CALCULUS
SINGLE VARIABLE
John Wiley & Sons, Inc
ACQUISITIONS EDITOR David Dietz
MARKETING MANAGER Jonathan Cottrell
SENIOR DEVELOPMENT EDITOR Mary O’Sullivan
PROJECT EDITOR Jennifer Brady
SENIOR DESIGNER Jim O’Shea
MEDIA EDITOR Melissa Edwards
PRODUCTION MANAGER Micheline Frederick
SENIOR PRODUCTION EDITOR Ken Santor
This book was set in LaTex by the authors and printed and bound by Courier Kendallville. The cover was printed by Courier Kendallville.
This book is printed on acid free paper.
Copyright © 2011 John Wiley & Sons, Inc. 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, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc. 222 Rosewood Drive, Danvers, MA 01923, website www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030-5774, 6011, fax (201)748-6008, website http//www.wiley.com/go/permissions.
“Evaluation copies are provided to qualifi ed academics and professionals for review purposes only, for use in their courses during the next academic year. These copies are licensed and may not be sold or transferred to a third party. Upon completion of the review period, please return the evaluation copy to Wiley. Return instructions and a free of charge return shipping label are available at www.wiley.com/go/returnlabel. Outside of the United States, please contact your local representative.”
Library of Congress Cataloging in Publication Data:
ISBN 13 978-0470-17930-7
Printed in the United States of America
CONTENTS
Chapter 1. Tools of the Trade 1
1.1. Power Functions 1 1.2. Polynomials 10
1.3. Rational Functions 18
1.4. Algebraic, Composite, and Piecewise Functions 26
1.5. Periodic Functions & Transformations 32
1.6. Exponential Functions 48
1.7. Logarithms 57
1.8. Inverse Functions 68
• Chapter Review 76
• Projects & Applications – Richardson’s Arms Race 83
Chapter 2. Limits and Continuity 89
2.1. Introduction to Limits 89
2.2. Calculating Limits 97
2.3. Limits at Infi nity 107
2.4. The Technical Defi nition of a Limit 116
2.5. Continuity 129
• Chapter Review 140
• Projects & Applications – The Apothem and Limits at Infi nity; Implementing the Bisection Algorithm 144
Chapter 3. The Derivative 147
3.1. The Derivative 148
3.2. Derivatives of Power Functions 160
3.3. Derivatives of Trigonometric and Exponential Functions 169
3.4. The Derivative as a Function 180
3.5. The Product Rule & Quotient Rule 192
3.6. The Chain Rule 202
3.7. Derivatives of Inverse Functions 213
3.8. Implicit and Logarithmic Differentiation 220
• Chapter Review 227
• Projects & Applications – Parabolic Refl ectors; The Cumulative Area Function 233
Chapter 4. Applications of the Derivative 236
4.1. Linear Approximation 236
4.2. Related Rates 251
4.3. Higher Order Derivatives 260
4.4. Local and Global Extrema 273
4.5. Mean Value Theorem & L’Hopital’s Rule 281
4.6. Curve Sketching 294
4.7. Optimization 301
• Chapter Review 312
• Projects & Applications – Implementing
Newton’s Method; Optimization and Apothem; Energy and Force; Quantum Mechanics 318
Chapter 5. Integration 325
5.1. Estimating Net Change 325
5.2. Area and the Defi nite Integral 343
5.3. Numerical Methods of Integration 359
5.4. Antiderivatives & the Fundamental Theorem 371
5.5. More About the Fundamental Theorem 384
• Chapter Review 394
• Projects & Applications – Numerical Integration with Spreadsheets; An Introduction to Stirling’s Approximation; Pollution and Pulmonary Distress 398
Chapter 6. Techniques of Integration 403
6.1. Substitution 403
6.2. Integration by Parts 419
6.3. Overview of Trigonometric Integrals 429
6.4. Trigonometric Substitution 437
6.5. Partial Fractions Decomposition 448
6.6. Improper Integrals 460
• Chapter Review 478
• Projects & Applications – The Gamma Function; The Seasons 482
CONTENTS
7.1. Arc Length & Surface Area 489
7.2. Solids of Revolution & the Method of Slicing 501
7.3. Volumes by Cylindrical Shells 516
7.4. Work, Energy, and Force 525
7.5. Center of Mass and Radius of Gyration 537
7.6. Hydrostatic Force 549
7.7. Average Value 557
7.8. Probability 563
• Chapter Review 576
• Projects & Applications – Ideal Gas, Energy, and the
n-Dimensional Ball; Asteroid Mining; Stopping on a Dime 581
Chapter 8. Sequences & Series 586
8.1. Sequences 586
8.2. Introduction to Series 602
8.3. Integral Test and Estimating Sums 617
8.4. Comparison Tests 628
8.5. Absolute Convergence, Ratio & Root Tests 639
8.6. Introduction to Power Series 649
8.7. Elementary Operations with Power Series 660
8.8. Taylor & Maclaurin Series 669
8.9. Applications of Taylor Series 685
• Chapter Review 696
• Projects & Applications – Revisiting Stirling’s Approximation; Weighing Exploitation Against Retaliation 702
Chapter 9. Parametric Curves, Polar
Coordinates, and Complex Numbers 706
9.1. Parametric Curves 706
9.2. Calculus with Parametric Curves 718
9.3. Polar Coordinates 730
9.4. Calculus in Polar Coordinates 745
9.5. Introduction to Complex Numbers 752
• Chapter Review 764
• Projects & Applications – Planetary Orbits 768
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Chapter 10. Differential Equations 771
10.1. Introduction to Differential Equations 771
10.2. Autonomous Equations and Equilibria 779
10.3. Slope Fields and Euler’s Method 787
10.4. Separable Equations 795
10.5. Linear Equations 804
10.6. Introduction to Dynamical Systems 819
• Chapter Review 831
• Projects & Applications – Crosswind Landing; Pursuit Curve 835
Appendix A Trigonometry 838
Appendix B Review of Algebraic Techniques 853
Appendix C Binomial Coeffi cients 862
Appendix D Conic Sections 869
Appendix E The World We Live In 884
Appendix F End Notes 890
Bibliography 895
Index 897
vii
TO THE STUDENT
During your study of calculus this year, you will likely have questions about why the mathematics works the way it does, where it arises in the real world, and how it’s used by professionals. Based on our experiences with our own students, we’ve tried to anticipate and answer as many of those questions as possible so that formulas and techniques will make sense to you. This is one part of a larger goal. We wrote this text to help you develop new skills and broaden your intuition, and to give you an idea of how mathematics is used to both describe and change our world. But in order to achieve that goal, we need your help. Specifi cally, we need you to read, so in the short space that we have here, we want to offer some advice about reading technical content.Effective reading strategies
The most important thing we can tell you is that learning technical content takes time, and often requires substantial mental effort, so fi rst and foremost, be patient.
Many people fi nd that a scan-then-dig strategy works well when reading technical writing. Your fi rst pass through a section is meant only to familiarize you with the general organization of topics and terms in the section so that you have a mental “road map” of the learning that you’ll do. The second pass is where you begin to dig into the main ideas and develop the relevant skills.
Note our use of the word “dig.” Learning technical material, like the physical act of digging, is an active endeavor that often requires signifi cant effort, even for professionals. Recently when reading a book about applications of physics to the medical sciences, one of us found himself saying, “That doesn’t seem right. That would mean the relationship is something like …” after which there was a lot of writing on scrap paper, a few occurrences of, “Oh …” and fi nally, “Now I get it.”
