CONTENTS

1 FUNCTIONS AND CHANGE

1.1 WHAT IS A FUNCTION?

1.2 LINEAR FUNCTIONS

1.3 AVERAGE RATE OF CHANGE AND RELATIVE CHANGE

1.4 APPLICATIONS OF FUNCTIONS TO ECONOMICS

1.5 EXPONENTIAL FUNCTIONS

1.6 THE NATURAL LOGARITHM

1.7 EXPONENTIAL GROWTH AND DECAY

1.8 NEW FUNCTIONS FROM OLD

1.9 PROPORTIONALITY AND POWER FUNCTIONS

1.10 PERIODIC FUNCTIONS

REVIEW PROBLEMS

STRENGTHEN YOUR UNDERSTANDING

PROJECTS: COMPOUND INTEREST, POPULATION CENTER OF THE US, MEDICAL CASE STUDY: ANAPHYLAXIS

2 RATE OF CHANGE: THE DERIVATIVE

2.1 INSTANTANEOUS RATE OF CHANGE

2.2 THE DERIVATIVE FUNCTION

2.3 INTERPRETATIONS OF THE DERIVATIVE

2.4 THE SECOND DERIVATIVE

2.5 MARGINAL COST AND REVENUE

REVIEW PROBLEMS

STRENGTHEN YOUR UNDERSTANDING

PROJECTS: ESTIMATING TEMPERATURE OF A YAM; TEMPERATURE AND ILLUMINATION; CHLOROFLUOROCARBONS IN THE ATMOSPHERE

FOCUS ON THEORY

LIMITS, CONTINUITY, AND THE DEFINITION OF THE DERIVATIVE

3 SHORTCUTS TO DIFFERENTIATION

3.1 DERIVATIVE FORMULAS FOR POWERS AND POLYNOMIALS

3.2 EXPONENTIAL AND LOGARITHMIC FUNCTIONS

3.3 THE CHAIN RULE

3.4 THE PRODUCT AND QUOTIENT RULES

3.5 DERIVATIVES OF PERIODIC FUNCTIONS

REVIEW PROBLEMS

STRENGTHEN YOUR UNDERSTANDING

PROJECTS: CORONER'S RULE OF THUMB; AIR PRESSURE AND ALTITUDE; RELATIVE GROWTH RATES: POPULATION, GDP, AND GDP PER CAPITA; KEELING CURVE: ATMOSPHERIC CARBON DIOXIDE

FOCUS ON THEORY

ESTABLISHING THE DERIVATIVE FORMULAS

FOCUS ON PRACTICE

4 USING THE DERIVATIVE

4.1 LOCAL MAXIMA AND MINIMA

4.2 INFLECTION POINTS

4.3 GLOBAL MAXIMA AND MINIMA

4.4 PROFIT, COST, AND REVENUE

4.5 AVERAGE COST

4.6 ELASTICITY OF DEMAND

4.7 LOGISTIC GROWTH

4.8 THE SURGE FUNCTION AND DRUG CONCENTRATION

REVIEW PROBLEMS

STRENGTHEN YOUR UNDERSTANDING

PROJECTS: AVERAGE AND MARGINAL COSTS, FIREBREAKS, PRODUCTION AND THE PRICE OF RAW MATERIALS, MEDICAL CASE STUDY: IMPACT OF ASTHMA ON BREATHING

