PREFACE
Calculus is one of the greatest achievements of the human intellect. Inspired by problems in astronomy, Newton and Leibniz developed the ideas of calculus 300 years ago. Since then, each century has demonstrated the power of calculus to illuminate questions in mathematics, the physical sciences, engineering, and the social and biological sciences.
Calculus has been so successful because of its extraordinary power to reduce complicated problems to simple rules and procedures. Therein lies the danger in teaching calculus: it is possible to teach the subject as nothing but the rules and procedures—thereby losing sight of both the mathematics and of its practical value. This edition of Applied Calculus continues our effort to promote courses in which understanding reinforces computation.
Mathematical Concepts and Modeling
The first stage in the development of mathematical thinking is the acquisition of a clear intuitive picture of the central ideas. In the next stage, the student learns to reason with the intuitive ideas in plain English. After this foundation has been laid, there is a choice of direction. All students benefit from both mathematical concepts and modeling, but the balance may differ for different groups of students. For instructors wishing to emphasize the connection between calculus and other fields, the text includes:
- A variety of problems and examples from the biological sciences and economics.
- Models from the health sciences and of population growth.
- New problems on sustainability.
- New case studies on medicine by David E. Sloane, MD.
Active Learning: Good Problems
As instructors ourselves, we know that interactive classrooms and well-crafted problems promote student learning. Since its inception, the hallmark of our text has been its innovative and engaging problems. These problems probe student understanding in ways often taken for granted. Praised for their creativity and variety, the influence of these problems has extended far beyond the users of our textbook.
The Fifth Edition continues this tradition. Under our approach, which we called the “Rule of Four,” ideas are presented graphically, numerically, symbolically, and verbally, thereby encouraging students with a variety of learning styles to expand their knowledge. This edition expands the types of problems available:
- End of Section Problems reinforce the ideas of that section and make connections with earlier sections; Chapter Review Problems ask students to review ideas from the whole chapter.
- ConcepTests promote active learning in the classroom. These can be used with or without clickers (personal response systems), and have been shown to dramatically improve student learning. ConcepTests are particularly useful to instructors teaching in a “flipped classroom.” Available in a book or on the web at www.wiley.com/college/hughes-hallett.
- Projects at the end of each chapter provide opportunities for a sustained investigation, often using skills from different parts of the course. These now include medical case studies based on clinical practice.
- True-False Strengthen Your Understanding questions at the end of every chapter enable students to check their progress.
- Spreadsheet Projects in the Appendix to the book provide the opportunity for students to develop their spreadsheet skills while deepening their understanding of functions and calculus.
- Focus on Practice exercises at the end of Chapter 3 and 6 (Differentives and Antiderivatives) build student skill and confidence.
- Online Problems available in WileyPLUS or WeBWorK, for example. Many problems are randomized, providing students with expanded opportunities for practice with immediate feedback.
Origin of the Text: A Community of Instructors
This text, like others we write, draws on the experience of a diverse group of authors and users. We have benefitted enormously from input from a broad spectrum of instructors—at research universities, four-year colleges, community colleges, and secondary schools. For Applied Calculus, the contributions of colleagues in biology, economics, medicine, business, and other life and social sciences have been equally central to the development of the text. It is the collective wisdom of this community of mathematicians, teachers, natural and social scientists that forms the basis for the new edition.
What Student Background is Expected?
This book is intended for students in business, the social sciences, and the life sciences. A background in trigonometry is not required; the sections involving trigonometry are optional.
We have found the material to be thought-provoking for well-prepared students while still accessible to students with limited algebra backgrounds. Providing numerical and graphical approaches as well as the algebraic gives students several ways of mastering the material. This approach encourages students to persist, thereby lowering failure rate; a pre-test over background material is available in the appendix to the book; An algebra refresher is avalable at the student book companion site at www.wiley.com/college/hughes-hallett.
Mathematical Skills: A Balance Between Symbolic Manipulation and Technology
To use calculus effectively, students need familiarity with both symbolic manipulation and the use of technology. The balance between them may vary, depending on the needs of the students and the wishes of the instructor. The book is adaptable to many different combinations.
The book does not require any specific software or technology. It has been used with graphing calculators, many types of software, including computer algebra systems. Any technology with the ability to graph functions and perform numerical integration will suffice. Students are expected to use their own judgment to determine where technology is useful.
