Applied Calculus for the Managerial, Life, and Social Sciences

895 Pages·2006·18.424 MB·English

by  Soo T. Tan

Learn calculus by seeing how it applies in your world.
Soo Tan’s text helps you succeed in applied calculus by using real-life examples,
clear explanations, and up-to-date technology. Real-world applications—such as
satellite radio subscriptions, Google’s revenue, job outsourcing, and the effects TAN
of smoking bans—are drawn from the areas of business and the behavioral, life,
7
and social sciences. “Portfolio” profiles give you a firsthand look at how real-
world professionals use applied calculus in their work. Exercises help you practice
and assess your problem-solving skills and your understanding of concepts.
Other study resources to help you succeed:
ThomsonNOW™ for Tan’s Applied Student Solutions Manual
Calculus for the Managerial, Life, 0-495-11936-9 A
and Social Sciences, Seventh Edition Contains fully worked-out solutions to the p
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Through ThomsonNOW™, you can access: You’ll find study hints, review material, instructions
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for using various graphing calculators, plus other
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16-W3979-EPF 11/2/06 3:09 PM Page EP2
ABOUT THE COVER
Navin Khaneja’s passion for mathematics has become a lifelong pursuit. With a background in
both electrical engineering and mathematics, he went on to receive his Ph.D. in applied mathe-
matics from Harvard University in 2000. Building on his interests in the area of control theory,
Navin set out to explore its many applications. Currently, he is developing methods for optimal
control of quantum systems. Many scientific tools require control over quantum phenomena,
including existing technologies such as nuclear magnetic resonance (NMR) spectroscopy (used
to understand the structure and dynamics of proteins) and future innovations such as quantum
computers (designed to significantly increase computational power).
Internationally recognized for this work, Navin most recently received a Friedrich Wilhelm Bessel
Research Prize from the Alexander von Humboldt Foundation. Each year, the foundation—a non-
profit established by the Federal Republic of Germany for the promotion of international research
cooperation—grants awards to young scientists and scholars from abroad who are already recog-
NAVIN KHANEJA
nized as outstanding researchers in their fields. In the spirit of international cooperation, Navin Applied Scientist
will also work on research projects of his own choice with colleagues in Germany.
In 2003, Navin was awarded an Alfred P. Sloan Fellowship, which is designed to stimulate groundbreaking research among young
scholars with outstanding promise. Additionally, in 2001 he received the National Science Foundation’s (NSF) Career Award. One
of the NSF’s most prestigious awards, the Faculty Early Career Development (CAREER) Program supports the activities of teacher–
scholars who most effectively integrate research and education within the context of the mission of their organization.
Navin Khaneja is presently an associate professor of electrical engineering in the Division of Engineering and Applied Sciences
(DEAS) at Harvard University.
Look for other featured applied researchers in the Tan applied mathematics series:
PETER BLAIR HENRY CHRIS SHANNON MARK VAN DER LAAN JONATHAN D. FARLEY
International Economist Mathematical Economist Biostatistician Applied Mathematician
Stanford University University of California, University of California, Massachusetts Institute of
Berkeley Berkeley Technology
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L A
IST OF PPLICATIONS
