Soo t tan college mathematics for the managerial, life, and social sciences, (7th edition) cengage learning (2007)

1.1 The Cartesian Coordinate System 21.2 Straight Lines 10 Using Technology: Graphing a Straight Line 24 1.3 Linear Functions and Mathematical Models 28 Using Technology: Evaluating a Fu

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BUSINESS AND ECONOMICS

Access to capital, 476

Accumulated value of an income stream, 1012

Accumulation years of baby boomers, 721

Adjustable-rate mortgage, 318

Advertising, 56, 180, 182, 183, 192, 195, 235, 254,

370, 488, 759, 1057, 1060

Ailing ﬁnancial institutions, 653, 671

Aircraft structural integrity, 783

Alternative energy sources, 975

Alternative minimum tax, 805, 861

Amusement park attendance, 720, 1011

Annual retail sales, 620, 678

Auto replacement parts market, 614

Average age of cars in U.S., 824

Average price of a commodity, 963, 1003

Balloon payment mortgage, 317

Banking, 116, 584, 671, 919

Bidding for contracts, 371

Bidding for rights, 549

Broadband Internet households, 37, 589

Broadband versus dial-up, 50

Budget deﬁcit and surplus, 596, 776

Business spending on technology, 805

Business travel expenses, 93

Buying trends of home buyers, 531

Effect of speed on operating cost of a truck, 759 Effect of TV advertising on car sales, 975 Efﬁciency studies, 693, 804, 952 Elasticity of demand, 729, 730, 732, 734, 735, 754 Electricity consumption, 290

E-mail services, 394 E-mail usage, 613 Employee education and income, 427 Energy conservation, 966, 974 Energy consumption and productivity, 654 Energy efficiency of appliances, 881 Establishing a trust fund, 1035 Expected auto sales, 465 Expected demand, 464 Expected home sales, 465 Expected product reliability, 464 Expected profit, 456, 464 Expected sales, 464 Expressway tollbooths, 1046 Factory workers’ wages, 505 Federal budget deficit, 596, 776 Federal debt, 620, 838 Female self-employed workforce, 833 Financial analysis, 213, 316 Financial planning, 305, 333 Financing a car, 301, 316, 317 Financing a college education, 989 Financing a home, 305, 316, 317, 318, 763, 765 Fisheries, 693, 697

Flex-time, 440 Forecasting commodity prices, 763 Forecasting proﬁts, 763, 805 Forecasting sales, 682, 930 Foreign exchange, 131 401(k) retirement plans, 131, 405, 838 Franchises, 989, 1012

Fuel consumption of domestic cars, 1024 Fuel economy of cars, 696, 772 Gasoline consumption, 541 Gasoline prices, 815 Gasoline sales, 114, 118, 120, 121, 165, 815, 1045 Gasoline self-service sales, 585

Gender gap, 588 Google’s revenue, 806 Gross domestic product, 351, 674, 741, 763, 800, 820, 835

Growth in health club memberships, 713 Growth of bank deposits, 584 Growth of HMOs, 605, 697, 808, 1004 Growth of managed services, 785 Growth of service industries, 1026 Growth of Web sites, 860 Health-care costs, 694, 921 Health-care plan options, 358 Health club membership, 682, 713, 756 Home affordability, 312, 475 Home equity, 310 Home mortgages, 310, 316, 333, 1054, 1059, 1060 Home prices, 835, 961

Home reﬁnancing, 317

Compact disc sales, 1003 Company sales, 68, 323, 326, 330 Complimentary commodities, 1067, 1072 Computer-aided court transcription, 540 Computer game sales, 1049

Computer resale value, 994 Computer sales projections, 930 Consolidation of business loans, 290 Construction costs, 616

Construction jobs, 604 Consumer demand, 608, 705, 763, 908 Consumer price index, 693, 797, 907 Consumption functions, 36, 589 Consumption of electricity, 949 Consumption of petroleum, 1023 Corporate bonds, 290 Cost of drilling, 329 Cost of laying cable, 4, 8 Cost of producing DVDs, 769, 820 Cost of producing guitars, 920 Cost of producing loudspeakers, 827 Cost of producing PDAs, 603 Cost of producing solar cell panels, 928 Cost of producing surfboards, 674 Cost of removing toxic waste, 705, 820 Cost of wireless phone calls, 769 Creation of new jobs, 719 Credit card debt, 613, 921 Credit cards, 333, 376 Cruise ship bookings, 506, 719 Custodial accounts, 989 Customer service, 387, 488 Customer surveys, 440, 451 Decision analysis, 45, 49 Demand for agricultural commodities, 763 Demand for butter, 1045

Demand for commodities, 615 Demand for computer software, 1049 Demand for digital camcorder tapes, 995 Demand for DVD players, 614, 908 Demand for electricity, 40, 41, 63, 949 Demand for perfume, 881

Demand for personal computers, 719, 901 Demand for RNs, 803

Demand for wine, 882 Demand for wristwatches, 705, 719 Depletion of Social Security funds, 839 Depreciation, 31, 589, 879, 962 Designing a cruise ship pool, 1020 Determining the optimal site, 1084 Dial-up Internet households, 37 Digital camera sales, 692 Digital TV sales, 804 Digital TV services, 22 Digital TV shipments, 620 Digital versus ﬁlm cameras, 50 Disability beneﬁts, 741 Disposable annual incomes, 611 Document management, 613 Double-declining balance depreciation, 327, 330 Downloading music, 405

Driving costs, 607, 638, 678 Drug spending, 805 Durable goods orders, 393 DVD sales, 700, 921 Economic surveys, 351 Effect of advertising on bank deposits, 802 Effect of advertising on hotel revenue, 805 Effect of advertising on proﬁt, 674, 763

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Home sales, 697

Home shopping industry, 659

Hotel chain growth, 972

Hotel occupancy rate, 603, 614, 718

Households with microwaves, 903

Life insurance premiums, 464

Life span of color television tubes, 1046

Life span of light bulbs, 1038, 1041

Linear depreciation, 31, 36, 68, 589

Loan amortization, 316, 319, 884, 1054

Loan delinquencies, 506

Loans at Japanese banks, 881

Locating a TV relay station, 1082

Marginal average cost function, 724, 725, 733, 734

Marginal cost function, 723, 733, 734, 952, 994, 995

Marginal productivity of labor and capital, 1066

Marginal productivity of money, 1094

Marginal proﬁt, 727, 733, 734, 951, 952

Marginal propensity to consume, 734

Marginal propensity to save, 734

Maximizing crop yield, 848

Maximizing oil production, 882

Product design, 847 Product reliability, 426, 428, 476, 505, 1046 Product safety, 392

Production costs, 732, 947 Production function, 1059 Production of steam coal, 1003 Production planning, 113, 126, 132, 133, 229, 235,

238, 254, 267 Production scheduling, 75, 89, 93, 176, 181, 182, 194,

210, 234, 235, 271, 272 Productivity fueled by oil, 882 Productivity of a country, 1071 Profit from sale of pagers, 603 Profit from sale of PDAs, 603 Profit functions, 33, 36, 68, 200 Profit of a vineyard, 616, 845 Projected retirement funds, 786 Projection TV sales, 994 Promissory notes, 290 Purchasing power, 291 Quality control, 370, 371, 376, 386, 393, 395, 399,

409, 412, 420, 421, 424, 427, 428, 430, 434, 435,

439, 440, 477, 485, 486, 489, 503, 506, 509, 573, 920

Racetrack design, 849 Rate comparisons, 290 Rate of bank failures, 744, 790, 838 Rate of change of cost functions, 722 Rate of change of DVD sales, 700 Rate of change of housing starts, 749 Rate of net investment, 1004 Rate of return on an investment, 290, 332 Real estate, 78, 92, 131, 287, 291, 942, 961, 1024 Real estate transactions, 131, 403, 463, 465 Recycling, 375

Reﬁnancing a home, 317, 318 Reliability of a home theater system, 428 Reliability of computer chips, 901 Reliability of microprocessors, 1046 Reliability of robots, 1046 Reliability of security systems, 428 Resale value, 901

Retirement planning, 290, 304, 315, 317, 333, 995 Revenue growth of a home theater business, 291 Revenue of a charter yacht, 848

Revenue projection, 465 Reverse annuity mortgage, 989 Robot reliability, 489 Royalty income, 303 Salary comparisons, 329, 330 Sales forecasts, 579 Sales growth, 23, 329 Sales of camera phones, 852 Sales of digital signal processors, 619, 693 Sales of digital TVs, 612

