Applications Biology and Life Sciences Hardy-Weinberg Law,976,985 Plant biology lab,1169 Health Plant growth,1060 Agriculture,495,767,777,995,1020 AIDS cases,1194 Population Air sacs in the lungs,28 blood oxygen level,108,113 of bees,901 Alligator length,360 blood pressure,1050 of bears,402 Animal shelter,986 body temperature,137,595 of deer,191,332,390,887 Antler spread of an elk,399 epidemic,854,889 of elk,332 Aquaculture,742 exposure to a carcinogen and of fish,339,390,801 Average recycling cost,333,339 mortality,1021 of ring-necked pheasants,926 Bacteria count,233,236,237,254 exposure to sun,742 of sparrows,263 Bacteria growth,351,372,390,392,400, infant mortality,995 of trout,832 401,402 U.S. HIV/AIDS epidemic,640 Population growth,596,606,752,789, Bacterial culture,616,759,832,889,986 and wellness,481 794,890,900 Biorhythms,1047,1050 velocity of air flow into and out of Predator-prey cycle,1046,1050,1051 Blue oak,height of,103 the lungs,1050,1070 Psychology Body mass index (BMI),414 Heart rate,450 Ebbinghaus Model,767 Botany,525,1231 Human height,78,137,188 human memory model,332,333,339, Calories burned by exercise,495 Hydroflourocarbon emissions,103 360,362,381,399,400,402,880 Carbon dioxide,99,426 Kidney donation,1164 intelligence quotient (IQ),1233 Cardiovascular device sales,1107 Lab practical,1169 IQ scores,136,403 Carnivorous plants,531 Litter of kittens,1230 learning curve,392,401,718,795 Cat cadavers,200 Liver transplants,268 learning theory,759,767,777,781, Cell division,759 Lung volume,217 786,1195,1205 Clinical trial,19 Medical science memory experiment,898,900,927 Comparing calories,88 drug concentration,803 Cricket chirps,1224 length of pregnancy,1206 migraine prevalence,580 Crop spraying mixture,437 surface area of a human body,1019 skill retention model,363 Diet supplement,450 velocity of air during coughing,668 sleep patterns,868 Dissections,1168 volume of air in the lungs,867 Stanford-Binet Test,(IQ test),967 E. coli bacterium,length,65 Medicine Ratio of reptiles,1089 Ecology,fencing a study plot,743 amount of drug in bloodstream,594,622 Research study,19 Endangered species population,393,800, bone graft procedures,379 Respiratory diseases,1174 889 days until recovery after a medical Stocking a lake with fish,392,976 Environment procedure,1206,1231 Suburban wildlife,381 contour map of the ozone hole,950 drug absorption,910 Systolic blood pressure,604 oxygen level in a pond,606,686 drug concentration in bloodstream, Tree growth,816 pollutant level,616 300,333,583,686,736,910,1150 Water pollution,332 pollutant removal,546,718,890 duration of an infection,976 Weight of a puppy,182,189 recycling,621,1179 effectiveness of a pain-killing drug,594 Weights of adult male rhesus monkeys, size of an oil slick,648 healing rate of a wound,353 1203 smokestack emissions,328,339,715 heart transplants,1232 Wheelchair ramp,181 Environmental cost,pollutant removal,557 multiple births,twins,685 Wildlife management,401,718,736 Erosion,38 Poiseuille’s Law,686 Zebrafish embryos,1229 Farming,697 spread of a virus,388,678,801 Fertility rates,668 temperature of a patient,1041 Business and Economics Fishing quotas,845 treatment of a bacterial infection,1019 Advertising,192 Forest yield,381 Metabolic rate,113 expenses,278,297,301,303,648, Forestry,624,795,1150 Nutrition,447,986 986,1150 Fruit tree maximum yield,696,740 Optimal area of an archaeological dig Galloping speeds of animals,370 site,336 Annual Gardening,825 Orthopedic implant sales,1107 operating cost,136 Genders of children,1181,1194,1229 Oxygen level,61 payroll of new car dealerships,1089 Genetically modified soybeans,438 Peregrine falcons,450 sales,78,251,338,352,414,450, Genetics,1229 Pest management