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

LIST OF APPLICATIONS 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 Airfone usage, 440 Airline safety, 396 Airport traffic, 1042 Air travel, 904 Allocation of funds 181, 194, 271 Allocation of services, 517 Alternative energy sources, 975 Alternative minimum tax, 805, 861 Amusement park attendance, 720, 1011 Annual retail sales, 620, 678 Annuities, 298, 300, 302, 304, 905, 984, 989 Assembly-time studies, 387, 394, 406, 802, 804, 897, 949 Asset allocation, 181, 182, 194, 235, 236, 405 ATM cards, 358 Auditing tax returns, 426 Authentication technology, 61 Auto financing, 959 Automobile leasing, 304, 333 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 Black Monday, 809 BlackBerry subscribers, 612 Book design, 616, 847 Bookstore inventories, 116 Box-office receipts, 78, 93, 131, 638, 705, 820 Brand selection, 413 Break-even analysis, 44, 53 Bridge loans, 289 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 Cable ad revenue, 620 Cable TV subscription, 465, 768, 932, 962 Calling cards, 61 Capital expenditures, 144, 316, 333 Capital value, 1035 Cargo volume, 790 Cash reserves at Blue Cross and Blue Shield, 806 CDs, 333 Cellular phone subscription, 786 Charter revenue, 610, 616, 848 Chip sales, 612 City planning, 515, 516, 642, 706 Coal production, 961, 1003 Cobb–Douglas production function, 1066, 1071, 1096, 1113 COLAs, 329, 589 Commissions, 654 Commodity prices, 654, 682, 882, 902, 1003 Common stock transactions, 165, 290, 386, 570 Commuter airlines, 1002 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, 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 Effect of advertising on revenue, 974 Effect of advertising on sales, 612, 693, 759, 763, 797, 853, 901, 975 Effect of housing starts on jobs, 719 Effect of inflation on salaries, 291 Effect of luxury tax on consumption, 718 Effect of mortgage rates on housing starts, 603, 763 Effect of price increase on quantity demanded, 754, 766 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 (continued) List of Applications (continued) Home sales, 697 Home shopping industry, 659 Hotel chain growth, 972 Hotel occupancy rate, 603, 614, 718 Households with microwaves, 903 Housing appreciation, 290 Housing loans, 427 Housing starts, 604, 719, 749 Illegal ivory trade, 613 Income distributions, 431, 987, 996 Incomes of American families, 873, 884 Income streams, 981, 982, 1012, 1033, 1035 Indian gaming industry, 618 Industrial accidents, 472, 506 Inflation, 291, 739 In-flight service, 405 Information security software sales, 59 Input–output analysis, 153, 155, 157, 158, 159, 161 Installment contract sales, 995 Installment loans, 304, 333 Insurance claims, 117 Insurance probabilities, 435, 464 Inventory control and planning, 109, 116, 464, 653, 845, 846, 849, 854 Investment analysis, 275, 275, 302, 305, 317, 352, 464, 465, 474, 475, 983, 989 Investment clubs, 78, 79, 92, 93, 147 Investment in technology, 405 Investment options, 288, 291, 350, 355, 367 Investment planning, 78, 92, 196, 290 Investment portfolios, 116 Investment returns, 764 Investments, 78, 92, 105, 130, 271, 290, 352, 376 IRAs, 288, 302, 317, 984, 989 Keogh accounts, 764, 995 Land prices, 1071, 1084, 1110 LCDs versus CRTs, 50 Leasing, 49, 53 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 Lorentz curves, 985, 986, 989, 1012 Machine scheduling, 165 Magazine circulation, 917 Management decisions, 79, 93, 104, 364, 370, 805, 983 Manufacturing capacity, 595, 697, 790, 808, 831 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 Marginal revenue, 727, 733, 734, 881, 994 Market equilibrium, 46, 47, 48, 50, 53, 69, 615, 681, 682, 980 Market for cholesterol-reducing drugs, 57 Market research, 196 Market share, 117, 518, 521, 532, 672, 918 Marketing surveys, 347 Markup on a car, 573 Maximizing crop yield, 848 Maximizing oil production, 882 Maximizing production, 184, 1096 Maximizing profit, 45, 176, 183, 184, 187, 194, 229, 232, 235, 275, 827, 833, 834, 852, 853, 1081, 1091, 1093, 1095 Maximizing revenue, 834, 835, 848, 881 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 Price of perfume, 881 Price of replacement automobile parts, 614 Price of wine, 882 Pricing, 147, 568 Prime interest rate, 654 Probability of engine failure, 489 Producers’ surplus, 978, 980, 988, 995, 1011, 1025, 1049 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 Sales promotions, 881 List of Applications (continued) Sales tax, 36, 589 Sampling, 376, 409 Satellite radio subscriptions, 920 Selling price of DVD recorders, 612, 717 Service-utilization studies, 395 Shadow prices, 205 Shoplifting, 395 Shopping habits, 1045 Shuttle bus usage, 387 Sickouts, 838 Sinking fund, 313, 316, 333, 985 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, 615, 692, 750, 763, 932 Switching jobs, 405 Tax planning, 302, 303, 305, 317, 333 Tax-deferred annuity, 302 Taxicab movement, 517, 529 Telemarketing, 506 Television commercials, 235 Television pilots, 450 Television programming, 370 Testing new products, 384, 391, 392, 741 Theater bookings, 506 Ticket revenue, 147 Time on the market, 809, 838 Tour revenue, 145 Tracking with GPS, 860 Transportation, 181, 210, 253 Transportation problem, 178, 195 Tread lives of tires, 1026 Trust funds, 279, 290, 316, 330, 1035 TV households, 403 TV-viewing patterns, 672, 717, 932 Unemployment rates, 464 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 Value of an investment, 602 VCR ownership, 1011 Venture-capital investment, 835 Violations of the building code, 488 Volkswagen’s revenue, 475 Wages, 466, 669 Waiting lines, 370, 446, 450, 454, 466, 1045 Warehouse problem, 179, 183, 249 