Sales of drugs, 60, 66 Sales of DVD players vs VCRs, 614 Sales of functional food products, 786 Sales of GPS equipment, 22, 60 Sales of loudspeakers, 994 Sales of mobile processors, 805 Sales of navigation systems, 22 Sales of prerecorded music, 588 Sales of security products, 703 Sales of vehicles, 476 Sales projections, 488

Maximizing sales, 1096 Meeting proﬁt goals, 573 Metal fabrication, 846 Minimizing average cost, 829, 834, 853 Minimizing construction costs, 846, 853, 1095, 1096 Minimizing container costs, 843, 853, 847, 1096 Minimizing costs of laying cable, 848 Minimizing heating and cooling costs, 1085 Minimizing mining costs, 181, 195, 275 Minimizing packaging costs, 847, 853 Minimizing production costs, 834 Minimizing shipping costs, 8, 182, 183, 195, 253, 254, 272

Money market mutual funds, 291 Money market rates, 450 Morning trafﬁc rush, 791 Mortgages, 310, 316, 317, 318, 333, 1010 Motorcycle sales, 117

Movie attendance, 383, 393 Multimedia sales, 744, 809 Municipal bonds, 290 Mutual funds, 290, 333 Net investment ﬂow, 962 Net-connected computers in Europe, 60 New construction jobs, 708

Newsmagazine shows, 932 Newspaper subscriptions, 352 Nielsen television polls, 658, 672 Nuclear plant utilization, 21 Nurses’ salaries, 60 Ofﬁce rents, 835 Oil production, 962, 974, 995, 1001 Oil production shortfall, 974 Oil spills, 682, 718, 754, 932, 1020, 1049 Online banking, 60, 880, 904

Online buyers, 692, 891 Online hotel reservations, 852 Online sales, 61, 291, 921 Online shopping, 61, 621 Online travel, 66 Operating costs of a truck, 759 Operating rates of factories, mines, and utilities, 831 Optimal charter ﬂight fare, 848

Optimal market price, 878 Optimal selling price, 849 Optimal selling time, 882 Optimal speed of a truck, 849 Optimal subway fare, 842 Optimizing production schedules, 194, 234, 236, 271 Optimizing proﬁt, 211, 232

Organizing business data, 109, 111 Organizing production data, 109, 111, 132 Organizing sales data, 108, 120 Outpatient service companies, 922 Outsourcing of jobs, 612, 717, 805 Packaging, 470, 499, 579, 616, 841, 843, 846, 853,

1084, 1085

PC shipments, 805 Pensions, 290, 291 Perpetual net income stream, 1035 Perpetuities, 1032, 1033, 1035, 1049 Personal consumption expenditure, 734 Personnel selection, 371, 412, 436 Petroleum consumption, 1023 Petroleum production, 165 Plans to keep cars, 405 Pocket computers, 952 Portable phone services, 692 Predicting sales ﬁgures, 16 Predicting the value of art, 16 Prefab housing, 183, 235 Present value of a franchise, 996, 1004 Present value of an income stream, 883, 984, 1033

List of Applications (continued)

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Sales tax, 36, 589

Sampling, 376, 409

Satellite radio subscriptions, 920

Selling price of DVD recorders, 612, 717

Social Security beneﬁciaries, 660

Social Security beneﬁts, 36

Social Security contributions, 21

Social Security wage base, 61

Solvency of the Social Security system, 823

Spam messages, 602

Spending on ﬁber-optic links, 786

Spending on Medicare, 693

Stafﬁng, 359

Starbucks’ annual sales, 66

Starbucks’ store count, 59, 66

Starting salaries, 476

Stock purchase, 570

Stock transactions, 122, 128, 165

Substitute commodities, 1067, 1072

Sum-of-the-years’-digits method of depreciation, 329

Supply and demand, 35, 37, 38, 48, 50, 69, 608, 609,

Union bargaining issues, 358

U.S daily oil consumption, 1025

U.S drug sales, 60

U.S ﬁnancial transactions, 50

U.S nutritional supplements market, 613

U.S online banking households, 60

U.S strategic petroleum reserves, 1025

Use of automated ofﬁce equipment, 541

Use of diesel engines, 838

World production of coal, 961, 995, 1003

Worldwide production of vehicles, 721 Yahoo! in Europe, 891

Zero coupon bonds, 290, 291

SOCIAL SCIENCES

Accident prevention, 392 Age distribution in a town, 479 Age distribution of renters, 436 Age of drivers in crash fatalities, 787 Aging drivers, 612

Aging population, 717, 742 Air pollution, 718, 786, 787, 791, 806, 834, 922, 930, 1025

Air puriﬁcation, 741, 953, 974 Alcohol-related trafﬁc accidents, 1003 Americans without health insurance, 476 Annual college costs, 66

Arrival times, 394 Arson for proﬁt, 1059 Auto-accident rates, 435, 464 Automobile pollution, 600 Bursts of knowledge, 648 Car theft, 427

Civil service exams, 505 Closing the gender gap in education, 589 College admissions, 22, 59, 69, 131, 427, 440, 500 College graduates, 489

College majors, 436, 521 Committee selection, 366 Commuter options, 357 Commuter trends, 350, 520, 530, 994 Commuting times, 461

Compliance with seat belt laws, 435 Computer security, 809

Conservation of oil, 970 Consumer decisions, 8, 289, 329 Consumer surveys, 347, 349, 350, 351, 404 Continuing education enrollment, 718 Correctional supervision, 395 Cost of removing toxic waste, 638, 702, 705, 820 Course enrollments, 404

Court judgment, 289 Crime, 350, 435, 741, 763, 781, 835 Cube rule, 590

Curbing population growth, 694 Decline of union membership, 595 Demographics, 902

Dependency ratio, 806 Disability beneﬁts, 741 Disability rates, 860 Disposition of criminal cases, 396 Dissemination of information, 902 Distribution of families by size, 450 Distribution of incomes, 573, 987, 989 Drivers’ tests, 371, 413

Driving age requirements, 474 Education, 505, 541 Educational level of mothers and daughters, 523 Educational level of senior citizens, 18 Educational level of voters, 426 Education and income, 427 Effect of budget cuts on crime rates, 804 Effect of smoking bans, 804

Elderly workforce, 786 Elections, 376, 435 Election turnout, 476 Endowments, 1033, 1035 Energy conservation, 966, 970 Energy needs, 949

Enrollment planning, 436, 521 Exam scores, 358, 371, 450, 466, 475, 489 Female life expectancy, 716, 932 Financing a college education, 290, 317 Food stamp recipients, 839

Foreign-born residents, 835 Gender gap, 588, 589 Global epidemic, 954 Global supply of plutonium, 603 Grade distributions, 393, 505 Growth of HMOs, 697, 808 Gun-control laws, 406 Health-care spending, 694, 697, 921 Highway speeds, 505

HMOs, 605 Homebuying trends, 531 Homeowners’ choice of energy, 521, 531 Hours worked in some countries, 475 Immigration, 614, 900

Income distributions, 987 Increase in juvenile offenders, 885 Intervals between phone calls, 1046 Investment portfolios, 122 IQs, 505

Jury selection, 370 Lay teachers at Roman Catholic schools, 902, 905 Learning curves, 648, 653, 705, 763, 897, 901, 932 Library usage, 448

Life expectancy, 117 Logistic curves, 899 Male life expectancy at 60, 769 Marijuana arrests, 621, 954 Marital status of men, 475 Marital status of women, 509 Married households, 860 Mass transit, 842 Mass-transit subsidies, 59 Medical school applicants, 786 Membership in credit unions, 962 Mortality rates, 117

Narrowing gender gap, 22 Network news viewership, 531 Oil used to fuel productivity, 882 One- and two-income families, 531 Opinion polls, 358, 393, 435, 437 Organizing educational data, 131, 344 Organizing sociological data, 450, 474, 475 Overcrowding of prisons, 603, 787 Ozone pollution, 922

Percentage of females in the labor force, 885 Percentage of population relocating, 880 Political polls, 358, 387, 396, 520 Politics, 344, 432, 434, 590 Population density, 1105, 1106, 1107, 1109 Population growth, 329, 638, 642, 682, 694, 706, 708,

922, 932, 954, 975, 996 Population growth in Clark County, 620, 806, 946 Population growth in the 21st century, 902, 905 Population over 65 with high-school diplomas, 18 Prison population, 603, 787

