**Single Variable Calculus: Early Transcendentals, 9th Edition**

By James Stewart, Daniel K. Clegg and Saleem Watson

Preface x

A Tribute to James Stewart xxii

About the Authors xxiii

Technology in the Ninth Edition xxiv

To the Student xxv

Diagnostic Tests xxvi

A Preview of Calculus 1

1 Functions and Models 7

1.1 Four Ways to Represent a Function 8

1.2

1.3 New Functions from Old Functions 36

1.4 Exponential Functions 45

1.5 Inverse Functions and Logarithms 54

Review 67

Principles of Problem Solving 70

2 Limits and Derivatives 77

2.1 The Tangent and Velocity Problems 78

2.2 The Limit of a Function 83

2.3 Calculating Limits Using the Limit Laws 94

2.4 The Precise Definition of a Limit 105

2.5 Continuity 115

2.6 Limits at Infinity; Horizontal Asymptotes 127

2.7 Derivatives and Rates of Change 140

writing project • Early Methods for Finding Tangents 152

2.8 The Derivative as a Function 153

Review 166

Problems Plus 171

3 Differentiation Rules 173

3.1 Derivatives of Polynomials and Exponential Functions 174

applied project • Building a Better Roller Coaster 184

3.2 The Product and Quotient Rules 185

3.3 Derivatives of Trigonometric Functions 191

3.4 The Chain Rule 199

applied project • Where Should a Pilot Start Descent? 209

3.5 Implicit Differentiation 209

discovery project • Families of Implicit Curves 217

3.6 Derivatives of Logarithmic and Inverse Trigonometric Functions 217

3.7 Rates of Change in the Natural and Social Sciences 225

3.8 Exponential Growth and Decay 239

applied project • Controlling Red Blood Cell Loss During Surgery 247

3.9 Related Rates 247

3.10 Linear Approximations and Differentials 254

discovery project • Polynomial Approximations 260

3.11 Hyperbolic Functions 261

Review 269

Problems Plus 274

4 Applications of Differentiation 279

4.1 Maximum and Minimum Values 280

applied project • The Calculus of Rainbows 289

4.2 The Mean Value Theorem 290

4.3 What Derivatives Tell Us about the Shape of a Graph 296

4.4 Indeterminate Forms and l’Hospital’s Rule 309

writing project • The Origins of l’Hospital’s Rule 319

4.5 Summary of Curve Sketching 320

4.6 Graphing with Calculus and Technology 329

4.7 Optimization Problems 336

applied project • The Shape of a Can 349

applied project • Planes and Birds: Minimizing Energy 350

4.8 Newton’s Method 351

4.9 Antiderivatives 356

Review 364

Problems Plus 369

5 Integrals 371

5.1 The Area and Distance Problems 372

5.2 The Definite Integral 384

discovery project • Area Functions 398

5.3 The Fundamental Theorem of Calculus 399

5.4 Indefinite Integrals and the Net Change Theorem 409

writing project • Newton, Leibniz, and the Invention of Calculus 418

5.5 The Substitution Rule 419

Review 428

Problems Plus 432

6 Applications of Integration 435

6.1 Areas Between Curves 436

applied project • The Gini Index 445

6.2 Volumes 446

6.3 Volumes by Cylindrical Shells 460

6.4 Work 467

6.5 Average Value of a Function 473

applied project • Calculus and Baseball 476

applied project • Where to Sit at the Movies 478

Review 478

Problems Plus 481

7 Techniques of Integration 485

7.1 Integration by Parts 486

7.2 Trigonometric Integrals 493

7.3 Trigonometric Substitution 500

7.4 Integration of Rational Functions by Partial Fractions 507

7.5 Strategy for Integration 517

7.6 Integration Using Tables and Technology 523

discovery project • Patterns in Integrals 528

7.7 Approximate Integration 529

7.8 Improper Integrals 542

Review 552

Problems Plus 556

8 Further Applications of Integration 559

8.1 Arc Length 560

discovery project • Arc Length Contest 567

8.2 Area of a Surface of Revolution 567

discovery project • Rotating on a Slant 575

8.3 Applications to Physics and Engineering 576

discovery project • Complementary Coffee Cups 587

8.4 Applications to Economics and Biology 587

8.5 Probability 592

Review 600

Problems Plus 602

9 Differential Equations 605

9.1 Modeling with Differential Equations 606