Warranties, 358, 400, 505 Waste generation, 66 Web hosting, 786 Wilson lot-size formula, 1060 Wireless subscribers, 61 Worker efficiency, 590, 611, 693, 804, 820, 853 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 (continued) List of Applications (continued) 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 Solar energy, 485, 521 Solar power, 612 Spending on medical devices, 612 Spread of rumor, 803 Student dropout rate, 351 Student enrollment, 426, 932 Student financial aid, 427 Student loans, 316 Student reading habits, 351 Student surveys, 351, 376 Study groups, 370 Switching Internet service providers (ISPs), 428 Teacher attitudes, 404 Teaching assistantships, 370 Television-viewing polls, 358, 450 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 Voter affiliation, 131 Voter registration, 1011 Voters, 488 Voter turnout, 437 Voting patterns, 435 Voting quorums, 372 Waiting times, 1045 Waste disposal, 963 Women in the professions, 531 Working mothers, 717, 742 Working-age population, 614 World energy consumption, 66 World population growth, 791, 835, 880, 884, 900 LIFE SCIENCES Absorption of drugs, 861, 863, 872, 882, 883, 903, 905, 908, 965 Absorption of light, 892 Administration of an IV solution, 654 Adult obesity, 768 Aids in Massachusetts, 965 Amount of rainfall, 642, 923 Animal nutrition, 196, 271 Anticipated rise in Alzheimer’s patients, 590, 594, 743 Arteriosclerosis, 714, 718 Average weights and heights of infants, 671, 932 Bacteria growth, 330 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 Body mass, 1058 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 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 Designing a Norman window, 616, 847 List of Applications (continued) Engine efficiency, 1072 Error measurement, 760, 761, 762 Estimating the amount of paint required, 762 Estimating the flow rate of a river, 1025, 1026 Exercise program, 329 Expected snowfall, 1045 Family food expenditure, 290 Fencing, 615, 840, 846 Flight of a rocket, 694, 785, 804, 830, 833, 922 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 Lightning deaths, 395 Lotteries, 305, 358, 464, 989 Magnitude of earthquakes, 871, 892 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 Stopping distance of a racing car, 693 Strength of a beam, 848 Surface area of the Central Park reservoir, 1025 Sweepstakes, 396 Team selection, 376 Temperature conversion, 21, 37 Terminal velocity, 819 Travel options, 356 Trial run of an attack submarine, 1024 Turbo-charged engine performance, 975 Vacation costs, 93 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 This page intentionally left blank College Mathematics for the Managerial, Life, and Social Sciences Seventh Edition S T TAN STONEHILL COLLEGE Australia • Brazil • Canada • Mexico • Singapore • Spain United Kingdom • United States College Mathematics for the Managerial, Life, and Social Sciences, 7e S T Tan Acquisitions Editor: Carolyn Crocket Development Editor: Danielle Derbenti Assistant Editor: Beth Gershman Editorial Assistant: Ashley Summers Technology Project Manager: Donna Kelley Marketing Manager: Joe Rogove Marketing Assistant: Jennifer Liang Marketing Communications Manager: Jessica Perry Project Manager, Editorial Production: Janet Hill Creative Director: Rob Hugel Print Buyer: Becky Cross Permissions Editor: Bob Kauser Production Service: Newgen–Austin—Jamie Armstrong Text Designer: Diane Beasley Photo Researcher: Kathleen Olson Cover Designer: Irene Morris Cover Image: Portrait of Jonathan D Farley by Peter Kiar Cover Printer: Coral Graphic Services Compositor: Newgen Printer: R R Donnelley—Willard © 2008, Thomson Brooks/Cole, a part of The Thomson Corporation Thomson, the Star logo, and Brooks/Cole are trademarks used herein under license Thomson Higher Education 10 Davis Drive Belmont, CA 94002-3098 USA ALL RIGHTS RESERVED No part of this work covered by the copyright hereon may be reproduced or used in any form or by any means—graphic, electronic, or mechanical, including photocopying, recording, taping, web distribution, information storage and retrieval systems, or in any other manner—without the written permission of the publisher Printed in the United States of America 11 10 09 08 07 ExamView ® and ExamView Pro ® are registered trademarks of FSCreations, Inc Windows is a registered trademark of the Microsoft Corporation used herein under license Macintosh and Power Macintosh are registered trademarks of Apple Computer, Inc Used herein under license © 2008 Thomson Learning, Inc All Rights Reserved Thomson Learning WebTutor™ is a trademark of Thomson Learning, Inc Library of Congress Control Number: 2006935209 Student Edition ISBN-13: 978-0-495-01583-3 ISBN-10: 0-495-01583-0 For more information about our products, contact us at: Thomson Learning Academic Resource Center 1-800-423-0563 For permission to use material from this text or product, submit a request online at http://www.thomsonrights.com Any additional questions about permissions can be submitted by e-mail to thomsonrights@thomson.com CONTENTS TO PAT, BILL, AND MICHAEL iii ANSWERS TO CHAPTER 14 ODD-NUMBERED EXERCISES a 60 words/min b 107 words/min c 136 words/min 1181 a D(t) 15 2000 1000 b 325 million c 76.84 million/decade a t 50 a 573 computers; 1177 computers; 1548 computers; 1925 computers b 2000 computers c 45 computers/mo 17 a 122.3 cm 19 a 86.1% b 14 cm/yr b 10.44%/yr 21 76.4 million 27 a ln ba bϪa c 1970 23 1080; 280 b 25 15 1b k g/cm3 bϪa c 0.237 g/cm3 b c 200 cm Using Technology Exercises 14.5, page 904 a e Chapter 14 Concept Review, page 906 Power; 0; 1; exponential a (Ϫϱ, ϱ); (0, ϱ) 29 b 5599 yr d 0.760 g/cm3 b (0, 1); (Ϫϱ, ϱ) a (0, ϱ); (Ϫϱ, ϱ); (1, 0) a x b Ͻ1; Ͼ1 b x a e f (x)fЈ(x) f ¿ 1x2 f 1x2 b a Initially; growth b Decay c Time; one half a Horizontal asymptote: C b Horizontal asymptote; A, carrying capacity Chapter 14 Review Exercises, page 907 b 12.146%/yr c 9.474%/yr y a and b a y = 2–x = ( 12 ) x x b 666 million, 818.8 million c 33.8 million/yr a log2/3 278 ϭ Ϫ3 x ϩ y ϩ z log16 0.125 ϭ Ϫ34 x ϩ 2y Ϫ z y ϩ 2z y y = log (x + 3) b 86.12%/yr c 10.44%/yr d 1970 x 15 x ϭ 1182 ANSWERS TO CHAPTER 14 ODD-NUMBERED EXERCISES y 10 y = log3 (x + 1) 39 Absolute maximum value: e 40 Absolute maximum value: ln ; absolute minimum value: x −1 10 41 9.58 yr 42 a Q(t) ϭ 2000e0.01831t −2 e 1x ϩ x ϩ x 14 et ϩ 1te t ϩ 1t 12 21e ϩ 2 15 23 26 28 11 ϩ e 18 3e 2x(1 ϩ e 2x)1/2 2xex r 1r Ϫ r ϩ r ϩ 12 e 25 4e e4x ϩ ϩ e 11 Ϫ x ln x2 x 27 11 ϩ r 2 12x2 ϩ 2x2 # ln x Ϫ e x x11 ϩ e x 2 x11 ϩ ln x2 32 Ϫ2 14x3 Ϫ 5x2 ϩ 2 1x2 Ϫ 2 1x Ϫ 2 t 19 (x ϩ 1)2e x 4x 24 (x ϩ 2)e x 36 y ϭ Ϫ1 1ln x 2 x1x ϩ 2 31 50 ln x Ϫ 22 ex ϩ x Ϫ x ln x ϩ 1 30 ᎏ x 34 21 2000 16 4xe 2x Ϫ2x 2 20 ln t ϩ 1te 4000 2t 2x 21 ϩ x2 17 (1 Ϫ 2x 2)eϪx Ϫ 4t 13 43 0.0004332 D(t) 44 11 (2x ϩ 1)e 2x b 161,992 29 Ϫ a 1175, 2540, 3289 b 4000 45 970 46 a 12.5/1000 live births; 9.3/1000 live births; 6.9/1000 live births y b 12 13x ϩ 2 33 6x(x ϩ 2)2(3x ϩ 2x ϩ 1) 35 y ϭ Ϫ(2x Ϫ 3)eϪ2 e t 10 20 47 a g/cm3 b 0.0361 g/cm3 (t) x d c 0.08 g/cm3 y 37 0.08 0.5 0.06 0.04 y = xe –2x x 0.02 t 20 40 60 80 100 y 38 Chapter 14 Before Moving On, page 908 10 Ϫ0.9589 e2x a x e1x 1x 4x2 ln 3x ϩ 4x Ϫ x2 ϩ ln b 8.7 ANSWERS TO CHAPTER 15 ODD-NUMBERED EXERCISES CHAPTER 15 65 s1t2 ϭ 43 t3/2 Exercises 15.1, page 918 71 a Ϫt ϩ 6t ϩ 45t b y ϭ 2x ϩ C 67 $3370 69 5000 units; $34,000 b 212 73 a Ϫ0.125t ϩ 1.05t ϩ 2.45t ϩ 1.5 y b 24.375 million 75 a Ϫ1.493t ϩ 34.9t ϩ 279.5t ϩ 2917 77 a 3.133t Ϫ 6.7t ϩ 14.07t ϩ 36.7 5 83 1.0974t Ϫ 0.0915t 87 a 9.3eϪ0.02t –5 89 y C=5 C = 10 30 20 10 2 k1R 11 x4 ϩ C Ϫ 13 Ϫ 3x3 ϩC 5/3 17 Ϫ 1/4 ϩ C 19 Ϫ ϩ C x ϩC x x 3/2 21 pt ϩ C 23 3x Ϫ x ϩ C 1 25 x3 ϩ x2 Ϫ ϩ C 27 4e x ϩ C 2x x ϩ ex ϩ C 31 x4 ϩ 33 x7/2 ϩ x5/2 Ϫ x2 ϩ C 37 u ϩ u Ϫ uϩC 3 41 x Ϫ 2x Ϫ ϩ C x 43 47 ϪxϩC x 35 x3/2 ϩ 61x ϩ C 39 3 t Ϫ t Ϫ 2t ϩ C 3 s ϩ s2 ϩ s ϩ C 1 x ϩ x Ϫ ln x Ϫ ϩ C x 49 ln x ϩ Ϫ ϩC x 1x 59 1x 2t tϩ4 95 a 2 in.; in b 99 True 61 e x ϩ 12 x ϩ 2 20 1x 17 57 x ϩ ln x 63 Branch A 23 1t ϩ 2 3/2 ϩ C ϩC Ϫ 10 ϩ C 11 Ϫ15 ln Ϫ x5 ϩ C 15 ln 3x2 Ϫ ϩ C ln10.3x2 Ϫ 0.4x ϩ 2 ϩ C 19 Ϫ12 eϪ2x ϩ C 25 e x ϩ eϪx ϩ C 29 2e1x ϩ C 27 ln(1 ϩ e x) ϩ C 31 Ϫ 23 Ϫ12 eϪx ϩ C 21 Ϫe 2Ϫx ϩ C 61e3x ϩ x3 2 ϩC 33 1e2x ϩ ϩ C 35 1ln 5x2 ϩ C 37 ln ln x ϩ C 39 1ln x2 3/2 ϩ C 41 x2 2e Ϫ 12 ln1x2 ϩ 2 ϩ C Ϫ ϩ 31 1x Ϫ 2 ϩ 81 1x Ϫ ϩ ln 1x Ϫ ϩ C 16x ϩ 1x Ϫ 12 ϩC 45 42 43 ᎏ3 1x 47 ϩ 1x Ϫ x Ϫ ln11 ϩ 1x2 ϩ C 49 Ϫ252 11 Ϫ √2 128√2 ϩ 7√ ϩ ϩ C 51 12x Ϫ ϩ 57 21,000 Ϫ 51 x ϩ x ϩ 1 55 x Ϫ ϩ x 212x2 ϩ 2 13 1x3 Ϫ 2x2 ϩ C 13 ln(x Ϫ 2)2 ϩ C 15 53 x ϩ 2x Ϫ x Ϫ 91 979 ft/sec2; 396 ft 15 14x ϩ ϩ C –10 – 20 t eϩ1 ϩC eϩ1 c 6619 Exercises 15.2, page 931 45 et ϩ Ϫ r2 97 True C=0 C = –5 x 29 x ϩ b $4264.11 85 Ϫt ϩ 96t ϩ 120t; 63,000 ft b 7030 93 0.924 ft/sec2 b y ϭ 13 x3 ϩ C 6x ϩ C b 103,201 81 21,960 x –4 b $9168 79 a y ϭ 4.096t Ϫ 75.2797t ϩ 695.23t ϩ 3142 –5 1183 53 eϪx 20,000 11 ϩ 0.2t ; 6858 ϩ1 59 61 301 12t ϩ Ϫ 2 ; 14,400p ft2 63 65.8794 ϩ 2.449eϪ0.3277t Ϫ1 ϩ 0.3; 56.22 in 55 17,341,000 250 216 ϩ x2 65 r 11 Ϫ eϪat a 1184 ANSWERS TO CHAPTER 15 ODD-NUMBERED EXERCISES Exercises 15.3, page 941 Exercises 15.5, page 961 4.27 sq units 10 y a sq units 11 y = 3x 13 18152 31 15 21 ln 23 27 33 Ϫ1 17 13 Ϫ 32154 25 2e Ϫ 2e Ϫ ln 35 1e4 Ϫ x 1.5 43 $40,339.50 49 c 5.25 sq units d Yes 37 71 False ϩ 56 29 1e4 Ϫ 41 Ϸ$2.24 million 47 1367 53 $14.78 67 a Ϫ1 65 2e 1eϪ4 Ϫ eϪ8 Ϫ 13 51 16 ft/sec million 69 True 17 45 49.69 ft/sec 63 Property y a sq units $6 13 2415 1ln 19 Ϫ ln 32 ᎏ21 39 120.3 billion metric tons 0.5 b 4.5 sq units 32 15 19 19 15 b 55 80.7% c Ϫ13 73 True y = – 2x Using Technology Exercises 15.5, page 965 7.71667 17.56487 10,140 60.45 mg/day x 0.5 b 4.8 sq units 1.5 c 4.4 sq units a 18.5 sq units c 18.66 sq units 108 sq units b 0.16 sq unit d Ϸ0.2 sq unit 15 0.95 sq unit sq units ln sq units 29 ln 39 19 12 31 56 33 256 15 35 45 51 sq units f 1t2 Ϫ g1t2 dt 49 30 ft/sec faster 51 21,850 53 True 223 Using Technology Exercises 15.6, page 977 47 a 0.2833t Ϫ 1.936t ϩ 5t ϩ 5.6 49 49.7 million 12 10,13313 ft 23 15 37 Ύ 2050 47 a A2 Ϫ A1 b The distance car is ahead of car after t sec 17 27 b 12.8% 53 False a a c 5.2% 55 False Using Technology Exercises 15.4, page 954 Ϫ0.5888 6.1787 0.7873 3.9973 11 46%; 24% 2.7044 13 333,209 29 713 Ύ 3g1x2 Ϫ f 1x2 dx 2010 b $900 b $219.20 45 Shortfall ϭ 11 1713 sq units 25 45 41 a $4100 43 a $2800 27 212 b 12 sq units 15 (e Ϫ 1) sq units 23 21 434 sq units 21 1823 19 switching to the new agency; S ϭ sq units 19 14 13 27 313 43 S is the additional revenue that Odyssey Travel could realize by 17 9400 sq ft Exercises 15.4, page 951 13 1814 sq units 17 1223 23 12 Ϫ ln 25 e Ϫ e Ϫ ln 31 33 e3 Ϫ ϩ 35 2056 e 71 37 39 41 18 12 a 25 sq units b 21.12 sq units c 19.88 sq units d Ϸ19.9 sq units 13 4.64 sq units Ϫ1 223 sq units units 11 313 15 2(e Ϫ e ) 2 sq 112 sq units d Yes b 18.64 sq units d Ϸ18.7 sq units 11 a 0.0625 sq unit c 0.2025 sq unit Exercises 15.6, page 972 15 903,213 b 1074.2857 b 0.9961 1185 ANSWERS TO CHAPTER 15 ODD-NUMBERED EXERCISES a a a Unknown b Function g¿1x2 dx; Ύ f 1u du Ύ a b f 1x2 dx b Minus a a F(b) Ϫ F(a); antiderivative Ύ b b f ¿1x2 dx a bϪa a b 5.4603 b 25.8549 a 11 a Ύ b Ύ b f 1x2 dx b Area; area a f 1x2 Ϫ g1x2 dx a Ύ a x D1x2 dx Ϫ p x Ύ b p x Ϫ Ύ T R1t2 eϪrt dt Ύ b b 3.5799 11 A ϭ 13 207.43 Exercises 15.7, page 988 $11,667 $6667 15 $76,615 11 $148,239 13 $52,203 17 $111,869 19 $20,964 y 21 a R1t2 eϪrt dt mP rT 1e Ϫ r Ύ 12 L