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1 FUNCTIONS AND CHANGE 1 1.1 WHAT IS A FUNCTION? 2 1.2 LINEAR FUNCTIONS 6 1.3 RATES OF CHANGE 14 1.4 APPLICATIONS OF FUNCTIONS TO ECONOMICS 22 1.5 EXPONENTIAL FUNCTIONS 33 1.6 THE NATURAL LOGARITHM 40 1.7 EXPONENTIAL GROWTH AND DECAY 44 1.8 NEW FUNCTIONS FROM OLD 52 1.9 PROPORTIONALITY, POWER FUNCTIONS, AND POLYNOMIALS 56 1.10 PERIODIC FUNCTIONS 64 REVIEW PROBLEMS 71 PROJECTS: COMPOUND INTEREST, POPULATION CENTER OF THE US 77 FOCUS ON MODELING 78 FITTING FORMULAS TO DATA 78 COMPOUND INTEREST AND THE NUMBER e 86 FOCUS ON THEORY 92 LIMITS TO INFINITY AND END BEHAVIOR 92 2 RATE OF CHANGE: THE DERIVATIVE 97 2.1 INSTANTANEOUS RATE OF CHANGE 98 2.2 THE DERIVATIVE FUNCTION 106 2.3 INTERPRETATIONS OF THE DERIVATIVE 111 2.4 THE SECOND DERIVATIVE 119 2.5 MARGINAL COST AND REVENUE 125 REVIEW PROBLEMS 130 PROJECTS: ESTIMATING THE TEMPERATURE OF A YAM, TEMPERATURE AND ILLUMINATION 133 FOCUS ON THEORY 135 LIMITS, CONTINUITY, AND THE DEFINITION OF THE DERIVATIVE 135 Preface xvii 3 SHORT–CUTS TO DIFFERENTIATION 141 3.1 DERIVATIVE FORMULAS FOR POWERS AND POLYNOMIALS 142 3.2 EXPONENTIAL AND LOGARITHMIC FUNCTIONS 148 3.3 THE CHAIN RULE 153 3.4 THE PRODUCT AND QUOTIENT RULES 158 3.5 DERIVATIVES OF PERIODIC FUNCTIONS 162 REVIEW PROBLEMS 166 PROJECTS: CORONER’S RULE OF THUMB, AIR PRESSURE AND ALTITUDE 169 FOCUS ON THEORY 170 ESTABLISHING THE DERIVATIVE FORMULAS 170 FOCUS ON PRACTICE 173 173 4 USING THE DERIVATIVE 175 4.1 LOCAL MAXIMA AND MINIMA 176 4.2 INFLECTION POINTS 182 4.3 GLOBAL MAXIMA AND MINIMA 188 4.4 PROFIT, COST, AND REVENUE 195 4.5 AVERAGE COST 202 4.6 ELASTICITY OF DEMAND 208 4.7 LOGISTIC GROWTH 213 4.8 THE SURGE FUNCTION AND DRUG CONCENTRATION 221 REVIEW PROBLEMS 227 PROJECTS: AVERAGE AND MARGINAL COSTS, FIREBREAKS, PRODUCTION AND THE PRICE OF RAW MATERIALS 232 5 ACCUMULATED CHANGE: THE DEFINITE INTEGRAL 235 5.1 DISTANCE AND ACCUMULATED CHANGE 236 5.2 THE DEFINITE INTEGRAL 242 5.3 THE DEFINITE INTEGRAL AS AREA 249 5.4 INTERPRETATIONS OF THE DEFINITE INTEGRAL 254 5.5 THE FUNDAMENTAL THEOREM OF CALCULUS 262 REVIEW PROBLEMS 266 PROJECTS: CARBON DIOXIDE IN POND WATER, FLOODING IN THE GRAND CANYON 269 xviii Preface FOCUS ON THEORY 271 THEOREMS ABOUT DEFINITE INTEGRALS 271 6 USING THE DEFINITE INTEGRAL 275 6.1 AVERAGE VALUE 276 6.2 CONSUMER AND PRODUCER SURPLUS 280 6.3 PRESENT AND FUTURE VALUE 286 6.4 INTEGRATING RELATIVE