BUSINESS AND ECONOMICS Cost of producing surfboards, 150 Forecasting sales, 158, 416
401(k) investors, 314 Cost of removing toxic waste, 181, 296 Franchises, 475, 506
Accumulated value of an income stream, 506 Cost of wireless phone calls, 245 Fuel capacity of a space shuttle, 483
Accumulation years of baby boomers, 197, 541 Creation of new jobs, 195 Fuel consumption of domestic cars, 518
Advertising, 86, 150, 239, 278, 329, 541, 573 Credit card debt, 88, 407 Fuel economy of cars, 172, 248
Ailing financial institutions, 129, 147 Crop yield, 148, 371 Gasoline prices, 291
Aircraft structural integrity, 259 Cruise ship bookings, 195 Gasoline self-service sales, 57
Airplane arrival times, 671 Demand for agricultural commodities, 239 Gas station sales, 679
Air travel, 390 Demand for butter, 551, 679 Gender gap, 60
Alternative energy sources, 461 Demand for CD players, 616 Google’s revenue, 282
Alternative minimum tax, 281, 337 Demand for commodities, 650 Gross domestic product, 150, 166, 217, 239, 276, 311
Amusement park attendance, 196, 505 Demand for computer software, 532 Growth of bank deposits, 56
Annual retail sales, 95, 154 Demand for digital camcorder tapes, 489 Growth of HMOs, 173, 498
Annuities, 359, 391, 475 Demand for DVDs, 574 Growth of managed services, 261
Approximating changes in revenue, 595 Demand for electricity, 580 Growth of service industries, 520, 648, 708
Assembly time of workers, 278, 383, 434 Demand for perfume, 367 Growth of Web sites, 336
Authentication technology, 578 Demand for personal computers, 195, 387 Health-care costs, 170, 407
Auto financing, 445 Demand for RNs, 279 Health club membership, 158, 189
Auto replacement parts market, 89 Demand for wine, 368 Home mortgages, 538, 543, 544, 766
Average age of cars in U.S., 300 Demand for wristwatches, 181, 195, 757, 766 Home sales, 173
Average daily TV-viewing time, 617 Depletion of Social Security funds, 315 Home-shopping industry, 135
Banking, 56, 147, 405 Depreciation, 88, 365, 448 Hotel occupancy rate, 75, 89, 194, 808
Blackberry subscribers, 87 Designing a cruise ship pool, 587 Households with microwaves, 389
Black Monday, 285 Determining the optimal site, 568 Housing prices, 358, 447
Book design, 91, 323 Digital camera sales, 168 Housing starts, 76, 195, 225
Box-office receipts, 114, 181, 296 Digital TV sales, 280 Illegal ivory trade, 88
Broadband Internet households, 61 Digital TV services, 44 Income distribution of a country, 490
Budget deficit and surplus, 68, 252 Digital TV shipments, 95 Incomes of American families, 370
Business cycles, 791, 795, 803 Digital versus film cameras, 88 Income streams, 468, 529
Business spending on technology, 281 Disability benefits, 217 Indian gaming industry, 94
Cable ad revenue, 96 Disposable annual incomes, 86 Inflation, 215
Cable TV subscription, 244, 407, 418, 448, 664 Document management, 87 Information security software sales, 576
Calling cards, 577 Driving costs, 81, 114, 154 Installment contract sales, 489
Capacity of man-made lake, 485 Driving range of an automobile, 10 Internal rate of return, 762, 765
Capital value, 529, 726 Drug spending, 281 Inventory control and planning, 129, 321, 322, 325,
Cargo volume, 266 DVD sales, 176, 407 329, 330
Cash reserves at Blue Cross and Blue Shield, 282 Effect of advertising on bank deposits, 278 Investment analysis, 359, 469, 475
CDs, 351, 489 Effect of advertising on hotel revenue, 281 Investment groups, 766
Cellular phone subscription, 262 Effect of advertising on profit, 150, 239 Investment options, 358
Charter-flight revenue, 84, 92 Effect of advertising on sales, 86, 169, 235, 273, 387, Investment returns, 240, 358, 394
Chip sales, 87 461 IRAs, 470
City planning, 118, 182 Effect of capital and labor on productivity, 598 Keogh accounts, 240, 489
