Calculus (12^th Ed.)
Single Variable

Calculus: Single Variable, 12th Edition, offers students a rigorous and intuitive treatment of single variable calculus, including the differentiation and integration of one variable. Using the Rule of Four, the authors present mathematical concepts from verbal, algebraic, visual, and numerical points of view. The book includes numerous exercises, applications, and examples that help readers learn and retain the concepts discussed within, and discusses polynomials, rational functions, exponentials, logarithms, and trigonometric functions late in the text.

Preface vii

Supplements ix

Acknowledgments xi

The Roots of Calculus xvi

1 Limits and Continuity 1

1.1 Limits (An Intuitive Approach) 1

1.2 Computing Limits 13

1.3 Limits at Infinity; End Behavior of a Function 21

1.4 Limits (Discussed More Rigorously) 30

1.5 Continuity 39

1.6 Continuity of Trigonometric Functions 50

2 The Derivative 58

2.1 Tangent Lines and Rates of Change 58

2.2 The Derivative Function 68

2.3 Introduction to Techniques of Differentiation 79

2.4 The Product and Quotient Rules 86

2.5 Derivatives of Trigonometric Functions 91

2.6 The Chain Rule 95

2.7 Implicit Differentiation 102

2.8 Related Rates 109

2.9 Local Linear Approximation; Differentials 116

3 The Derivative in Graphing and Applications 128

3.1 Analysis of Functions I: Increase, Decrease, and Concavity 128

3.2 Analysis of Functions II: Relative Extrema; Graphing Polynomials 137

3.3 Analysis of Functions III: Rational Functions, Cusps, and Vertical Tangents 146

3.4 Absolute Maxima and Minima 154

3.5 Applied Maximum and Minimum Problems 162

3.6 Rectilinear Motion 175

3.7 Newton’s Method 182

3.8 Rolle’s Theorem; Mean-Value Theorem 188

4 Integration 200

4.1 An Overview of the Area Problem 200

4.2 The Indefinite Integral 205

4.3 Integration by Substitution 214

4.4 The Definition of Area as a Limit; Sigma Notation 220

4.5 The Definite Integral 231

4.6 The Fundamental Theorem of Calculus 239

4.7 Rectilinear Motion Revisited Using Integration 251

4.8 Average Value of a Function and Its Applications 259

4.9 Evaluating Definite Integrals by Substitution 263

5 Applications of the Definite Integral in Geometry, Science, and Engineering 273

5.1 Area Between Two Curves 273

5.2 Volumes by Slicing; Disks and Washers 281

5.3 Volumes by Cylindrical Shells 290

5.4 Length of a Plane Curve 297

5.5 Area of a Surface of Revolution 302

5.6 Work 307

5.7 Moments, Centers of Gravity, and Centroids 315

5.8 Fluid Pressure and Force 323

6 Exponential, Logarithmic, and Inverse Trigonometric Functions 331

6.1 Exponential and Logarithmic Functions 331

6.2 Derivatives and Integrals Involving Logarithmic Functions 342

6.3 Derivatives of Inverse Functions; Derivatives and Integrals Involving Exponential Functions 349

6.4 Graphs and Applications Involvig Logarithmic and Exponential Functions 355

6.5 L’Hoˆ pital’s Rule; Indeterminate Forms 362

6.6 Logarithmic and Other Functions Defined by Integrals 371

6.7 Derivatives and Integrals Involving Inverse Trigonometric Functions 382

6.8 Hyperbolic Functions and Hanging Cables 392

7 Principles of Integral Evaluation 406

7.1 An Overview of Integration Methods 406

7.2 Integration by Parts 409

7.3 Integrating Trigonometric Functions 417

7.4 Trigonometric Substitutions 424

7.5 Integrating Rational Functions by Partial Fractions 430

7.6 Using Computer Algebra Systems and Tables of Integrals 437

7.7 Numerical Integration; Simpson’s Rule 446

7.8 Improper Integrals 458

8 Mathematical Modeling with Differential Equations 471

8.1 Modeling with Differential Equations 471

8.2 Separation of Variables 477

8.3 Slope Fields; Euler’s Method 488

8.4 First-Order Differential Equations and Applications 494

9 Infinite Series 504

9.1 Sequences 504

9.2 Monotone Sequences 513

9.3 Infinite Series 520

9.4 Convergence Tests 528

9.5 The Comparison, Ratio, and Root Tests 534

9.6 Alternating Series; Absolute and Conditional Convergence 540

9.7 Maclaurin and Taylor Polynomials 549

9.8 Maclaurin and Taylor Series; Power Series 559

9.9 Convergence of Taylor Series 567

9.10 Differentiating and Integrating Power Series; Modeling with Taylor Series 575

10 Parametric and Polar Curves; Conic Sections 588

10.1 Parametric Equations; Tangent Lines and Arc Length for Parametric Curves 588

10.2 Polar Coordinates 600

10.3 Tangent Lines, Arc Length, and Area for Polar Curves 613

10.4 Conic Sections 622

10.5 Rotation of Axes; Second-Degree Equations 639

10.6 Conic Sections in Polar Coordinates 644

A Appendices

A Trigonometry Review (Summary) App-1

B Functions (Summary) App-8

C New Functions From Old (Summary) App-11

D Families of Functions (Summary) App-16

E Inverse Functions (Summary) App-23

Ready Reference RR-1

Answers to Odd-numbered Exercises Ans-1

Index Ind-1

Web Appendices (online only)

Available in WileyPLUS

A Trigonometry Review

B Functions

C New Functions From Old

D Families of Functions

E Inverse Functions

F Real Numbers, Intervals, and Inequalities

G Absolute Value

H Coordinate Planes, Lines, and Linear Functions

I Distance, Circles, and Quadratic Equations

J Solving Polynomial Equations

K Graphing Functions Using Calculators and Computer Algebra Systems

L Selected Proofs

M Early Parametric Equations Option

N Mathematical Models

O The Discriminant

P Second-Order Linear Homogeneous Differential Equations

Chapter Web Projects: Expanding the Calculus Horizon (online only)

Available in WileyPLUS

Robotics — Chapter 2

Railroad Design — Chapter 7

Iteration and Dynamical Systems — Chapter 9

Comet Collision — Chapter 10