9.2 Direction Fields and Euler’s Method 612

9.3 Separable Equations 621

applied project • How Fast Does a Tank Drain? 630

9.4 Models for Population Growth 631

9.5 Linear Equations 641

applied project • Which Is Faster, Going Up or Coming Down? 648

9.6 Predator-Prey Systems 649

Review 656

Problems Plus 659

10 Parametric Equations and Polar Coordinates 661

10.1 Curves Defined by Parametric Equations 662

discovery project • Running Circles Around Circles 672

10.2 Calculus with Parametric Curves 673

discovery project • Bézier Curves 684

10.3 Polar Coordinates 684

discovery project • Families of Polar Curves 694

10.4 Calculus in Polar Coordinates 694

10.5 Conic Sections 702

10.6 Conic Sections in Polar Coordinates 711

Review 719

Problems Plus 722

11 Sequences, Series, and Power Series 723

11.1 Sequences 724

discovery project • Logistic Sequences 738

11.2 Series 738

11.3 The Integral Test and Estimates of Sums 751

11.4 The Comparison Tests 760

11.5 Alternating Series and Absolute Convergence 765

11.6 The Ratio and Root Tests 774

11.7 Strategy for Testing Series 779

11.8 Power Series 781

11.9 Representations of Functions as Power Series 787

11.10 Taylor and Maclaurin Series 795

discovery project • An Elusive Limit 810

writing project • How Newton Discovered the Binomial Series 811

11.11 Applications of Taylor Polynomials 811

applied project • Radiation from the Stars 820

Review 821

Problems Plus 825

12 Vectors and the Geometry of Space 829

12.1 Three-Dimensional Coordinate Systems 830

12.2 Vectors 836

discovery project • The Shape of a Hanging Chain 846

12.3 The Dot Product 847

12.4 The Cross Product 855

discovery project • The Geometry of a Tetrahedron 864

12.5 Equations of Lines and Planes 864

discovery project • Putting 3D in Perspective 874

12.6 Cylinders and Quadric Surfaces 875

Review 883

Problems Plus 887

13 Vector Functions 889

13.1 Vector Functions and Space Curves 890

13.2 Derivatives and Integrals of Vector Functions 898

13.3 Arc Length and Curvature 904

13.4 Motion in Space: Velocity and Acceleration 916

applied project • Kepler’s Laws 925

Review 927

Problems Plus 930

14 Partial Derivatives 933

14.1 Functions of Several Variables 934

14.2 Limits and Continuity 951

14.3 Partial Derivatives 961

discovery project • Deriving the Cobb-Douglas Production Function 973

14.4 Tangent Planes and Linear Approximations 974

applied project • The Speedo LZR Racer 984

14.5 The Chain Rule 985

14.6 Directional Derivatives and the Gradient Vector 994

14.7 Maximum and Minimum Values 1008

discovery project • Quadratic Approximations and Critical Points 1019

14.8 Lagrange Multipliers 1020

applied project • Rocket Science 1028

applied project • Hydro-Turbine Optimization 1030

Review 1031

Problems Plus 1035

15 Multiple Integrals 1037

15.1 Double Integrals over Rectangles 1038

15.2 Double Integrals over General Regions 1051

15.3 Double Integrals in Polar Coordinates 1062

15.4 Applications of Double Integrals 1069

15.5 Surface Area 1079

15.6 Triple Integrals 1082

discovery project • Volumes of Hyperspheres 1095

15.7 Triple Integrals in Cylindrical Coordinates 1095

discovery project • The Intersection of Three Cylinders 1101

15.8 Triple Integrals in Spherical Coordinates 1102

applied project • Roller Derby 1108

15.9 Change of Variables in Multiple Integrals 1109

Review 1117

Problems Plus 1121

16 Vector Calculus 1123

16.1 Vector Fields 1124

16.2 Line Integrals 1131

16.3 The Fundamental Theorem for Line Integrals 1144

16.4 Green’s Theorem 1154

16.5 Curl and Divergence 1161

16.6 Parametric Surfaces and Their Areas 1170

16.7 Surface Integrals 1182

16.8 Stokes’ Theorem 1195

16.9 The Divergence Theorem 1201

16.10 Summary 1208

Review 1209

Problems Plus 1213

Appendixes A1

A Numbers, Inequalities, and Absolute Values A2

B Coordinate Geometry and Lines A10

C Graphs of Second-Degree Equations A16

D Trigonometry A24

E Sigma Notation A36

F Proofs of Theorems A41

