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1 FUNCTIONS AND CHANGE 1 1.1 WHAT IS A FUNCTION? 2 1.2 LINEAR FUNCTIONS 8 1.3 AVERAGE RATE OF CHANGE AND RELATIVE CHANGE 16 1.4 APPLICATIONS OF FUNCTIONS TO ECONOMICS 28 1.5 EXPONENTIAL FUNCTIONS 39 1.6 THE NATURAL LOGARITHM 46 1.7 EXPONENTIAL GROWTH AND DECAY 51 1.8 NEW FUNCTIONS FROM OLD 60 1.9 PROPORTIONALITY AND POWER FUNCTIONS 65 1.10 PERIODIC FUNCTIONS 71 REVIEW PROBLEMS 78 STRENGTHEN YOUR UNDERSTANDING 84 PROJECTS: COMPOUND INTEREST, POPULATION CENTER OF THE US, MEDICAL CASE STUDY: ANAPHYLAXIS 86 2 RATE OF CHANGE: THE DERIVATIVE 89 2.1 INSTANTANEOUS RATE OF CHANGE 90 2.2 THE DERIVATIVE FUNCTION 97 2.3 INTERPRETATIONS OF THE DERIVATIVE 103 2.4 THE SECOND DERIVATIVE 113 2.5 MARGINAL COST AND REVENUE 119 REVIEW PROBLEMS 125 STRENGTHEN YOUR UNDERSTANDING 130 PROJECTS: ESTIMATING TEMPERATURE OF A YAM; TEMPERATURE AND ILLUMINATION;CHLOROFLUOROCARBONS IN THE ATMOSPHERE 131 FOCUS ON THEORY 133 LIMITS, CONTINUITY, AND THE DEFINITION OF THE DERIVATIVE 133 3 SHORTCUTS TO DIFFERENTIATION 137 3.1 DERIVATIVE FORMULAS FOR POWERS AND POLYNOMIALS 138 3.2 EXPONENTIAL AND LOGARITHMIC FUNCTIONS 145 3.3 THE CHAIN RULE 150 3.4 THE PRODUCT AND QUOTIENT RULES 156 3.5 DERIVATIVES OF PERIODIC FUNCTIONS 161 REVIEW PROBLEMS 165 STRENGTHEN YOUR UNDERSTANDING 168 PROJECTS: CORONER’S RULE OF THUMB; AIR PRESSURE AND ALTITUDE; RELATIVE GROWTH RATES: POPULATION, GDP, AND GDP PER CAPITA; KEELING CURVE: ATMOSPHERIC CARBON DIOXIDE 169 FOCUS ON THEORY 171 ESTABLISHING THE DERIVATIVE FORMULAS 171 FOCUS ON PRACTICE 174 FOCUS ON PRACTICE 174 4 USING THE DERIVATIVE 175 4.1 LOCAL MAXIMA AND MINIMA 176 4.2 INFLECTION POINTS 183 4.3 GLOBAL MAXIMA AND MINIMA 189 4.4 PROFIT, COST, AND REVENUE 194 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 228 STRENGTHEN YOUR UNDERSTANDING 235 PROJECTS: AVERAGE AND MARGINAL COSTS, FIREBREAKS, PRODUCTION AND THE PRICE OF RAW MATERIALS, MEDICAL CASE STUDY: IMPACT OF ASTHMA ON BREATHING 237 5 ACCUMULATED CHANGE: THE DEFINITE INTEGRAL 241 5.1 DISTANCE AND ACCUMULATED CHANGE 242 5.2 THE DEFINITE INTEGRAL 250 5.3 THE DEFINITE INTEGRAL AS AREA 255 5.4 INTERPRETATIONS OF THE DEFINITE INTEGRAL 260 5.5 TOTAL CHANGE AND THE FUNDAMENTAL THEOREM OF CALCULUS 268 5.6 AVERAGE VALUE 272 REVIEW PROBLEMS 276 STRENGTHEN YOUR UNDERSTANDING 281 PROJECTS: CARBON DIOXIDE IN POND WATER, FLOODING IN THE GRAND CANYON 283 FOCUS ON THEORY 286 FOCUS ON THEORY 287 THEOREMS ABOUT DEFINITE INTEGRALS 287 6 ANTIDERIVATIVES AND APPLICATIONS 291 6.1 ANALYZING ANTIDERIVATIVES GRAPHICALLY AND NUMERICALLY 292 6.2 ANTIDERIVATIVES AND THE INDEFINITE INTEGRAL 297 6.3 USING