you should learn calculus. Your understanding should be deep; you ought to feel it in your bones. Your understanding should be portable; you ought to be able to take it with you and apply it in a variety of contexts. Mathematicians find mathematics exciting and beautiful, and this book may, I hope, provide you with a window through which to see, appreciate, and even come to share this excitement. In some PART I Functions: An Introduction 1 CHAPTER Functions Are Lurking Everywhere 1.1 FUNCTIONS ARE EVERYWHERE Each of us attempts to make sense out of his or her environment; this is a fundamental human endeavor. We think about the variables characterizing our world; we measure these variables and observe how one variable affects another. For instance, a child, in his first years of life, names and categorizes objects, people, and sensations and looks for predictable relationships. As a child discovers that a certain phenomenon precipitates a predictable outcome, the child learns. The child learns that the position of a switch determines whether a lamp is on or off, and that the position of a faucet determines the flow of water into a sink. The novice musician learns that hitting a piano key produces a note, and that which key is hit determines which note is heard. The deterministic relationship between the piano key hit and the resulting note is characteristic of the input-output relationship that is the object of our study in this first chapter. 1 2 CHAPTER 1 Functions Are Lurking Everywhere Mathematical modeling involves constructing mathematical machines that mimic im- portant characteristics of commonly occurring phenomena. Chemists, biologists, environ- mental scientists, economists, physicists, engineers, computer scientists, students, and parents all search for relationships between measurable variables. 1 A chemist might be interested in the relationship between the temperature and the pressure of a gas, an environ- mental scientist in the relationship between use of pesticides and mortality rate of songbirds, a physician in the relationship between the radius of a blood vessel and blood pressure, an economist in the relationship between the quantity of an item purchased and its price, a grant manager in the relationship between funds allocated to a program and results achieved. A thermometer manufacturer must know the relationship between the temperature and the volume of a gram of mercury in order to calibrate a thermometer. The list is endless. As human beings trying to make sense of a complex world, we instinctively try to identify relationships between variables. We will concern ourselves here with relationships that can be structured as input- output relationships with the special characteristic that the input completely determines the output. For example, consider the relationship between the temperature and the volume of a gram of mercury. We can structure this relationship by considering the input variable to be temperature and the output variable to be volume. A specific temperature is the input; the output is the volume of one gram of mercury at that specific temperature. The temperature determines the volume. As another example, consider a hot-drink machine. If your inputs are inserting a dollar bill and pressing the button labeled “hot chocolate,” the output will be a cup of hot cocoa and 55¢ in change. In such a machine the input completely determines the output. The mathematical machine used to model such relations is called a function. Mathematicians define a function as a relationship of inputs and outputs in which each input is associated with exactly one output. Notice that the mathematical use of the word “function” and its use in colloquial English are not identical. In colloquial English we might say, “The number of hours it takes to drive from Boston to New York City is a function of the time one departs Boston.” By this we mean that the trip length depends on the time of departure. But the trip length is not uniquely determined by the departure time; holiday traffic, accidents, and road construction play roles. Therefore, in a mathematical sense the length of the trip is not a function of the departure time. Think about the task of calibrating a bottle, marking it so that it can subsequently be used for measuring. The calibration function takes a volume as input and gives a height as output. For any 1.2 What Are Functions? Basic Vocabulary and Notation 11 (g) The rule assigns to every number the square of that number. (h) The rule assigns to every nonzero number the reciprocal of that number. 3. (a) Which of the following maps are maps of functions? I II III IV aaaaaa bbbbb cc c x b b c c dddccd eezde (b) For each function, determine the range and the domain. 4. Which of the functions in Problem 3 are 1-to-1? 5. There are infinitely many prime numbers. This has been known for a long time; Euclid proved it sometime between 300 b.c. and 200 b.c. 6 Number theorists (mathematicians who study the theory and properties of numbers) are interested in the distribution of prime numbers. Let P (n) = number of primes less than or equal to n, where n is a positive number. Is P (n) a function? Explain. 6. Writing: We would like to tailor this course to your needs and interests; therefore we’d like to find out more about what these needs and interests are. On a sheet of paper separate from the rest of your homework, please write a paragraph or two telling us a bit about yourself by addressing the following questions. (a) What are you interested in studying in the future, both in terms of math and otherwise? (b) Are there things you have found difficult or confusing in mathematics in the past? If so, what? (c) What was your approach to studying mathematics in the past? Did it work well for you? (d) What are your major extracurricular activities or interests? (e) What do you hope to get out of this course? For Problems 7 through 9 determine whether the relationship described is a function. If the relationship is a function, (a) what is the domain? the range? (b) is the function 1-to-1? 6 The proof went like this: Assume there are finitely many primes: 2, 3, 5, 7 ,N, where N is the largest prime. Let M = 2 · 3 · 5 · 7 · N +1; i.e., let M be 1 bigger than the product of all the primes. Then M is not divisible by any of the primes on the list. But that means the list doesn’t include all of the prime numbers. Whoops! We have a contradiction. Therefore, our assumption that there are finitely many primes is false. So there must be infinitely many primes. 12 CHAPTER 1 Functions Are Lurking Everywhere 7. Input Output 8. Input Output 9. Input Output 02 √ 22 √ 20 13 √ 33 2 √ 20 22 √ 55 3 √ 20 33 √ 66 4 √ 20 42 10. Express each of the following rules for obtaining the output of a function using func- tional notation. (a) Square the input, add 3, and take the square root of the result. (b) Double the input, then add 7. (c) Take half of 3 less than the input. (d) Increase the input by 10, then cube the result. 11. Let C be a circle of radius 1 and let A(n) be the area of a regular n-gon inscribed in the circle. For instance, A(3) is the area of an equilateral triangle inscribed in circle C, A(4) is the area of a square inscribed in circle C, and A(5) is the area of a regular pentagon inscribed in circle C. (A polygon inscribed in a circle has all its vertices lying on the circle. A regular polygon is a polygon whose sides are all of equal length and whose angles are all of equal measure.) (a) Find A(4). (b) Is A(n) a function? If it is, answer the questions that follow. (c) What is the natural domain of A(n)? (d) As n increases, do you think that A(n) increases, or decreases? This is hard to justify rigorously, but what does your intuition tell you? (e) Will A(n) increase without bound as n increases, or is there a lid above which the values of A(n) will never go? If there is such a lid (called an upper bound) give one. What is the smallest lid possible? Rigorous justification is not requested. 12. Some friends are taking a long car trip. They are traveling east on Route 66 from Flagstaff, Arizona, through New Mexico and Texas and into Oklahoma. Let f be the function that gives the number of miles traveled t hours into the trip, where t = 0 denotes the beginning of the trip. For instance, f(7)is the mileage 7 hours into the trip. If the travelers set an ... functions and discuss their graphical behavior in [link], [link], and [link] 14/30 Rates of Change and Behavior of Graphs 15/30 Rates of Change and Behavior of Graphs 16/30 Rates of Change and Behavior. .. instruction and practice with rates of change • Average Rate of Change Key Equations Average rate of change Δy Δx = ( ) ( ) f x2 − f x1 x2 − x1 19/30 Rates of Change and Behavior of Graphs Key... 5/30 Rates of Change and Behavior of Graphs Now we compute the average rate of change Average rate of change = f(4) − f(2) 4−2 63 −2 = 4−2 = = 49 49 Try It Find the average rate of change of f(x)