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Calculus
. . . . . . . . . . .
An Integrated Approach to Functions and Their Rates of Change
P R E L I M I N A R Y
E D I T I O N
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Calculus
. . . . . . . . . . .
An Integrated Approach to Functions and Their Rates of Change
P R E L I M I N A R Y
E D I T I O N
ROBIN J. GOTTLIEB
H A R V A R D
U N I V E R S I T Y
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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
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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.
2 3 4 5 6 7 8 9 10—CRS—04 03 02 01
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To my family,
especially my grandmother,
Sonia Gottlieb.
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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 connections between functions and the world around us are important to fostering a conceptual 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.
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viii
Preface
Developing Mathematical Reasoning
Mathematical reasoning skills are developed by learning to make conjectures and convincing 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 arguments, 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
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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 manipulation. In the sense that mathematics is a language, these problems are analogous to 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 formal 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 quantity 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
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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 bx is bx . Part III also takes up polynomials and optimization.
Part IV deals more fully with logarithmic and exponential functions and their derivatives. 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 manipulations. For instance, to differentiate ln(3x 7) the student must rewrite the expression
as ln 3 + 7 ln x . 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. (Convergence 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
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Preface
xi
course. Sections on algebra, the theoretical basis of calculus, including Rolle’s Theorem
and the Mean Value Theorem, induction, conics, l’Hˆopital’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
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 also want 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
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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 sense mathematics is a language—a way to communicate. You can think of
some of your mathematics work as a language lab. Learning any language requires active
practice; it requires drill; it requires expressing your own thoughts in that language. But
mathematics is more than simply a language. Mathematics is born from inquiry. New
mathematics arises from problem solving and from pushing out the boundaries of what is
known. Questioning leads to the expansion of knowledge; it is the heart of academic pursuit.
From one question springs a host of other questions. Like a branching road, a single inquiry
can lead down multiple paths. A path may meander, may lead to a dead-end, detour into
fascinating terrain, or steer a straight course toward your destination. The art of questioning,
coupled with some good, all-purpose problem-solving skills, may be more important than
any neatly packaged set of facts you have tucked under your arm as you stroll away from
your studies at the end of the year. For this reason, the text is not a crisp, neatly packed
and ironed set of facts. But because you will want to carry away something you can use for
reference in the future, this book will supply some concise summaries of the conclusions
reached as a result of the investigations in it.
We, the author and your instructors, would like you to leave the course equipped with
a toolbox of problem-solving skills and strategies—skills and strategies that you have tried
and tested throughout the year. We encourage you to break down the complex problems you
tackle into a sequence of simpler pieces that can be put together to construct a solution. We
urge you to try out your solutions in simple concrete cases and to use numerical, graphical,
and analytic methods to investigate problems. We ask that you think about the answers you
get, compare them with what you expect, and decide whether your answers are reasonable.
Many students will use the mathematics learned in this text in the context of another
discipline: biology, medicine, environmental science, physics, chemistry, economics, or
one of the social sciences. Therefore, the text offers quite a bit of mathematical modeling—
working in the interface between mathematics and other disciplines. Sometimes modeling
is treated as an application of mathematics developed, but frequently practical problems
from other disciplines provide the questions that lead to the development of mathematical
ideas and tools.
To learn mathematics successfully you need to actively involve yourself in your studies
and work thoughtfully on problems. To do otherwise would be like trying to learn to be a
good swimmer without getting in the water. Of course, you’ll need problems to work on.
But you’re in luck; you have a slew of them in front of you. Enjoy!