The Fifth Edition
Because different users often choose very different topics to cover in a one-semester applied calculus course, we have designed this book for either a one-semester course (with much flexibility in choosing topics) or a two-semester course. Sample syllabi are provided in the Instructor's Manual.
The fifth edition has the same vision as previous editions. In preparing this edition, we solicited comments from a large number of mathematics instructors who had used the text. We continued to discuss with our colleagues in client disciplines the mathematical needs of their students. We were offered many valuable suggestions, which we have tried to incorporate, while maintaining our original commitment to a focused treatment of a limited number of topics. The changes we have made include:
- Updated data and fresh applications throughout the book, including
- New problems on sustainability.
- New case studies on medicine by David E. Sloane, MD.
- Many new problems have been added, designed to build student confidence with basic concepts and to reinforce skills.
- The material on integration has been streamlined and reorganized.
- In Chapter 5, Sections 5.1-5.5 have been streamlined.
- Section 5.6 on Average Value is the former Section 6.1.
- Chapters 6,7 have been rearranged and combined, putting an introduction to antiderivatives before the applications to consumer surplus and present value. This gives instructors the choice of evaluating definite integrals numerically or using the Fundamental Theorem of Calculus.
- New projects have been added in Chapters 1, 2, 3, 4, 5, and 10.
- As in the previous edition, a Pre-test is included for students whose skills may need a refresher prior to taking the course.
Content
This content represents our vision of how applied calculus can be taught. It is flexible enough to accommodate individual course needs and requirements. Topics can easily be added or deleted, or the order changed.
Chapter 1: Functions and Change
Chapter 1 introduces the concept of a function and the idea of change, including the distinction between total change, rate of change, and relative change. All elementary functions are introduced here. Although the functions are probably familiar, the graphical, numerical, verbal, and modeling approach to them is likely to be new. We introduce exponential functions early, since they are fundamental to the understanding of real-world processes. The trigonometric functions are optional.
A brief introduction to elasticity has been added to Section 1.3.
Chapter 2: Rate of Change: The Derivative
Chapter 2 presents the key concept of the derivative according to the Rule of Four. The purpose of this chapter is to give the student a practical understanding of the meaning of the derivative and its interpretation as an instantaneous rate of change. Students will learn how the derivative can be used to represent relative rates of change. After finishing this chapter, a student will be able to approximate derivatives numerically by taking difference quotients, visualize derivatives graphically as the slope of the graph, and interpret the meaning of first and second derivatives in various applications. The student will also understand the concept of marginality and recognize the derivative as a function in its own right.
Focus on Theory: This section discusses limits and continuity and presents the symbolic definition of the derivative.
Chapter 3: Short-Cuts to Differentiation
The derivatives of all the functions in Chapter 1 are introduced, as well as the rules for differentiating products, quotients, and composite functions. Students learn how to find relative rates of change using logarithms.
Focus on Theory: This section uses the definition of the derivative to obtain the differentiation rules.
Focus on Practice: This section provides a collection of differentiation problems for skill-building.
Chapter 4: Using the Derivative
The aim of this chapter is to enable the student to use the derivative in solving problems, including optimization and graphing. It is not necessary to cover all the sections.
Chapter 5: Accumulated Change: The Definite Integral
Chapter 5 presents the key concept of the definite integral, in the same spirit as Chapter 2.
The purpose of this chapter is to give the student a practical understanding of the definite integral as a limit of Riemann sums, and to bring out the connection between the derivative and the definite integral in the Fundamental Theorem of Calculus. We use the same method as in Chapter 2, introducing the fundamental concept in depth without going into technique. The student will finish the chapter with a good grasp of the definite integral as a limit of Riemann sums, and the ability to approximate a definite integral numerically and interpret it graphically. The chapter includes applications of definite integrals in a variety of contexts, including the average value of a function.
Chapter 5 can be covered immediately after Chapter 2 without difficulty.
The introduction to the definite integral has been streamlined. Average values, formerly in Section 6.1, are now in Section 5.6.
Focus on Theory: This section presents the Second Fundamental Theorem of Calculus and the properties of the definite integral.
Chapter 6: Antiderivatives and Applications
This chapter combines the former Chapter 6 and 7. It covers antiderivatives from a graphical, numerical, and algebraic point of view. The Fundamental Theorem of Calculus is used to evaluate definite integrals.