G The Logarithm Defined as an Integral A53

H Answers to Odd-Numbered Exercises A61

Index A143

A Tribute to James Stewart xxii

About the Authors xxiii

Technology in the Ninth Edition xxiv

To the Student xxv

Diagnostic Tests xxvi

A Preview of Calculus 1

1 Functions and Models 7

1.1 Four Ways to Represent a Function 8

1.2

**Mathematical**Models: A Catalog of Essential Functions 211.3 New Functions from Old Functions 36

1.4 Exponential Functions 45

1.5 Inverse Functions and Logarithms 54

Review 67

Principles of Problem Solving 70

2 Limits and Derivatives 77

2.1 The Tangent and Velocity Problems 78

2.2 The Limit of a Function 83

2.3 Calculating Limits Using the Limit Laws 94

2.4 The Precise Definition of a Limit 105

2.5 Continuity 115

2.6 Limits at Infinity; Horizontal Asymptotes 127

2.7 Derivatives and Rates of Change 140

writing project • Early Methods for Finding Tangents 152

2.8 The Derivative as a Function 153

Review 166

Problems Plus 171

3 Differentiation Rules 173

3.1 Derivatives of Polynomials and Exponential Functions 174

applied project • Building a Better Roller Coaster 184

3.2 The Product and Quotient Rules 185

3.3 Derivatives of Trigonometric Functions 191

3.4 The Chain Rule 199

applied project • Where Should a Pilot Start Descent? 209

3.5 Implicit Differentiation 209

discovery project • Families of Implicit Curves 217

3.6 Derivatives of Logarithmic and Inverse Trigonometric Functions 217

3.7 Rates of Change in the Natural and Social Sciences 225

3.8 Exponential Growth and Decay 239

applied project • Controlling Red Blood Cell Loss During Surgery 247

3.9 Related Rates 247

3.10 Linear Approximations and Differentials 254

discovery project • Polynomial Approximations 260

3.11 Hyperbolic Functions 261

Review 269

Problems Plus 274

4 Applications of Differentiation 279

4.1 Maximum and Minimum Values 280

applied project • The Calculus of Rainbows 289

4.2 The Mean Value Theorem 290

4.3 What Derivatives Tell Us about the Shape of a Graph 296

4.4 Indeterminate Forms and l’Hospital’s Rule 309

writing project • The Origins of l’Hospital’s Rule 319

4.5 Summary of Curve Sketching 320

4.6 Graphing with Calculus and Technology 329

4.7 Optimization Problems 336

applied project • The Shape of a Can 349

applied project • Planes and Birds: Minimizing Energy 350

4.8 Newton’s Method 351

4.9 Antiderivatives 356

Review 364

Problems Plus 369

5 Integrals 371

5.1 The Area and Distance Problems 372

5.2 The Definite Integral 384

discovery project • Area Functions 398

5.3 The Fundamental Theorem of Calculus 399

5.4 Indefinite Integrals and the Net Change Theorem 409

writing project • Newton, Leibniz, and the Invention of Calculus 418

5.5 The Substitution Rule 419

Review 428

Problems Plus 432

6 Applications of Integration 435

6.1 Areas Between Curves 436

applied project • The Gini Index 445

6.2 Volumes 446

6.3 Volumes by Cylindrical Shells 460

6.4 Work 467

6.5 Average Value of a Function 473

applied project • Calculus and Baseball 476

applied project • Where to Sit at the Movies 478

Review 478

Problems Plus 481

7 Techniques of Integration 485

7.1 Integration by Parts 486

7.2 Trigonometric Integrals 493

7.3 Trigonometric Substitution 500

7.4 Integration of Rational Functions by Partial Fractions 507

7.5 Strategy for Integration 517

7.6 Integration Using Tables and Technology 523

discovery project • Patterns in Integrals 528

7.7 Approximate Integration 529

7.8 Improper Integrals 542

Review 552

Problems Plus 556

8 Further Applications of Integration 559

8.1 Arc Length 560

discovery project • Arc Length Contest 567

8.2 Area of a Surface of Revolution 567

discovery project • Rotating on a Slant 575

8.3 Applications to Physics and Engineering 576

discovery project • Complementary Coffee Cups 587

8.4 Applications to Economics and Biology 587

8.5 Probability 592

Review 600

Problems Plus 602

9 Differential Equations 605