Professional women, 531 Psychology experiments, 357, 520, 530 Public housing, 413

Quality of environment, 785, 853 Recycling programs, 1011 Registered vehicles in Massachusetts, 590 Research funding, 147

Restaurant violations of the health code, 488 Ridership, 78, 92

Rising median age, 590 Risk of an airplane crash, 406 Rollover deaths, 405 Safe drivers, 596 Same-sex marriage, 394 SAT scores, 59, 351, 398 Seat-belt compliance, 435 Selection of Senate committees, 371 Selection of Supreme Court judges, 436 Senior citizens, 953

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Senior workforce, 833, 853

Single female-headed households with children, 952

Small-town revival, 520

Social ladder, 436

Socially responsible funds, 717

Social programs planning, 182, 195

Thurstone learning models, 682, 718

Time intervals between phone calls, 1046

Tracking with GPS, 860

Trafﬁc studies, 394, 694, 718

Trafﬁc-ﬂow analysis, 101, 105

Transcription of court proceedings, 540

Trends in auto ownership, 532, 568

TV viewing patterns, 658, 717, 932

UN Security Council voting, 368, 370

Urban–suburban population ﬂow, 515, 516, 521

U.S birth rate, 474

U.S Census, 995

U.S nursing shortage, 787

U.S population by age, 450

U.S population growth, 791

U.S senior citizens, 962

Use of public transportation, 78, 92

World energy consumption, 66

World population growth, 791, 835, 880, 884, 900

Birth weights of infants, 499

Birthrate of endangered species, 602

Birthrates, 474

Blood alcohol level, 881, 882

Blood ﬂow in an artery, 922, 953

Blood pressure, 871

Blood types, 386, 393, 394, 489

Obese children in the United States, 613 Obesity in America, 694, 741 Organizing medical data, 116 Outpatient visits, 61 Over-100 population, 880 Oxygen-restoration rate in a pond, 655, 702, 820, 834 Ozone pollution, 922

Photosynthesis, 639 Poiseuille’s law, 590, 1058 Polio immunization, 881 Preservation of species, 688 Prevalence of Alzheimer’s patients, 590, 594, 743, 785 Probability of transplant rejection, 428

Pulse rates, 718, 974 Radioactive decay, 895, 901, 903, 907 Rate of growth of a tumor, 905 Reaction of a frog to a drug, 613 Reaction to a drug, 836 Reliability of medical tests, 434, 435 Rising median age, 590

Senior citizens’ health care, 614 Serum cholesterol levels, 501 Smoking and emphysema, 423 Speed of a chemical reaction, 639 Spread of contagious disease, 853 Spread of ﬂu epidemic, 819, 882, 899, 902, 908 Spread of HIV, 697

Storing radioactive waste, 848 Strain of vertebrae, 891 Success of heart transplants, 486 Surface area of a horse, 763 Surface area of a lake, 1049 Surface area of a single-celled organism, 588 Surface area of the human body, 922, 1059, 1072 Surgeries in physicians’ ofﬁces, 790, 809 Testosterone use, 612

Time rate of growth of a tumor, 905 Toxic pollutants, 638

Unclogging arteries, 762 U.S infant mortality rate, 908 Velocity of blood, 692, 835 Veterinary science, 196, 271 Violations of the health code, 488 Von Bertanlanffy growth function, 902 Waiting times, 1045

Walking versus running, 613 Water pollution, 802 Weber–Fechner law, 892 Weight of whales, 22 Weights of children, 882 Weiss’s law, 654 Whale population, 688, 962 Yield of an apple orchard, 616

GENERAL INTEREST

Automobile options, 341 Automobile selection, 358 Ballast dropped from a balloon, 921 Birthday problem, 410, 414 Blackjack, 357, 413 Blowing soap bubbles, 755 Boston Marathon, 783 Boyle’s law, 590 Car pools, 370 Carrier landing, 923 Coast Guard patrol search mission, 755 Code words, 358

Coin tosses, 357, 509 Computer dating, 358 Computing phone bills, 132 Crossing the ﬁnish line, 923 Designing a grain silo, 848

Cancer survivors, 590 Carbon-14 dating, 896, 901 Carbon monoxide in the air, 697, 717, 896, 901, 922 Cardiac output, 1013, 1019

Chemical reactions, 902 Child obesity, 693 Cholesterol levels, 116 Clark’s rule, 68, 681 Color blindness, 416 Concentration of a drug in an organ, 933 Concentration of a drug in the bloodstream, 638, 705,

786, 820, 882, 960, 963, 1003 Concentration of glucose in the bloodstream, 903 Conservation of species, 688, 694, 740 Contraction of the trachea during a cough, 828 Corrective lens use, 395

Cost of hospital care, 290 Cowling’s rule, 37 Cricket chirping and temperature, 37 Crop planning, 181, 195, 210, 235, 269, 271 Crop yield, 672, 761, 848, 885, 965 Dietary planning, 79, 93, 132, 147, 182, 195, 254, 272 Diet-mix problems, 147, 254

Diffusion, 1004 Doomsday situation, 638 Drug dosages, 37, 611, 705 Drug effectiveness, 509 Drug testing, 489, 506 Effect of bactericide, 672, 705 Effect of enzymes on chemical reactions, 820 Energy expended by a ﬁsh, 654, 836 Environment of forests, 785 Epidemic models, 882, 902, 908, 1010 Eradication of polio, 881

Extinction situation, 884 Female life expectancy at 60, 932 Fertilizer mixtures, 78, 92, 147 Flights of birds, 849 Flow of blood in an artery, 963, 922 Flu epidemic, 882, 908

Forensic science, 872 Forestry, 671, 785 Formaldehyde levels, 705 Friend’s rule, 589 Gastric bypass surgeries, 921 Genetically modiﬁed crops, 920 Genetics, 531, 538

Global epidemic, 954 Gompertz growth curve, 903 Groundﬁsh population, 693, 697 Growth of a cancerous tumor, 588, 762 Growth of a fruit ﬂy population, 630, 902, 905, 1011 Growth of bacteria, 674, 894, 900, 907

Harbor cleanup, 590 Heart transplant survival rate, 503 Heights of children, 891, 922 Heights of trees, 872 Heights of women, 479, 509 Ideal heights and weights for women, 22 Importance of time in treating heart attacks, 707 Index of environmental quality, 853

Length of a hospital stay, 1049 Lengths of ﬁsh, 872, 901, 905 Lengths of infants, 1026 Life span of a plant, 1045 Male life expectancy, 60, 65, 769 Measuring cardiac output, 1026 Medical diagnoses, 436 Medical records, 501 Medical research, 428, 435 Medical surveys, 423 Nuclear fallout, 901 Nutrition, 104, 147, 177, 188, 254

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Engine efﬁciency, 1072

Error measurement, 760, 761, 762

Estimating the amount of paint required, 762

Estimating the ﬂow rate of a river, 1025, 1026

Flight path of a plane, 659

Frequency of road repairs, 1045

Gambler’s ruin, 536, 537

Game shows, 386

IQs, 1058

Keeping with the trafﬁc ﬂow, 595

Launching a ﬁghter aircraft, 923

License plate numbers, 358

Velocity of a car, 673, 917, 919, 920, 962 Velocity of a dragster, 1003

VTOL aircraft, 741 Wardrobe selection, 357 Windchill factor, 1072 Women’s soccer, 806 Zodiac signs, 414

Manned bomber research, 432 Menu selection, 358 Meteorology, 393 Motion of a maglev, 622, 738, 920 Newton’s law of cooling, 872, 904, 962 Parking fees, 654

Period of a communications satellite, 766 Poker, 366, 376, 413

Postal regulations, 590, 653, 847 Rafﬂes, 406, 457

Reaction time of a motorist, 1045 Rings of Neptune, 760, 766 Rocket launch, 751 Roulette, 396, 413, 458, 459, 465 Safe drivers, 596

Saving for a college education, 291, 333 Slot machines, 359, 413

Sound intensity, 872 Speedboat racing, 952 Sports, 370, 372, 387, 466, 488, 506, 509

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College Mathematics for the Managerial, Life, and Social Sciences, 7e

S T Tan

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CONTENTS

TO PAT, BILL, AND MICHAEL

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1.1 The Cartesian Coordinate System 2

1.2 Straight Lines 10

Using Technology: Graphing a Straight Line 24

1.3 Linear Functions and Mathematical Models 28

Using Technology: Evaluating a Function 39

1.4 Intersection of Straight Lines 42

Using Technology: Finding the Point(s) of Intersection of Two Graphs 52

*1.5 The Method of Least Squares 54

Using Technology: Finding an Equation of a Least-Squares Line 63

Chapter 1 Summary of Principal Formulas and Terms 67 Chapter 1 Concept Review Questions 67

Chapter 1 Review Exercises 68 Chapter 1 Before Moving On 69

2.1 Systems of Linear Equations: An Introduction 72

2.2 Systems of Linear Equations: Unique Solutions 80

Using Technology: Systems of Linear Equations: Unique Solutions 94

2.3 Systems of Linear Equations: Underdetermined and Overdetermined Systems 97

Using Technology: Systems of Linear Equations: Underdetermined and

Overdetermined Systems 106

2.4 Matrices 108

Using Technology: Matrix Operations 118

2.5 Multiplication of Matrices 121

Using Technology: Matrix Multiplication 134

2.6 The Inverse of a Square Matrix 136

Using Technology: Finding the Inverse of a Square Matrix 150

*2.7 Leontief Input–Output Model 153

Using Technology: The Leontief Input–Output Model 160

Note: Sections marked with an asterisk are not prerequisites for later material.

