Applied Calculus

1 Functions and Change 1

1.1 What is a Function? 2

1.2 Linear Functions 9

1.3 Average Rate of Change and Relative Change 18

1.4 Applications of Functions to Economics 31

1.5 Exponential Functions 42

1.6 The Natural Logarithm 51

1.7 Exponential Growth and Decay 56

1.8 New Functions from Old 66

1.9 Proportionality and Power Functions 72

1.10 Periodic Functions 78

Chapter 1 Summary Digital

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Projects: Compound Interest, Population Center of The Us, Medical Case Study: Anaphylaxis 85

2 Rate of Change: The Derivative 87

2.1 Instantaneous Rate of Change 88

2.2 The Derivative Function 97

2.3 Interpretations of the Derivative 103

2.4 The Second Derivative 114

2.5 Marginal Cost and Revenue 120

Chapter 2 Summary Digital

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Projects: Estimating Temperature of a Yam; Temperature and Illumination; Chlorofluorocarbons in the Atmosphere 126

Focus On Theory: Limits and the Definition of the Derivative 127

Limits, Continuity, and The Definition of the Derivative 128

3 Shortcuts to Differentiation 135

3.1 Derivative formulas for Powers and Polynomials 136

3.2 Exponential and Logarithmic Functions 144

3.3 The Chain Rule 150

3.4 The Product and Quotient Rules 156

3.5 Derivatives of Periodic Functions 161

Chapter 3 Summary Digital

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Projects: Coroner’s Rule of Thumb; Air Pressure and Altitude; Relativegrowthrates: Population, GDP, and GDP Per Capita; Keelingcurve: Atmospheric Carbon Dioxide 166

Focus On Theory: Establishing the Derivative formulas 168

Establishing the Derivative formulas 168

Focus On Practice 172

4 Using The Derivative 173

4.1 Local Maxima and Minima 174

4.2 Inflection Points 181

4.3 Global Maxima and Minima 187

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 223

Chapter 4 Summary Digital

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Projects: Average and Marginal Costs, Firebreaks, Production and the Price of Raw Materials, Medical Case Study: Impact of asthma On Breathing 229

5 Accumulated Change: The Definite Integral 233

5.1 Distance and Accumulated Change 234

5.2 The Definite Integral 243

5.3 The Definite Integral as Area 249

5.4 Interpretations of the Definite Integral 255

5.5 total Change and the Fundamental Theorem of Calculus 264

5.6 Average Value 268

Chapter 5 Summary Digital

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Projects: Carbon Dioxide in Pond Water, Flooding in the Grand Canyon 273

Focus On Theory: The Second Fundamental Theorem of Calculus 276

Theorems About Definite Integrals 277

6 Antiderivatives and Applications 281

6.1 Analyzing Antiderivatives Graphically and Numerically 282

6.2 Antiderivatives and the Indefinite Integral 288

6.3 Using the Fundamental Theorem to Find Definite Integrals 293

6.4 Application: Consumer and Producer Surplus 297

6.5 Application: Present and Future Value 303

6.6 Integration by Substitution 308

6.7 Integration by Parts 314

Chapter 6 Summary Digital

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Projects: Quabbin Reservoir, Distribution of Resources, Yield from an Apple Orchard 317

Focus On Practice 319

7 Probability 321

7.1 Density Functions 322

7.2 Cumulative Distribution Functions and Probability 326

7.3 The Median and the Mean 333

Chapter 7 Summary Digital

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Projects: Triangular Probability Distribution 338

8 Functions of Several Variables 339

8.1 Understanding Functions of Two Variables 340

8.2 Contour Diagrams 345

8.3 Partial Derivatives 357

8.4 Computing Partial Derivatives Algebraically 364

8.5 Critical Points and Optimization 370

8.6 Constrained Optimization 376

Chapter 8 Summary Digital

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Projects: A Heater in a Room, Optimizing Relative Prices for Adults and Children, Maximizing Production and Minimizing Cost: “Duality” 384

Focus On Theory: Deriving the formula for Regression Lines 385

Deriving the formula for a Regression Line 386

9 Mathematical Modeling Using Differential Equations 391

9.1 Mathematical Modeling: Setting Up a Differential Equation 392

9.2 Solutions of Differential Equations 396

9.3 Slope Fields 400

9.4 Exponential Growth and Decay 406

9.5 Applications and Modeling 411

9.6 Modeling the Interaction of Two Populations 421

9.7 Modeling the Spread of a Disease 427

Chapter 9 Summary Digital

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Projects: Harvesting and Logistic Growth, Population Genetics, The Spread of Sars 431

Focus On Theory: Separation of Variables 434

Separation of Variables 434

10 Geometric Series Digital

10.1 Geometric Series Digital

10.2 Applications to Business and Economics Digital

10.3 Applications to the Natural Sciences Digital

Chapter 10 Summary Digital

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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 Digital

Appendices Digital

A Fitting formulas to Data Digital

B Compound Interest and the Number E Digital

C Spreadsheet Projects Digital

1. Malthus: Population Outstrips Food Supply Digital

2. Credit Card Debt Digital

3. Choosing a Bank Loan Digital

4. Comparing Home Mortgages Digital

5. Present Value of Lottery Winnings Digital

6. Comparing Investments Digital

7. Investing for the Future: Tuition Payments Digital

8. New or Used? Digital

9. Verhulst: The Logistic Model Digital

10. The Spread of Information: A Comparison of Two Models Digital

11. The Flu in World War I Digital

Answers to Odd-Numbered Problems 439

Index 461