in a forest,191 1107 Gestation period of rabbits,557 Physiology Average cost,205,206,332,339,677, Growth of a red oak tree,686 blood flow,845 705,714,716,718,840,1150 Gypsy moths,525 body surface area,736 Average cost and profit,742 Average monthly retail sales,155 Earnings per share,sales,and shareholder Market stabilization,1114 Average production,1012 equity,PepsiCo,996 Marketing,889 Average profit,718,1010 Economics,595 Maximum production level,980,981, Average revenue,1012 equation of exchange,1018 1019,1021 Average weekly demand,1197 gross domestic product,736 Maximum profit,218,666,701,705, Average weekly profit,1012 investment,1195 706,719,972,982 Book value per share,205,217 marginal benefits and costs,816 Maximum revenue,698,700,705,706,764 Break-even analysis,136,152,410,411, present value,926 Mean and median useful lifetimes of a 414,440,462 revenue,740 product,1200 Budget analysis,1116 Elasticity of demand,706,719,741 Media selection for advertising,459 Budget deficit,853 Elasticity and revenue,703 Minimum average cost,699,705,719, Budget variance,7,9 Equimarginal Rule,985 785,786 Cable television companies,496 Expected sales,1185 Minimum cost,695,696,697,706,740, Capital accumulation,845 Factory production,494,530 977,1019 Capital campaign,880 Federal Mobile homes manufactured,205 Capitalized cost,921,928 cost of food stamps,207 Monthly cost,103,114 Cash flow,825 debt,438 Monthly profit,87 Cash flow per share,Harley-Davidson,791 financial aid awarded,254 Monthly sales,92 Charter bus fares,206 government expenses,19 Monthly flight cost,124 Cobb-Douglas production function,640, Pell Grants,47,414 National defense budget,205 952,955,966,980,1012 Perkins Loans,414 National defense outlays,332 Compact disc shipments,685 student aid,47 National deficit,675 Company profits,146,153 Finance,cyclical stocks,1051 Negotiating a price,606 Comparing profits,235 Flour production,125 Number of Kohl’s stores,901 Comparing sales,236,254 Fuel cost,851 Office space,986 Optimal Competing restaurants,1169 Furniture production,449 cost,267,336,456,459 Complementary and substitute products, Gold prices,169,217,267 profit,267,336,455,458,459,460, 966 Hotel pricing,494 465,466 Construction,986 Increasing production,646 revenue,266,336,459,465 Consumer and producer surplus,446,449, Increasing profit,143 Owning a franchise,557 464,850,853,854,868,869,900 Insurance,1194 Patents issued,200 Contract bonuses,495 Inventory Payroll mix-up,1181 Cost,188,235,247,552,580,607,658, cost,677,741 Point of diminishing returns,675,677 668,707,718,728,813,815,816, of digital cameras,1230 Point of equilibrium,422,425,440,462,466 825,845,865,866,976,985,1070, of kayaks,449 Price-earning (P/E) ratio,237 1150,1206 levels,494,530,533 Price of a product,153 Cost-benefit model,328 of liquefied petroleum gases,1069 Production,264,640,865,952,955,985 Cost equation,111 management,557,596 cost,102 Cost,revenue,and profit,204,236,624, of movie players,464 limit,91 647,649,854 replenishment,607 Productivity,677 Daily morning newspapers,number of,993 Job applicants,1168,1170,1229 of a new employee,363 Defective units,1178,1181,1196 Labor/wage requirements,495,507,530 Profit,154,155,170,227,247,268, Demand,589,590,595,596,606,607, Least-Cost Rule,985 288,297,303,321,337,403,506, 638,640,707,736,742,758,776, Lifetime of a product,1191 595,596,624,625,645,648,658, 785,800,815,832,879,995,1195, Making a sale,1179,1181,1182 668,697,707,728,735,740,741, 1205,1206 Managing a store,607 795,803,816,839,867,900,955, Manufacturing,1206 975,1019,1107,1156 Demand function,302,351,372,381, Marginal analysis,731,732,736,742, Profit analysis,656,658 400,825,961 845,909 Projected expenses,90 Depreciation,37,616,750,767,803, Marginal