Optional application sections are included on consumer and producer surplus and on present and future value; the integrals in these sections can be evaluated numerically or using the Fundamental Theorem. The chapter concludes with optional sctions on integration by substitution and integration by parts.
Section 6.1, on graphical and numerical antiderivatives, is based on the former Section 7.5. Section 6.2, on symbolic antiderivatives, is based on the former Section 7.1. Using the Fundamental Theorem to find definite integrals is in Section 6.3, formerly Section 7.3. Sections 6.4 and 6.5 are the former Sections 6.2 and 6.3. Sections 6.6 and 6.7 are the former Sections 7.2 and 7.4.
Focus on Practice: This section provides a collection of integration problems for skill-building.
Chapter 7: Probability
This chapter covers probability density functions, cumulative distribution functions, the median and the mean.
Chapter 7 is the former Chapter 8.
Chapter 8: Functions of Several Variables
This chapter introduces functions of two variables from several points of view, using contour diagrams, formulas, and tables. It gives students the skills to read contour diagrams and think graphically, to read tables and think numerically, and to apply these skills, along with their algebraic skills, to modeling. The idea of the partial derivative is introduced from graphical, numerical, and symbolic viewpoints. Partial derivatives are then applied to optimization problems, ending with a discussion of constrained optimization using Lagrange multipliers.
Chapter 8 is the former Chapter 9.
Focus on Theory: This section uses optimization to derive the formula for the regression line.
Chapter 9: Mathematical Modeling Using Differential Equations
This chapter introduces differential equations. The emphasis is on modeling, qualitative solutions, and interpretation. This chapter includes applications of systems of differential equations to population models, the spread of disease, and predator-prey interactions.
Chapter 9 is the former Chapter 10.
Focus on Theory: This section explains the technique of separation of variables.
Chapter 10: Geometric Series
This chapter covers geometric series and their applications to business, economics, and the life sciences.
Chpater 10 is the former Chapter 11.
Appendices
The first appendix introduces the student to fitting formulas to data; the second appendix provides further discussion of compound interest and the definition of the number e. The third appendix contains a selection of spreadsheet projects.
Supplementary Materials
Supplements for the instructor can be obtained by sending a request on your institutional letterhead to Mathematics Marketing Manager, John Wiley & Sons Inc., 111 River Street, Hoboken, NJ 07030-5774, or by contacting your local Wiley representative. The following supplementary materials are available.
- Instructor's Manual (ISBN 978-1-118-71506-2) containing teaching tips, sample syllabii, calculator programs, and overhead transparency masters.
- Instructor's Solution Manual (ISBN 978-1-118-71498-0) with complete solutions to all problems.
- Student's Solution Manual (ISBN 978-1-118-71499-7) with complete solutions to half the odd-numbered problems.
- Additional Material for Instructors, elaborating specially marked points in the text, as well as password protected electronic versions of the instructor ancillaries, can be found on the web at www.wiley.com/college/hughes-hallett.
- Additional Material for Students, at the student book companion site at www.wiley.com/college/hughes-hallett, includes an algebra refresher and web quizzes.
Getting Started Technology Manual Series:
- Getting Started with Mathematica, 3rd edn, by C-K. Cheung, G.E. Keough, Robert H. Gross, and Charles Landraitis of Boston College (ISBN 978-0-470-45687-3)
- Getting Started with Maple, 3rd edn, by C-K. Cheung, G.E. Keough, both of Boston College, and Michael May of St. Louis University (ISBN 978-0-470-45554-8)
ConcepTests
ConcepTests (ISBN 978-1-118-71494-2), or clicker questions, modeled on the pioneering work of Harvard physicist Eric Mazur, are questions designed to promote active learning during class, particularly (but not exclusively) in large lectures. Evaluation data shows that students taught with ConcepTests outperformed students taught by traditional lecture methods 73% versus 17% on conceptual questions, and 63% versus 54% on computational problems.1 A supplement to Applied Calculus, 5th edn, containing ConcepTests by section, is available from your Wiley representative.
Wiley Faculty Network
The Wiley Faculty Network is a peer-to-peer network of academic faculty dedicated to the effective use of technology in the classroom. This group can help you apply innovative classroom techniques and implement specific software packages. Visit www.wherefacultyconnect.com or ask your Wiley representative for details.
WileyPLUS
WileyPLUS, Wiley's digital learning environment, is loaded with all of the supplements above, and also features:
- E-book, which is an exact version of the print text, but also features hyperlinks to questions, definitions, and supplements for quicker and easier support.