The point is that in order to really learn something and get good at, you have to do it. Keep a pencil and paper handy when you read so you can check calculations (that
will help you get used to the ways that various terms interact and develop a basic understanding of what they mean), and work through the “Try It Yourself ” examples that we provide so that your reading is an active, not a passive endeavor. If your answer disagrees with ours, and you can’t fi gure out why, we’ve posted full solutions online.
Effective practice
When taking a quiz or exam, and more importantly, later, when you’re actually using calculus to solve problems in the real world, you’ll have to think creatively about how to combine the ideas and techniques that you learn here. You won’t develop that creative mathematical thinking by watching someone else do it. You have to practice it yourself. We’ve already mentioned this kind of thing as part of effective reading strategies, but here we mean to offer advice about making homework exercises effective learning tools.
Specifi cally, many students fi nd that having a solutions manual is helpful, and we have worked with Dr. Justin Young to provide a good one for this text. Our advice is to use it as a stepping stone, rather than a crutch. Each time you complete an exercise by reading part or all of its solution in the manual, try to do a similar exercise on your own. Otherwise, someone else is doing the creative “heavy lifting” for you, and you may end up in the situation of saying, “I don’t understand why I got all of the homework exercises right but did poorly on the exam.”
In closing
Our last piece of advice is to talk about what you’re learning. The best way to solidify and synthesize ideas is often to articulate them to other people. And if “other people” includes your teacher, he or she can help you by hearing and responding to the way that you describe ideas and techniques.
viii
PREFACE
This project started when a former student spoke to us about calculus texts. He said, “The texts that are out there are good reference books, but they don’t actually teach you anything. If I couldn’t get it from the teacher, there’s no way I could learn it from the text.” After a review of the most popular texts on the market today, we understood what he meant. While correct and concise, these texts are written in a style that’s diffi cult to parse for readers who are unaccustomed to technical writing, and they often neglect to provide motivation of their content at a practical level that speaks to modern students. The result is that both students and teachers tend to skim these texts rather than actually read them. Although they make great reference books in later years, during the actual learning process these texts function as little more than very heavy exercise sets.
A calculus text should spark curiosity in its readers and it should hold students’ attention. It should motivate the material in relevant and interesting ways, and answer the spectrum of students’ questions from “How do we do this?” to “Why do we do this?” to “How can I make sense of this?” As a complement to their classroom experience, it should engage students in the discussion and discovery of ideas, develop their intuition, and help them learn about the power, beauty, and versatility of calculus. This book is our effort to provide such a text.
Engaging students
This text is written primarily for students who are interested in physical and medical sciences, mathematics, engineering, or computer science, but also provides inlets to the subject and exercises for students whose interests lie in business or social science. Most students who take calculus are interested in applications, so we often use applications in our examples and in the preliminary discussion of a new topic.
For example, the idea of a limit is introduced using a thought experiment in which we track the position of a planet that passes between us and the sun; the concept of the derivative is introduced using clinical data about the radius of an artery; derivatives of trigonometric functions are motivated by a vibrating guitar string; Riemann sums are introduced to calculate the net displacement of an accelerating vehicle based on data from a radar gun; the idea of a sequence is motivated by concentrations of medication in a patient’s bloodstream; and the list goes on.
We recognize that the applications must be presented at an elementary level, with broad generalizations, so that people who are new to a subject can quickly get a rough understanding without being drawn away from the mathematical topic at hand. At the same time, we have found that students feel a connection to the subject when it
is presented in context, and that they appreciate a glimpse of the rich tapestry that lies beyond what they are learning at the moment.
Calling on students’ experience
Part of engaging students is helping them learn by connecting the ideas of calculus to their experience and observations. Doing so provides them with a way to understand the material in terms of familiar facets of life, so calculus makes sense to them rather than being a random assortment of facts and formulas. Here are some examples that we use to accomplish this:
Newton’s method: We begin our discussion
of Newton’s method by asking the question, “Suppose a model rocket is coming back down, after its fl ight. It’s descending at 6 feet per second, and is 30 feet above the ground. How much longer before it lands?” Students always respond with the expected answer of fi ve seconds based on their experience with motion. Pedagogically, what’s important about their answer is that they’ve assumed a constant rate of change. That’s Newton’s method in a nutshell.
Mean Value Theorem: We begin by asking
you to think about a race against a friend. The friend promises to run at a steady, constant rate from start to fi nish, and though you may speed up and slow down at will, by some fl uke you tie. Could you have been running faster than your friend at each and every moment, or slower the whole time? No, because one of you would have won the race uncontested. If you start and fi nish together, there must have been some time at which you were running at the same speed. This discussion opens the door to the Mean Value Theorem.
Substitution: When we apply the substitution
technique to defi nite integrals, we begin by determining a change in altitude in two different ways––one calculation based on time, and the other based on position. This allows us to help students view the substitution technique as a change in our point of view, from initial and terminal times to starting and stopping locations. By drawing on students’ experience with the world, we help them internalize the ideas of calculus, solidify their understanding, and develop their intuition.
ix
mathematics done rigorously and well. On the other, we are aware that our readers have little or no experience reading technical documents. For this reason, we have chosen to write in a colloquial style during the development of ideas, so that by the time students read the formal language of a theorem, they already know what it says.We also use color to help students follow longer or more complicated calculations, and to reinforce similarities across examples. We make common use of underbraces to provide guidance as students read equations, and often include margin notes to answer common questions without interrupting the fl ow of the presentation.
Promoting active reading
Some years ago, we attended a seminar in which Dr. P.K. Imbrie of Purdue University spoke about ways of engaging students, even in large classes. He suggested breaking up lectures with small “try it yourself ” moments that keep students actively involved in the learning process. There are other ways, of course, but Dr. Imbrie’s idea made a lot of sense to us as authors: students who are inexperienced at learning mathematics from a book often read in a passive way, as if it’s a novel, but learning mathematics is an active endeavor. To promote active learning while students read, we often include a “Try It Yourself ” example immediately after one whose solution is fully laid out. The answers to these Try It Yourself examples are provided in the text, and if students need to see a full solution they can access it online. These examples act as pedagogical “stepping stones,” half-way between examples and exercises, allowing students to try something for themselves but with a virtual safety net.
Reading v. Spelling in mathematics
For many students, a mathematical equation is just a collection of symbols that they are expected to manipulate in order to arrive at an answer, but they see no meaning in the equation itself. So starting early on, we teach students how to interpret the equations they see, and to assess the ones they write themselves so that mathematics becomes a communication of ideas.
We consider this skill to be a kind of quantitative literacy, closely related to English literacy. When reading, we don’t view each symbol (letter) separately, but look at groups of symbols to derive meaning. For example, the following sentence would be confusing if read letter by letter (try it, out loud):
“T-h-e-n-e-x-t-t-i-m-e-I-s-l-e-e-p-,-i-t-w-o-n-’-t- b-e-o-n-m-y-c-a-l-c-u-l-u-s-b-o-o-k-.-It-d-i-d-n-’- t-h-e-l-p-l-a-s-t-t-i-m-e-a-n-d-n-o-w-I-h-a-v-e-a-c-r-i-c-k-i-n-m-y-n-e-c-k-.”
“The next time I sleep, it won’t be on my calculus book. It didn’t help last time and now I have a crick in my neck.”
Each grouping (word) conveys a concept, and together they communicate an idea. We help students move beyond spelling to reading mathematics by demonstrating this skill in the presentation and by including exercises that ask students to make the transition from English to mathematical notation and back.
Additionally, we often use dimensional analysis to check that calculations of physical objects and phenomena make sense. When done at an elementary level, this helps students internalize the meaning of the difference quotient, which has units such as meters per second, and understand why Riemann sums approximate the net change in, for example, position (when summands have the form rate X time). Dimensional analysis also helps students to remember formulas such as those for cylindrical shells and washers, which are used when calculating volumes of solids of revolution.