Coal production, 447, 489 Effect of housing starts on jobs, 195 Land prices, 555, 568, 613
Cobb–Douglas production function, 550, 555, 588, Effect of inflation on salaries, 359 Life span of car batteries, 694
595, 617 Effect of inventory and floor space on profit, 597 Life span of color television tubes, 664, 679
COLAs, 61 Effect of luxury tax on consumption, 194 Life span of light bulbs, 658, 669
Commissions, 130 Effect of mortgage rates on housing starts, 75, 239 Loan amortization, 370, 544
Commodity prices, 130, 158, 368, 497 Effectofpriceincreaseonquantitydemanded,239,242 Loan consolidation, 358
Common stock transactions, 4, 358 Effect of production on profit, 597 Loans at Japanese banks, 367
Commuter airlines, 496 Effect of speed on operating cost of a truck, 235 Locating a TV relay station, 566
Compact disc sales, 497 Effect of tax cut on spending, 729, 770 Lorentz curves, 472, 475, 506
Comparison of bank rates, 359 Effect of TV advertising on car sales, 461 Magazine circulation, 403
Complementary commodities, 551, 556 Efficiency studies, 169, 280, 438 Mail-order phone sales, 664
Computer game sales, 532 Elasticity of demand, 205, 208, 210, 211, 230 Management decisions, 281, 469
Computer resale value, 488 E-mail usage, 87 Manufacturing capacity, 67, 173, 266, 284
Construction jobs, 76, 103, 184 Energy conservation, 452, 460 Manufacturing capacity operating rate, 307
Consumer decisions, 30 Energy consumption and productivity, 130, 358 Marginal average cost function, 200, 201, 209, 210
Consumer demand, 168, 181, 245, 394 Energy efficiency of appliances, 367 Marginal cost function, 198, 199, 209, 210, 437, 488,
Consumer price index, 169, 273, 394 Establishing a trust fund, 529, 770 489
Consumers’ surplus, 467, 473, 474, 476, 477, 489, Estimating changes in profit, 617 Marginal productivity of labor and capital, 550
505, 510, 519, 532 Expected demand, 394 Marginal productivity of money, 589
Consumption function, 61 Expressway tollbooths, 664 Marginal profit, 203, 209, 210
Consumption of electricity, 435 Factory worker wages, 690, 691, 757 Marginal propensity to consume, 210
Consumption of petroleum, 519 Federal budget deficit, 68, 252 Marginal propensity to save, 210
Cost of laying cable, 26, 30 Federal debt, 96, 314 Marginal revenue, 203, 209, 210, 311, 367, 480
Cost of producing calculators, 329 Female self-employed workforce, 309 Market equilibrium, 83, 91, 95, 157, 158, 466, 757
Cost of producing guitars, 406 Financing a college education, 359, 475 Market for cholesterol reducing drugs, 78
Cost of producing loudspeakers, 303 Financing a home, 239, 241 Market for drugs, 577, 581
Cost of producing PDAs, 75 Forecasting commodity prices, 239 Market share, 148, 404
Cost of producing solar cell panels, 414 Forecasting profits, 239, 281 Markup on a car, 10
((ccoonnttiinnuueedd))
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List of Applications (continued)
Mass transit subsidies, 576 Projected retirement funds, 262 Waiting times, 664, 666
Maximizing crop yield, 323 Projection TV sales, 488 Web hosting, 262
Maximizing oil production, 368 Purchasing power, 358 Wilson lot size formula, 544
Maximizing production, 591 Quality control, 10, 406, 715 Worker efficiency, 62, 86, 169, 280, 329
Maximizing profit, 303, 309, 310, 328, 329, 565, 568, Racetrack design, 325 World production of coal, 447, 489
574, 586, 590 Rate of bank failures, 220, 266, 314 Worldwide production of vehicles, 197
Maximizing revenue, 310, 324, 367, 617 Rate of change of DVD sales, 176 Yahoo! in Europe, 377
Maximizing sales, 591 Rate of change of housing starts, 225
Meeting profit goals, 10 Rate of net investment, 625 SOCIAL SCIENCES
Meeting sales targets, 10 Rate of return on investment, 358, 498 Age of drivers in crash fatalities, 263