- Homework management tools, which easily enable the instructor to assign and automatically grade questions, using a rich set of options and controls.
- QuickStart pre-designed reading and homework assignments. Use them as-is or customize them to fit the needs of your classroom.
- Guided Online (GO) Exercises, which prompt students to build solutions step-by-step. Rather than simply grading an exercise answer as wrong, GO problems show students precisely where they are making a mistake.
- Algebra & Trigonometry Refresher quizzes, which provide students with an opportunity to brush-up on material necessary to master calculus, as well as to determine areas that require further review.
- Graphing Calculator Manual, to help students get the most out of their graphing calculator, and to show how they can apply the numerical and graphing functions of their calculators to their study of calculus.
Acknowledgements
First and foremost, we want to express our appreciation to the National Science Foundation for their faith in our ability to produce a revitalized calculus curriculum and, in particular, to Louise Raphael, John Kenelly, John Bradley, Bill Haver, and James Lightbourne. We also want to thank the members of our Advisory Board, Benita Albert, Lida Barrett, Bob Davis, Lovenia DeConge-Watson, John Dossey, Ron Douglas, Don Lewis, Seymour Parter, John Prados, and Steve Rodi for their ongoing guidance and advice.
In addition, we want to thank all the people across the country who encouraged us to write this book and who offered so many helpful comments. We would like to thank the following people, for all that they have done to help our project succeed: Ruth Baruth, Graeme Bird, Jeanne Bowman, Lucille Buonocore, Scott Clark, Jeff Edmunds, Sunny Fawcett, Lynn Garner, Sheldon P. Gordon, Ole Hald, Jenny Harrison, Adrian Iovita, Thomas Judson, Christopher Kennedy, Donna Krawczyk, Suzanne Lenhart, Madelyn Lesure, Georgia Kamvosoulis Mederer, Nolan Miller, Andrew Pasquale, Richard D. Porter, Laurie Rosatone, Kenneth Santor, Anne Scanlan-Rohrer, Alfred Schipke, Virginia Stallings, “Suds” Sudholz, Ralph Teixeira, Joe B. Thrash, J. Jerry Uhl, Rachel Deyette Werkema, Hannah Winkler, and Hung-Hsi Wu.
Reports from the following reviewers were most helpful in shaping the fifth edition:
Anthony Barcellos, Catherine Benincasa, Bill Blubagh, Carol Demas, Darlene Diaz, Lauren Fern, Wesley Griffith, Juill Guerra, Molly Martin, Rebecca McKay, Barry Peratt, Karl Schaffer, Randy Scott, Paul Vicknair, Tracy Whelan, P. Jay Zeltner.
Reports from the following reviewers were most helpful in shaping the third edition:
Victor Akatsa, Carol Blumberg, Mary Ann Collier, Murray Eisenberg, Donna Fatheree, Dan Fuller, Ken Hannsgen, Marek Kossowski, Sheri Lehavi, Deborah Lurie, Jan Mays, Jeffery Meyer, Bobra Palmer, Barry Peratt, Russ Potter, Ken Price, Maijian Qian, Emily Roth, Lorenzo Traldi, Joan Weiss, Christos Xenophontos.
Reports from the following reviewers were most helpful in shaping the second edition:
Victor Akatsa, Carol Blumberg, Jennifer Fowler, Helen Hancock, Ken Hannsgen, John Haverhals, Mako E. Haruta, Linda Hill, Thom Kline, Jill Messer Lamping, Dennis Lewandowski, Lige Li, William O. Martin, Ted Marsden, Michael Mocciola, Maijian Qian, Joyce Quella, Peter Penner, Barry Peratt, Emily Roth, Jerry Schuur, Barbara Shabell, Peter Sternberg, Virginia Stover, Bruce Yoshiwara, Katherine Yoshiwara.