Helping students avoid common errors
We have found that students often benefi t from learning about common mistakes, so we pause to discuss them at the end of each section. For example, students sometimes forget whether the recursion formula in Newton’s
method requires the derivative in the numerator or the denominator; and they sometimes make use of the same letter for the substitution as the original variable of an integral, which invariably leads to errors. So throughout the book we discuss ways to remember formulas and avoid common errors.
Exercises and projects
Since different schools require different levels of rigor, we’ve designed the exercise sets to include various levels of diffi culty. We’ve also sorted them according to the particular skills that are required, and we’ve separated skill exercises from those that develop concepts or demonstrate relevance by way of application. At the end of each chapter you’ll fi nd a Chapter Review, and projects that have been distilled from those we’ve assigned to teams of students in our own classes through the years. Some focus on purely mathematical ideas, while others ask students to use their skill and understanding in context.
Helping students learn the “art”
of calculus –– decision making
When working with integration or infi nite series, students have to decide which technique is appropriate to each problem. Knowing when to use the ratio test, for example, is just as important as knowing how to use it,
x
to help students develop that skill, we include designated groups of exercises called Mixed Practice sets that require students to use techniques from previous sections on integration techniques and convergence tests, respectively. This helps students learn to analyze problems and decide on appropriate methods at the same time that they are learning the mechanics that each choice entails.Proofs
Proofs lie at the heart of deeper mathematical
understanding, but fi rst-year students tend to fi nd formal proofs unsatisfactory as answers to the question, “Why?” Indeed, from their point of view, formal proofs obfuscate facts more often than they illuminate them. With this in mind, we frequently precede proofs with informal discussions that focus on the salient ideas students are about to see, and put them in a context that allows students to intuit the relationship between quantities. This often makes the formal proofs easier for students to follow, and leaves them more satisfi ed with the proof and the role it plays.
Technology
Generally speaking, today’s students are technologically savvy. They are familiar with chat rooms, have pages on Facebook, maintain blogs, send emails and text messages, and are typically comfortable interacting online. It’s important to understand that their mathematical education, like the rest of their lives, has been affected by the
presence of technology. Many of today’s students begin a math problem by reaching for their graphing calculator because they’ve been trained to think in terms of graphs, which have always been readily accessible. Using computational technology is as normal to them as driving a car.
In principle, we believe that computational technology is a powerful tool that can help expedite the problem solving process, but only if the user knows what to look for––much in the same way that your car’s maximum speed is irrelevant if you’re lost. So we tend to focus on ideas and concepts, and bring in technology where we believe it’s appropriate.
We also believe that numerical techniques should not
for engineers, scientists, and applied mathematicians, so we regularly ask students to use them to approximate quantities that are otherwise impossible for them to calculate. To facilitate students’ use of these algorithms, we have provided instructions for implementing them on a generic spreadsheet program (a tool that is often used by engineers and scientists, but isn’t as common in mathematics education) in several places throughout the book.
In closing
In a discussion with Dr. George Thurston, he said (we paraphrase),
There seems to be a certain momentum that draws the teaching community away from the original motivation and understanding of a topic—away from the why of things, toward the what and how of it.
This was our own experience learning calculus, and based on conversations with students, it is the experience that many people still have today. In this book we strive to focus students’ attention not only on what’s true but also on why it’s true, so they can both do the calculus and understand the calculus.
We’ve worked to produce a text that is interesting to students by providing some context to the subject, and by using students’ intuition and life experience as a framework in which to present ideas. Simultaneously, we’ve focused on making the text accessible to students by supporting their reading with a healthy sprinkling of margin notes, and by couching much of the presentation in an informal style that’s comfortable for many students, while preserving the rigor of the mathematics itself. We hope that after digging through this text a little, you’ll fi nd ways that it can complement your classroom teaching, and that you’ll ask your students to crack the cover, and honestly try reading it.
Since this is a preliminary edition, we welcome your feedback–––especially ideas about other ways to help students. You can email your comments to us at lutzer@ wiley.com. We look forward to hearing from you. Thanks!
xi
IN-TEXT FEATURES
• Chapter Openers.
Each chapter sets the stage with a unique,
real-world example from a varied assortment of topics.• Try It Yourself. Many of the examples that are worked out in the text are followed immediately by a similar example that students are asked to work out for themselves. An answer is provided in the text, and the full solution is provided at the book’s companion website and in WileyPLUS.
• You Should Know/You Should be Able to. Appearing at the
end of each section a boxed lists that serve as checklists for students to make sure they understand key concepts and can demonstrate key skills.
• Exercises. Each section includes relevant skill-development
exercises, and exercises devoted to applications from the sciences, business, economics, engineering, physics, and mathematics.
• Projects and Applications. These are chapter-ending
cumulative reviews that require students to apply the concepts and demonstrate the skills they learned in the chapter.
xii
DIGITAL RESOURCES
Wiley PLUS is an innovative,
research-based, online environment for
effective teaching and learning.
What do students receive
with WileyPLUS?
A Research-based Design. WileyPLUS provides an online environment that integrates relevant resources, including the entire digital textbook, in an easy-to-navigate framework that helps students study more effectively.
• WileyPLUS adds structure by organizing textbook content into smaller, more manageable “chunks”. • Related media, examples, and sample practice items
reinforce the learning objectives.
• Innovative features such as calendars, visual progress tracking and self-evaluation tools improve time management and strengthen areas of weakness.
One-on-one Engagement. With WileyPLUS Lutzer & Goodwill/ Calculus students receive 24/7 access to resources that promote positive learning outcomes. Students engage with related examples (in various media) and sample practice items, including:
• Algorithmically generated exercises, with randomized numeric values and an answer-entry palette for symbolic notation, are provided on-line through WileyPLUS. Students can click on “help” buttons for hints, link to the relevant section of the text, show their work or query their instructor using a white board, or see a step-by-step solution (depending on instructor-selecting settings). • Applets are available on-line through WileyPLUS.
We have found that there are specifi c times in the teaching of calculus when students benefi t from exploring a given topic with a Java applet, Maple worksheet, or spreadsheet.
Measurable Outcomes. Throughout each study session, students can assess their progress and gain immediate feedback. WileyPLUS provides precise reporting of strengths and weaknesses, as well as individualized quizzes, so that students are confi dent they are spending their time on the right things. With WileyPLUS, students always know the exact outcome of their efforts.
What do instructors receive with
WileyPLUS ?
WileyPLUS provides reliable, customizable resources that reinforce course goals inside and outside of the classroom as well as visibility into individual student progress. Pre-created materials and activities help instructors optimize their time:
Customizable Course Plan: WileyPLUS comes with
a pre-created Course Plan designed by a subject matter expert uniquely for this course. Simple drag-and-drop tools make it easy to assign the course plan as-is or modify it to refl ect your course syllabus.
Pre-created Activity Types Include:
◊ Assignments with algorithmic exercises ◊ Readings and resources
◊ Print Tests
Course Materials and Assessment
Content:
◊ Lecture Notes PowerPoint Slides
◊ Classroom Response System (Clicker) Questions ◊ Instructor’s Solutions Manual
◊ Gradable Reading Assignment Questions (embedded with online text)
◊ Assignments: auto-graded end-of-chapter problems coded algorithmically with randomized numeric values. Instructor sets availability of student hints, links to text, whiteboard/”Show Work” feature, and worked-out solution
◊ Test Bank
Gradebook: WileyPLUSprovides instant access to reports on trends in class performance, student use of course materials and progress towards learning objectives, helping inform decisions and drive classroom discussions.