Metal fabrication, 322 Real estate, 355, 359, 428, 447, 518 Aging drivers, 86
Minimizing construction costs, 322, 329, 590 Real estate investment groups, 766 Aging population, 193, 218, 815
Minimizing container costs, 319, 323, 329, 591 Reliability of computer chips, 387 Air pollution, 194, 262, 263, 267, 282, 408, 519
Minimizing costs of laying cable, 324 Reliability of microprocessors, 664, 717 Air purification, 217
Minimizing fencing costs, 617 Reliability of robots, 664 Alcohol-related traffic accidents, 497
Minimizing heating and cooling costs, 569 Resale value, 387, 650 Annual college costs, 581
Minimizing packaging costs, 323, 329 Restaurant revenue, 795, 803 Arson for profit, 543
Minimizing production costs, 310 Retirement planning, 358, 359, 489 Bursts of knowledge, 124
Minimizing shipping costs, 29 Revenue growth of a home theater business, 358 Civil service exams, 691
Morning traffic rush, 267 Revenue of a charter yacht, 324 Closing the gender gap in education, 61
Mortgage rates, 504 Revenue of a travel agency, 598 College admissions, 43, 576
MP3 player reliability, 757 Reverse annuity mortgage, 475 Commuter trends, 488
Multimedia sales, 220, 285 Sales forecasts, 51 Continuing education enrollment, 194, 708
Multiplier effect, 726 Sales growth and decay, 44 Cost of removing toxic waste, 114, 178, 181, 296
Navigation systems, 44, 48 Sales of a sporting good store, 38 Crime, 217, 239, 257, 311
Net investment flow, 448 Sales of digital signal processors, 95, 169 Cube rule, 62
Net sales, 576 Sales of digital TVs, 86 Curbing population growth, 170
New construction jobs, 184 Sales of drugs, 577 Decline of union membership, 67
Newsmagazine shows, 418 Sales of DVD players versus VCRs, 89 Demographics, 388
Nielsen television polls, 134, 147 Sales of functional food products, 262 Dependency ratio, 282
Office rents, 311 Sales of GPS equipment, 577 Disability benefits, 217
Oil production, 448, 460, 489, 495 Sales of loudspeakers, 488 Disability rates, 336
Oil spills, 230, 514, 532 Sales of mobile processors, 281 Dissemination of information, 388
Online ad sales, 407 Sales of pocket computers, 438 Distribution of families by size, 666
Online banking, 366, 390, 577 Sales of prerecorded music, 60 Distribution of incomes, 10, 360, 473, 475
Online buyers, 168, 377 Sales promotions, 367 Driving age requirement, 681
Online hotel reservations, 328 Sales tax, 61 Educational level of senior citizens, 40, 575
Online retail sales, 358 Satellite radio subscriptions, 406 Effect of budget cuts on crime rate, 280
Online sales of used autos, 577 Selling price of DVD recorders, 87, 193 Effect of immigration on population growth, 641
Online shopping, 96 Shopping habits, 679 Effect of smoking bans, 280
Online spending, 96, 577 Sickouts, 314 Elderly workforce, 262
Online travel, 581 Sinking funds, 471, 641 Endowments, 527, 529, 729
Operating costs of a truck, 235 Social Security beneficiaries, 136 Energy conservation, 456
Operating rates of factories, mines, and utilities, 307 Social Security contributions, 43 Energy needs, 435
Optimal charter flight fare, 324 Social Security wage base, 577 Families’ annual income, 360
Optimal market price, 364 Solvency of the Social Security system, 299, 315 Female life expectancy, 192, 418, 617
Optimal selling price, 368 Spending on Medicare, 169 Food stamp recipients, 315
Optimal speed of a truck, 325 Starbucks’ annual sales, 580 Foreign-born residents, 311
Optimal subway fare, 318 Starbucks’ store count, 576 Gender gap, 60
Outpatient service companies, 408 Stock prices, 794, 801, 803, 804 Global epidemic, 440
Outsourcing of jobs, 87, 193, 281 Stock purchase, 4 Global supply of plutonium, 75
Ownership of portable phones, 168 Substitute commodities, 551, 556 Grade distributions, 691