Deborah Hughes-Hallett
Patti Frazer Lock
Daniel E. Flath
Andrew M. Gleason
Eric Connally
Selin Kalaycıoğlu
Brigitte Lahme
David O. Lomen
David Lovelock
Guadalupe I. Lozano
William G. McCallum
Brad G. Osgood
Cody L. Patterson
Douglas Quinney
Karen Rhea
Adam Spiegler
Jeff Tecosky-Feldman
Thomas W. Tucker
Aaron D. Wooton
APPLICATIONS INDEX
Business and Economics
Admission fees 36
Advertising 7, 79, 111, 116, 357, 364, 374
Aircraft landing/takeoff 45
Airline capacity and revenue 119–120, 346, 354, 363, 369, 375
Annual interest rate 54–55, 59, 60, 85, 86, 109, 111, 126, 148, 155, 314–315, 325, 326, 357, 395, 428–429, 437, 451, 472, 479, 480
Annual yield 329
Apple Apps downloads 112
Average cost 202–207, 232, 236, 237, 297
Bank account 55–57, 85, 148, 155, 166, 167, 312–313, 315, 354, 357, 381, 395, 412–413, 425–426, 428–429, 437, 451–454, 468, 472, 473, 479
Beef consumption 213, 357, 374
Billionaires in US 26
Break–even point 29–32, 35, 36
Budget constraints 34, 38, 387–389, 391–393, 398–401
Business revenue
Apple 82
General Motors 25
Car payments 55, 111, 396, 479
Car rental 13, 79, 355, 356, 381
Cartel pricing 309
Chemical costs 95, 106, 107, 123
Cobb–Douglas production function 201, 207, 361–362, 381, 393, 397, 398, 400
Coffee 57, 58, 79, 126, 166, 270, 374, 386, 398, 413, 435
College savings account 315
Competing businesses 443
Compound interest 23, 43, 49, 53–57, 59, 60, 82, 86, 109, 126, 148, 167, 312–316, 325, 354–355, 425–426, 428–429, 437, 451–454, 465–466, 470–473, 479, 480
Consols 479
Consumer surplus 297, 306–311, 325
Consumption
alternative fuels 59
calorie 69, 144, 373, 374, 400, 452
CFC 7
E85 fuel 59
fossil fuel 81, 247, 264, 279, 475–477, 480
gas in car 126, 151, 167–168, 190–191, 249, 397
wind power 46
Consumption smoothing 275
Contract negotiation 59–60, 479
Cost function 28–29, 31, 35, 36, 62–63, 80, 85, 118–119, 121, 144, 150, 196, 200, 202–207, 231–232, 236, 269–270, 279, 282, 297, 302, 386, 393, 399
Cost overruns 342
Coupon 473
Crop yields 79, 144, 193, 329, 336, 429
Demand curve 31–32, 35–38, 68–69, 71, 81, 85, 144, 149, 157, 209, 297, 302, 306–311, 325
Depreciation 4, 8, 31, 59, 81, 414
Doubling time 51, 53–55, 57–59, 85
Duality 402
Economy 27, 71–72, 362, 399, 471, 473, 479, 481
Economy of scale 118
Elasticity of demand 22, 208–213, 236
Energy output and consumption 44, 57, 107, 126, 247, 264, 276, 305, 323, 477, 478
Equilibrium prices 32–34, 36–39, 81, 85, 297, 306–311, 325
Equilibrium solution 430, 433–434, 437–441, 450, 452, 454, 477
Facebook subscribers 118
Fertilizer use 7, 79, 106, 144, 342
Fixed cost 28–30, 35–37, 62, 80, 118, 197, 199, 201, 203, 207, 231–232, 269–270, 279, 282, 302
Future value 55–56, 59, 60, 85, 312–316, 421
Gas mileage 7, 151, 167, 190–191, 249, 397, 400
Gold production and reserves 109
Government spending 34, 109, 471, 473, 479
Gross Domestic Product 3, 27, 43, 108, 169–170
Gross World Product 50
Harrod–Hicks model 481
Heating costs 273
Households
with PCs 125
Inflation 27, 45, 55, 149, 156, 165
Interest 7, 23, 49, 53–54, 57, 59, 60, 70, 82, 86, 109, 111, 126, 148, 155, 167, 312–316, 325, 326, 354, 357, 374, 375, 381, 395–397, 412–413, 425–426, 428–429, 437, 451–454, 465–466, 470–473, 479, 480
Investments 7, 44, 55, 59, 180, 268, 315, 325, 354, 362, 398, 399, 414, 452–453, 466
Job satisfaction 34
Labor force 25, 362, 366, 378, 393–394, 398
Lifetime
of a machine 336
of a transistor 343
Loan payments 59, 60, 70, 86, 111, 275, 326, 375, 396
Manufacturing 28, 36, 95, 200, 238–239, 383–384, 393, 394, 398, 399
Marginal cost 28–29, 31, 35, 36, 80, 85, 105, 118–123, 129, 144, 149, 150, 194–200, 202–207, 213, 231, 236, 237, 269–270, 279, 282, 302