WileyPLUS. Learn More.www.wileyplus.com.
Powered by proven technology and built on a foundation of cognitive research, WileyPLUS has enriched the education of millions of students, in over 20 countries around the world.
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In addition to those listed below who patiently wadedthrough draft chapters and provided useful feedback, we wish to thank the staff at John Wiley & Sons who expertly guided us through the process from prospectus to publication, especially David Dietz, Mary O’Sullivan, and Leslie Lahr. We also wish to express our gratitude to Joseph DeLorenzo, Ray Hodges, Jeffery McBeth, David Ross, and George Thurston for their support and contributions to this effort.
Darry Andrews, Ohio State University Scott Barnett, Henry Ford Community College William Basener, Rochester Institute of Technology Phil Bowers, Florida State University
David Bradley, University of Maine Mark Bridger, Northeastern University Ray Cannon, Baylor University
Kristen Chatas, Washtenaw Community College Min Chen, Purdue University
David Collingwood, University of Washington Randy Combs, West Texas A&M University Ted Dobson, Mississippi State University Michael Dorff, Brigham Young University Gregory Dresden, Washington & Lee University David Finn, Rose–Hulman Institute of Technology Ralph Grimaldi, Rose–Hulman Institute of Technology Huiqiang Jiang, University of Pittsburgh
Joe Lackey, New Mexico State University Hong-Jian Lai, West Virginia University Glenn Ledder, University of Nebraska–Lincoln
Sergey Lvin, University of Maine Vania Masconi, Ball State University
Philip McCartney, Northern Kentucky University Jie Miao, Arkansas State University
Mona Mocanasu, Metropolitan State College of Denver Aaron Montgomery, Central Washington University Karen Mortensen, University of Illinois at Urbana-Champaign
John Orr, University of Nebraska–Lincoln Emma Previato, Boston University
Dennis Reissig, Suffolk County Community College Peter Roper, University of Utah
Amit Savkar, University of Connecticut Chris Storm, Adelphi University
Barrett Walls, Georgia Perimeter College Scott Wilde, Baylor University
Brock Williams, Texas Tech University
Robin Wilson, California State Polytechnic University, Pomona
Tzu-Yi Alan Yang, Columbus State Community College Last, but certainly not least, we take a moment to thank our families. They have sacrifi ced a great deal of time with us so that we could pursue this effort to help others, and have done so lovingly without complaint. Without their sacrifi ce and support this project could never have come to fruition.
--Carl Lutzer and Tim Goodwill September, 2010
xiv
PHOTO CREDITS
Icons
Icon page 5 © Evgeny Rannev/iStockphoto; Icon page 18 © Purestock / SuperStock; Icon page 43 ©kemie/iStockphoto.
Chapter 1
UN 1.1 page 4 Photo Researchers Inc.; Figure 2.2 page 12, Carl Lutzer; UN 1.2 page 5, © Tom England/iStockphoto; Figure 5.6 page 34, Carl Lutzer; UN 1.3 page 35, © music Alan King/Alamy; Figure 5.16 page 40, Carl Lutzer; UN 1.4 page 49, Nick Wass/©AP/Wide World Photos; UN 1.6 page 83, © Janne Ahvo/iStockphoto; UN 1.7 page 83, © Frank van den Bergh/iStockphoto.
Chapter 2
UN 2.1 page 89,© Michiel de Boer/iStockphoto; UN 2.2 page 107, © cloki/ iStockphoto; UN 2.3 page 137, “Phil Walter/Getty Images, Inc.”.
Chapter 3
UN page 160, © Jan Kratochvila/iStockphoto; UN 3.2 page 167, © Greg Nicholas/ iStockphoto; UN 3.3 page 169, © Neil Sullivan/iStockphoto; UN 3.4 page 171, © Kris Hanke/iStockphoto; UN 6.1 page 202, © Royalty-Free/CORBIS; UN 6.2 page 202, moodboard RF/Photolibrary; UN 3.5 page 211, © Pogorelov Vladimir/iStockphoto; UN 3.6 page 211, © Shannon Stent/iStockphoto; Figure 7.1 page 213, Copyright © Yoav Levy /Phototake; Figure 7.5 page 219,”VGL/amanaimagesRF /Getty Images, Inc.”; Figure 7.7 page 219, © Glyn Jones/©Corbis; Figure 7.7 page 219, © Frank Leung/iStockphoto; Figure 7.8 page 219, “VGL/amanaimagesRF /Getty Images, Inc.”; UN 3.7 page 226, © Cristian Andrei Matei/iStockphoto; UN 3.8 page 233, © Mike Norton/iStockphoto.
Chapter 4
Figure 1.1 page 236, Bill Aron /PhotoEdit; Figure 1.3 page 238, Carl Lutzer; Figure 1.9 page 245, ©Planetpix/Alamy; UN 4.1 page 251, „David Madison / Getty Images, Inc.“; Figure 2.1 page 251, ©Mooneydriver/iStockphoto; Figure 2.2 page 252, © Lawrence Manning/©Corbis; Figure 2.2 page 252, © brentmelissa/ iStockphoto; Figure 2.2 page 252, © Mark Ross/iStockphoto; Figure 2.3 page 253, ©Mooneydriver/iStockphoto; Figure 2.12 page 258, Photo by Rachel Goldstein; Figure 3.3 page 263, ©Mark Ross/iStockphoto; Figure 3.4 page 264, SMC Images /Getty Images, Inc.; Figure 3.8 page 270, ©Pogorelov Vladimir/iStockphoto; Figure 7.1 page 301, © oversnap/iStockphoto; Figure 7.1 page 301, „Joe Michl / Getty Images, Inc.“; Figure 7.1 page 301, Ralph A Clevenger/ Flirt Collection/ Photolibrary; UN 4.3 page 306, „rubberball /Getty Images, Inc.“; Figure 7.12 page 307, ©peter zelei/iStockphoto; Figure 7.2 page 310, ©Mlenny Photograhy | Alexander Hafemann/iStockphoto; Figure 9.4 page 322, „SSPL via Getty Images, Inc.“; Figure 9.5 page 322, ©Mary Evans Picture Library/The Image Works; Figure 7.14 page 308, Bela Szandelszky/©AP/Wide World Photos; Figure 8.2 page 315, Carl Lutzer.
Chapter 5
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Chapter 6
Figure 8.2 page 483, Photo Researchers Inc.; Figure 8.3 page 484, Photo Researchers Inc.; Figure 4.1 page 447, © DNY59/iStockphoto.
Chapter 7
UN 7.1 page 489, © www.gerardbrown.co.uk/Alamy; Figure 1.1 page 489; © Maurice van der Velden/iStockphoto; Figure 5.1 page 537, Carl Lutzer; Figure 5.2 page 537, © Andrew Aitchison/Alamy; Figure 6.1 page 549, “Sami Sarkis /Getty Images, Inc.”; Figure 8.3 page 564, “Burazin/Getty Images, Inc.”; Figure 7.1b page 564, © Perttu Sironen/iStockphoto.
Chapter 9
UN 9.1 page 730, “Steve Hathaway/Getty Images, Inc.”.
Chapter 10
UN 10.1 page 774, © sgame/iStockphoto; UN10.2 page 804, © dem10/
iStockphoto; UN 10.3 page 809, © Jorgen Udvang/iStockphoto; UN 10.4 page 810, © NoDerog/iStockphoto; Figure 4.1 page 819, ©camilla wisbauer/iStockphoto; Figure 6.2 page 836, © Paul Mayall /Alamy; Figure 6.2 page 836, © Paul Mayall/ Alamy; Figure 6.2 page 836, “Mai /Getty Images, Inc.”.