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3.1 Graphing Systems of Linear Inequalities in Two Variables 168

3.2 Linear Programming Problems 176

3.3 Graphical Solution of Linear Programming Problems 185

*3.4 Sensitivity Analysis 198

PORTFOLIO: Morgan Wilson 206

Chapter 3 Summary of Principal Terms 212 Chapter 3 Concept Review Questions 212 Chapter 3 Review Exercises 213

Chapter 3 Before Moving On 214

4.1 The Simplex Method: Standard Maximization Problems 216

Using Technology: The Simplex Method: Solving Maximization Problems 238

4.2 The Simplex Method: Standard Minimization Problems 243

Using Technology: The Simplex Method: Solving Minimization Problems 255

*4.3 The Simplex Method: Nonstandard Problems 260

Chapter 4 Summary of Principal Terms 274 Chapter 4 Concept Review Questions 274 Chapter 4 Review Exercises 274 Chapter 4 Before Moving On 275

5.1 Compound Interest 278

Using Technology: Finding the Accumulated Amount of an Investment, the

Effective Rate of Interest, and the Present Value of an Investment 292

5.2 Annuities 296

Using Technology: Finding the Amount of an Annuity 306

5.3 Amortization and Sinking Funds 309

PORTFOLIO: Mark Weddington 313

Using Technology: Amortizing a Loan 319

*5.4 Arithmetic and Geometric Progressions 322

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CONTENTS vii

CONTENTS

6.1 Sets and Set Operations 336

6.2 The Number of Elements in a Finite Set 346

6.3 The Multiplication Principle 353

PORTFOLIO: Stephanie Molina 356

6.4 Permutations and Combinations 359

Using Technology: Evaluating n!, P(n, r), and C(n, r) 373

7.1 Experiments, Sample Spaces, and Events 380

7.2 Definition of Probability 388

7.3 Rules of Probability 397

PORTFOLIO: Todd Good 401

7.4 Use of Counting Techniques in Probability 407

7.5 Conditional Probability and Independent Events 414

8.1 Distributions of Random Variables 444

Using Technology: Graphing a Histogram 451

8.2 Expected Value 454

PORTFOLIO: Ann-Marie Martz 461

8.3 Variance and Standard Deviation 467

Using Technology: Finding the Mean and Standard Deviation 478

8.4 The Binomial Distribution 480

8.5 The Normal Distribution 490

8.6 Applications of the Normal Distribution 499

CHAPTER 6

CHAPTER 7

CHAPTER 8

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Markov Chains 511

9.1 Markov Chains 512

Using Technology: Finding Distribution Vectors 522

9.2 Regular Markov Chains 523

Using Technology: Finding the Long-Term Distribution Vector 533

9.3 Absorbing Markov Chains 535

10.1 Exponents and Radicals 548

10.2 Algebraic Expressions 552

10.3 Algebraic Fractions 560

10.4 Inequalities and Absolute Value 568

Chapter 10 Summary of Principal Formulas and Terms 574 Chapter 10 Review Exercises 574

11.1 Functions and Their Graphs 578

Using Technology: Graphing a Function 592

11.2 The Algebra of Functions 596

11.3 Functions and Mathematical Models 604

Using Technology: Finding the Points of Intersection of Two Graphs and Modeling 617

11.4 Limits 621

Using Technology: Finding the Limit of a Function 640

11.5 One-Sided Limits and Continuity 643

Using Technology: Finding the Points of Discontinuity of a Function 656

11.6 The Derivative 659

Using Technology: Graphing a Function and Its Tangent Line 676

D ifferentiation 683

12.1 Basic Rules of Differentiation 684

Using Technology: Finding the Rate of Change of a Function 695

12.2 The Product and Quotient Rules 698

Using Technology: The Product and Quotient Rules 707

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12.3 The Chain Rule 709

Using Technology: Finding the Derivative of a Composite Function 720

12.4 Marginal Functions in Economics 721

12.5 Higher-Order Derivatives 736

PORTFOLIO: Steve Regenstreif 737

Using Technology: Finding the Second Derivative of a Function at a Given

Point 743

*12.6 Implicit Differentiation and Related Rates 745

12.7 Differentials 756

Using Technology: Finding the Differential of a Function 764

13.1 Applications of the First Derivative 772

Using Technology: Using the First Derivative to Analyze a Function 788

13.2 Applications of the Second Derivative 791

Using Technology: Finding the Inflection Points of a Function 807

14.1 Exponential Functions 856

Using Technology 862

14.2 Logarithmic Functions 863

14.3 Differentiation of Exponential Functions 873

PORTFOLIO: Robert Derbenti 874

Using Technology 884

14.4 Differentiation of Logarithmic Functions 885

*14.5 Exponential Functions as Mathematical Models 893

Using Technology: Analyzing Mathematical Models 903

ix

CONTENTS

CHAPTER 13

CHAPTER 14

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Integration 909

15.1 Antiderivatives and the Rules of Integration 910

15.2 Integration by Substitution 924

15.3 Area and the Definite Integral 934

15.4 The Fundamental Theorem of Calculus 943

Using Technology: Evaluating Definite Integrals 954

15.5 Evaluating Definite Integrals 955

Using Technology: Evaluating Definite Integrals for Piecewise-Defined

Functions 964

15.6 Area between Two Curves 966

Using Technology: Finding the Area between Two Curves 977

*15.7 Applications of the Definite Integral to Business and Economics 978

Using Technology: Business and Economic Applications 990

17.1 Functions of Several Variables 1052

17.2 Partial Derivatives 1061

Using Technology: Finding Partial Derivatives at a Given Point 1074

17.3 Maxima and Minima of Functions of Several Variables 1075

PORTFOLIO: Kirk Hoiberg 1078

17.4 Constrained Maxima and Minima and the Method of Lagrange Multipliers 1086

17.5 Double Integrals 1097

Chapter 17 Summary of Principal Terms 1111 Chapter 17 Concept Review Questions 1111 Chapter 17 Review Exercises 1112

Chapter 17 Before Moving On 1114

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The System of Real Numbers 1115

Table 1: Binomial Probabilities 1119

Table 2:The Standard Normal Distribution 1123

Answers to Odd-Numbered Exercises 1125 Index 1191

xi

CONTENTS

APPENDIX A

APPENDIX B

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Trang 20

to realizing our goal.

General Approach

■ Coverage of Topics Since the book contains more than enough material forthe usual two-semester or three-quarter course, the instructor may be flexible inchoosing the topics most suitable for his or her course The following chart onchapter dependency is provided to help the instructor design a course that is mostsuitable for the intended audience

11

Functions, Limitsand the Derivative

17

Calculus of SeveralVariables

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■ Custom Publishing Due to the flexible nature of the topic coverage, instructorscan easily design a custom text containing only those topics that are most suitablefor their course Please see your sales representative for more information onBrooks/Cole’s custom publishing options.

