Applied Calculus

The 5 th Edition of Applied Calculus continues to exhibit the same strengths from earlier editions including a focus on creative conceptual and modeling problems and the "Rule of Four", an emphasis on concepts and modeling, exposition that teaches a flexible approach to technology. This issue provides readers with deeper skills needed to apply calculus on the job and highlights connections with real-world concerns. The problems and exercises are challenging and provoke deeper thinking to help apply math in new ways. The material is presented in a way to help readers decide when to use technology, which empowers them to learn what calculators/computers can and cannot do.

The 5 th Edition of Applied Calculus continues to exhibit the same strengths from earlier editions including a focus on creative conceptual and modeling problems and the "Rule of Four", an emphasis on concepts and modeling, exposition that teaches a flexible approach to technology.

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 CO