cost,594,595,596,624,705, Projected profit,151 845,1155 716,833,966,1019 Projected revenue,90,151 Diminishing returns,675,677 Marginal productivity,966 Property tax,190,251 Dividends for Coca-Cola,206 Marginal profit,588,592,594,595,596, Purchasing power of the dollar,193,900 Dollar value,251 624,705 Quality control,9,606,921,1230 Dow Jones Industrial Average,596,678 Marginal revenue,591,594,595,624, Raw materials,505,531 Earnings-dividend ratio,Wal-Mart 705,966,1019 Real estate,1020 Stores,247 Market analysis,1195 Reimbursed expenses,191 Earnings per share,170,929,956 Market research,122,125,152 Returning phone calls,1225 Revenue,181,189,192,218,253,321, Trade deficit,593 Chemistry and Physical Science 338,401,463,566,577,580,595, Transportation cost,300 Acceleration,629,649 707,735,740,795,800,801,804, U.S. currency,381 Acceleration due to gravity,630 832,853,865,868,880,890,900, Useful life,1195,1205,1206,1231 Accuracy of a measurement,133,137,152 975,996,1019,1098,1194,1230 Wages,191,1206,1232 Acid mixture,462 Revenue per share,153,192,300,438, Weekly demand,1190 Acid solution,87 577,613,685,795 Worker’s productivity,395 Acidity of rainwater,1018 Salary contract,557,621 Years of service for employees,1176 Airplane speed,421,424 Sales,136,174,181,251,382,425, Automobile aerodynamics,227 648,678,684,792,795,832,926, Interest Rates Automobile crumple zones,382 1041,1050,1051,1052,1116, Annuity,842,845,867 Average velocity,584 1155,1157,1230 Balance in an account,25,28,39,65, Biomechanics,Froude number,1018 Avon Products,759,868 67,252,372,378,390,402,754, Boiling temperature of water,785 Bausch & Lomb,625 756,1159 Bouncing ball,1116 of concert tickets,449,464 Becoming a millionaire,28 Capacitance in series circuits,92 of e-commerce companies,786 Bond investment,505 Carbon dating,386,392,777,794 of exercise equipment,795 Borrowing money,124,436,481,529 Catenary,763 of gasoline,596 Cash advance,124,152 Charge of an electron,28 Home Depot,622,623 Cash settlement,351 Chemical reaction,395 of insect control products,1089 Certificate of deposit,759 Chemistry experiment,1229 of movie tickets,425 Charitable foundation,921 Circuit analysis,505 of petroleum and coal products,217 College tuition fund,880 Comet orbit,A25,A28 PetSmart,869 Comparing investment returns,90 Dating organic material,746 of prescription drugs by mail order,153 Compound interest,26,47,91,121, Diesel mechanics,247 Procter & Gamble,707 124,146,153,347,348,349,351, Earth and its shape,947,977 Safeway,1077 381,398,546,554,557,616,758, Earthquake magnitude,Richter scale, Scotts Miracle-Gro,567,580 767,768,775,776,790,794,801, 389,395,401,786 of shoes,425 845,867,1089,1104,1106,1155, Electricity,684 sporting goods,88 1157,1159 Electron microscopes,28 Starbucks,750 Credit card rate,616 Escape velocity,35,38 Sales analysis,607 Doubling time,774,776,804 Estimating speed,38 Sales commission,351 Doubling and tripling an investment,378 Estimating the time of death,395 Sales,equity,and earnings per share, Effective rate of interest,755,758,801 Falling object,97,101,112,114,1098 Johnson & Johnson,1018 Effective yield,794 Eiffel tower,111 Sales growth,677 Endowment,921 Grand Canyon,151 Sales per share,113,206,302,340,392, Finance,777 instantaneous rate of change,585 439,581,613,707 present value,926 on the moon,151 Sales price and list price,191 Future value,758,880 the owl and the mouse,101 Seasonal sales,1057,1069,1070,1076, Inflation rate,26,750,768,803 Royal Gorge Bridge,101 1077,1078 Investment,382,449,463,466,956,967 Fluid flow,A28 Shareholder’s equity,Wal-Mart,956,967 mix,90,414 Geology Social Security Trust Fund,854 plan,398 contour map of seismic amplitudes,956 State income tax,183 