Appendix
Appendix A Figure C.1 page 865, © Peter Steiner /Alamy; Appendix D Figure D.14 page 881, ©Sebastian Kaulitzki/iStockphoto; Appendix E Figure E.1 page 886, © Oleksiy Maksymenko /Alamy.
Tools of the Trade
You learned at an early age that planets orbit the sun, and our sun orbits the center of the Milky Way because of a force called gravity. Many people today find that fact unremarkable, but for most of human history celestial motion was some-thing of a mystery. It wasn’t until the early 1600s that our modern understanding of the universe really began to take shape. That’s when the telescope was invented, Galileo Galilei saw the moons of Jupiter, and Johannes Kepler published his finding that planets move around the sun in elliptical orbits. After that, people knew what happened, and turned their attention to why. Men like Robert Hooke, Edmund Halley, and Christopher Wren struggled with the problem until 1684, when Halley paid a visit to Isaac Newton in Cambridge.
Halley asked if a force that obeyed an inverse square law would result in the
It’s okay if the term inverse square law is unfamiliar to you. We’ll talk about it in more de-tail in a moment.
orbits that Kepler had described. Newton’s ready response was, yes. In fact, he’d already made the calculations . . . though he had misplaced them (see [29]). Newton promised to find or redevelop the calculations, and three years later, he published his famous Philosophiae Naturalis Principia Mathematica in which he introduced the calculus. Since that time, it has become not only a tool for making calculations, but a way of understanding and talking about the world we live in.
This chapter is intended as a review of the basic functions that you will see in calculus—tools of the trade, as it were. The point of such a review is not to rehash the mechanics of how these functions work, but rather to help you begin thinking about mathematics in a new way.
• Whereas algebra and precalculus traditionally focus on the value of a function, calculus is fundamentally about how functions change, so our discussion will often come back to the topic of change.
• We’ll emphasize equations as a communication of ideas rather than as a col-lection of symbols to be manipulated. Part of this emphasis will involve a technique called dimensional analysis, which provides deeper understanding of functions and change based on the units involved in a problem.
• While there are a lot of mathematical ideas that are intriguing, or beautiful, or just plain fun, humans have been driven by the need to describe and un-derstand the world we live in. So our discussion will often include situations in which particular types of functions or important ideas arise.
As you read, you’ll see that examples in the presentation are often followed by “try it yourself” examples, to which an answer is provided here but not the full solution. Our hope is that you’ll try these examples, and learn by doing as you go. If you find that you need help, you can find a full solution on-line.
1.1
Power Functions
1.1 Power Functions . . . 21.2 Polynomial Functions . . . 10
1.3 Rational Functions . . . 18
1.4 Algebraic, Composite and Piecewise Functions . . . 26
1.5 Periodic Functions . . . 32
1.6 Exponential Functions . . . 48
1.7 Logarithms . . . 57
1.8 Inverse Functions . . . 68
Chapter Review . . . 76
Projects & Applications . . . 83
In the chapter opener, Halley asked Newton about a force that obeyed an inverse square law. What he had in mind was a force, F , that changes with distance like
As often as possible, we de-sign notation to help us remem-ber what variables mean. In this case, F reminds us of the word force, and r reminds of the word radial.
F = (some constant)
r2 , (1.1)
where r is the radial distance (i.e., straight-line distance) between the sun and a planet. This equation tells us that F gets smaller when r increases. Alternatively, you could say that r gets smaller as F increases. The important thing to understand is that the quantities F and r change together.
In equation (1.1) we wrote the numerator as “some constant,” so that we could focus on the relationship between F and r. More specifically, the relationship between distance and gravitational force is
F = Gm1m2
r2 , (1.2)
where
G is called the gravitational constant. Its value doesn’t depend on the objects, the distance between them, or anything else. It is the same from situation to situation, from moment to moment. m1 and m2 are the masses of the objects involved.
Notice how subscripts are used to distinguish between the masses. This is common prac-tice when you have more than one occurence of the same kind of quantity.
Although F and r change together, we typically think of gravitational attraction as depending on distance. When we think of the relationship in this way, we write F as a function of r:
F (r) = Gm1m2 r2 .
The number r is called the argument of the function F . We could also describe this function by writing
The name of the function z}|{ F : r |{z} The argument of the function 7−→ Gm1m2r−2 | {z } The output of the function . (1.3)
The “mapping arrow” in this notation (7−→) emphasizes the idea that a function takes an input, does something with it, and arrives at an output. By way of analogy, a function is like a verb that connects two nouns (the input and the output)—it’s an action.
As you might guess from the name of this section, F is an example of a power
function, which takes an input and raises it to a specified (nonzero) power. You The specified power could be 1,in which case we usually don’t write it.
can multiply the resulting value by a constant, and it’s still called a power function.
Power functions and arrow notation
Example 1.1. The circumference and area of a circle whose radius is r are given
by the power functions These formulas are derived inAppendix A.
A(r) = πr2 and C(r) = 2πr.
Write these functions with the “arrow” notation that you see in (1.3). Solution: A : r 7−→ πr2 and C : r 7−→ 2πr.
Power functions are a great place to begin studying calculus because they change in a straightforward way. We’ll begin with the simplest kind of exponents: integers.
Parity of an integer exponent
The word parity refers to whether an integer is even or odd. When a power function has an even exponent, the function value changes in the same way whether we move left or right from x = 0 (see the graph of f2(x) = x2 in Figure 1.1). By contrast,
power functions with odd exponents change in opposite ways when we move left or right from x = 0 (see the graph of f3(x) = x3 in Figure 1.1).
Since f1 and f2 are the
same kind of thing—power functions with positive, inte-ger exponents—we’re using the same letter to denote them, and distinguishing between them with subscripts, much as we did with masses a moment ago. Note that the subscripts have been chosen according to the exponent.
| {z }
The way f2(x) = x2changes
| {z }
The way f3(x) = x3changes
Figure 1.1: The way that a power function changes depends on the parity of its exponent.
Note: Calculus is about change, so we’re talking in those terms. We could also use the language of geometry, and focus on symmetry: When the exponent of a power function is even, its graph is symmetric about the y-axis. When its exponent is odd, its graph is symmetric about the origin.
Polarity of an integer exponent
The word polarity refers to whether a number is positive or negative. In the case of power functions, the polarity of the exponent determines whether the mag-nitude (i.e. size) of the function value grows or shrinks as we move away from x = 0. When the exponent is positive, the size of the function value grows when the
x g1(x) g2(x) 1 1.0000 1.0000 3 0.3333 0.1111 5 0.2000 0.0400 7 0.1429 0.0204 9 0.1111 0.0123 11 0.0909 0.0083 13 0.0769 0.0059 15 0.0667 0.0044 17 0.0588 0.0035 | {z } Table 1 x g1(x) g2(x) 1 1 1 .5 2 4 .25 4 16 .125 8 64 .0625 16 256 .0312 32 1024 .0156 64 4096 | {z } Table 2
size of the argument grows, as seen in Figure 1.1. However, when the exponent is negative, the size of the function value shrinks as the size of the argument grows. For example, consider the functions
g1(x) = x−1 and g2(x) = x−2.
As you see in Tables 1 and 2, the numbers g1(x) and g2(x) decrease as x moves
farther away from zero, and the function values grow in size when x heads toward zero. You can also see this happen in Figure 1.2, which depicts the graphs of g1
and g2.
Note: When the exponent of a power function is negative and x is near zero, small changes in the argument (x) cause large changes in the function value (y).