ϭ 3x Ϫ f 1x2 dx b 0.175; 0.816 1 4x ϩ 23 x3 Ϫ 12 x2 ϩ C 5x Ϫ 12 x4 Ϫ 1x ϩ C 3x 10 x4 ϩ 25 x5/2 ϩ C 33 ϩC x y b 0.104; 0.504 18 15 Ϫ 23 x3/2 ϩ 4x ϩ C Ϫ 13 x3 ϩ 23 x3/2 Ϫ x ϩ C 38 13x2 Ϫ 2x ϩ 4/3 ϩ C 11 Ϫ 2x ϩ ϩ C 2 ln1x 1ln x2 ϩ C 15 Ϫ 23 x3 ϩ 8x ϩ C 13 12 ex ϩxϩ1 ϩ C 12 ϪeϪ2x ϩ C 4/3 4x 7/2 7x 12 x Ϫ 12 x2 ϩ ln x ϩ 5x ϩ C 1x3 ϩ 2 11 y = f (x) ᎏ13 12x ϩ 3/2 ϩ C 23 a T Chapter 15 Review Exercises, page 993 $11,667 Consumers’ surplus: $13,333; producers’ surplus: $11,667 $824,200 S1x2 dx 10 a erT b 10.5144 x 16 (ln x)2 ϩ C 13x Ϫ 2 1x ϩ 3/2 ϩ C 14 17 19 ϩC eϪx ϩ x 111x2 Ϫ 1x2 ϩ 11 264 ϩC 1x ϩ 1x Ϫ ϩ C 20 21x Ϫ 22 1x ϩ ϩ C y = f (x) 25 Ϫ80 24 242 30 Ϫ x 1 e2 31 26 21 132 eϪ1 211 ϩ e2 33 f(x) ϭ x Ϫ 2x ϩ x ϩ Using Technology Exercises 15.7, page 990 22 Ϫ6 27 ln 32 36 f 1x2 ϭ 12 1ln x2 Ϫ 35 f(x) ϭ x ϩ eϪx ϩ 37 Ϫ4.28 Consumers’ surplus: $33,120; producers’ surplus: $2880 39 a Ϫ0.015x ϩ 60x; Investment A 40 V(t) ϭ 1900(t Ϫ 10) ϩ 10,000; $40,400 38 $6740 b p ϭ Ϫ0.015x ϩ 60 Chapter 15 Concept Review, page 993 a FЈ(x) ϭ f (x) b F(x) ϩ C b Ύ f 1x2 dx Ϯ Ύ g1x2 dx 41 3000t Ϫ 50,000(1 Ϫ eϪ0.04t ); 16,939 42 N1t2 ϭ 15,000 11 ϩ 0.4t ϩ 85,000; 112,659 43 26,027 44 $3100 15 34 f 1x2 ϭ 2x2 ϩ 1 Consumers’ surplus: $18,000,000; producers’ surplus: $11,700,000 a c Ύ f 1x2 dx 28 23 523 45 37.7 million 29 1186 ANSWERS TO CHAPTER 16 ODD-NUMBERED EXERCISES 46 19.4 billion metric tons 48 1e Ϫ sq units sq units 50 sq units 10 sq units 53 sq units 49 51 (e Ϫ 3) sq units 54 234,500 barrels 47 15 sq units 52 55 56 26°F sq units 60 $98,973 ln a 11 61 $505,696 13 y 62 a 81 x a ϩ 2x2 b ϩ x2 Ϫ ln a x ϩ ϩ x2 b d ϩ C B4 128 B4 57 $270,000 58 Consumers’ surplus: $2083; producers’ surplus: $3333 59 $197,652 c 11 ϩ 4x Ϫ b ϩC 11 ϩ 4x ϩ x 29 Ϫ x2 21 0.2 0.4 0.6 0.8 1.0 b 0.1017; 0.3733 ϩ 24 Ϫ x ` ϩC x 19 ln͉ln(1 ϩ x)͉ ϩ C 1 c ϩ ln11 ϩ 3e x d ϩ C ϩ 3e x 23 6[e(1/2)x Ϫ ln(1 ϩ e(1/2)x )] ϩ C 25 19 12 ϩ ln x Ϫ ln ϩ ln x ϩ C c 0.315 63 70,784 27 e Ϫ Chapter 15 Before Moving On, page 996 12 x4 ϩ 23 x3/2 ϩ ln x Ϫ 41x ϩ C ex ϩ 12 x2 ϩ 13 12 12 Ϫ 2x2 ϩ ϩ C ln x 12x2 Ϫ 2x2 Ϫ Ϫ ln x ϩ 2x2 Ϫ ϩ C 17 14 12x Ϫ e2x ϩ C x ϩC 15 24 Ϫ x Ϫ ln ` 1.0 0.8 0.6 0.4 0.2 9 sq units 29 x3 13 ln x Ϫ ϩ C 31 x[(ln x)3 Ϫ 3(ln x)2 ϩ ln x Ϫ 6] ϩ C 33 Ϸ$2329 35 27,136 37 44; 49 39 26,157 41 $418,444 CHAPTER 16 Exercises 16.3, page 1024 Exercises 16.1, page 1003 1 2x e 12x 2x 2e 13 17 Ϫ 12 ϩ C Ϫ 21x Ϫ 2ex ϩ 13 x3 ϩ C 21x ϩ 2 1x ϩ ϩC 11 Ϫ 52 x 321ln Ϫ 52 r1 ϩ 3 32 23 a 3.6 b 0.0002848 31 21.65 mpg 35 1922.4 ft3/sec b $51,708 41 Ϸ50% 39 474.77 million barrels 43 $131,324 45 101,606 d ϩ c2 moles/L ln r1 Ϫ ln r2 ϩ 3x Ϫ ln ϩ 3x ϩ C 11 ϩ 2x 2 Ϫ 411 ϩ 2x2 ϩ ln ϩ 2x ϩ C 49 True 45 False b 0.0324 b 0.00043 37 a $51,558 Exercises 16.2, page 1011 21 3.7757; 3.7625 33 17.1 million barrels 39 2.04 mg/mL r2 Ϫ r1 19 0.8806; 0.8818 29 52.84 mi 27 ln Ϫ 15 1.1170; 1.1114 17 1.3973; 1.4052 27 a 0.0078125 35 ln Ϫ 41 Ϫ20eϪ0.1t(t ϩ 10) ϩ 200 13 0.3837; 0.3863; Ϸ0.3863 25 a 0.013 23 Ϫ(x ϩ 2x ϩ 2)eϪx ϩ C 33 Ϫ12 xeϪ2x Ϫ 14 eϪ2x ϩ 134 ϩC 0.5090; 0.5004; 12 11 0.6336; 0.6321; Ϸ0.6321 5.2650; 5.3046; 163 5/2 31 14 13e4 ϩ 12 29 ln Ϫ 47 1c2 Ϫ c1 c Ϫ 15 1x x4 14 ln x Ϫ 12 ϩ C 16 19 Ϫ 1ln x ϩ 12 ϩ C x x2 Ϫ ln x ϩ 14 ϩ C 37 1485 ft 3/2 0.2656; 0.2500; 14 0.6970; 0.6933; Ϸ0.6931 xe x ϩ C 15 ln 1x Ϫ ϩ C 21 x(ln x Ϫ 1) ϩ C 25 x1x x2 12 ln 2x Ϫ ϩ C 3/2 x 13 2.7037; 2.6667; 223 4(x Ϫ 4)e x/4 ϩ C 43 6.42 L/min 47 True Exercises 16.4, page 1034 ᎏ23 sq sq 13 a 21 Ϫ18 unit unit 3/2 3b sq unit 2e 15 23 sq units sq units 11 sq unit 17 25 19 Divergent 27 29 Divergent ANSWERS TO CHAPTER 17 ODD-NUMBERED EXERCISES 31 Ϫ1 33 Divergent 39 Divergent 45 r2 Chapter 16 Before Moving On, page 1050 37 41 Convergent 10,000r ϩ 4000 49 False 35 13 x3 ln x Ϫ 19 x3 ϩ C 43 $18,750 3.00358 47 True dollars 51 b $83,333 15 a f 1x2 ϭ Ϫx/15 15 e 19 2500 lb 21 27 0.18 b .06 16 b .2676 c .37 23 yr 29 False 17 313 25 0.37 x ϩ 1; 2x dx; (27) ¢x f 1x0 ϩ f 1x1 ϩ p ϩ f 1xnϪ1 ϩ f 1xn 4; M1b Ϫ a2 12n2 ¢x f 1x0 ϩ f 1x1 ϩ f 1x2 ϩ f 1x3 ϩ f 1x4 M1b Ϫ a2 ϩ p ϩ 4f 1xnϪ1 ϩ f 1xn 4; even; 180n4 Ύ aSϪϱ b f 1x2 dx; lim bSϱ a Ύ b b 31 32 f 1x2 dx; Ύ c f 1x2dx ϩ Ϫϱ a f(0, 0) ϭ Ϫ4; f(1, 0) ϭ Ϫ2; f(0, 1) ϭ Ϫ1; f(1, 2) ϭ 4; f(2, Ϫ1) ϭ Ϫ3 f(1, 2) ϭ 7; f(2, 1) ϭ 9; f(Ϫ1, 2) ϭ 1; f(2, Ϫ1) ϭ h(1, e) ϭ 1; h(e, 1) ϭ Ϫ1; h(e, e) ϭ Product: u√ Ϫ ͐ √ du; u; easy to integrate lim 2e2 6.3367 g11, 2 ϭ ϩ 12; g12, ϭ ϩ 12; g10, ϭ 2; g(4, 9) ϭ 56 31 False Chapter 16 Concept Review, page 1048 28 ϩ 2x2 ϩC 8x Exercises 17.1, page 1058 13 a k ϭ ln42 b Ϫ CHAPTER 17 Exercises 16.5, page 1044 11 a k ϭ 18 1187 Ύ g(1, 1, 1) ϭ e; g(1, 0, 1) ϭ 1; g(Ϫ1, Ϫ1, Ϫ1) ϭ Ϫe 11 All real values of x and y 13 All real values of u and √ except those satisfying the equation u ϭ √ 15 All real values of r and s satisfying rs Ն 17 All real values of x and y satisfying x ϩ y Ͼ ϱ f 1x2 dx y 19 y 21 c Chapter 16 Review Exercises, page 1048 Ϫ2(1 ϩ x)eϪx ϩ C x(ln 5x Ϫ 1) ϩ C 14 11 ϩ 3e4 Ϫ3x Ϫ 16 14x Ϫ 2e4x ϩ C 4 ln Ϫ ln Ϫ Ϫ3x 9e ϩ 9 c ϩ 2x Ϫ Ϫ ln ϩ 2x d ϩ C ϩ 2x 14 x ln 2x Ϫ ϩ C 15 25 2x Ϫ 25 18 21 0.8421; 0.8404 24 8.1310; 8.041 28 a k ϭ 19 32 e4x 18x2 Ϫ 4x ϩ ϩ C 10 16 y x 25 a 0.002604 23 2.2379; 2.1791 b 0.000033 25 9p ft3 b 0.6495 30 a k ϭ 31 a 0.22 b 0.39 33 48,761 34 $41,100 36 7850 sq ft 23 20 22 1.491; 1.464 b 0.3178 c days 37 $111,111 z=2 z=1 z = –2 2x2 Ϫ ϩC x 13 17 Divergent 29 a 11 ϩC –1 10 23 1x Ϫ 12x ϩ ϩ C x 14 –2 z=0 z = –1 12 Ϫ 14 11 Ϫ 3eϪ2 1x1ln x Ϫ 2 ϩ Ϫ13 xe x x 27 b 0.4815 32 $1,157,641 35 274,000 ft2; 278,667 ft2 27 a 24.69 b 81 kg 29 a Ϫ15 x2 Ϫ 14 y2 Ϫ 15 xy ϩ 200x ϩ 160y b The set of all points (x, y) satisfying 200 Ϫ 15 x Ϫ 101 y Ն 0, 160 Ϫ 101 x Ϫ 14 y Ն 1188 ANSWERS TO CHAPTER 17 ODD-NUMBERED EXERCISES 31 a Ϫ0.005x Ϫ 0.003y Ϫ 0.002xy ϩ 20x ϩ 15y b The set of