## TABLE OF CONTENTS

**1 Functions And Change 1**

1.1 What Is A Function? 2

1.2 Linear Functions 10

1.3 Average Rate of Change and Relative Change 21

1.4 Applications of Functions to Economics 34

1.5 Exponential Functions 47

1.6 The Natural Logarithm 57

1.7 Exponential Growth and Decay 63

1.8 New Functions from Old 75

1.9 Proportionality and Power Functions 81

1.10 Periodic Functions 87

Projects: Compound interest, Population center of the US, Medical case Study: Anaphylaxis 95

**2 Rate of Change: The Derivative 97**

2.1 Instantaneous Rate of Change 98

2.2 The Derivative Function 108

2.3 Interpretations of the Derivative 115

2.4 The Second Derivative 128

2.5 Marginal Cost and Revenue 134

Projects: Estimating Temperature of a Yam; Temperature and Illumination; Chlorofluorocarbons in the Atmosphere 140

Focus On Theory: Limits and the Definition of the Derivative 142

**3 Shortcuts to Differentiation 149**

3.1 Derivative Formulas for Powers and Polynomials 150

3.2 Exponential and Logarithmic Functions 158

3.3 The Chain Rule 165

3.4 The Product and Quotient Rules 171

3.5 Derivatives of Periodic Functions 177

Projects: Coroner’s rule of Thumb; Air pressure and altitude; Relative growth Rates: Population, GDP, And GDP Per Capita; Keeling Curve: Atmospheric Carbon Dioxide 181

Focus on Theory: Establishing the Derivative Formulas 184

Focus On Practice 188

**4 Using the Derivative 189**

4.1 Local Maxima and Minima 190

4.2 Inflection Points 197

4.3 Global Maxima and Minima 204

4.4 Profit, Cost, and Revenue 212

4.5 Average Cost 220

4.6 Elasticity of Demand 226

4.7 Logistic Growth 232

4.8 The Surge Function and Drug Concentration 241

Projects: Average And marginal Costs, Firebreaks, Production And The Price Of Raw Materials, Medical Case Study: Impact Of Asthma on Breathing 248

**5 Accumulated Change: The Definite Integral 251**

5.1 Distance and Accumulated Change 252

5.2 The Definite Integral 261

5.3 The Definite Integral as Area 268

5.4 Interpretations of the Definite Integral 273

5.5 Total Change and the Fundamental Theorem of Calculus 283

5.6 Average Value 288

Projects: Carbon Dioxide in Pond Water, Flooding In the Grand Canyon, Medical Case Studies: Flux Of Fluid from a Capillary, Testing For Kidney Disease 294

Focus On Theory: The Second Fundamental Theorem of Calculus 296

**6 Antiderivatives and Applications 301**

6.1 Analyzing Antiderivatives Graphically and Numerically 302

6.2 Antiderivatives and the Indefinite Integral 308

6.3 Using the Fundamental Theorem to Find Definite Integrals 314

6.4 Application: Consumer and Producer Surplus 319

6.5 Application: Present and Future Value 325

6.6 Integration by Substitution 331

6.7 Integration by Parts 337

Projects: Quabbin Reservoir, Distribution of Resources, Yield from an Apple Orchard 340

Focus On Practice 343

**7 Probability 345**

7.1 Density Functions 346

7.2 Cumulative Distribution Functions and Probability 350

7.3 The Median and the Mean 357

Projects: Triangular Probability Distribution 363

**8 Functions of Several Variables 365**

8.1 Understanding Functions of Two Variables 366

8.2 Contour Diagrams 371

8.3 Partial Derivatives 383

8.4 Computing Partial Derivatives Algebraically 392

8.5 Critical Points and Optimization 398

8.6 Constrained Optimization 405

Projects: A Heater in a Room, Optimizing Relative Prices for Adults and Children, Maximizing Production and Minimizing Cost: “Duality” 412

Focus on Theory: Deriving the Formula for Regression Lines 413

**9 Mathematical Modeling Using Differential Equations 419**

9.1 Mathematical Modeling: Setting Up a Differential Equation 420

9.2 Solutions of Differential Equations 424

9.3 Slope Fields 428

9.4 Exponential Growth and Decay 434

9.5 Applications and Modeling 440

9.6 Modeling the Interaction of Two Populations 450

9.7 Modeling the Spread of a Disease 456

Projects: Harvesting and Logistic Growth, Population Genetics, the Spread of Sars 462

Focus On Theory: Separation of Variables 465

**10 Geometric Series ****Online**

10.1 Geometric Series Online

10.2 Applications to Business and Economics Online

10.3 Applications to the Natural Sciences Online

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 Online

**Appendix ****Online**

A Fitting Formulas to Data Online

B Compound Interest and the Number E Online

C Spreadsheet Projects Online

1. Malthus: Population Outstrips Food Supply Online

2. Credit Card Debt Online

3. Choosing a Bank Loan Online

4. Comparing Home Mortgages Online

5. Present Value of Lottery Winnings Online

6. Comparing Investments Online

7. Investing For the Future: Tuition Payments Online

8. New or Used? Online

9. Verhulst: The Logistic Model Online

10. The Spread of Information: A Comparison Of Two Models Online

11. The Flu in World War I Online

Answers to Odd-Numbered Problems 469

Index 493