Magnitude
The word magnitude means the “size” or “extent” of something. When it comes to power functions, those whose exponents have larger magnitude tend to “do things faster.” For example, in Figure 1.2 you an see that g2(x) tends to zero faster than
g1(x) as |x| gets large. And as |x| approaches zero, g2(x) grows faster than g1(x).
Later, in our study of improper integrals and infinite series, this simple idea will be very im-portant.
Figure 1.2: The graphs of power functionsy = g1(x)andy = g2(x).
z Fractional Exponents
In the chapter opener, we mentioned Kepler’s discovery that the planets of our solar system orbit the sun along elliptical trajectories. Another of his discoveries was that the time it takes a planet to complete one orbit around the sun, say T (measured in Earth-years), depends on the shape of its orbit according to
Recall, the semi-major axis of an ellipse is half of the major axis (i.e., the longer one).
T (a) = a3/2, (1.4)
where a denotes the length of the semi-major axis of the planet’s orbit, measured in astronomical units. When a power function has a fractional exponent, pq (in lowest terms), parity and polarity still determine the function’s behavior.
Parity: When q is odd, the parity of p affects behavior in exactly the same way it does for integer exponents (see p. 3). When q is even, as in x1/2 or x−5/6, negative values of x aren’t in the domain of the function (see Figure 1.3), so the parity of p is irrelevant.
Polarity: The polarity of the exponent affects behavior in exactly the same way it does for integer exponents (see p. 3).
| {z }
The graphs ofy = x1/2andy = x−5/6 The graphs of| y = x{z1/3andy = x}−6/17
Figure 1.3: Graphs of various power functions with rational exponents.
How do fractional exponents affect a power function?
Example 1.2. Discuss the behavior of h(x) = x2/3.
Solution: The exponent in h(x) is positive, so |h(x)| will grow as |x| does, and will head toward zero as |x| does. The denominator of the exponent is odd, so the do-main of h includes x < 0. Since the numerator is even, the value of h(x) will change in the same way whether we move right or left of zero.
Try It Yourself: How do fractional exponents affect a power function?
Example 1.3. Without using a graphing utility, discuss the behavior of j(x) = −3x9/5, and k(x) = x−13/8.
Answer: Discussion of j(x) and k(x) is found on-line.
z Introduction to Dimensional Analysis
An equation that says “kilograms = meters” is nonsensical, so it’s good practice to check that both sides of an equation describe the same kind of quantity (mass, length, force, etc.) before you begin working with it. The act of checking is called dimensional analysis, and is done by accounting for the units that are associated with each term in the equation. Comparing units can also be a handy way of re-membering formulas.
Dimensional analysis
Example 1.4. Use dimensional analysis to determine which of πr2 and 2πr
de-scribes the area, and which the circumference of a circle. Solution: Suppose we measure length in meters (m). Then
area has units of square meters (m2), and circumference
| {z }
This is a length
has units of meters (m).
The radius, which is a length, will also have units of meters (m). Now let’s look at
Units can help you reason your way to/through a formula when you get mixed up.
the formulas:
πr2 2πr
↑
z }| {
Two factors of r, each with units of meters. This formula leaves us with units of m2.
↑
z }| {
One factor of r. This formula leaves us with units of m.
Based on the units involved, we know that A 6= 2πr because the left-hand side has units of m2but the right-hand side has units of m. So it must be that A = πr2and C = 2πr.
Try It Yourself: Dimensional analysis
Example 1.5. Use dimensional analysis to determine which of 43πr3 and 4πr2 describes the surface area, and which the volume of a sphere.
Answer: See solution on-line.
When a factor doesn’t change the kind of thing that’s being described, such as the number π in A = πr2 and C = 2πr, we say it’s dimensionless. Not all
constant factors are dimensionless, as seen in the next example.
Constants with units
Example 1.6. Suppose we measure force in newtons, denoted by N . When mass
When we use these units, the numerical value of G is 6.673 × 10−11. If we measure mass and
distance differently, the units and value of G will change ac-cordingly (not because gravity has changed, but because we’ve changed the way we’re measur-ing its effect).
is measured in kilograms (kg), and distance in meters (m), what are the units associated with the gravitational constant in equation (1.2)?
Solution: The “=” in equation (1.2) means more than just numerical equality. From a physical point of view, it means that they are the same kind of thing. The left-hand side is a force, so the right-left-hand side must be also, and must have units of force. If we write the units of G as2 (meaning “temporarily unknown”), equation (1.2) has the form
You might feel uneasy about this kind of “algebraic” work with units, but it’s the same thing you do intuitively when you say that you travel 10 kilo-meters by driving at 5 km/hr for 2 hours (rate × time = dis-tance). F = Gm1m2 r2 → force =2 (mass)(mass) (distance)2 → N =2 (kg)2 (m)2.
By “solving” for 2, we find that G must have units of (N)(m2)/(kg)2. Note that without the units of G, equation (1.2) would say that force = (mass/distance)2, which is nonsensical. The factor of G on the right-hand side of the equation changes the kind of thing that you see on the right-hand side.
z Common Difficulties with Power Functions
One of the most common mistakes, in one form or another, is distributing exponents over sums and differences. For example, you know that
(a + b)26= a2+ b2 since (a + b)2= (a + b)(a + b) = a2+ 2ab + b2.
The mistake of distributing powers over sums is more common when the power is negative or fractional. Always keep in mind that
1 a + b6= 1 a+ 1 b because
Exponents don’t distribute over sums
z }| { (a + b)−16= a−1+ b−1, √ a + b 6=√a + √ b because (a + b)1/26= a1/2+ b1/2. | {z }
Exponents don’t distribute over sums
This second bullet is particu-larly important because it’s at the heart of a skill that many beginners lack: the ability to tell when something is wrong. You should know
• the terms argument, parity, and polarity;
• the units on the left- and right-hand sides of an equation must be the same (e.g., an equation that says meters = kilograms can’t possibly be right).
You should be able to
• explain how the parity of an integer exponent is related to the symmetry of a power function;
• explain how the polarity of an exponent is related to the way a power function changes;
• determine the behavior of a power function;
• discern the parity and polarity of a power function’s exponent based on a graph;
z 1.1 Skill Exercises
Use your understanding of parity, polarity, rational, and integer exponents to match
Figure 1.4: For #1–4.
the graphs in Figure 1.4 with the functions in #1–4.
1. f (x) = x1/6 2. f (x) = x−2 3. f (x) = x−13/3 4. f (x) = x3/7
In each of 5–10 use the parity, polarity, and magnitude that you see in the exponent to compare and contrast the way that f (x) and g(x) change when x moves away from zero. Write your answer in complete sentences, and avoid using pronouns.
5. f (x) = x4/3, g(x) = x−4/3 6. f (x) = x5/3, g(x) = x−5/3 7. f (x) = x5/4, g(x) = x−5/4 8. f (x) = x7/8, g(x) = x−7/8 9. f (x) = x6/11, g(x) = x7/11 10. f (x) = x8/13, g(x) = x7/13
11. Which graph in Figure 1.5 is y = t2and which is y = t4? How can you tell?
Figure 1.5: For #11.
Answer in complete sentences (avoid using pronouns).
12. Which graph in Figure 1.6 is y = t3and which is y = t5? How can you tell? Answer in complete sentences (avoid using pronouns).
Figure 1.6: For #12.
13. If an object is h(t) meters above sea level after t seconds:
(a) Write the equation that says, “the object was 23.4 meters above sea level after 7 seconds.”
(b) Write the inequality that says, “the object was below sea level after t seconds.
14. In business, the revenue function relates the number of items sold (x) and the amount of money earned from sales (R). Suppose R(x) = 0.25x. The number R(x) has units of dollars.