THE FUNDAMENTAL THEOREM TO FIND DEFINITE INTEGRALS 302 6.4 APPLICATION: CONSUMER AND PRODUCER SURPLUS 306 6.5 APPLICATION: PRESENT AND FUTURE VALUE 312 6.6 INTEGRATION BY SUBSTITUTION 316 6.7 INTEGRATION BY PARTS 321 REVIEW PROBLEMS 324 STRENGTHEN YOUR UNDERSTANDING 326 PROJECTS: QUABBIN RESERVOIR, DISTRIBUTION OF RESOURCES, YIELD FROM AN APPLE ORCHARD 328 FOCUS ON PRACTICE 330 7 PROBABILITY 331 7.1 DENSITY FUNCTIONS 332 7.2 CUMULATIVE DISTRIBUTION FUNCTIONS AND PROBABILITY 336 7.3 THE MEDIAN AND THE MEAN 343 REVIEW PROBLEMS 348 STRENGTHEN YOUR UNDERSTANDING 350 PROJECTS: TRIANGULAR PROBABILITY DISTRIBUTION 351 8 FUNCTIONS OF SEVERAL VARIABLES 353 8.1 UNDERSTANDING FUNCTIONS OF TWO VARIABLES 354 8.2 CONTOUR DIAGRAMS 358 8.3 PARTIAL DERIVATIVES 369 8.4 COMPUTING PARTIAL DERIVATIVES ALGEBRAICALLY 376 8.5 CRITICAL POINTS AND OPTIMIZATION 381 8.6 CONSTRAINED OPTIMIZATION 387 REVIEW PROBLEMS 394 STRENGTHEN YOUR UNDERSTANDING 399 PROJECTS: A HEATER IN A ROOM, OPTIMIZING RELATIVE PRICES FOR ADULTS AND CHILDREN, MAXIMIZING PRODUCTION AND MINIMIZING COST: “DUALITY” 401 FOCUS ON THEORY 403 DERIVING THE FORMULA FOR A REGRESSION LINE 403 9 MATHEMATICAL MODELING USING DIFFERENTIAL EQUATIONS 409 9.1 MATHEMATICAL MODELING: SETTING UP A DIFFERENTIAL EQUATION 410 9.2 SOLUTIONS OF DIFFERENTIAL EQUATIONS 414 9.3 SLOPE FIELDS 418 9.4 EXPONENTIAL GROWTH AND DECAY 424 9.5 APPLICATIONS AND MODELING 430 9.6 MODELING THE INTERACTION OF TWO POPULATIONS 439 9.7 MODELING THE SPREAD OF A DISEASE 445 REVIEW PROBLEMS 450 STRENGTHEN YOUR UNDERSTANDING 452 PROJECTS: HARVESTING AND LOGISTIC GROWTH, POPULATION GENETICS, THE SPREAD OF SARS 455 FOCUS ON THEORY 458 SEPARATION OF VARIABLES 458 10 GEOMETRIC SERIES 463 10.1 GEOMETRIC SERIES 464 10.2 APPLICATIONS TO BUSINESS AND ECONOMICS 470 10.3 APPLICATIONS TO THE NATURAL SCIENCES 474 REVIEW PROBLEMS 479 STRENGTHEN YOUR UNDERSTANDING 480 PROJECTS: DO YOU HAVE ANY COMMON ANCESTORS?, HARROD–HICKS MODEL OF AN EXPANDING NATIONAL ECONOMY, PROBABILITY OFWINNING IN SPORTS, MEDICAL CASE STUDY: DRUG DESENSITIZATION SCHEDULE 481 APPENDIX 483 A FITTING FORMULAS TO DATA 484 B COMPOUND INTEREST AND THE NUMBER e 492 C SPREADSHEET PROJECTS 497 1. MALTHUS: POPULATION OUTSTRIPS FOOD SUPPLY 497 2. CREDIT CARD DEBT 498 3. CHOOSING A BANK LOAN 499 4. COMPARING HOME MORTGAGES 500 5. PRESENT VALUE OF LOTTERYWINNINGS 501 6. COMPARING INVESTMENTS 501 7. INVESTING FOR THE FUTURE: TUITION PAYMENTS 502 8. NEW OR USED? 502 9. VERHULST: THE LOGISTIC MODEL 503 10. THE SPREAD OF INFORMATION: A COMPARISON OF TWO MODELS 504 11. THE FLU IN WORLD WAR I 504 ANSWERS TO ODD–NUMBERED PROBLEMS 507 PRETEST 535 INDEX 539