GROWTH RATES 289 REVIEW PROBLEMS 294 PROJECTS: DISTRIBUTION OF RESOURCES, YIELD FROM AN APPLE ORCHARD 296 7 ANTIDERIVATIVES 299 7.1 CONSTRUCTING ANTIDERIVATIVES ANALYTICALLY 300 7.2 INTEGRATION BY SUBSTITUTION 304 7.3 USING THE FUNDAMENTAL THEOREM TO FIND DEFINITE INTEGRALS 308 7.4 ANALYZING ANTIDERIVATIVES GRAPHICALLY AND NUMERICALLY 313 REVIEW PROBLEMS 319 PROJECTS: QUABBIN RESERVOIR 321 FOCUS ON PRACTICE 322 322 8 PROBABILITY 323 8.1 DENSITY FUNCTIONS 324 8.2 CUMULATIVE DISTRIBUTION FUNCTIONS AND PROBABILITY 328 8.3 THE MEDIAN AND THE MEAN 335 REVIEW PROBLEMS 340 PROJECTS: TRIANGULAR PROBABILITY DISTRIBUTION 341 9 FUNCTIONS OF SEVERAL VARIABLES 343 9.1 UNDERSTANDING FUNCTIONS OF TWO VARIABLES 344 Preface xix 9.2 CONTOUR DIAGRAMS 348 9.3 PARTIAL DERIVATIVES 360 9.4 COMPUTING PARTIAL DERIVATIVES ALGEBRAICALLY 368 9.5 CRITICAL POINTS AND OPTIMIZATION 373 9.6 CONSTRAINED OPTIMIZATION 378 REVIEW PROBLEMS 387 PROJECTS: A HEATER IN A ROOM, OPTIMIZING RELATIVE PRICES FOR ADULTS AND CHILDREN, MAXIMIZING PRODUCTION AND MINIMIZING COST: “DUALITY” 391 FOCUS ON THEORY 392 DERIVING THE FORMULA FOR A REGRESSION LINE 392 10 MATHEMATICAL MODELING USING DIFFERENTIAL EQUATIONS 397 10.1 MATHEMATICAL MODELING: SETTING UP A DIFFERENTIAL EQUATION 398 10.2 SOLUTIONS OF DIFFERENTIAL EQUATIONS 402 10.3 SLOPE FIELDS 406 10.4 EXPONENTIAL GROWTH AND DECAY 412 10.5 APPLICATIONS AND MODELING 417 10.6 MODELING THE INTERACTION OF TWO POPULATIONS 427 10.7 MODELING THE SPREAD OF A DISEASE 432 REVIEW PROBLEMS 436 PROJECTS: HARVESTING AND LOGISTIC GROWTH, POPULATION GENETICS, THE SPREAD OF SARS 438 FOCUS ON THEORY 441 SEPARATION OF VARIABLES 441 11 GEOMETRIC SERIES 445 11.1 GEOMETRIC SERIES 446 11.2 APPLICATIONS TO BUSINESS AND ECONOMICS 451 11.3 APPLICATIONS TO LIFE SCIENCES 455 REVIEW PROBLEMS 460 PROJECTS: DO YOU HAVE ANY COMMON ANCESTORS?, HARROD–HICKS MODEL OF AN EXPANDING NATIONAL ECONOMY, PROBABILITY OF WINNING IN SPORTS 461 APPENDIX: SPREADSHEET PROJECTS 463 SPREADSHEET PROJECTS 464 1. MALTHUS: POPULATION OUTSTRIPS FOOD SUPPLY 464 2. CREDIT CARD DEBT 465 3. CHOOSING A BANK LOAN 466 4. COMPARING HOME MORTGAGES 467 5. PRESENT VALUE OF LOTTERY WINNINGS 468 6. COMPARING INVESTMENTS 468 7. INVESTING FOR THE FUTURE: TUITION PAYMENTS 469 8. NEW OR USED? 469 9. VERHULST: THE LOGISTIC MODEL 470 10. THE SPREAD OF INFORMATION: A COMPARISON OF TWO MODELS 471 11. THE FLU IN WORLD WAR I 471 INDEX 501