all ordered pairs (x, y) for which 20 Ϫ 0.005x Ϫ 0.001y Ն 15 Ϫ 0.001x Ϫ 0.003y Ն 49 a Ϸ20°F b ϷϪ0.3°F 33 a The set of all ordered pairs (P, T ), where P and T are positive numbers b 11.10 L 51 0.039 L/degree; Ϫ0.015 L/mm of mercury The volume increases by 0.039 L when the temperature increases by degree (beyond 300 K) and the pressure is fixed at 800 mm of mercury The volume decreases by 0.015 L when the pressure increases by mm of mercury (beyond 800 mm) and the temperature is fixed at 300 K 35 $7200 billion 55 False 37 103 39 a $733.76; $877.57 b $836.44 41 40.28 times gravity Using Technology Exercises 17.2, page 1074 43 False 45 False Ϫ 4x; 2√ 1u ϩ √2 ;Ϫ 4y ; x3 x2 (0, 0); relative maximum value: f(0, 0) ϭ 1u ϩ √2 (1, 2); saddle point: f(1, 2) ϭ 3(2s Ϫ t)(s Ϫ st ϩ t ) ; 3(2t Ϫ s)(s Ϫ st ϩ t ) 4y 4x 11 ; 13 ye xyϩ1; xe xyϩ1 31x2 ϩ y2 1/3 31x2 ϩ y2 1/3 2 y x 15 ln y ϩ ; ϩ ln x x y 17 eu ln √; 2 23 fx (1, 2) ϭ 8; fy (1, 2) ϭ 27 fx 11, 2 ϭ 12 ; fy 11, 22 ϭ Ϫ14 25 fx (2, 1) ϭ 1; fy (2, 1) ϭ (8, Ϫ6); relative minimum value: f(8, Ϫ6) ϭ Ϫ41 (1, 2) and (2, 2); saddle point: f(1, 2) ϭ Ϫ1; relative minimum value: f(2, 2) ϭ Ϫ2 1Ϫ13 , 113 and (1, 5); saddle point: f 1Ϫ13 , 113 ϭ Ϫ319 27 ; relative minimum value: f(1, 5) ϭ Ϫ13 eu √ 19 yz ϩ y ϩ 2xz; xz ϩ 2xy ϩ z 2; xy ϩ 2yz ϩ x 21 ste rst; rte rst; rse rst Ϫ1.8889; 0.7778 Exercises 17.3, page 1083 2u 1.3124; 0.4038 Ϫ0.3863; Ϫ0.8497 Exercises 17.2, page 1071 2; 57 True 11 (0, 0) and (1, 1); saddle point: f(0, 0) ϭ Ϫ2; relative minimum value: f(1, 1) ϭ Ϫ3 13 (2, 1); relative minimum value: f(2, 1) ϭ 29 fx (1, 1) ϭ e; fy (1, 1) ϭ e 15 (0, 0); saddle point: f(0, 0) ϭ Ϫ1 31 fx (1, 0, 2) ϭ 0; fy (1, 0, 2) ϭ 8; fz (1, 0, 2) ϭ 17 (0, 0); relative minimum value: f(0, 0) ϭ 33 fxx ϭ 2y; fxy ϭ 2x ϩ 3y ϭ fyx ; fyy ϭ 6xy 19 (0, 0); relative minimum value: f(0, 0) ϭ 35 fxx ϭ 2; fxy ϭ fyx ϭ Ϫ2; fyy ϭ 21 200 finished units and 100 unfinished units; $10,500 2 37 fxx ϭ fyy ϭ 39 fxx ϭ y 1x ϩ y2 x2 ; f ϭ fyx ϭ Ϫ 3/2 xy xy 1x2 ϩ y 2 ; 3/2 1x ϩ y 2 3/2 y x eϪx/y; fxy ϭ 23 Price of land ($200/ft2) is highest at 12 , 12 25 (0, 1) gives desired location 27 10Љ ϫ 10Љ ϫ 5Љ; 500 in.3 yϪx y x Ϫx/y fyy ϭ a Ϫ b e y y 41 a fx ϭ 7.5; fy ϭ 40 eϪx/y ϭ fyx; 29 30Љ ϫ 40Љ ϫ 10Љ; $7200 31 False Exercises 17.4, page 1095 Min of 34 at 34 , 14 b Yes 43 px ϭ 10—at (0, 1), the price of land is changing at the rate of $10/ft2/mile change to the right; py ϭ 0—at (0, 1), the price of land is constant/mile change upward 45 Complementary commodities 47 $30/unit change in finished desks; Ϫ$25/unit change in unfinished desks The weekly revenue increases by $30/unit for each additional finished desk produced (beyond 300) when the level of production of unfinished desks remains fixed at 250; the revenue decreases by $25/unit when each additional unfinished desk (beyond 250) is produced and the level of production of finished desks remains fixed at 300 Max of Ϫ74 at 12, 72 Min of at 12,12/2 and 1Ϫ12, Ϫ12/2 Max of Ϫ34 at 32 , 12 Max of 213 at 13/3, Ϫ16 and 13/3, 162 11 Max of at 1212, 2122 and 1Ϫ2 12, Ϫ2 12 ; of Ϫ8 at 12 12, Ϫ2 122 and 1Ϫ2 12, 2122 13 Max.: 213 13 ; min.: Ϫ 9 15 Min of 187 at 97 , 67 , 37 ANSWERS TO CHAPTER 17 ODD-NUMBERED EXERCISES 17 140 finished and 60 unfinished units 2x ϩ 3y ϭ z y 19 10 12 ft ϫ 40 12 ft 21 r ϭ 18 18 in.; h ϭ in Bp 3Bp 3 23 32 ϫ 23 9ϫ x 25 1500 units on labor and 250 units of capital 27 False −2 Exercises 17.5, page 1107 11 13 188 23 14 1e4 Ϫ 29 31 (e Ϫ 1)(1 Ϫ eϪ2) 412 84 15 17 25 32 1e Ϫ 223 z ϭ y Ϫ x y 27 48 35 16 39 ln 17 41 43 Ϫ 122 1 45 Ϫ 47 19 ln Ϫ 42 49 Ϸ2166 people/mi2 e 51 $10,460/wk 53 True 55 True 37 1e 21 12 13 Ϫ e 19 33 5(1 Ϫ 2eϪ2 ϩ eϪ4) 19 1 64 Chapter 17 Concept Review, page 1111 10 z ϭ 2x2 ϩ y2 xy; ordered pair; real number; f (x, y) y 2 Independent; dependent; value z ϭ f (x, y); f ; surface f (x, y) ϭ c; level curve; level curves; c Fixed number; x x x Slope; (a, b, f (a, b)); x; b Յ; (a, b); Յ; domain Domain; fx(a, b) ϭ and f y(a, b) ϭ 0; exist; candidate g(x, y) ϭ 0; f (x, y) ϩ lg(x, y); Fx ϭ 0; Fy ϭ 0; Fl ϭ 0; extrema 10 Volume; solid 11 Iterated; Ύ Ύ 12x ϩ y dx dy 11 z ϭ e xy y z=3 Chapter 17 Review Exercises, page 1112 0, 0, 12 ; no e, e2 2e , ; no ϩ ln ϩ ln z=1 x 2, Ϫ(e ϩ 1), Ϫ(e ϩ 1) The set of all ordered pairs (u, √) such that u √ The set of all ordered pairs (x, y) such that y Ϫx The set of all ordered pairs (x, y) such that x Յ and y Ն The set of all triplets (x, y, z) such that z Ն and x and z 1, y 1, x 12 fx ϭ 2xy ϩ 3y ϩ ; fy ϭ 3x 2y ϩ 6xy Ϫ y y y x 13 fx ϭ 1y ϩ ; fy ϭ ϩ 1x 1x 1y 14 fu ϭ √2 Ϫ 2 2u√2 Ϫ 2u ; f√ ϭ u√ 2u√2 Ϫ 2u 1189 1190 ANSWERS TO CHAPTER 17 ODD-NUMBERED EXERCISES 15 fx ϭ 3y 1y ϩ 2x2 2 y1y Ϫ x 16 gx ϭ 1x ϩ y 2 2 ; gy ϭ 32 1Ϫ13 , 133 2, 13,11 ; saddle point at f 1Ϫ13 , 133 ϭ 1Ϫ445 27 ; relative minimum value: f(3, 11) ϭ Ϫ35 3x ; fy ϭ Ϫ 1y ϩ 2x2 x1x2 Ϫ y 2 33 (0, 0); relative minimum value: f(0, 0) ϭ 1x2 ϩ y 2 34 (1, 1); relative minimum value: f(1,1) ϭ ln 17 hx ϭ 10y(2xy ϩ 3y 2)4; hy ϭ 10(x ϩ 3y)(2xy ϩ 3y 2)4 18 fx ϭ ey 21xe y ϩ 1/2 ; fy ϭ 19 fx ϭ 2x(1 ϩ x ϩ y 2)e x 20 fx ϭ 21 fx ϭ 4x ϩ 2x ϩ 4y 2x x ϩy 2 179 36 f 221 , 21 22 ϭ 44 21xe y ϩ 12 1/2 ϩy ; fy ϭ ; fy ϭ Ϫ 20 32 35 f 12 11 , 11 ϭ Ϫ11 xe y ; fy ϭ 2y(1 ϩ x ϩ y 2)e x ϩy 16y3 ϩ 2x2 ϩ 4y4 37 Relative maximum value: f(5, Ϫ5) ϭ 26; relative minimum value: f(Ϫ5, 5) ϭ Ϫ24 12 12 38 Relative maximum value: f a ,Ϫ b ϭ e12; relative 2 minimum value: f a Ϫ 2x2 y1x ϩ y 2 12 12 , b ϭ eϪ12 2 40 12 1eϪ2 Ϫ 2 42 41 13 Ϫ ln 22 39 48 23 fxx ϭ 12x ϩ 4y 2; fxy ϭ 8xy ϭ fyx ; fyy ϭ 4x Ϫ 12y 43 24 fxx ϭ 12(2x ϩ 3y 2)(10x ϩ 3y 2); fxy ϭ 144xy(2x ϩ 3y 2) ϭ fyx; fyy ϭ 18(2x ϩ 3y 2)(2x ϩ 15y 2) 46 a R(x, y) ϭ Ϫ0.02x Ϫ 0.2xy Ϫ 0.05y ϩ 80x ϩ 60y b The set of all points satisfying 