portfolio,424,425,433,437,460,463 crystals,938 State sales tax,183,190 Rule of 70,794 Hot air balloon,112 Straight-line depreciation,186,190,191, strategy,986 Hydrogen orbitals,1232 Ideal Gas Law,91 251,255 time,362,399 Kinetic energy,91 Sugar production,125 Monthly payments,61,953,956 Lensmaker’s equation,92 Supply and demand,425,462 Present value,352,756,758,801,876, Measurement errors,734,736 Supply function,825 877,880,901,909,921,926,928 Metallurgy,1205 Surplus,446,449,464,850,853,854, of a perpetual annuity,919 Meteorology 868,869,900 Savings plan,44,47 amount of rainfall,1195 Tax liability,285 Scholarship fund,921 annual snowfall in Reno,Nevada,193 Testing for defective units,1226 Simple interest,84,89,91,135,150,181, atmospheric pressure,956 Total cost,87 190,407,466 average monthly precipitation Total profit,1098 Stock mix,90 for Bismarck,North Dakota,1070 Total revenue,87,102,113,114,151,154 Tripling time,776 for Sacramento,California,1069 Total sales,112,147,205,1095,1098 Trust fund,758 for San Francisco,California,1077 (continued on back endsheets) College Algebra and Calculus An Applied Approach R O N L A R S O N The Pennsylvania State University The Behrend College A N N E V. H O D G K I N S Phoenix College with the assistance of DAVID C. FALVO The Pennsylvania State University The Behrend College Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States College Algebra and Calculus: An Applied Approach © 2010, Brooks/Cole, Cengage Learning Ron Larson and Anne V. Hodgkins ALL RIGHTS RESERVED. 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For your course and learning solutions, visit www.cengage.com Purchase any of our products at your local college store or at our preferred online store www.ichapters.com Printed in Canada 1 2 3 4 5 6 7 12 11 10 09 08 Contents iii Contents A Word from the Authors (Preface) viii Textbook Features x 0 Fundamental Concepts of Algebra 1 0.1 Real Numbers: Order and Absolute Value 2 0.2 The Basic Rules of Algebra 10 0.3 Integer Exponents 20 0.4 Radicals and Rational Exponents 29 Mid-Chapter Quiz 39 0.5 Polynomials and Special Products 40 0.6 Factoring 48 0.7 Fractional Expressions 55 Chapter Summary and Study Strategies 62 Review Exercises 64 Chapter Test 67 1 Equations and Inequalities 68 1.1 Linear Equations 69 1.2 Mathematical Modeling 79 1.3 Quadratic Equations 93 1.4 The Quadratic Formula 104 Mid-Chapter Quiz 114 1.5 Other Types of Equations 115 1.6 Linear Inequalities 126 1.7 Other Types of Inequalities 138 Chapter Summary and Study Strategies 148 Review Exercises 150 Chapter Test 154 Cumulative Test: Chapters 0–1 155 2 Functions and Graphs 156 2.1 Graphs of Equations 157 2.2 Lines in the Plane 171 2.3 Linear Modeling and Direct Variation 182 2.4 Functions 194 Mid-Chapter Quiz 207 2.5 Graphs of Functions 208 2.6 Transformations of Functions 219 2.7 The Algebra of Functions 228 2.8 Inverse Functions 238 Chapter Summary and Study Strategies 248 Review Exercises 250 Chapter Test 255 iv Contents 3 Polynomial and Rational Functions 256 3.1 Quadratic Functions and Models 257 3.2 Polynomial Functions of Higher Degree 269 3.3 Polynomial Division 279 3.4 Real Zeros of Polynomial Functions 289 Mid-Chapter Quiz 303 3.5 Complex Numbers 304 3.6 The Fundamental Theorem of Algebra 314 3.7 Rational Functions 322 Chapter Summary and Study Strategies 334 Review Exercises 336 Chapter Test 340 4 Exponential and Logarithmic Functions 341 4.1 Exponential Functions 342 4.2 Logarithmic Functions 354 4.3 Properties of Logarithms 364 Mid-Chapter Quiz 372 4.4 Solving Exponential and Logarithmic Equations 373 4.5 Exponential and Logarithmic Models 383 Chapter Summary and Study Strategies 396 Review Exercises 398 Chapter Test 402 