■ Level of Presentation Our approach is intuitive, and we state the results mally However, we have taken special care to ensure that this approach does notcompromise the mathematical content and accuracy

infor-PREFACE

xiv

APPLIED EXAMPLE 4 Enclosing an Area The owner of the Rancho Los Feliz has 3000 yards of fencing with which to enclose a rectangular piece

of grazing land along the straight portion of a river Fencing is not required along

the river Letting x denote the width of the rectangle, ﬁnd a function f in the able x giving the area of the grazing land if she uses all of the fencing (Figure 57)

vari-Solution

1 This information was given.

2 The area of the rectangular grazing land is A xy Next, observe that the amount of fencing is 2x y and this must be equal to 3000 since all the fenc-

ing is used; that is,

x

FIGURE 20

The rectangular grazing land has width

x and length y.

Guidelines for Constructing Mathematical Models

1 Assign a letter to each variable mentioned in the problem If appropriate,

draw and label a ﬁgure.

2 Find an expression for the quantity sought.

3 Use the conditions given in the problem to write the quantity sought as a

function f of one variable Note any restrictions to be placed on the domain

of f from physical considerations of the problem.

■ Approach A problem-solving

approach is stressed throughout

the book Numerous examples

and applications are used to

illustrate each new concept and

result in order to help the

stu-dents comprehend the material

presented An emphasis is

placed on helping the students

formulate, solve, and interpret

the results of the problems

involving applications Very

early on in the text, students are

given practice in setting up word

problems (Section 1.3) and

developing modeling skills

Later when the topic of linear

programming is introduced, one

entire section is devoted to

mod-eling and setting up the

prob-lems (Section 3.2) Also, in

calculus, guidelines are given

for constructing mathematical

models (Section 11.3) As

an-other example, when

optimiza-tion problems are covered the

problems are presented in two

sections First students are asked

to solve optimization problems

in which the objective function

to be optimized is given

(Sec-tion 13.4) and then students are

asked to solve problems where

they have to formulate the

opti-mization problems to be solved

(Section 13.5)

A Maximization Problem

As an example of a linear programming problem in which the objective function is

to be maximized, let’s consider the following simpliﬁed version of a production problem involving two variables.

APPLIED EXAMPLE 1 A Production Problem Ace Novelty wishes to produce two types of souvenirs: type A and type B Each type-A souvenir will result in a proﬁt of $1, and each type-B souvenir will result in a proﬁt of

$1.20 To manufacture a type-A souvenir requires 2 minutes on machine I and

1 minute on machine II A type-B souvenir requires 1 minute on machine I and

3 minutes on machine II There are 3 hours available on machine I and 5 hours available on machine II for processing the order How many souvenirs of each type should Ace make in order to maximize its proﬁt?

Solution As a ﬁrst step toward the mathematical formulation of this problem,

we tabulate the given information, as shown in Table 1.

Let x be the number of type-A souvenirs and y be the number of type-B souvenirs

to be made Then, the total proﬁt P (in dollars) is given by

Trang 22

PREFACE

Source: Wall Street Journal

The graph is decreasing rapidly from t 0 to t 1, reﬂecting the sharp drop

in the index in the ﬁrst hour of trading The point (1, 2047) is a relative minimum

point of the function, and this turning point coincides with the start of an aborted recovery The short-lived rally, represented by the portion of the graph that is

increasing on the interval (1, 2), quickly ﬁzzled out at t 2 (10:30 a.m.) The

rela-tive maximum point (2, 2150) marks the highest point of the recovery The function

is decreasing in the rest of the interval The point (4, 2006) is an inﬂection point of the function; it shows that there was a temporary respite at t 4 (12:30 p.m.) However, selling pressure continued unabated, and the DJIA continued to fall until the closing bell Finally, the graph also shows that the index opened at the high of

the day [ f (0) 2247 is the absolute maximum of the function] and closed at the low

of the day [ fÓ 12Ô 1739 is the absolute minimum of the function], a drop of 508

points!*

y

2200 2100 2000 1900 1800 1700 0

(1, 2047) (2, 2150)

t 0 corresponds to 8:30 a.m., when the market was open for business, and t 7.5

corresponds to 4 p.m., the closing time The following information may be gleaned from studying the graph.

Motivation

Illustrating the practical value of mathematics in applied areas is an important tive of our approach What follows are examples of how we have implemented thisrelevant approach throughout the text

objec-■ Real-life Applications Current

and relevant examples and

exer-cises are drawn from the fields

of business, economics, social

and behavioral sciences, life

sci-ences, physical scisci-ences, and

other fields of interest In the

examples, these are highlighted

with new icons that illustrate the

various applications

■ Intuitive Introduction to

Concepts Mathematical

cepts are introduced with

con-crete real-life examples,

wher-ever appropriate Our goal here

is to capture students’ interest

and show the relevance of

math-ematics to their everyday life

For example, curve-sketching

(Section 13.3) is introduced in

the manner shown here

of $4000 for an automobile, Murphy paid $400 per month for 36 months withinterest charged at 12% per year compounded monthly on the unpaid balance.What was the original cost of the car? What portion of Murphys total car pay-ments went toward interest charges?

Solution The loan taken up by Murphy is given by the present value of theannuity

P 400[1

0

(0

11.01)36]

 400a36 0.01

Trang 23

■ Developing Modeling Skills We believe that one of the most

important skills a student can acquire is the ability to translate a

real problem into a model that can provide insight into the problem

Many of the applications are based on mathematical models

(func-tions) that the author has constructed using data drawn from various

sources, including current newspapers, magazines, and data obtained

through the Internet Sources are given in the text for these applied

problems In Sections 1.3 and 11.3, the modeling process is discussed

and students are asked to use models (functions) constructed from

real-life data to answer questions about the Market for

Cholesterol-Reducing Drugs, HMO Membership, and the Driving Costs for a

Ford Taurus

■ Connections One example (the

maglev example) is used as a

common thread throughout the

development of calculus—from

limits through integration The

goal here is to show students the

connection between the concepts

presented—limits, continuity,

rates of change, the derivative,

the definite integral, and so on

Utilizing Tools Students Use

■ Technology Technology is used to explore mathematical ideas and as a tool tosolve problems throughout the text

■ Exploring with Technology

Questions Here technology is

used to explore mathematical

concepts and to shed further

light on examples in the text

These optional questions appear

throughout the main body of the

text and serve to enhance the

stu-dent’s understanding of the

con-cepts and theory presented Often

the solution of an example in the

text is augmented with a

graphi-cal or numerigraphi-cal solution

Com-plete solutions to these exercises

are given in the Instructor’s

Solution Manual.

PREFACE

xvi

15 B LACK B ERRY S UBSCRIBERS According to a study conducted in

2004, the number of subscribers of BlackBerry, the handheld e-mail devices manufactured by Research in Motion Ltd.,

is expected to be

N(t ) 0.0675t4 0.5083t3 0.893t2 0.66t 0.32

(0 t 4)

where N(t ) is measured in millions and t in years, with

t 0 corresponding to the beginning of 2002.

a How many BlackBerry subscribers were there at the

beginning of 2002?

b What is the projected number of BlackBerry subscribers

at the beginning of 2006?

Source: ThinkEquity Partners

Suppose we want to ﬁnd the velocity of the maglev at t 2 This is just the velocity of the maglev as shown on its speedometer at that precise instant of time Offhand, calculating this quantity using only Equation (1) appears to be an impossi-

ble task; but consider what quantities we can compute using this relationship Obviously, we can compute the position of the maglev at any time t as we did earlier for some selected values of t Using these values, we can then compute the aver-

age velocity of the maglev over an interval of time For example, the average

veloc-ity of the train over the time interval [2, 4] is given by

EXPLORING WITH TECHNOLOGY

Investments that are allowed to grow over time can increase in value surprisingly fast Consider the potential growth of $10,000 if earnings are reinvested More specifically,

invest-ment of $10,000 over a term of t years and earning interest at the rate of 4%, 6%, 8%,

10%, and 12% per year compounded annually.

1 Find expressions for A1(t), A2(t), , A5(t).

2 Use a graphing utility to plot the graphs of A1, A2, , A5 on the same set of

3 Use TRACEto find A1(20), A2(20), , A5 (20) and then interpret your results.

EXPLORING WITH TECHNOLOGY

Trang 24

PREFACE

■ Using Technology These are

optional subsections that appear

after the exercises They can be

used in the classroom if desired

or as material for self-study by

the student Here the graphing

calculator and Excel

spread-sheets are used as a tool to

solve problems The subsections

are written in the traditional

example-exercise format with

answers given at the back of

the book Illustrations showing

graphing calculator screens are

extensively used In keeping with

the theme of motivation through

real-life examples, many sourced

applications are again included

Students can construct their own

models using real-life data in

Using Technology Section 11.3

These include models for the

growth of the Indian gaming

industry, the population growth

in the fastest growing

metro-politan area in the U.S., and the

growth in online spending,

among others In Using

Tech-nology Section 13.3, students

are asked to predict when the

assets of the Social Security

“trust fund” (unless changes are

made) will be exhausted

APPLIED EXAMPLE 3 Indian Gaming Industry The following data gives the estimated gross revenues (in billions of dollars) from the Indian

gaming industries from 1990 (t 0) to 1997 (t 7).