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Contents
Preface vii
PART I
CHAPTER 1
Functions: An Introduction 1
Functions Are Lurking Everywhere 1
1.1
◆
1.2
1.3
CHAPTER 2
4
Characterizing Functions and Introducing Rates of Change 49
2.1
CHAPTER 3
Functions Are Everywhere 1
EXPLORATORY PROBLEMS FOR CHAPTER 1: Calibrating Bottles
What Are Functions? Basic Vocabulary and Notation 5
Representations of Functions 15
2.2
2.3
Features of a Function: Positive/Negative, Increasing/Decreasing,
Continuous/Discontinuous 49
A Pocketful of Functions: Some Basic Examples 61
Average Rates of Change 73
◆
2.4
2.5
EXPLORATORY PROBLEMS FOR CHAPTER 2: Runners 82
Reading a Graph to Get Information About a Function 84
The Real Number System: An Excursion 95
Functions Working Together 101
3.1
3.2
3.3
◆
3.4
Combining Outputs: Addition, Subtraction, Multiplication, and Division of
Functions 101
Composition of Functions 108
Decomposition of Functions 119
EXPLORATORY PROBLEMS FOR CHAPTER 3: Flipping, Shifting, Shrinking, and
Stretching: Exercising Functions 123
Altered Functions, Altered Graphs: Stretching, Shrinking, Shifting,
and Flipping 126
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Contents
PART II
CHAPTER 4
Rates of Change: An Introduction to the Derivative 139
Linearity and Local Linearity 139
4.1
4.2
4.3
Making Predictions: An Intuitive Approach to Local Linearity
Linear Functions 143
Modeling and Interpreting the Slope 153
◆
EXPLORATORY PROBLEM FOR CHAPTER 4: Thomas Wolfe’s Royalties for
The Story of a Novel 158
Applications of Linear Models: Variations on a Theme 159
4.4
CHAPTER 5
CHAPTER 6
The Derivative Function 169
5.1
5.2
5.3
Calculating the Slope of a Curve and Instantaneous Rate of Change
The Derivative Function 187
Qualitative Interpretation of the Derivative 194
◆
5.4
EXPLORATORY PROBLEMS FOR CHAPTER 5: Running Again
Interpreting the Derivative: Meaning and Notation 208
CHAPTER 8
169
206
The Quadratics: A Profile of a Prominent Family of Functions
6.1
6.2
◆
6.3
6.4
CHAPTER 7
139
217
A Profile of Quadratics from a Calculus Perspective 217
Quadratics From A Noncalculus Perspective 223
EXPLORATORY PROBLEMS FOR CHAPTER 6: Tossing Around Quadratics
Quadratics and Their Graphs 231
The Free Fall of an Apple: A Quadratic Model 237
226
The Theoretical Backbone: Limits and Continuity 245
7.1
7.2
7.3
7.4
Investigating Limits—Methods of Inquiry and a Definition 245
Left- and Right-Handed Limits; Sometimes the Approach Is Critical 258
A Streetwise Approach to Limits 265
Continuity and the Intermediate and Extreme Value Theorems 270
◆
EXPLORATORY PROBLEMS FOR CHAPTER 7: Pushing the Limit
275
Fruits of Our Labor: Derivatives and Local Linearity Revisited 279
8.1
◆
8.2
8.3
Local Linearity and the Derivative 279
EXPLORATORY PROBLEMS FOR CHAPTER 8: Circles and Spheres 286
The First and Second Derivatives in Context: Modeling Using Derivatives 288
Derivatives of Sums, Products, Quotients, and Power Functions 290
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Contents
PART III
CHAPTER 9
Exponential, Polynomial, and Rational Functions—
with Applications 303
Exponential Functions
Exponential Growth: Growth at a Rate Proportional to Amount
Exponential: The Bare Bones 309
Applications of the Exponential Function 320
◆
EXPLORATORY PROBLEMS FOR CHAPTER 9: The Derivative of the
Exponential Function 328
The Derivative of an Exponential Function 334
303
Optimization 341
10.1
10.2
10.3
◆
CHAPTER 11
303
9.1
9.2
9.3
9.4
CHAPTER 10
xv
Analysis of Extrema 341
Concavity and the Second Derivative 356
Principles in Action 361
EXPLORATORY PROBLEMS FOR CHAPTER 10: Optimization
A Portrait of Polynomials and Rational Functions
11.1 A Portrait of Cubics from a Calculus Perspective
11.2 Characterizing Polynomials 379
11.3 Polynomial Functions and Their Graphs 391
365
373
373
EXPLORATORY PROBLEMS FOR CHAPTER 11: Functions and Their Graphs:
Tinkering with Polynomials and Rational Functions 404
11.4 Rational Functions and Their Graphs 406
◆
PART IV
CHAPTER 12
Inverse Functions: A Case Study of Exponential
and Logarithmic Functions 421
Inverse Functions: Can What Is Done Be Undone? 421
12.1 What Does It Mean for f and g to Be Inverse Functions?
12.2 Finding the Inverse of a Function 429
12.3 Interpreting the Meaning of Inverse Functions 434
◆
CHAPTER 13
421
EXPLORATORY PROBLEMS FOR CHAPTER 12: Thinking About the Derivatives of
Inverse Functions 437
Logarithmic Functions
439
13.1 The Logarithmic Function Defined 439
13.2 The Properties of Logarithms 444
13.3 Using Logarithms and Exponentiation to Solve Equations
449
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Contents
CHAPTER 14
◆
EXPLORATORY PROBLEM FOR CHAPTER 13: Pollution Study
13.4 Graphs of Logarithmic Functions: Theme and Variations 462
458
Differentiating Logarithmic and Exponential Functions
467
14.1 The Derivative of Logarithmic Functions 467
◆
EXPLORATORY PROBLEM FOR CHAPTER 14: The Derivative of the
Natural Logarithm 468
14.2 The Derivative of bx Revisited 473
14.3 Worked Examples Involving Differentiation 476
CHAPTER 15
Take It to the Limit 487
15.1 An Interesting Limit 487
15.2 Introducing Differential Equations
◆
PART V
CHAPTER 16
497
EXPLORATORY PROBLEMS FOR CHAPTER 15: Population Studies
Adding Sophistication to Your Differentiation 513
Taking the Derivative of Composite Functions
16.1 The Chain Rule 513
16.2 The Derivative of x n where n is any Real Number
16.3 Using the Chain Rule 523
◆
CHAPTER 17
CHAPTER 18
513
521
EXPLORATORY PROBLEMS FOR CHAPTER 16: Finding the Best Path
Implicit Differentiation and its Applications 535
17.1
17.2
17.3
17.4
PART VI
507
Introductory Example 535
Logarithmic Differentiation 538
Implicit Differentiation 541
Implicit Differentiation in Context: Related Rates of Change
An Excursion into Geometric Series 559
Geometric Sums, Geometric Series 559
18.1
18.2
18.3
18.4
18.5
Geometric Sums 559
Infinite Geometric Series 566
A More General Discussion of Infinite Series 572
Summation Notation 575
Applications of Geometric Sums and Series 579
550
528
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Contents
PART VII
CHAPTER 19
Trigonometric Functions 593
Trigonometry: Introducing Periodic Functions
19.1
19.2
19.3
19.4
CHAPTER 20
593
The Sine and Cosine Functions: Definitions and Basic Properties
Modifying the Graphs of Sine and Cosine 603
The Function f (x) = tan x 615
Angles and Arc Lengths 619
594
Trigonometry—Circles and Triangles 627
20.1
20.2
20.3
20.4
20.5
Right-Triangle Trigonometry: The Definitions 627
Triangles We Know and Love, and the Information They Give Us 635
Inverse Trigonometric Functions 645
Solving Trigonometric Equations 651
Applying Trigonometry to a General Triangle: The Law of Cosines and the
Law of Sines 657
20.6 Trigonometric Identities 667
20.7 A Brief Introduction to Vectors 671
CHAPTER 21
Differentiation of Trigonometric Functions
683
21.1 Investigating the Derivative of sin x Graphically, Numerically, and
Using Physical Intuition 683
21.2 Differentiating sin x and cos x 688
21.3 Applications 695
21.4 Derivatives of Inverse Trigonometric Functions 703
21.5 Brief Trigonometry Summary 707
PART VIII
CHAPTER 22
Integration: An Introduction 711
Net Change in Amount and Area: Introducing the
Definite Integral 711
22.1
22.2
22.3
22.4
CHAPTER 23
Finding Net Change in Amount: Physical and Graphical Interplay 711
The Definite Integral 725
The Definite Integral: Qualitative Analysis and Signed Area 731
Properties of the Definite Integral 738
The Area Function and Its Characteristics 743
x
23.1 An Introduction to the Area Function a f (t) dt
23.2 Characteristics of the Area Function 747
23.3 The Fundamental Theorem of Calculus 757
743
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xviii
Contents
CHAPTER 24
The Fundamental Theorem of Calculus
761
24.1 Definite Integrals and the Fundamental Theorem 761
24.2 The Average Value of a Function: An Application of the Definite Integral
PART IX
CHAPTER 25
775
Applications and Computation of the Integral 783
Finding Antiderivatives—An Introduction to Indefinite
Integration 783
25.1 A List of Basic Antiderivatives 783
25.2 Substitution: The Chain Rule in Reverse 787
25.3 Substitution to Alter the Form of an Integral 798
CHAPTER 26
Numerical Methods of Approximating Definite Integrals
805
26.1 Approximating Sums: Ln, Rn, Tn, and Mn 805
26.2 Simpson’s Rule and Error Estimates 820
CHAPTER 27
Applying the Definite Integral: Slice and Conquer 827
27.1 Finding “Mass” When Density Varies 827
27.2 Slicing to Find the Area Between Two Curves
CHAPTER 28
843
More Applications of Integration 853
28.1 Computing Volumes 853
28.2 Arc Length, Work, and Fluid Pressure: Additional Applications of the Definite
Integral 865
CHAPTER 29
Computing Integrals
29.1
29.2
29.3
29.4
PART X
CHAPTER 30
877
Integration by Parts—The Product Rule in Reverse 877
Trigonometric Integrals and Trigonometric Substitution 886
Integration Using Partial Fractions 898
Improper Integrals 903
Series 919
Series 919
30.1
30.2
30.3
30.4
30.5
Approximating a Function by a Polynomial 919
Error Analysis and Taylor’s Theorem 934
Taylor Series 941
Working with Series and Power Series 952
Convergence Tests 964
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Contents
CHAPTER 31
Differential Equations
31.1
31.2
31.3
31.4
31.5
31.6
983
Introduction to Modeling with Differential Equations 983
Solutions to Differential Equations: An Introduction 991