(a) What are the units of the variable x?
(b) Based on part (a), what units must the number 0.25 have?
(c) Interpret those units using plain, simple language. What does the num-ber 0.25 mean to us in this context?
15. When we model the behavior of something that swings back and forth under the force of gravity (e.g., the arm of a pendulum), we find that T = 2πpL/g, where L denotes the length of the pendulum, T denotes the period of its motion, and g is the acceleration due to gravity (with units of secm2). Perform
The word “acceleration” means “change in velocity,” so its units are
m
sec per second, which we write as
m/sec sec or, more commonly, as
m sec2.
a dimensional analysis to verify that the equation makes sense.
16. A company’s customer service center processes q(t) calls per month (in hun-dreds) t months after the national debut of a product line.
(a) In a complete sentence, interpret the equation q(4) = 2.5 in terms of months and the number of calls per month.
(b) Write an equation that says, “The number of calls per month 3 months after the debut is larger than the number of calls-per-month 7 months after the debut.”
(c) Write an equation that says, “The number of calls per month t1months
after the debut is larger than the number of calls-per-month t2 months
17. A famous equation in modern physics is m = m0/
q
1 − sc2
, where m and m0 denote mass, measured in kilograms, and c is the speed of light, measured
in m/sec. What have to be the dimensions of s in order for this equation to make sense?
18. The momentum of an object is denoted by p, and is related to both its mass (m) and its velocity (v) by p = mv (see Appendix E). What are the units of momentum?
19. Coulomb’s force law says that the force between two ions, F , is related to their charges (q1 and q2) and the distance between them, r, according to
F = kq1q2/r2, where k is a constant. If force is measured in newtons, charge
in coulombs, and distance in meters, what have to be the units of k in order for the equation to make sense?
20. Add together functions you know in order to make a function whose graph is
Figure 1.7: For #20.
like the one in Figure 1.7.
z 1.1 Concept and Application Exercises
21. Equation (1.2) formulates gravitational force according to an inverse square Physics
law. The gravitational potential function, U (r), is slightly different: F (r) = Gm1m2
r2 and U (r) = −
Gm1m2
r .
Answer the following without using a graphing utility:
(a) Which function values gets smaller (in magnitude) faster as r > 0 grows large?
(b) Which function values gets larger (in magnitude) faster as r > 0 shrinks toward zero?
22. A flywheel is often used to store energy in the form of rotational motion. Mechanical Engineering
Suppose that the disk shown in the margin is spinning, and that it makes 8π radians per second.
(a) Point A is 1 cm from the center. How far does the point travel in 1 second? (Hint: what’s the circumference of the circle around which it’s traveling?)
(b) Points B, C and D are each 1 cm farther from the center. Determine how far each travels in 1 second.
(c) Write an equation that relates a point’s distance from the center, r, to the distance it travels in one second.
(d) How does your formula change if the wheel rotates at ω radians per second, instead of 8π?
23. When an object is rotating freely, a quantity called angular momentum, de-noted by L, is conserved (i.e., is constant). In simple situations, the formula for angular momentum is L = mr2ω, where m is the object’s mass, r is its
Figure 1.8: The formula in #23 is really the magnitude of the angular momentum in a special case.
distance to the axis of rotation, and ω is its angular velocity (in radians per second).
(a) What does this formula say happens to ω when the object moves farther away from the center of the rotation (assuming no change in mass)?
(b) Suppose we decrease angular momentum by applying torque (“rotational force”) to the system. If ω remains constant, what must happen? 24. The Ideal Gas Law describes the relationship among pressure (P ), volume
(V ), the number of molecules in a gas (N ), and the temperature (T ) as P V = N kBT , where kB is a number called Boltzmann’s constant. When distance is
measured in meters, force in newtons, and temperature in kelvins (so T ≥ 0), Boltzmann’s constant is kB= 1.3806503 × 10−23. Suppose N = 1023. Then
P = 1.3806503 T
V . (1.5)
When temperature is held constant, equation (1.5) defines pressure as a func-tion of volume. The graph of P is called an isotherm.
The prefix “iso” means “same,” and the ending “therm” refers to thermal energy.
(a) Draw isotherms for two different temperatures. (b) Label the curves as T0 and T1, so that T0< T1.
(c) Pick a point on the T1-isotherm, somewhere near the left side, and call
it A. Then pick a point on the T0-isotherm, somewhere near the right
side, and call it B.
(d) Label the V -coordinates of A and B as V1and V0, respectively (V1< V0),
and mark them on the V -axis.
(e) During an adiabatic compression, the point (V, P ) moves from B to A
The word adiabatic means that no heat flows into or out of the system, perhaps because com-pression (or expansion) hap-pens too quickly, or because of insulation or isolation.
along a smooth curve called an adiabat. Connect A to B by such a curve. It should look a lot like a parabola that opens up, or like a slightly tilted isotherm.
25. The exact relationship between pressure and volume in an adiabatic change of an ideal gas (see #24) is P = cV−(d+2)/d, where c is a constant that depends on initial relationship between pressure and volume, and d is the number of so-called “degrees of freedom” for the gas. (Near room temperature, the number d is 3 for a monatomic gas, and 5 for a diatomic gas.)
(a) For a fixed value of c, which adiabat (monatomic or diatomic) heads to zero faster for large values of V ?
(b) For a fixed value of c, which adiabat (monatomic or diatomic) is steeper when V is near zero?
(c) Let’s interpret your finding from part (b) in terms of the real world: he-lium is a monatomic gas, and hydrogen is a diatomic gas. Suppose these two gases are held separately with the same initial conditions.
Accord-The phrase same initial condi-tions means “the same number of particles, same initial start-ing temperature, same initial pressure, and same initial vol-ume (which we’ll assvol-ume to be very small).”
ing to the Ideal Gas Law, which of them will have a greater temperature increase during an adiabatic compression?
1.2
Polynomials
1.1 Power Functions . . . 21.2 Polynomial Functions . . . 10
1.3 Rational Functions . . . 18 1.4 Algebraic, Composite
and Piecewise Functions . . . 26 1.5 Periodic Functions . . . 32 1.6 Exponential Functions . . . 48 1.7 Logarithms . . . 57 1.8 Inverse Functions . . . 68 Chapter Review . . . 76 Projects & Applications . . . 83
A polynomial function is a sum of power functions whose exponents are non-negative integers. The largest power is called the degree, and the summand with that power is called the leading term. For example,
f (t) = 4 + 2t + 10t2+ 2t7 and g(t) = −4t3+ t
are polynomial functions of degree 7 and 3, respectively; and the leading terms are 2t7and −4t3, respectively.
z Linear Polynomials
Polynomial functions of degree 1 are called linear functions and in Chapter 3 you will see that they are very important in our study of calculus because the rate at which they change is constant. In the physical world, they often appear
We’ll make the phrase “rate of change” technically precise in Chapter 3. For now, the fol-lowing examples and Figure 2.1 provide numerical and graphi-cal descriptions of the meaning.
when two quantities are proportional. For example, you know that an elastic object (such as a spring) extends farther when you pull with greater force. If we denote your force by F and the object’s extension beyond its natural length by x (x < 0 means compression), the simplest relationship in which larger force results in greater extension is
Notice that for a given force, the extension of the object de-pends on k. When k is large, the spring is “stiff,” and the ex-tension is small.
x = 1
kF, or equivalently F = kx,
where k is a positive number called the spring constant. This is a famous and important equation called Hooke’s Law. Instead of thinking of extension as the consequence of a given force, we can say that F is the force required to hold the spring at a given extension, x. Writing this in function notation, F (x) = kx.