Revenue 0.5 0.7 1.6 2.6 3.4 4.8 5.6 6.8

a Use a graphing utility to ﬁnd a polynomial function f of degree 4 that models the

data.

b Plot the graph of the function f, using the viewing window [0, 8] [0, 10].

c Use the function evaluation capability of the graphing utility to compute f (0),

f (1), , f (7) and compare these values with the original data.

Source: Christiansen/Cummings Associates

Solution

a Choosing P 4 REG(fourth-order polynomial regression) from the STAT CALC tistical calculations) menu of a graphing utility, we ﬁnd

(sta-f (t) 0.00379t4 0.06616t3 0.41667t2 0.07291t 0.48333

b The graph of f is shown in Figure T3.

c The required values, which compare favorably with the given data, follow:

f (t) 0.0129t4 0.3087t3 2.1760t2 62.8466t 506.2955 (0 t 35)

where f (t) is measured in millions of dollars and t is measured in years, with t 0 corresponding to 1995.

a Use a graphing calculator to sketch the graph of f.

b Based on this model, when can the Social Security system be expected to go

broke?

Source: Social Security Administration

Solution

a The graph of f in the window [0, 35] [1000, 3500] is shown in Figure T5.

b Using the function for ﬁnding the roots on a graphing utility, we ﬁnd that y 0

when t 34.1, and this tells us that the system is expected to go broke around 2029.

FIGURE T5

The graph of f ( t )

Trang 25

Exercise Sets The exercise sets are designed to help students understand and applythe concepts developed in each section Three types of exercises are included inthese sets.

■ Self-Check Exercises offer

stu-dents immediate feedback on

key concepts with worked-out

solutions following the section

exercises

■ New Concept Questions are

designed to test students’

under-standing of the basic concepts

discussed in the section and at

the same time encourage students

to explain these concepts in their

own words

■ Exercises provide an ample set

of problems of a routine

compu-tational nature followed by an

extensive set of

d Find an equation of the tangent line to the graph of f at the

point (0, 3).

e Sketch the graph of f and the tangent line to the curve at the

point (0, 3).

2 The losses (in millions of dollars) due to bad loans extended

chieﬂy in agriculture, real estate, shipping, and energy by the Franklin Bank are estimated to be

A f(t) t2 10t 30 (0 t 10)

where t is the time in years (t 0 corresponds to the ning of 1994) How fast were the losses mounting at the beginning of 1997? At the beginning of 1999? At the beginning of 2001?

begin-Solutions to Self-Check Exercises 11.6 can be found on page 676

4 Under what conditions does a function fail to have a derivative

at a number? Illustrate your answer with sketches.

11.6 Exercises

1 A VERAGE W EIGHT OF AN I NFANT The following graph shows the weight measurements of the average infant from the time of

birth (t 0) through age 2 (t 24) By computing the slopes

of the respective tangent lines, estimate the rate of change of

the average infant’s weight when t 3 and when t 18 What

is the average rate of change in the average infant’s weight over the ﬁrst year of life?

t

y

30

22.5 20

10 7.5

2 4 6 8

10 12 14 16 18 20 22 24 3

T1

T2

5 7.5

6 3.5

Months

2 F ORESTRY The following graph shows the volume of wood

produced in a single-species forest Here f (t) is measured in cubic meters/hectare and t is measured in years By comput-

ing the slopes of the respective tangent lines, estimate the rate at which the wood grown is changing at the beginning of year 10 and at the beginning of year 30.

Source: The Random House Encyclopedia

t

y

30 25 20 15 10 5

y = f(t )

Trang 26

2.{x x is a letter of the word TALLAHASSEE}

3 The set whose elements are the even numbers between 3

and 11

4.{x (x 3)(x 4) 0; x, a negative integer}

C {a, d, e} In Exercises 13–16, verify the equation by direct computation.

■ Summary of Principal

Formulas and Terms highlights

important equations and terms

with page numbers given for

quick review

■ New Concept Review

Questions give students a

chance to check their knowledge

of the basic definitions and

con-cepts given in each chapter

■ Review Exercises offer routine

computational exercises followed

by applied problems

Review Sections These sections are designed to help students review the material

in each section and assess their understanding of basic concepts as well as solving skills

problem-Fill in the blanks.

1 A well-defined collection of objects is called a/an

These objects are also called of the

2 Two sets having exactly the same elements are said to be

8 An arrangement of a set of distinct objects in a definite order

is called a/an ; an arrangement in which the order

is not important is a/an

TERMS

) 3 ( e l p i c i r p o i a i p i u m )

7 ( t e s y t p m e )

6 ( t e s

) 4 ( e l p i c i r p o i a i p i u m d i a r e g )

8 ( t e s l a s e i n )

6 ( t e s a f o t n m e l e

) 9 ( n i a t u m r e )

8 ( m a r g i d n V )

6 ( n i a t o r e t s o r

) 4 ( n i a i b m o )

9 ( n i c s e t n i e s )

6 ( y t a e t e s

) 9 ( n i a t n m e l p m o t e s )

7 ( t e s b s

Trang 27

■ Explore & Discuss are optional

questions appearing throughout

the main body of the text that can

be discussed in class or assigned

as homework These questions

generally require more thought

and effort than the usual

exer-cises They may also be used to

add a writing component to the

class or as team projects

Com-plete solutions to these exercises

are given in the Instructor’s

So-lutions Manual.

■ New Portfolios The real-life

ex-periences of a variety of

profes-sionals who use mathematics in

the workplace are related in these

interviews Among those

inter-viewed are a Process Manager

who uses differential equations

and exponential functions in his

work (Robert Derbenti at Linear

Technology Corporation) and

an Associate on Wall Street

who uses statistics and calculus

in writing options (Gary Li at

JPMorgan Chase & Co.)

PREFACE

xx

EXPLORE & DISCUSS

The average price of gasoline at the pump over a 3-month period, during which there

was a temporary shortage of oil, is described by the function f deﬁned on the interval

[0, 3] During the ﬁrst month, the price was increasing at an increasing rate Starting with the second month, the good news was that the rate of increase was slowing down, although the price of gas was still increasing This pattern continued until the

1 Describe the signs of f

(2, 3).

2 Make a sketch showing a plausible graph of f over [0, 3].

C {b, c, e, f } Find

2 Let A, B, and C be subsets of a universal set U and suppose

n(U) 120, n(A) 20, n(A B) 10, n(A C) 11,

n(B C) 9, and n(A B C) 4 Find n[A (B C) c].

3 In how many ways can four compact discs be selected from

six different compact discs?

4 From a standard 52-card deck, how many 5-card poker hands

can be dealt consisting of 3 deuces and 2 face cards?

5 There are six seniors and five juniors in the Chess Club at

Madison High School In how many ways can a team ing of three seniors and two juniors be selected from the mem- bers of the Chess Club?

