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