Marginal product of labor 202
Marginal profit 31, 36, 85, 196, 236, 270
Marginal revenue 31, 35, 36, 85, 119–123, 129, 144, 155, 157, 194–200, 202, 212, 231, 236, 271, 302
Market stabilization point 472, 473
Maximum profit 194–197, 200–202, 212, 231, 236, 355, 383–384, 399
Maximum revenue 68–69, 198, 200, 210
Mortgage payments 70, 126, 357, 374
Multiplier, fiscal policy 109
Multiplier, Lagrange 388–389, 391–394, 399, 401
Net worth of a company 271, 410–411, 432–433, 443
Oil production 112, 270, 271, 302
Phone rates 13, 21, 24, 211, 350
Photocopy reduction 45
Point of diminishing returns 216, 219, 233
Present value 55–56, 59, 85, 86, 312–316, 325, 361, 470–473, 479, 480
Production costs 36, 393–394, 398
Production function 201, 238–239, 361–362, 378, 381, 387, 391–394, 397–402
Production workers 97
Productivity 78
Profit function 30–31, 35, 36, 195–198, 384
Railway passengers 44
Relative change 21–22, 26, 85, 107
Relative rate of change 26, 41, 85, 107–108, 129, 143, 149, 154–156, 160, 165, 170
Revenue function 29–30, 35, 36, 61, 68–69, 71, 85, 118–119, 123, 129, 144, 157, 198–200, 211, 212, 231, 236, 271, 302, 354, 363
Sales forecasts 7, 108, 116, 217, 219, 315
Sales of jeans 22
Stock market 26
Supply curve 31–34, 37, 38, 62–63, 81, 297, 306, 308, 310, 311, 325
Tax cut or rebate 471, 473, 479
Textbook prices 51
Tobacco production 24
Total cost 28, 35, 36, 78, 79, 111, 121–123, 195–197, 199, 201, 202, 204–207, 231–232, 236, 237, 269–270, 273, 279, 282, 302, 355, 365, 386, 393, 399, 479
Total profit 36, 194–195, 198–200, 231–232, 383–384
Total revenue 29–30, 35, 36, 79, 121, 155, 160, 195–197, 199–201, 212, 231–232, 271, 302, 383–384
Total utility 117
Variable cost 28, 35–37, 80, 201, 203, 269, 270
Vehicles per person 51
Wage, real 202
Wages, human height and 16
Warehouse storage 201, 275, 279
Water supply charges 79
World production
automobile 24
bicycle 23
coal 263
gold 109
grain 14
meat 110
milk 14
tobacco 24
zinc 37
Yield, annual 329
Life Sciences and Ecology
AIDS 58
Algae population growth 6, 275
Anaphylaxis 87
Asthma 239
Bacterial colony growth 82, 244–245, 261, 266, 377–378
Birds and worms 234, 439–442, 444
Birth and death rates 266
Blood pressure 87, 194, 275, 284–285, 367, 375, 479
Body mass of a mammal 66, 69, 154
Carbon dioxide levels 283
Carrying capacity 114, 216, 219, 236, 306, 412
Clutch size 194
Cornea curvature 367
Crows and whelks 193
Decomposition of leaves 452, 453
Deforestation 43
Dolphin speed 69
Drug concentrations 7, 40, 57, 58, 65, 91, 99–100, 105–106, 149, 160, 167, 193, 218–227, 233, 237, 262–263, 267, 282, 294, 306, 323, 349, 355–356, 366, 374, 375, 411, 428–432, 438, 439, 451–454, 464, 469, 474–480
Drug desensitization 482
Drug saturation curve 65
Endocrinology 266
Energy (calorie) expenditure 69, 374, 452
Environmental Protection Agency (EPA) 51–52, 148, 279–280
Exponential growth and decay 39–40, 43, 44, 48–60, 85, 407, 424, 425, 459–460
Eye 367
Firebreaks and forest fires 237–238, 264
Fish harvest 25, 339–341, 347, 410, 414, 453, 455
Fish population 50, 143, 149, 193, 248, 410, 414, 417, 453, 456
Foraging time 193
Global warming 360
Gompertz growth equation 235, 423
Ground contamination 58, 78, 411
Growth of a tumor 96, 423, 451
Half–life and decay 53, 57, 58, 82, 85, 221, 428, 429, 452, 469, 474–475, 477–480
Heart rate 7, 15, 26, 111, 266
Hematocrit 193
HIV–AIDS 58
Insect population 413
Ion channel 221
Kleiber's Law 68
Lizard loping 26
Loading curve (in feeding birds) 234
Logistic growth 114, 213–221, 233, 236, 412, 455
Lotka–Volterra equations 439–440, 442
Money supply 381
Muscle contraction 25, 110–111
Nicotine 6, 25, 57, 110, 221–222, 225–226, 438, 469, 478