Cumulative Test: Chapters 2–4 403 5 Systems of Equations and Inequalities 404 5.1 Solving Systems Using Substitution 405 5.2 Solving Systems Using Elimination 415 5.3 Linear Systems in Three or More Variables 427 Mid-Chapter Quiz 440 5.4 Systems of Inequalities 441 5.5 Linear Programming 451 Chapter Summary and Study Strategies 461 Review Exercises 462 Chapter Test 466 6 Matrices and Determinants 467 6.1 Matrices and Linear Systems 468 6.2 Operations with Matrices 482 6.3 The Inverse of a Square Matrix 497 Mid-Chapter Quiz 507 6.4 The Determinant of a Square Matrix 508 6.5 Applications of Matrices and Determinants 518 Chapter Summary and Study Strategies 527 Review Exercises 529 Chapter Test 533 Contents v 7 Limits and Derivatives 534 7.1 Limits 535 7.2 Continuity 547 7.3 The Derivative and the Slope of a Graph 558 7.4 Some Rules for Differentiation 569 Mid-Chapter Quiz 581 7.5 Rates of Change: Velocity and Marginals 582 7.6 The Product and Quotient Rules 597 7.7 The Chain Rule 608 Chapter Summary and Study Strategies 619 Review Exercises 621 Chapter Test 625 8 Applications of the Derivative 626 8.1 Higher-Order Derivatives 627 8.2 Implicit Differentiation 634 8.3 Related Rates 641 Mid-Chapter Quiz 649 8.4 Increasing and Decreasing Functions 650 8.5 Extrema and the First-Derivative Test 659 8.6 Concavity and the Second-Derivative Test 669 Chapter Summary and Study Strategies 681 Review Exercises 683 Chapter Test 687 9 Further Applications of the Derivative 688 9.1 Optimization Problems 689 9.2 Business and Economics Applications 698 9.3 Asymptotes 708 Mid-Chapter Quiz 719 9.4 Curve Sketching: A Summary 720 9.5 Differentials and Marginal Analysis 729 Chapter Summary and Study Strategies 739 Review Exercises 740 Chapter Test 743 10 Exponential and Logarithmic Functions 744 10.1 Exponential Functions 745 10.2 Natural Exponential Functions 751 10.3 Derivatives of Exponential Functions 760 Mid-Chapter Quiz 768 10.4 Logarithmic Functions 769 10.5 Derivatives of Logarithmic Functions 778 10.6 Exponential Growth and Decay 787 Chapter Summary and Study Strategies 798 Review Exercises 800 Chapter Test 804 vi Contents 11 Integration and Its Applications 805 11.1 Antiderivatives and Indefinite Integrals 806 11.2 Integration by Substitution and The General Power Rule 817 11.3 Exponential and Logarithmic Integrals 826 Mid-Chapter Quiz 833 11.4 Area and the Fundamental Theorem of Calculus 834 11.5 The Area of a Region Bounded by Two Graphs 846 11.6 The Definite Integral as the Limit of a Sum 855 Chapter Summary and Study Strategies 863 Review Exercises 865 Chapter Test 869 12 Techniques of Integration 870 12.1 Integration by Parts and Present Value 871 12.2 Partial Fractions and Logistic Growth 881 12.3 Integration Tables 891 Mid-Chapter Quiz 901 12.4 Numerical Integration 902 12.5 Improper Integrals 911 Chapter Summary and Study Strategies 924 Review Exercises 926 Chapter Test 929 13 Functions of Several Variables 930 13.1 The Three-Dimensional Coordinate System 931 13.2 Surfaces in Space 939 13.3 Functions of Several Variables 948 13.4 Partial Derivatives 957 13.5 Extrema of Functions of Two Variables 968 Mid-Chapter Quiz 977 13.6 Lagrange Multipliers 978 13.7 Least Squares Regression Analysis 987 13.8 Double Integrals and Area in the Plane 997 13.9 Applications of Double Integrals 1005 Chapter Summary and Study Strategies 1015 Review Exercises 1017 Chapter Test 1021 14 Trigonometric Functions (online)* 14.1 Radian Measure of Angles 14.2 The Trigonometric Functions 14.3 Graphs of Trigonometric Functions Mid-Chapter Quiz 14.4 Derivatives of Trigonometric Functions 14.5 Integrals of Trigonometric Functions Chapter Summary and Study Strategies Review Exercises Chapter Test