Qualitative Analysis of Solutions to Autonomous Differential Equations 1002
Solving Separable First Order Differential Equations 1018
Systems of Differential Equations 1024
Second Order Homogeneous Differential Equations with Constant Coefficients
1045
Appendices 1051
APPENDIX A
Algebra 1051
A.1
A.2
A.3
Introduction to Algebra: Expressions and Equations
Working with Expressions 1056
Solving Equations 1070
1051
APPENDIX B
Geometric Formulas 1085
APPENDIX C
The Theoretical Basis of Applications of the Derivative 1087
APPENDIX D
Proof by Induction 1095
APPENDIX E
Conic Sections
E.1
E.2
E.3
APPENDIX F
1099
Characterizing Conics from a Geometric Viewpoint
Defining Conics Algebraically 1101
The Practical Importance of Conic Sections 1106
1100
ˆ
L’Hopital’s
Rule: Using Relative Rates of Change to
Evaluate Limits 1111
F.1
Indeterminate Forms
1111
APPENDIX G
Newton’s Method: Using Derivatives to Approximate Roots
APPENDIX H
Proofs to Accompany Chapter 30, Series 1127
Index 1133
1121
xix
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P A R T
Functions: An Introduction
I
C
H
1
A
P
T
E
R
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 rst 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 ow 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 rst chapter.
1
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CHAPTER 1
Functions Are Lurking Everywhere
Mathematical modeling involves constructing mathematical machines that mimic important characteristics of commonly occurring phenomena. Chemists, biologists, environmental 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 environmental 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 inputoutput 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 speci c temperature is the
input; the output is the volume of one gram of mercury at that speci c 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 de ne 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
traf c, 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 particular bottle we can say that the height of the liquid in the bottle is a
function of the volume of the liquid; that is, height (output) is completely determined by
volume (input). We use pictures to illustrate the relationship. Tracking the input variable
along the horizontal axis and the output variable along the vertical axis is a mathematical
convention for displaying graphs of functions.
1 While physicists hope to uncover physical laws, economists and other social scientists often aim for some working understanding that can be applied appropriately.
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1.1
Functions Are Everywhere
3
Height (output)
Volume
(input)
Figure 1.1
Calibration of conical ask
The concept of a function is important and versatile. We can attach many different
mental pictures and representations to it; our choice of representation will often depend
upon context. Upon rst exposure, the notation and representations may be confusing, but
as you use the language of functions the nuances will become as natural as nuances in the
English language.
Aside: In any discipline accurate communication is critical; in mathematics a great deal of
care goes into de nitions because the eld is structured so that the validity of arguments rests on
logic, de nitions, and a few postulates. De nitions arise because they are needed in order to make
precise, unambiguous statements. Sometimes a formal de nition can initially seem unnatural to
you, usually because the person who made the de nition wanted to be sure to include (or exclude)
a certain situation that hasn t yet crossed your mind. To circumvent this problem, in this text we
will sometimes begin with an informal de nition and formalize it later.
Just in case you are not kindly disposed to learning the language of functions, we ll take a
brief foray into English word usage to put mathematical language in perspective. Consider the
word subway, meaning an underground railway. The word conveys a meaning; we associate
it with the physical object. A Bostonian, a New Yorker, and a Tokyo commuter might each have
a slightly different mental image but the essence is similar. We have convenient shorthand
notations for subway. In Boston, people refer to the subway as the T. If there is a T symbol
with stairs going downward, that indicates a subway stop; if you see a T symbol on a street
sign, with no stairs in sight, chances are that you re at a bus stop. The symbol takes on a life of
its own when a Bostonian says, I ll take the ‘T downtown. On the other hand, New Yorkers
look for an M (for Metropolitan Transit Authority) when they want to nd a subway. But a
New Yorker never says, I ll take the ‘M downtown. To make matters muddier, in London
a subway refers to an underground walkway, while the underground rail is referred to as the
tube. Adding to the general zaniness of usage is a chain of subway shops selling submarine
(hero, or grinder) sandwiches. And if you think you can clarify everything by switching to the
term underground railroad, think again: Harriet Tubman s underground railroad was something
altogether different. Compared with this murky tangle, mathematical notation and usage may
provide lucid relief.
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CHAPTER 1
Functions Are Lurking Everywhere
Exploratory Problems for Chapter 1
Calibrating Bottles
From The Language of Functions and Graphs: An Examination Module for Secondary
Schools, 1985, Shell Centre for Mathematical Education.