Nitrous oxide levels 51
Photosynthesis 190, 192, 283, 295
Plant growth 190, 193, 261–262, 278, 283, 342
Pollutant levels 17, 51, 83, 117, 263, 277, 279–280, 399, 411, 413, 426–427, 429, 453
Pulmonologist 109
Rats and formaldehyde 370–371, 375
Relative change 21–22, 26, 85, 107
Relative rate of change 26, 41, 85, 107–108, 129, 143, 149, 154–156, 160, 165, 170
Respiratory deaths 51
Ricker curve 193
Species diversity 6, 15, 67–69, 76, 164
Sperm count 79
Spread of a disease 194, 215, 220, 230, 445–449, 457–458, 464
Sustainable yield 455
Tiger population 58
Tree growth 125, 193, 265, 329, 335, 341
Urology 295
Waste generation 10–11, 266, 279–280, 399
Water flow 188, 229, 271, 281, 328, 414, 426–427
Water pollution 83, 248, 263, 279–280, 411, 426–427, 429
Wolf population 41
Yeast population 118
Zebra mussel population 44, 143
Physical Sciences
Acceleration 106, 117, 127, 248, 249, 263, 267, 281
Altitude 169
Beam strength 69
Biofuel production 112
Brightness of a star 76
Carbon dioxide concentration 5, 77, 81, 170, 264, 283, 302
Carbon dioxide emissions 249
Chemical reactions 108, 194, 429
Chlorofluorocarbons (CFCs) 7, 53, 112, 130
Climate change 168
Distance 5, 13, 14, 18, 20–21, 24, 26, 45, 51, 66, 69, 78, 87, 95, 102, 104, 108, 124, 125, 129–130, 136, 151, 155, 164, 166–168, 170, 193, 212, 233, 234, 236, 237, 242–243, 246–249, 261, 264, 265, 267, 277–279, 325, 343, 366, 372, 374, 398, 399, 403–405, 470
Elevation 6, 45, 85, 109, 126, 335, 359–360
Exponential growth and decay 39–40, 43, 44, 48–60, 85, 407, 424, 425, 459–460
Fog 357
Gravitational force 69
Greenland Ice Sheet 112
Half–life and decay 53, 57, 58, 82, 85, 221, 428, 429, 451, 469, 470, 474–475, 477–478, 480
Height of a ball 144, 165, 278, 470
Hybrid vehicles 249
Hydrochlorofluorocarbons 264
Isotherms 358
Keeling Curve 170
Missile range 386
Newton's laws of cooling and heating 167, 435, 438, 439
Radioactive decay 51–52, 57, 82, 150, 166, 267, 280, 413, 451
Relative change 21–22, 26, 85, 107
Relative rate of change 26, 41, 85, 107–108, 129, 143, 149, 154–156, 160, 165, 170
sea level changes 6, 8, 117–118
Specific heat 70
Temperature changes 2–3, 6, 45, 71, 72, 76–78, 83, 109, 117, 125, 126, 129–130, 144, 149, 166–169, 181, 188, 270, 272, 275, 357, 358, 360–365, 372–374, 386, 395, 397, 401, 412–413, 435–439, 450, 451
Topographical maps 359, 364, 365, 367
Velocity, average 20–21, 24, 26, 69, 79, 85, 90–91, 95, 117, 124
Velocity, instantaneous 90–91, 95, 124
Velocity of a ball 144, 165, 278
Velocity of a bicycle 243, 278
Velocity of a bungee jumper 264
Velocity of a car 7, 20–21, 126, 151, 242–243, 246–249, 261, 277–278, 325, 348, 412, 450
Velocity of a mouse 280
Velocity of a particle 85, 95, 117, 124, 127, 264, 278
Velocity of a rocket 394
Velocity of a runner 246–247, 277
Velocity vs speed 20
Volcanic explosion 357
Volume of a hot air balloon 65
Volume of air in the lungs 76, 109, 164
Volume of a tank 126, 264, 281
Volume of water 80, 186–188, 225, 247, 248, 282, 426–427
Social Sciences
Age distribution 332–334, 336–337
Ancestors 481
Baby boom 217
Birth and death rates 266
Commuting 375
Distribution of resources 118, 213, 328–329, 475–476
DuBois formula 83
Ebbinghaus model for forgetting 438
GPAs 341
Happiness 365
Health care 332
Human body weight 15, 83, 110, 136, 278, 373, 374, 400, 452
Human height and wages 16
Human height prediction 266
Infant mortality rates and health care 348
IQ scores 348
Job satisfaction 34
Monod growth curve 161
Normal distribution 346–347, 351
Okun's Law 15
Olympic records 8–9, 16, 41, 82
Population, United States 26, 87, 96, 97, 168, 213–217, 220, 332–334, 336–339, 344–345, 350
Population, world 24, 25, 43, 59, 96, 112, 165, 169–170, 248, 325