Packaging, 52, 91, 317, 319, 329, 568, 569, 688 Supply and demand, 83, 90, 168, 226, 230, 418, 625, Growth of HMOs, 173, 284
PC shipments, 281 632 Health-care spending, 73, 170, 407, 578
Pensions, 358, 359 Tax planning, 358 HMOs, 79
Perpetual net income stream, 529 Television pilots, 666 Immigration, 89, 386
Perpetuities, 529, 532, 729 Testing new products, 217 Income distributions, 473
Personal consumption expenditure, 210 Time on the market, 285, 314 Increase in juvenile offenders, 371
Petroleum consumption, 517 Training personnel, 641 Intervals between phone calls, 532
Portable phone services, 168, 578 Tread lives of tires, 520 Lay teachers at Roman Catholic schools, 388, 391
Present value of a franchise, 490, 498 Truck leasing, 61 Learningcurves,124,129,181,239,387,418,641,650
Present value of an income stream, 469, 527 Trust funds, 529 Logistic curves, 385
Price earnings ratio, 598 TV viewing patterns, 134, 193 Male life expectancy, 245, 578
Price of replacement automobile parts, 89 U.S. daily oil consumption, 519 Marijuana arrests, 96, 440
Prime interest rate, 130 U.S. drug sales, 577 Married households, 336
Producers’ surplus, 467, 473, 474, 476, 505, 519, 532 Use of diesel engines, 314 Married households with children, 168
Product design, 323 U.S. nutritional supplements market, 88 Mass transit, 318, 576
Production costs, 208, 209, 433 U.S. online banking households, 577 Medical school applicants, 262
Production function, 543 U.S. strategic petroleum reserves, 519 Membership in credit unions, 448
Production of steam coal, 497 Value of an art object, 39 Narrowing gender gap, 44
Productivity fueled by oil, 368 Value of an investment, 74 Nuclear plant utilization, 43
Productivity of a country, 555 VCR ownership, 505 Oil spills, 158, 194, 230, 418, 514, 532
Product reliability, 664, 691 Venture-capital investment, 311 Overcrowding of prisons, 75, 263
Profit of a vineyard, 92, 325 Wages, 145 Ozone pollution, 408
(continued on back endpaper)
00-W3979-FM 11/2/06 10:55 AM Page i
Applied
Calculus
for the Managerial, Life,
and Social Sciences
Seventh Edition
S. T. TAN
STONEHILL COLLEGE
Australia • Brazil • Canada • Mexico • Singapore • Spain
United Kingdom • United States
00-W3979-FM 11/2/06 10:55 AM Page ii
Applied Calculus
for the Managerial, Life, and Social Sciences,7e
S. T. Tan
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Student Edition
ISBN-13: 978-0-495-01582-6
ISBN-10: 0-495-01582-2
00-W3979-FM 11/2/06 10:55 AM Page iii
CONTENTS iii
TO PAT, BILL, AND MICHAEL
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Contents
Preface x
CHAPTER 1 Preliminaries 1
1.1 Precalculus Review I 2
1.2 Precalculus Review II 11
1.3 The Cartesian Coordinate System 23
1.4 Straight Lines 31
Chapter 1 Summary of Principal Formulas and Terms 46
Chapter 1 Concept Review Questions 46
Chapter 1 Review Exercises 47
Chapter 1 Before Moving On 48
CHAPTER 2 Functions, Limits, and the Derivative 49
2.1 Functions and Their Graphs 50
Using Technology:Graphing a Function 64
2.2 The Algebra of Functions 68
2.3 Functions and Mathematical Models 76
Using Technology:Finding the Points of Intersection of Two Graphs and Modeling 93
2.4 Limits 97
Using Technology:Finding the Limit of a Function 116
2.5 One-Sided Limits and Continuity 119
Using Technology:Finding the Points of Discontinuity of a Function 132
2.6 The Derivative 135
Using Technology:Graphing a Function and Its Tangent Line 152
Chapter 2 Summary of Principal Formulas and Terms 155
Chapter 2 Concept Review Questions 155
Chapter 2 Review Exercises 156
Chapter 2 Before Moving On 158
CHAPTER 3 Differentiation 159
3.1 Basic Rules of Differentiation 160
Using Technology:Finding the Rate of Change of a Function 171
Note:Sections marked with an asterisk are not prerequisites for later material.
