Brief Calculus and Its Applications

Improved from previous editions based on reader suggestions, but focused on the same goals: to begin the calculus as soon as possible, present it in an intuitive and intellectually satisfying way, and to illustrate the many application of calculus to social, biological, and management sciences. Previous edition: c1999. DLC: Calculus.

Preface

ix

Introduction

1

(2)

Functions

3

(67)

Functions and Their Graphs

3

(15)

Some Important Functions

18

(9)

The Algebra of Functions

27

(7)

Zeros of Functions---The Quadratic Formula and Factoring

34

(8)

Exponents and Power Functions

42

(7)

Functions and Graphs in Applications

49

(21)

Graphing Functions Using Technology

62

(6)

Chapter Project

68

(2)

The Derivative

70

(74)

The Slope of a Straight Line

72

(11)

The Slope of a Curve at a Point

83

(6)

The Derivative

89

(12)

Limits and the Derivative

101

(10)

Differentiability and Continuity

111

(5)

Some Rules for Differentiation

116

(7)

More About Derivatives

123

(5)

The Derivative as a Rate of Change

128

(16)

Chapter Project

143

(1)

Applications of the Derivative

144

(70)

Describing Graphs of Functions

144

(12)

The First and Second Derivative Rules

156

(10)

Curve Sketching (Introduction)

166

(7)

Curve Sketching (Conclusion)

173

(7)

Optimization Problems

180

(9)

Further Optimization Problems

189

(9)

Applications of Derivatives to Business and Economics

198

(16)

Chapter Project

213

(1)

Techniques of Differentiation

214

(29)

The Product and Quotient Rules

214

(8)

The Chain Rule and the General Power Rule

222

(6)

Implicit Differentiation and Related Rates

228

(15)

Chapter Project

242

(1)

The Exponential and Natural Logarithm Functions

243

(31)

Exponential Functions

243

(4)

The Exponential Function ex

247

(5)

Differentiation of Exponential Functions

252

(5)

The Natural Logarithm Function

257

(5)

The Derivative of ln x

262

(4)

Properties of the Natural Logarithm Function

266

(8)

Chapter Project

272

(2)

Applications of the Exponential and Natural Logarithm Functions

274

(39)

Exponential Growth and Decay

274

(10)

Compound Interest

284

(8)

Applications of the Natural Logarithm Function to Economics

292

(6)

Further Exponential Models

298

(15)

Chapter Project

312

(1)

The Definite Integral

313

(76)

Antidifferentiation

314

(9)

Areas and Riemann Sums

323

(9)

Definite Integrals and the Fundamental Theorem

332

(12)

Areas in the xy-Plane

344

(10)

Applications of the Definite Integral

354

(9)

Techniques of Integration

363

(9)

Improper Integrals

372

(7)

Applications of Calculus to Probability

379

(10)

Chapter Project

388

(1)

Functions of Several Variables

389

(63)

Examples of Functions of Several Variables

389

(7)

Partial Derivatives

396

(9)

Maxima and Minima of Functions of Several Variables

405

(8)

Lagrange Multipliers and Constrained Optimization

413

(11)

The Method of Least Squares

424

(7)

Nonlinear Regression

431

(12)

Double Integrals

443

(9)

Chapter Project

451

(1)

The Trigonometric Functions

452

(1)

Radian Measure of Angles

452

(4)

The Sine and the Cosine

456

(8)

Differentiation and Integration of sin t and cos t

464

(10)

The Tangent and Other Trigonometric Functions

474

(7)

Chapter Project

481

Appendix A: Calculus and the TI-82 Calculator

A1

Appendix B: Calculus and the TI-83/TI-83 Plus Calculators

A5

Appendix C: Calculus and the TI-85 Calculator

A9

Appendix D: Calculus and the TI-86 Calculator

A13

Answers to Exercises

A19

Index

I1

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Brief Calculus and Its Applications

Summary

Table of Contents

Supplemental Materials