Population density 365–366, 397–398
Population growth 5, 13, 21, 25, 39–41, 43, 44, 46, 48, 50–53, 57–59, 70, 84, 85, 87, 96, 97, 102, 112, 114, 126, 129, 143, 146–147, 149, 161, 165, 168–170, 184–185, 193, 213–217, 219–221, 230, 233, 236, 244, 248, 261–263, 266, 270, 273, 275, 278, 306, 325, 407, 410, 412–415, 417, 421, 425, 429, 439–444, 452, 454–458, 481
Poverty line 117
Relative change 21–22, 26, 85, 107
Relative rate of change 26, 41, 85, 107–108, 129, 143, 149, 154–156, 160, 165, 170
Rituals 398
Scholarship funds 472
Test success rates 343
Traffic patterns 8, 102, 126, 136
Waiting times 334–335, 338, 340, 348, 350
To Students: How to Learn from this Book
- This book may be different from other math textbooks that you have used, so it may be helpful to know about some of the differences in advance. At every stage, this book emphasizes the meaning (in practical, graphical or numerical terms) of the symbols you are using. There is much less emphasis on “plug-and-chug” and using formulas, and much more emphasis on the interpretation of these formulas than you may expect. You will often be asked to explain your ideas in words or to explain an answer using graphs.
- The book contains the main ideas of calculus in plain English. Success in using this book will depend on reading, questioning, and thinking hard about the ideas presented. It will be helpful to read the text in detail, not just the worked examples.
- There are few examples in the text that are exactly like the homework problems, so homework problems can't be done by searching for similar–looking “worked out” examples. Success with the homework will come by grappling with the ideas of calculus.
- For many problems in the book, there is more than one correct approach and more than one correct solution. Sometimes, solving a problem relies on common sense ideas that are not stated in the problem explicitly but which you know from everyday life.
- Some problems in this book assume that you have access to a graphing calculator or computer. There are many situations where you may not be able to find an exact solution to a problem, but you can use a calculator or computer to get a reasonable approximation.
- This book attempts to give equal weight to four methods for describing functions: graphical (a picture), numerical (a table of values), algebraic (a formula), and verbal (words). Sometimes it's easier to translate a problem given in one form into another. For example, you might replace the graph of a parabola with its equation, or plot a table of values to see its behavior. It is important to be flexible about your approach: if one way of looking at a problem doesn't work, try another.
- Students using this book have found discussing these problems in small groups helpful. There are a great many problems which are not cut-and-dried; it can help to attack them with the other perspectives your colleagues can provide. If group work is not feasible, see if your instructor can organize a discussion session in which additional problems can be worked on.
- You are probably wondering what you'll get from the book. The answer is, if you put in a solid effort, you will get a real understanding of one of the crowning achievements of human creativity—calculus—as well as a real sense of the power of mathematics in the age of technology.
1”Peer Instruction in Physics and Mathematics” by Scott Pilzer in Primus, Vol XI, No 2, June 2001. At the start of Calculus II, students earned 73% on conceptual questions and 63% on computational questions if they were taught with ConcepTests in Calculus I; 17% and 54% otherwise.