0.02x ϩ 0.1y Յ 80, 0.1x ϩ 0.05y Յ 60, x Ն 0, y Ն c 15,300; the revenue realized from the sale of 100 16-speed and 300 10-speed electric blenders is $15,300 25 gxx ϭ gyy ϭ Ϫ2y2 ; gxy ϭ 1x ϩ y 2x13y Ϫ x2 2y1x Ϫ y2 1x ϩ y2 44 1045 cu units 45 ϭ gyx; 47 Complementary 1x ϩ y 2 34 41 63 22 fxx ϭ 6x Ϫ 4y; fxy ϭ Ϫ4x ϭ fyx; fyy ϭ 26 gxx ϭ 2(1 ϩ 2x 2)e x ϩy ; gxy ϭ 4xye x 2 gyy ϭ 2(1 ϩ 2y 2)e x ϩy ϩy ϭ gyx; 1 27 hss ϭ Ϫ ; hst ϭ hts ϭ 0; htt ϭ s t 28 fx (1, 1, 0) ϭ 3; fy(1, 1, 0) ϭ 3; fz (1, 1, 0) ϭ Ϫ2 48 The company should spend $11,000 on advertising and employ 14 agents in order to maximize its revenue 49 337.5 yd ϫ 900 yd 50 75 units on labor; 25 units on capital Chapter 17 Before Moving On, page 1114 All real values of x and y satisfying x Ն 0, x 1, y Ն 0, y 29 (2, 3); relative minimum value: f(2, 3) ϭ Ϫ13 fx ϭ 2xy ϩ ye ; fxx ϭ 2y ϩ y e ; fxy ϭ 2x ϩ (xy ϩ 1)e ϭ fyx; fy ϭ x ϩ xe xy; fyy ϭ x 2e xy 30 (8, Ϫ2); saddle point at f(8, Ϫ2) ϭ Ϫ8 Rel min.: (1, 1, Ϫ7) 31 10, and 32 , 94 ; saddle point at f(0, 0) ϭ 0; relative minimum value: f 32 , 94 ϭ Ϫ27 16 xy f 12 , 12 ϭ 52 xy xy Index Abscissa, Absolute extrema, 824–826, 1075 Absolute value, 571 Absorbing state, 535 Absorbing stochastic matrix, 535 Addition principle, 391, 398 Amortization, 309–312 formula, 309 schedule, 310 Annuity, 296, 984–985 certain, 296 compound amount factor, 298 future value, 298 ordinary, 296 present value, 300 simple, 296 term, 296 Antiderivative, 910 Antidifferentiation, 912 Area under a curve, 934–939, 958 between curves, 966–971 Arithmetic progression, 322–324 common difference, 322 nth term, 322 sum, 323 Asymptotes, 631, 810–813 Augmented matrix, 83 Average, 454 Average cost function, 724 Average rate of change, 662 Average value of an exponential density function, 1042 of a function of one variable, 958–959 of a function of two variables, 1105 Axes, Basic variables, 218 Bayes’ theorem, 429–432 Bernoulli trials, 480–487 Binding constraints, 207 Binomial distribution, 480–487 approximation by a normal distribution, 501–504 binomial trials, 480 mean, 484 standard deviation, 484 variance, 484 Binomial experiment, 480 Binomial random variable, 483 Binomial trials, 480 Birthday problem, 410–411 Book value, 31, 327 Bounded solution set, 172 Break-even analysis, 43–45 Break-even point, 43 Carbon-14 dating, 895–896 Cartesian coordinate system, 3–4, 1054 abscissa, axes, ordered pair, ordinate, quadrants, Central limit theorem, 502 Chain rule, 709–713 for exponential functions, 876 for logarithmic functions, 887 for powers of functions, 711–713 Change of variable, 956–957 Chebychev’s inequality, 471–473 Circle, Closed interval, 568 Cobb–Douglas production function, 1066 Column matrix, 109 Column vector, 109 Combinations, 364–368 Common logarithm, 906 Complementary commodities, 1067 Complement of a set, 339 Composite function, 599 Compound interest, 279–282 See also Interest Concavity, 791–794 Conditional probability, 414–419 Constant of integration, 912 Constrained extrema, 1086 Constrained optimization, 1086–1094 Consumers’ surplus, 979 Continuity, 644–646 Continuous compound interest, 282–283 Continuous probability distribution, 490 Continuous random variable, 445, 491 Contour map, 1055 Convergence, 1029 Coordinates, Corner point, 187 Cost function, 32, 722–726 Counting problem, 346 Critical number, 778 Critical point, 1077 Current state, 512 Curve sketching, 809–814 Decay constant, 895 Decreasing function, 772 Definite integral, 938 geometric interpretation, 940–941 properties, 955 Demand function, 38 De Morgan’s laws, 340 Dependent system, 74 Dependent variable, 31, 579 Depreciation book value, 31 double-declining balance, 326–328 linear, 31–32 simple, 31–32 sum-of-the-years’-digits method, 329 Derivative definition, 663 first partial, 1062 higher-order, 736–738 as a rate of change, 666–669 second, 736 Derivative rules, 684–686 chain, 709–712 constant, 684 constant multiple of a function, 686 for exponential functions, 873, 876 for logarithmic functions, 885, 887 power rule, 685 product rule, 698 quotient rule, 699 sum rule, 686 Deviations, 471 Difference quotient, 662 Differentiable function, 669 Differential equations, 916–917 Differentials, 756–759 Discontinuous function, 645 Disjoint sets, 339 Disjunction, 573 Distance formula, Distribution vector, 525 Divergence, 1029 Domain, 30, 1052 Double integral, 1098–1106 Dual problem, 244 e, 858 Effective rate of interest, 284 1191 1192 INDEX Elastic demand, 730 Elasticity of demand, 728–730 Elementary event, 389 Empty set, 337 Equation of a circle, Equations of straight lines, 18 Equilibrium price, 46 quantity, 46 Equivalent system, 80 Error analysis, 1022–1023 Events, 380–382 independent, 422–423 mutually exclusive, 382, 389 probability of, 388–392 simple, 389 Expected value of a binomial random variable, 483 of an exponential density function, 1042 of a random variable, 455 Experiment, 380 event, 380 outcome, 380 sample point, 380 sample space, 380 Exponent, 548–549 Exponential decay, 895–896 Exponential density function, 1039 Exponential function, 858 derivative of, 873, 876 indefinite integral of, 915 properties of, 858 Exponential growth, 894 Factorial, 361 Factoring polynomials, 556–558 Fair game, 458 Feasible set, 185 Feasible solution, 185, 217 Finite interval, 568 Finite sample space, 380 Finite stochastic process, 419 First derivative test, 779 Fixed costs, 33, 598 Function, 29, 578 composite, 709 cost, 32 decreasing, 772 demand, 33 differentiable, 669 discontinuous, 645 domain, 30, 578 exponential, 856 increasing, 772 independent variable, 30 inverse, 868 linear, 30 logarithmic, 863 objective, 176 probability, 389 probability density, 491 profit, 