Linear functions and change
Example 2.1. Discuss how the force in Hooke’s Law varies as we increase the extension beyond x0.
A subscript of 0 often indi-cates an initial (starting) value of some kind. Sometimes it’s also used to indicate that a particular variable has been fixed for the purposes of discus-sion, which is what’s happening here.
Solution: Let’s begin by investigating the change in F that occurs when we increase x from x0 to x0+ 1. Using Hooke’s Law for F , we see that
∆F |{z}
= F (x0+ 1)−F (x0)=k(x0+ 1)−kx0= k. ∆ is the upper-case Greek letter delta,
and is often used to mean “change in”
Similarly, the change in F that results from increasing the extension from x0 to
x0+ 0.12 is
∆F = F (x0+ 0.12) − F (x0) = 0.12k.
And in general, extending the spring by an additional ∆x requires F to increase by ∆F = k∆x. Note that ∆F does not depend on x0. This is what we mean by
saying that the rate of change in the linear function F is constant.
Try It Yourself: Linear functions and change
Example 2.2. Suppose y(t) = 4t. Show that raising t from t0 to t0+ ∆t increases
y by 4∆t, regardless of t0.
Answer: See on-line solution.
When there is some kind of initial or resting value, it’s often true that the graph of a linear function no longer passes through the origin. For example, as the heart
Figure 2.1: When a function is linear, increasing its argument by 1 always results in the same change in value, no matter where we start.
fills with blood its volume changes in response to internal pressure. If we denote the internal pressure by P and the heart’s volume by V , and model the elastic properties of the heart wall with Hooke’s Law, we see
V = V0
|{z} + P
k |{z}
where k > 0 is some constant.
This is the initial volume, when there is no excess internal pressure.
This is the change in volume, due to internal pressure. Linear functions and change
Example 2.3. Discuss how the heart’s volume varies as the internal pressure in-creases beyond P0.
Solution: Since we’re thinking of volume as a function of internal pressure, let’s write V (P ) = V0+ P/k. Then by increasing the pressure from P0 to P0+ 1, the
heart’s volume increases by
∆V =V (P0+ 1)−V (P0)= V0+ P0+ 1 k − V0+ P0 k = 1 k.
Moreover, if we increase the internal pressure by some fraction of 1, the heart’s volume will change by exactly that fraction of 1/k. For example, the change in V that occurs when pressure is increased from P0to P0+ 0.27 is
∆V = V (P0+ 0.27) − V (P0) = 0.27
1 k
.
Note that ∆V does not depend on P0. The current pressure has nothing to do with
the response of V to a change in pressure. This is what we mean by saying that the rate of change in the linear function V is constant.
Try It Yourself: Linear functions and change
Example 2.4. Suppose x(t) = 3 − 14t. Show that x changes by −14∆t when t rises from t0 to t0+ ∆t, regardless of t0.
Answer: See on-line solution.
z Nonlinear Polynomials
Polynomial functions whose degree is larger than 1 are said to be nonlinear. They occur naturally in a variety of applications, from micro-mechanical systems to hu-man biology to imaging science.
Nonlinear polynomials and imaging
Example 2.5. In a black-and-white photo each picture element (pixel) has an associated intensity, x ∈ [0, 1], where x = 0 means black and x = 1 means white.
Suppose we change the intensity at each pixel from x to f (x) = 3x − 5x2+ 4x3− x4. There would be no change atall if we had used f (x) = x in-stead.
Graph the function f and explain how its application changes the original image. Solution: The graph of f is shown in Figure 2.2 with the line y = x. Note that f (x) > x except at x = 0 and x = 1, so almost all the pixels are becoming lighter. Further, you can see that f (x) ≈ x when x > 0.85 but that f (x) is substantially larger than x when x ∈ (0.1, 0.7), so the effect on pixels with dark and mid-range intensities is more pronounced than it is on lighter pixels.
Figure 2.2: (left) The graphs of y = x and y = f (x) for Example 2.5; (middle) original image; (right) image filtered by the nonlinear polynomial f (x).
When working with a polynomial function, it’s often helpful to know that its qualitative behavior is determined by its leading term when the argument is large. This fact is discussed further in the next example.
Large-scale behavior of polynomials
Example 2.6. Discuss f (t) = 4 + 2t + 10t2+ 2t7 when |t| is large.
Solution: The graph of f is shown in Figure 2.3, and you can see that its behavior is similar to y = 2t7 when |t| is large. To understand why this happens, think in
terms of percentages. Suppose you and a friend want to buy a $20 pizza, but he only has $8. If you chip in $12, you’ve contributed
Figure 2.3: The graphs ofy = f (t)andy = 2t7.
(your contribution)
(total value)
= 12
12 + 8 = 0.6 (i.e., 60%) of the money.
Similarly, we can determine what percentage of f (t) comes from the leading term by calculating
(leading term)
(total value) =
2t7
4 + 2t + 10t2+ 2t7. (2.1)
It’s easier to interpret (2.1) if we factor 2t7 out of the denominator:
(leading term) (total value) = 2t 7 2t7 4 2t7 + 2t 2t7 + 10t2 2t7 + 1 = 1 2 t7 + 1 t6 + 5 t5 + 1 .
Now when t is very large, the fraction of f (t) that comes from 2t7is (leading term) (total value) = 1 2 t7 + 1 t6 + 5 t5 + 1 ≈ 1 (i.e., 100%), | {z }
which is why 2t7 dominates the behavior of f (t) when |t| is large.
If we want to discuss the way a polynomial function changes when its argument is small, a factored form is extremely helpful.
Behavior of polynomials near their roots
Example 2.7. Suppose f (t) = 2(t − 1)3(t − 2)(t − 3)2. Discuss its behavior (a)
when |t| is large, and (b) when t is near a root of f .
Solution: (a) If you were to multiply the factors together you’d see a 6th-degree
Figure 2.4: The curvesy = 2t6
andy = f (t).
polynomial whose leading term is 2t6. Therefore, f (t) will behave like 2t6 when |t| is large (see Figure 2.4).
(b) Let’s begin by discussing what happens when t ≈ 1. We know that f (1) = 0 because of the (t − 1)3 factor. The key to understanding the behavior of f near
time t = 1 is recognizing that the other factors play no role in forcing the function value toward zero. Specifically, (t − 2) ≈ −1 and (t − 3)2 ≈ 4 when t is near 1,
so f (t) ≈ −8(t − 1)3. Because f (t) behaves like a cubic near t = 1, its graph will
“snake” through the t-axis there, just like the graph of a cubic function. We can perform the same kind of qualitative analysis near the other roots . . .
t f (t) = 2(t − 1)3(t − 2)(t − 3)2 Behavior
≈ 1 ≈ 2(t − 1)3(−1)(4) = −8(t − 1)3 cubic
≈ 2 ≈ 2(1)(t − 2)(1) = 2(t − 2) linear
≈ 3 ≈ 2(8)(1)(t − 3)2 = 16(t − 3)2 quadratic
Since f (t) is like a first-degree power function near t = 2, the graph of f will “crash”
Figure 2.5: Zooming in on the graph of f (t) in Example 2.7.
through the t-axis there, just as a line would. And since f (t) is like a quadratic near t = 3, the graph of f will rebound off of the t-axis just like a parabola at its vertex.
Try It Yourself: Polynomials in factored form
Example 2.8. Use the technique of Example 2.7 to analyze the behavior of g(t) = t − 4t3, both when |t| is large, and near its roots.
Answer: See on-line solution.
You should know
• the terms polynomial, degree, and leading term; • that linear functions have a constant rate of change.
You should be able to
• discuss a polynomial’s behavior when its argument is large; • calculate and discuss change in linear functions.