iv
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CONTENTS v
3.2 The Product and Quotient Rules 174
Using Technology:The Product and Quotient Rules 183
3.3 The Chain Rule 185
Using Technology:Finding the Derivative of a Composite Function 196
3.4 Marginal Functions in Economics 197
3.5 Higher-Order Derivatives 212
PORTFOLIO:Steve Regenstreif 213
Using Technology:Finding the Second Derivative of a Function at a Given Point 219
*3.6 Implicit Differentiation and Related Rates 221
3.7 Differentials 232
Using Technology:Finding the Differential of a Function 240
Chapter 3 Summary of Principal Formulas and Terms 242
Chapter 3 Concept Review Questions 243
Chapter 3 Review Exercises 243
Chapter 3 Before Moving On 245
CHAPTER 4 Applications of the Derivative 247
4.1 Applications of the First Derivative 248
Using Technology:Using the First Derivative to Analyze a Function 264
4.2 Applications of the Second Derivative 267
Using Technology:Finding the Inflection Points of a Function 283
4.3 Curve Sketching 285
Using Technology:Analyzing the Properties of a Function 298
4.4 Optimization I 300
Using Technology:Finding the Absolute Extrema of a Function 313
4.5 Optimization II 315
Chapter 4 Summary of Principal Terms 327
Chapter 4 Concept Review Questions 327
Chapter 4 Review Exercises 328
Chapter 4 Before Moving On 330
CHAPTER 5 Exponential and Logarithmic Functions 331
5.1 Exponential Functions 332
Using Technology 338
5.2 Logarithmic Functions 339
5.3 Compound Interest 347
5.4 Differentiation of Exponential Functions 360
PORTFOLIO:Robert Derbenti 361
Using Technology 370
5.5 Differentiation of Logarithmic Functions 371
*5.6 Exponential Functions as MathematicalModels 379
Using Technology:Analyzing Mathematical Models 389
Chapter 5 Summary of Principal Formulas and Terms 392
Chapter 5 Concept Review Questions 392
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vi CONTENTS
Chapter 5 Review Exercises 393
Chapter 5 Before Moving On 394
CHAPTER 6 Integration 395
6.1 Antiderivatives and the Rules of Integration 396
6.2 Integration by Substitution 410
6.3 Area and theDefinite Integral 420
6.4 The Fundamental Theorem of Calculus 429
Using Technology:Evaluating Definite Integrals 440
6.5 Evaluating Definite Integrals 441
Using Technology:Evaluating Definite Integrals for Piecewise-Defined Functions 450
6.6 Area betweenTwo Curves 452
Using Technology:Finding the Area between Two Curves 463
*6.7 Applications of the Definite Integral to Business and Economics 464
Using Technology:Business and Economic Applications 476
*6.8 Volumes of Solids of Revolution 478
Chapter 6 Summary of Principal Formulas and Terms 485
Chapter 6 Concept Review Questions 487
Chapter 6 Review Exercises 487
Chapter 6 Before Moving On 490
CHAPTER 7 Additional Topics in Integration 491
7.1 Integration byParts 492
*7.2 Integration Using Tables of Integrals 499
7.3 Numerical Integration 506
7.4 Improper Integrals 521
Chapter 7 Summary of Principal Formulas and Terms 530
Chapter 7 Concept Review Questions 531
Chapter 7 Review Exercises 531
Chapter 7 Before Moving On 533
CHAPTER 8 Calculus of Several Variables 535
8.1 Functions of Several Variables 536
8.2 Partial Derivatives 545
Using Technology:Finding Partial Derivatives at a Given Point 558
8.3 Maxima and Minima of Functions of Several Variables 559
PORTFOLIO:Kirk Hoiberg 562
8.4 The Method of Least Squares 570
Using Technology:Finding an Equation of a Least-Squares Line 579
8.5 Constrained Maxima and Minima and the Method of Lagrange Multipliers 581
*8.6 Total Differentials 592
*8.7 Double Integrals 599
*8.8 Applications of Double Integrals 606