32 range, 30, 578 revenue, 32 Functions of several variables, 1052–1057 maxima and minima, 1075–1080 partial derivative of, 1061–1064 saddle point, 1076 second derivative test, 1078 Fundamental theorem of calculus, 943– 944, 950 Fundamental theorem of duality, 245 Future value, 286 Gauss–Jordan elimination method, 80–90, 97–103 equivalent system, 80 row operations, 85 steps in, 87 Generalized multiplication principle, 354 Geometric progression, 324–326 common ratio, 324 nth term, 324 sum, 326 Gompertz growth curve, 903 Graph of an equation, 584 of a function, 581 Graphing linear inequalities, 168–172 Growth constant, 895 Half-life, 895 Half-open interval, 568 Half-planes, 168 Higher-order derivative, 736–738 Histogram, 447–448 Horizontal asymptote, 631, 812–813 Identity matrix, 126 Implicit differentiation, 745–747 Improper integral, 1027–1030 convergent, 1029 divergent, 1029 Income stream, 981–982 Inconsistent system, 74 Increasing function, 772 Increment, 756 Indefinite integral, 912 of a constant, 912 of a constant multiple of a function, 913 of an exponential function, 915 power rule, 913 sum rule, 914 Independent events, 422–423 Independent variable, 31, 579 Indeterminate form, 628 Inelastic demand, 730 Inequalities, 569 Infinite interval, 568 Inflection point, 794 Initial value problem, 916 Input–output analysis, 153–157 Instantaneous rate of change, 662–663 Integers, 1116 Integral change of variable, 956 definite, 938–939 improper, 1027–1030 indefinite, 912 properties of, 955 tables, 1006–1007 Integrand, 912 Integration constant of, 912 limits of, 938, 956 by parts, 998–1001 rules, 912–915 by substitution, 924–927 See also Integral Intercepts, 15 Interest compound, 279–282 formula, 280 future value, 286 present value, 286 principal, 280 continuous compound, 282–283 conversion period, 280 rate effective, 284 nominal, 279 simple, 278–279 accumulated amount, 278 formula, 278 Intermediate value theorem, 648 Intersection of lines, 42–48 Intersection of sets, 339 Intervals, 568–569 Inventory control, 845–846 Inverse functions, 868 Inverse of a matrix, 137 INDEX Irrational number, 1116 Isoprofit line, 186 Isoprofit surface, 1057 Lagrange multipliers, 1088 Law of exponents, 549, 856 Laws of logarithms, 864 Learning curves, 648, 897 Least-squares line, 54 Leontief, Wassily, 71, 153 Leontief input–output model, 155–157 gross production matrix, 154 input–output matrix, 153 internal consumption matrix, 155 Level curves, 1055 Limits, 621–634 of a function, 624 at infinity, 631 of integration, 956 left-hand, 643 properties of, 626–627 right-hand, 643 Linear depreciation, 31–32 Linear equations, 13–18 general form, 17 intercepts, 15 in n variables, 76 point-slope form, 13 slope, 10 slope-intercept form, 15 vertical lines, 13, 18 Linear function(s), 30–35, 606 break-even analysis, 43–45 demand and supply curves, 33–35 simple depreciation, 31–32 Linear inequality, 168–171 Linear programming corner point, 187 feasible set, 185 feasible solution, 185, 217 graphical solution, 185–191 linear constraints, 176 maximization problems, 176–177 method of corners, 187 minimization problems, 177–180 objective function, 176 optimal solution, 185 problem, 176, 216 basic solution, 218 dual, 244 existence of a solution, 187 nonstandard, 260–261, 265 primal, 244 standard, 216, 260 simplex method, 216–232, 243–251 basic variables, 218 nonbasic variables, 218 nonstandard problems, 260 optimal solution(s), 222 pivot element, 220 simplex tableau, 221 slack variables, 217 steps in, 222, 265 theorem, 186, 187 Line of complete equality, 986 Lines See Linear equations Logarithmic differentiation, 888–889 Logarithmic functions, 866–868 derivative of, 885–887 properties of, 867 Logarithms, 863–867 Logistic curve, 898–899 Logistic growth function, 898 Lorentz curves, 985 Lower limit of integration, 938 Marginal analysis, 721–728 average cost function, 723 cost function, 723 productivity, 1066 productivity of money, 1094 profit function 727 revenue function, 727 Market equilibrium, 46–48 Markov chains, 512–516 absorbing, 535 state, 512 stochastic matrix, 515, 526 current state, 512 regular, 526 state, 512 steady-state distribution vector, 525, 527 steady-state matrix, 525, 537 stochastic matrix absorbing, 535 regular, 526 transition matrix, 514 transition probabilities, 513 Mathematical model, 29, 604 Matrix, 109 absorbing stochastic, 515, 526 addition, 111 associative law, 112 commutative law, 112 augmented, 83 coefficient, 83 column, 108 cost, 127 dimension, 109 elements, 108 entries, 108 equality, 110 equation, 113 gross production, 154 identity, 126 ijth entry (aij), 109 input–output, 153 internal consumption, 155 inverse, 137 inverse of a ϫ matrix, 141 multiplication, 121–128 associative law, 125 distributive law, 125 nonsingular, 138 pivot element, 86 product, 122 regular stochastic, 526 row, 109 row-reduced form, 84 scalar product, 113 singular, 138 size, 109 square, 109 steady-state, 525, 537 subtraction, 111 transition, 514 transpose, 113 zero, 112 Maxima and minima absolute, 824, 1075 constrained, 1086–1094 relative, 777, 1075 Mean, 454 Median, 461 Method of bisection, 650 Method of corners, 187 Method of integration by substitution, 924–928, 955–957 Method of least squares, 54–58 least-squares line, 54 normal equations, 55 principle, 54 regression line, 54 scatter diagram, 54 Minima See Maxima and minima Mixed constraints, 261 Mode, 461 Modeling, 28, 604–606 Multiplication principle, 353–356 Mutually exclusive events, 382, 389 1193 1194 INDEX nth root, 548 Natural logarithm, 864 Natural numbers, 1116 Net change, 946 Nonbasic variables, 218 Nonstandard linear programming problems, 260–269 Normal