■ New Before Moving On

Exercises give students a chance

to see if they have mastered the

basic computational skills

de-veloped in each chapter If they

solve a problem incorrectly,

they can go to the companion

Web site and try again In fact,

they can keep on trying until

they get it right If students need

step-by-step help, they can

uti-lize the ThomsonNOW Tutorials

that are keyed to the text and

work out similar problems at

their own pace

Gary Li

TITLE Associate INSTITUTION JPMorgan Chase

As one of the leading ﬁnancial institutions in the world, JPMorgan Chase & Co depends on a wide range of mathematical disciplines from statistics to linear programming to calculus Whether assessing the credit worthiness

of a borrower, recommending portfolio investments or pricing an exotic derivative, quantitative understanding is

a critical tool in serving the ﬁnancial needs of clients

I work in the Fixed-Income Derivatives Strategy group A derivative in ﬁnance is an instrument whose value depends on the price of some other underlying instrument A simple type of derivative is the forward con- tract, where two parties agree to a future trade at a speci- ﬁed price In agriculture, for instance, farmers will often pledge their crops for sale to buyers at an agreed price before even planting the harvest Depending on the weather, demand and other factors, the actual price may

with interest rates With trillions of dollars in this form, especially government bonds and mortgages, fixed- income derivatives are vital to the economy As a strategy group, our job is to track and anticipate key drivers and developments in the market using, in significant part, quantitative analysis Some of the derivatives we look at are of the forward kind, such as interest-rate swaps, where over time you receive fixed-rate payments in exchange for paying a floating-rate or vice-versa A whole other class of derivatives where statistics and calculus are especially relevant are options

Whereas forward contracts bind both parties to a future trade, options give the holder the right but not the obligation to trade at a speciﬁed time and price Similar to

an insurance policy, the holder of the option pays an upfront premium in exchange for potential gain Solving this pricing problem requires statistics, stochastic calculus and enough insight to win a Nobel prize Fortunately for

PORTFOLIO

Trang 28

PREFACE

■ New Skillbuilder Videos,

avail-able through ThomsonNOW

and Enhanced WebAssign, offer

hours of video instruction from

award-winning teacher Deborah

Upton of Stonehill College

Watch as she walks students

through key examples from the

text, step by step—giving them a

foundation in the skills that they

need to know Each example

available online is identified

by the video icon located in the

margin

Other Changes in the Seventh Edition

■ New Applications More than 150 new real-life applications have been duced Among these applications are Sales of GPS Equipment, Broadband Inter-net Households, Switching Internet Service Providers, Digital vs Film Cameras,Online Sales of Used Autos, Financing College Expenses, Balloon Payment Mort-gages, Nurses Salaries, Revenue Growth of a Home Theater Business, Same-SexMarriage, Rollover Deaths, Switching Jobs, Downloading Music, Americans with-out Health Insurance, Access to Capital, Volkswagen’s Revenue, Cancer Survi-vors, Spam Messages, Global Supply of Plutonium, Testosterone Use, BlackBerrySubscribers, Outsourcing of Jobs, Spending on Medicare, Obesity in America, U.S.Nursing Shortage, Effects of Smoking Bans, Google’s Revenue, Computer Secu-rity, Yahoo! in Europe, Satellite Radio Subscriptions, Gastric Bypass Surgeries,and the Surface Area of the New York Central Park Reservoir

intro-■ Two New Sections on Linear Programming Sensitivity Analysis is now

cov-ered in Section 3.4 and The Simplex Method: Nonstandard Problems is nowcovered in Section 4.3

■ Expanded Coverage of Markov Chains Markov Chains is now covered in

Chapter 9 in three sections—9.1 Markov Chains, 9.2 Regular Markov Chains, and9.3 Absorbing Markov Chains

■ Using Technology subsections have been updated for Office 2003 and new

dia-log boxes are now shown

■ A Revised and Expanded Student Solutions Manual Problem-solving gies and additional algebra steps and review for selected problems have beenadded to this supplement

strate-■ Other Changes In Functions and Mathematical Models (Section 11.3), a newmodel describing the membership of HMOs is now discussed by using a scatterplot of the real-life data and the graph of a function that describes that data Anothermodel describing the driving costs of a Ford Taurus is also presented in this samefashion A discussion of the median and the mode has been added to Section 8.6

APPLIED EXAMPLE 7 Oxygen-Restoration Rate in a Pond When organic waste is dumped into a pond, the oxidation process that takes place reduces the pond’s oxygen content However, given time, nature will restore the

oxygen content to its natural level Suppose the oxygen content t days after

organic waste has been dumped into the pond is given by

f (t) 100t

t

2 2

1 2

0 0

t t

1 1

0 0

percent of its normal level.

Trang 29

In Section 13.2 an example calling for the interpretation of the first and second rivatives to help sketch the graph of a function has been added In Section 15.4,the definite integral as a measure of net change is now discussed along with a newexample giving the Population Growth in Clark County Also, Section 16.2, Inte-gration Using Tables of Integrals has been added.

de-Teaching Aids

■ Instructor’s Solutions Manual includes complete solutions for all exercises in

the text, as well as answers to the Exploring with Technology and Explore & cuss questions.

Dis-■ Instructor’s Suite CD includes the Instructor’s Solutions Manual and Test Bank in formats compatible with Microsoft Ofﬁce®

■ Printed Test Bank, which includes test questions (including multiple-choice) and

sample tests for each chapter of the text, is available to adopters of the text

■ Enhanced WebAssign offers an easy way for instructors to deliver, collect, grade, and record assignments via the web Within WebAssign you will ﬁnd:

■ 1500 problems that match the text’s end-of-section exercises

■ Active examples integrated into each problem that allow students to work by-step at their own pace

step-■ Links to Skillbuilder Videos that provide further instruction on each problem

■ Portable PDFs of the textbook that match the assigned section

■ ExamView ®

Computerized Testing allows instructors to create, deliver, and

customize tests and study guides (both print and online) in minutes with this

easy-to-use assessment and tutorial system, which contains all questions from the Test Bank in electronic format.

■ JoinIn™ on Turning Point ®

offers instructors text-specific JoinIn content forelectronic response systems Instructors can transform their classroom and assessstudents’ progress with instant in-class quizzes and polls Turning Point softwarelets instructors pose book-specific questions and display students’ answers seam-lessly within Microsoft PowerPoint lecture slides, in conjunction with a choice of

“clicker” hardware Enhance how your students interact with you, your lecture,and one another

Learning Aids

■ Student Solutions Manual contains complete solutions for all odd-numbered

ex-ercises in the text, plus problem-solving strategies and additional algebra stepsand review for selected problems 0-495-11974-1

■ ThomsonNOW ™ for Tan’s College Mathematics for the Managerial, Life, and Social Sciences, Seventh Edition, designed for self-study, offers text-speciﬁc tu-

torials that require no setup or involvement by instructors (If desired, instructorscan assign the tutorials online and track students’ progress via an instructor grade-book.) Students can explore active examples from the text as well as SkillbuilderVideos that provide additional reinforcement Along the way, they can check theircomprehension by taking quizzes and receiving immediate feedback

■ vMentor™ allows students to talk (using their own computer microphones) to

tu-tors who will skillfully guide them through a problem using an interactive board for illustration Up to 40 hours of live tutoring a week is available and PREFACE

white-xxii

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Kimberly Jordan Burch

Montclair State University

Northern Arizona University

Sandra Wray McAfee

Trang 31

of this edition I also thank Jamie Armstrong and the staff at Newgen for their duction services Finally, a special thanks to the mathematicians—Chris Shannonand Mark van der Lann at Berkeley, Peter Blair Henry at Stanford, Jonathan D Far-ley at MIT, and Navin Khaneja at Harvard for taking time off from their busy sched-ules to describe how mathematics is used in their research Their pictures appear onthe covers of my applied mathematics series.

pro-S T Tan

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About the AuthorSOO T TAN received his S.B degree from Massachusetts

Institute of Technology, his M.S degree from the sity of Wisconsin–Madison, and his Ph.D from the Univer-sity of California at Los Angeles He has published numer-ous papers in Optimal Control Theory, Numerical Analysis,and Mathematics of Finance He is currently a Professor ofMathematics at Stonehill College

Univer-“By the time I started writing the first of what turned out

to be a series of textbooks in mathematics for students in the managerial, life, and social sciences, I had quite a few years of experience teaching mathe-

matics to non-mathematics majors One of the most important lessons I learned from my early

experience teaching these courses is that many of the students come into these courses with

some degree of apprehension This awareness led to the intuitive approach I have adopted in all

of my texts As you will see, I try to introduce each abstract mathematical concept through an

example drawn from a common, real-life experience Once the idea has been conveyed, I then

proceed to make it precise, thereby assuring that no mathematical rigor is lost in this intuitive

treatment of the subject Another lesson I learned from my students is that they have a much

greater appreciation of the material if the applications are drawn from their fields of interest

and from situations that occur in the real world This is one reason you will see so many

exer-cises in my texts that are modeled on data gathered from newspapers, magazines, journals, and

other media Whether it be the market for cholesterol-reducing drugs, financing a home, bidding

for cable rights, broadband Internet households, or Starbucks’ annual sales, I weave topics of

current interest into my examples and exercises to keep the book relevant to all of my readers.”