5 ACCUMULATED CHANGE: THE DEFINITE INTEGRAL

5.1 DISTANCE AND ACCUMULATED CHANGE

5.2 THE DEFINITE INTEGRAL

5.3 THE DEFINITE INTEGRAL AS AREA

5.4 INTERPRETATIONS OF THE DEFINITE INTEGRAL

5.5 TOTAL CHANGE AND THE FUNDAMENTAL THEOREM OF CALCULUS

5.6 AVERAGE VALUE

REVIEW PROBLEMS

STRENGTHEN YOUR UNDERSTANDING

PROJECTS: CARBON DIOXIDE IN POND WATER, FLOODING IN THE GRAND CANYON

FOCUS ON THEORY

THEOREMS ABOUT DEFINITE INTEGRALS

6 ANTIDERIVATIVES AND APPLICATIONS

6.1 ANALYZING ANTIDERIVATIVES GRAPHICALLY AND NUMERICALLY

6.2 ANTIDERIVATIVES AND THE INDEFINITE INTEGRAL

6.3 USING THE FUNDAMENTAL THEOREM TO FIND DEFINITE INTEGRALS

6.4 APPLICATION: CONSUMER AND PRODUCER SURPLUS

6.5 APPLICATION: PRESENT AND FUTURE VALUE

6.6 INTEGRATION BY SUBSTITUTION

6.7 INTEGRATION BY PARTS

REVIEW PROBLEMS

STRENGTHEN YOUR UNDERSTANDING

PROJECTS: QUABBIN RESERVOIR, DISTRIBUTION OF RESOURCES, YIELD FROM AN APPLE ORCHARD

FOCUS ON PRACTICE

7 PROBABILITY

7.1 DENSITY FUNCTIONS

7.2 CUMULATIVE DISTRIBUTION FUNCTIONS AND PROBABILITY

7.3 THE MEDIAN AND THE MEAN

REVIEW PROBLEMS

STRENGTHEN YOUR UNDERSTANDING

PROJECTS: TRIANGULAR PROBABILITY DISTRIBUTION

8 FUNCTIONS OF SEVERAL VARIABLES

8.1 UNDERSTANDING FUNCTIONS OF TWO VARIABLES

8.2 CONTOUR DIAGRAMS

8.3 PARTIAL DERIVATIVES

8.4 COMPUTING PARTIAL DERIVATIVES ALGEBRAICALLY

8.5 CRITICAL POINTS AND OPTIMIZATION

8.6 CONSTRAINED OPTIMIZATION

REVIEW PROBLEMS

STRENGTHEN YOUR UNDERSTANDING

PROJECTS: A HEATER IN A ROOM, OPTIMIZING RELATIVE PRICES FOR ADULTS AND CHILDREN, MAXIMIZING PRODUCTION AND MINIMIZING COST: “DUALITY”

FOCUS ON THEORY

DERIVING THE FORMULA FOR A REGRESSION LINE

9 MATHEMATICAL MODELING USING DIFFERENTIAL EQUATIONS

9.1 MATHEMATICAL MODELING: SETTING UP A DIFFERENTIAL EQUATION

9.2 SOLUTIONS OF DIFFERENTIAL EQUATIONS

9.3 SLOPE FIELDS

9.4 EXPONENTIAL GROWTH AND DECAY

9.5 APPLICATIONS AND MODELING

9.6 MODELING THE INTERACTION OF TWO POPULATIONS

9.7 MODELING THE SPREAD OF A DISEASE

REVIEW PROBLEMS

STRENGTHEN YOUR UNDERSTANDING

PROJECTS: HARVESTING AND LOGISTIC GROWTH, POPULATION GENETICS, THE SPREAD OF SARS

FOCUS ON THEORY

SEPARATION OF VARIABLES

10 GEOMETRIC SERIES

10.1 GEOMETRIC SERIES

10.2 APPLICATIONS TO BUSINESS AND ECONOMICS

10.3 APPLICATIONS TO THE NATURAL SCIENCES

REVIEW PROBLEMS

STRENGTHEN YOUR UNDERSTANDING

PROJECTS: DO YOU HAVE ANY COMMON ANCESTORS?, HARROD-HICKS MODEL OF AN EXPANDING NATIONAL ECONOMY, PROBABILITY OF WINNING IN SPORTS, MEDICAL CASE STUDY: DRUG DESENSITIZATION SCHEDULE

APPENDIX

A FITTING FORMULAS TO DATA

B COMPOUND INTEREST AND THE NUMBER e

C SPREADSHEET PROJECTS

1. MALTHUS: POPULATION OUTSTRIPS FOOD SUPPLY

2. CREDIT CARD DEBT

3. CHOOSING A BANK LOAN

4. COMPARING HOME MORTGAGES

5. PRESENT VALUE OF LOTTERY WINNINGS

6. COMPARING INVESTMENTS

7. INVESTING FOR THE FUTURE: TUITION PAYMENTS

8. NEW OR USED?

9. VERHULST: THE LOGISTIC MODEL

10. THE SPREAD OF INFORMATION: A COMPARISON OF TWO MODELS

11. THE FLU IN WORLD WAR I

ANSWERS TO ODD-NUMBERED PROBLEMS

PRETEST

INDEX