approximation of binomial distribution, 501–503 Normal curve, 491, 1039 Normal density function, 1039 Normal distribution, 491 Normal equations, 55 Number line, Numerical integration, 1012–1023 error analysis, 1022–1023 Simpson’s rule, 1017–1021 trapezoidal rule, 1014–1016 Objective function, 176 Odds, 459–460 One-sided limits, 643 Open interval, 568 Optimal solution, 185 Optimization, 824–831, 839–845 Ordered pair, 2, 581 Ordered triple, 1054 Ordinary annuity, 296 Ordinate, Origin, Outcome, 380 Parabola, 583 Parallel lines, 12 Parameter, 74 Partial derivative first-order, 1062 second-order, 1068 Partition, 429 Permutations, 359–364 Perpendicular lines, 14 Perpetuity, 1032 Piecewise-defined function, 583 Pivot element, 220 Pivoting, 85 Point-slope form, 13 Polynomial, 559 Population density, 1106 Power function, 607 Power rule, 685 Present value, 286 Primal problem, 244 Principle of least squares, 54 Probability addition principle, 391, 398 a posteriori, 429 a priori, 429 Bayes’ theorem, 430 Bernoulli trials, 480–487 binomial distribution, 483 binomial experiment, 480 conditional, 415–419 continuous probability distribution, 490 counting techniques, 407–410 density function, 491, 1036 distribution, 389, 445–446 elementary event, 389 empirical, 389 of an event, 389–391 expected value, 455 experiment, 380 event, 380 outcome, 380 finite stochastic process, 419 function, 389 independent event, 423 mutually exclusive events, 382, 389 partition, 429 product rule, 417 properties, 397–400 relative frequency, 389 rule of complements, 400 sample point, 380 sample space, 380 finite, 380 infinite, 384 reduced, 415 uniform, 390 simple event, 389 transition, 514 tree diagram, 353–355, 419–421 Producers’ surplus, 979 Product rule, 417, 698 Profit function, 32 Proper subset, 337 Quadrant, Quadratic formula, 558 Quadratic function, 606 Quotient rule, 699 Radicals, 550–551 Radioactive decay, 548–549, 895–896 Random variable, 444 binomial, 480 continuous, 445, 490 expected value, 455 finite discrete, 445 infinite discrete, 445 probability distribution of, 445 standard deviation of, 469 standard normal, 493 variance, 468, 484 Range, 30, 578 Rate of change, 662–663 average, 662 instantaneous, 662–663 Rational expressions, 560 Rational function, 606 Rationalization, 550–566 Rational numbers, 1116 Real number line, Real number system, 1116–1117 Regular Markov chain, 526 Regular stochastic matrix, 526 Related rates, 749 Relative frequency, 389 Relative maximum, 777, 1075 Relative minimum, 777, 1075 Revenue function, 32 Riemann sum, 937, 1097 Roots of an equation, 558 Roulette, 458 Row matrix, 109 Row operations, 85–86 Row-reduced form, 84 Saddle point, 1076 Sample space, 380 finite, 380 infinite, 384 reduced, 415 uniform, 390 Scalar, 113 Scalar product, 113 Scatter diagram, 54 Secant line, 661 Second derivative test, 798, 1078 Second-order partial derivative, 1068 Sensitivity analysis, 198–208 binding constraints, 207 changes in the objective function coefficients, 199–203 changes in the right-hand-side constraints, 203–204 shadow prices, 205 Set(s), 336–342 complement, 339 De Morgan’s laws, 340 disjoint, 339 elements, 336 empty, 337 equal, 336 intersection, 339 members, 336 INDEX notation, 336 number of elements in, 346–347 operations, 340–341 proper subset, 337 subset, 337 union, 339 universal, 338 Venn diagrams, 338–342 Shadow price, 205–208 Simple depreciation, 31–32 Simple interest, 278–279 Simplex method See Linear programming Simplex tableau, 220 Simplification, 553 Simpson’s rule, 1017–1019 Singular matrix, 138 Sinking fund, 313 Slack variables, 217 Slope, 10 Slope-intercept form, 15 Solution set of a system of linear inequalities, 171 bounded, 172 unbounded, 172 Square matrix, 109 Standard deviation, 469 Standard linear programming problem, 176, 216 Standard normal curve, 493 Standard normal variable, 493 Standard viewing window, 24 Steady-state distribution vector, 525 Steady-state matrix, 525 Stochastic matrix, 515 Stochastic process, 512, 535 Straight-line (linear) depreciation, 31–32 Subset, 337 Substitute commodities, 1067 Supply function, 34 Systems of linear equations, 72–77 dependent, 74 equivalent, 80 Gauss–Jordan elimination method, 80–90, 97–103 inconsistent, 74 solution by inverses, 142–145 solution by substitution, 73 three variables, 75 two variables, 72 Systems of linear inequalities, 171–172 half-planes, 168 procedure for graphing, 168 Tangent line, 661 Trace, 1055 Transition matrix, 514 Transitional probabilities, 514 Trapezoidal rule, 1014–1016 Transportation problem, 178–179 Transpose of a matrix, 113 Tree diagram, 353–355, 419–421 Triangle inequality, 571 1195 Unbounded solution set, 172 Uniform sample space, 390 Union of sets, 339 Unit column, 85 Unitary demand, 730 Universal set, 338 Upper limit of integration, 938 Value of a game, 548 Variable(s) basic, 218 dependent, 30 independent, 30 nonbasic, 218 random, 444 slack, 217 Variable costs, 33 Variance, 468, 484 Velocity average, 622, 667, 675 instantaneous, 622, 667, 675 Venn diagrams, 338–339 Vertical asymptote, 811 Vertical-line test, 584 Volume of a solid, 1102 Warehouse problem, 179–180, 249–251 Whole numbers, 1116 Zero matrix, 112 ... 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 student’s understanding... using the definition of the derivative b Find the slope of the tangent line to the graph of f at the point (0, 3) c Find the rate of change of f when x ϭ d Find an equation of the tangent line to the. .. textbooks in mathematics for students in the managerial, life, and social sciences, I had quite a few years of experience teaching mathematics to non -mathematics majors One of the most important