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of real numbers This in turn enables us to compute the distance between

two points algebraically We also study straight lines Linear functions,

whose graphs are straight lines, can be used to describe many relationshipsbetween two quantities These relationships can be found in fields of study

as diverse as business, economics, the social sciences, physics, and cine In addition, we see how some practical problems can be solved byfinding the point(s) of intersection of two straight lines Finally, we learnhow to find an algebraic representation of the straight line that “best” fits

medi-a set of dmedi-atmedi-a points thmedi-at medi-are scmedi-attered medi-about medi-a strmedi-aight line

company use? Robertson Controls

Company must decide between two

manufacturing processes for its

Model C electronic thermostats In

Example 4, page 44, you will see

how to determine which process will

be more profitable

1

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The Cartesian Coordinate System

The real number system is made up of the set of real numbers together with the usualoperations of addition, subtraction, multiplication, and division We assume that youare familiar with the rules governing these algebraic operations (see Appendix A).Real numbers may be represented geometrically by points on a line This line is

called the real number, or coordinate, line We can construct the real number line

as follows: Arbitrarily select a point on a straight line to represent the number 0 This

point is called the origin If the line is horizontal, then choose a point at a

conve-nient distance to the right of the origin to represent the number 1 This determines

the scale for the number line Each positive real number x lies x units to the right of

0, and each negative real number x lies x units to the left of 0.

In this manner, a one-to-one correspondence is set up between the set of realnumbers and the set of points on the number line, with all the positive numbers lying

to the right of the origin and all the negative numbers lying to the left of the origin(Figure 1)

In a similar manner, we can represent points in a plane (a two-dimensionalspace) by using the Cartesian coordinate system, which we construct as follows:Take two perpendicular lines, one of which is normally chosen to be horizontal

These lines intersect at a point O, called the origin (Figure 2) The horizontal line is called the x-axis, and the vertical line is called the y-axis A number scale is set up

along the x-axis, with the positive numbers lying to the right of the origin and the

negative numbers lying to the left of it Similarly, a number scale is set up along the

y-axis, with the positive numbers lying above the origin and the negative numbers

lying below it

Note The number scales on the two axes need not be the same Indeed, in many

applications different quantities are represented by x and y For example, x may resent the number of cell phones sold and y the total revenue resulting from the sales.

rep-In such cases it is often desirable to choose different number scales to represent thedifferent quantities Note, however, that the zeros of both number scales coincide atthe origin of the two-dimensional coordinate system

We can represent a point in the plane uniquely in this coordinate system by an

ordered pairof numbers—that is, a pair (x, y), where x is the first number and y the second To see this, let P be any point in the plane (Figure 3) Draw perpendiculars from P to the x-axis and y-axis, respectively Then the number x is precisely the number that corresponds to the point on the x-axis at which the perpendicular through P hits the x-axis Similarly, y is the number that corresponds to the point on the y-axis at which the perpendicular through P crosses the y-axis.

p

1 2

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Conversely, given an ordered pair (x, y) with x as the first number and y the ond, a point P in the plane is uniquely determined as follows: Locate the point on the x-axis represented by the number x and draw a line through that point parallel to the y-axis Next, locate the point on the y-axis represented by the number y and draw

sec-a line through thsec-at point psec-arsec-allel to the x-sec-axis The point of intersection of these two lines is the point P (Figure 3).

In the ordered pair (x, y), x is called the abscissa, or x-coordinate, y is called the ordinate, or y-coordinate, and x and y together are referred to as the coordinatesof the point P The point P with x-coordinate equal to a and y-coordinate equal

to b is often written P(a, b).

The points A(2, 3), B(2, 3), C(2, 3), D(2, 3), E(3, 2), F(4, 0), and G(0,5) are plotted in Figure 4

Quadrant I (+, +)

y

Quadrant II (–, +)

Quadrant III (–, –)

Quadrant IV (+, –)

x

y

4 2

–3 –1

D(2, – 3) C(–2, –3)

– 2 – 4 – 6

F(4, 0) E(3, 2)

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The Distance Formula

One immediate benefit that arises from using the Cartesian coordinate system is thatthe distance between any two points in the plane may be expressed solely in terms

of the coordinates of the points Suppose, for example, (x1, y1) and (x2, y2) are any

two points in the plane (Figure 6) Then the distance d between these two points is,

by the Pythagorean theorem,

For a proof of this result, see Exercise 45, page 9

In what follows, we give several applications of the distance formula

Solution Let P1(4, 3) and P2(2, 6) be points in the plane Then we have

x1 4 and y1 3

Using Formula (1), we have

rep-resents the position of a power relay station located on a straight coastal

high-way, and M shows the location of a marine biology experimental station on a

nearby island A cable is to be laid connecting the relay station with the mental station If the cost of running the cable on land is $1.50 per running footand the cost of running the cable underwater is $2.50 per running foot, find thetotal cost for laying the cable

experi-y (feet)

O

x(feet)

S(10,000, 0) Q(2000, 0)

Refer to Example 1 Suppose

we label the point (2, 6) as P1

and the point (4, 3) as P2

(1) Show that the distance d

between the two points is the

same as that obtained earlier

(2) Prove that, in general, the

distance d in Formula (1) is

independent of the way we label

the two points

FIGURE 7

The cable will connect the relay station

S to the experimental station M.

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Solution The length of cable required on land is given by the distance from S

to Q This distance is (10,000 2000), or 8000 feet Next, we see that the length

of cable required underwater is given by the distance from Q to M This distance is

or approximately 3605.55 feet Therefore, the total cost for laying the cable is

1.5(8000) 2.5(3605.55) 21,013.875

or approximately $21,014

cen-ter C(h, k) (Figure 8) Find a relationship between x and y.

Solution By the definition of a circle, the distance between C(h, k) and P(x, y) is r Using Formula (1), we have

which, upon squaring both sides, gives an equation

(x h)2 (y k)2 r2

that must be satisfied by the variables x and y.

A summary of the result obtained in Example 3 follows

(1, 3) and (b) radius 3 and center located at the origin

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(a) The circle with radius 2 and center (b) The circle with radius 3 and center

– 1

2 (–1, 3) 1

1 STRAIGHT LINES AND LINEAR FUNCTIONS

6

FIGURE 9

1.1 Self-Check Exercises

1 a Plot the points A(4, 2), B(2, 3), and C(3, 1).

b Find the distance between the points A and B, between B

and C, and between A and C.

c Use the Pythagorean theorem to show that the triangle

with vertices A, B, and C is a right triangle.

2 The accompanying figure shows the location of cities A, B,

and C Suppose a pilot wishes to fly from city A to city C but must make a mandatory stopover in city B If the single-

engine light plane has a range of 650 mi, can the pilot make

the trip without refueling in city B?

x (miles)

100 200 300 400 500 600 700

y (miles)

300 200 100

B (200, 50)

A (0, 0)

C (600, 320)

Solutions to Self-Check Exercises 1.1 can be found on page 9.

1 Use the distance formula to help you describe the set of points in the xy-plane

satisfying each of the following inequalities

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THE CARTESIAN COORDINATE SYSTEM

1 What can you say about the signs of a and b if the point

P(a, b) lies in (a) the second quadrant? (b) The third quadrant?

(c) The fourth quadrant?

2 a What is the distance between P1(x1, y1) and P2(x2, y2)?

b When you use the distance formula, does it matter which

point is labeled P1and which point is labeled P2? Explain

In Exercises 1–6, refer to the accompanying figure and

determine the coordinates of the point and the quadrant in

which it is located.

1 A 2 B 3 C

4 D 5 E 6 F

In Exercises 7–12, refer to the accompanying figure.

7 Which point has coordinates (4, 2)?

8 What are the coordinates of point B?

9 Which points have negative y-coordinates?

10 Which point has a negative x-coordinate and a negative

11 Which point has an x-coordinate that is equal to zero?

12 Which point has a y-coordinate that is equal to zero?

In Exercises 13–20, sketch a set of coordinate axes and then plot the point.

25 Find the coordinates of the points that are 10 units away from

the origin and have a y-coordinate equal to 6

26 Find the coordinates of the points that are 5 units away from

the origin and have an x-coordinate equal to 3.

27 Show that the points (3, 4), (3, 7), (6, 1), and (0, 2)form the vertices of a square

28 Show that the triangle with vertices (5, 2), (2, 5), and(5,2) is a right triangle

In Exercises 29–34, find an equation of the circle that isfies the given conditions.

sat-29 Radius 5 and center (2, 3)

30 Radius 3 and center (2, 4)

31 Radius 5 and center at the origin

32 Center at the origin and passes through (2, 3)

33 Center (2, 3) and passes through (5, 2)

34 Center (a, a) and radius 2a

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