9.1 Modeling with Differential Equations 606

9.2 Direction Fields and Euler’s Method 612

9.3 Separable Equations 621

applied project • How Fast Does a Tank Drain? 630

9.4 Models for Population Growth 631

9.5 Linear Equations 641

applied project • Which Is Faster, Going Up or Coming Down? 648

9.6 Predator-Prey Systems 649

Review 656

Problems Plus 659

10 Parametric Equations and Polar Coordinates 661

10.1 Curves Defined by Parametric Equations 662

discovery project • Running Circles Around Circles 672

10.2 Calculus with Parametric Curves 673

discovery project • Bézier Curves 684

10.3 Polar Coordinates 684

discovery project • Families of Polar Curves 694

10.4 Calculus in Polar Coordinates 694

10.5 Conic Sections 702

10.6 Conic Sections in Polar Coordinates 711

Review 719

Problems Plus 722

11 Sequences, Series, and Power Series 723

11.1 Sequences 724

discovery project • Logistic Sequences 738

11.2 Series 738

11.3 The Integral Test and Estimates of Sums 751

11.4 The Comparison Tests 760

11.5 Alternating Series and Absolute Convergence 765

11.6 The Ratio and Root Tests 774

11.7 Strategy for Testing Series 779

11.8 Power Series 781

11.9 Representations of Functions as Power Series 787

11.10 Taylor and Maclaurin Series 795

discovery project • An Elusive Limit 810

writing project • How Newton Discovered the Binomial Series 811

11.11 Applications of Taylor Polynomials 811

applied project • Radiation from the Stars 820

Review 821

Problems Plus 825

12 Vectors and the Geometry of Space 829

12.1 Three-Dimensional Coordinate Systems 830

12.2 Vectors 836

discovery project • The Shape of a Hanging Chain 846

12.3 The Dot Product 847

12.4 The Cross Product 855

discovery project • The Geometry of a Tetrahedron 864

12.5 Equations of Lines and Planes 864

discovery project • Putting 3D in Perspective 874

12.6 Cylinders and Quadric Surfaces 875

Review 883

Problems Plus 887

13 Vector Functions 889

13.1 Vector Functions and Space Curves 890

13.2 Derivatives and Integrals of Vector Functions 898

13.3 Arc Length and Curvature 904

13.4 Motion in Space: Velocity and Acceleration 916

applied project • Kepler’s Laws 925

Review 927

Problems Plus 930

14 Partial Derivatives 933

14.1 Functions of Several Variables 934

14.2 Limits and Continuity 951

14.3 Partial Derivatives 961

discovery project • Deriving the Cobb-Douglas Production Function 973

14.4 Tangent Planes and Linear Approximations 974

applied project • The Speedo LZR Racer 984

14.5 The Chain Rule 985

14.6 Directional Derivatives and the Gradient Vector 994

14.7 Maximum and Minimum Values 1008

discovery project • Quadratic Approximations and Critical Points 1019

14.8 Lagrange Multipliers 1020

applied project • Rocket Science 1028

applied project • Hydro-Turbine Optimization 1030

Review 1031

Problems Plus 1035

15 Multiple Integrals 1037

15.1 Double Integrals over Rectangles 1038

15.2 Double Integrals over General Regions 1051

15.3 Double Integrals in Polar Coordinates 1062

15.4 Applications of Double Integrals 1069

15.5 Surface Area 1079

15.6 Triple Integrals 1082

discovery project • Volumes of Hyperspheres 1095

15.7 Triple Integrals in Cylindrical Coordinates 1095

discovery project • The Intersection of Three Cylinders 1101

15.8 Triple Integrals in Spherical Coordinates 1102

applied project • Roller Derby 1108

15.9 Change of Variables in Multiple Integrals 1109

Review 1117

Problems Plus 1121

16 Vector Calculus 1123

16.1 Vector Fields 1124

16.2 Line Integrals 1131

16.3 The Fundamental Theorem for Line Integrals 1144

16.4 Green’s Theorem 1154

16.5 Curl and Divergence 1161

16.6 Parametric Surfaces and Their Areas 1170

16.7 Surface Integrals 1182

16.8 Stokes’ Theorem 1195

16.9 The Divergence Theorem 1201

16.10 Summary 1208

Review 1209

Problems Plus 1213

Appendixes A1

A Numbers, Inequalities, and Absolute Values A2

B Coordinate Geometry and Lines A10

C Graphs of Second-Degree Equations A16

D Trigonometry A24

E Sigma Notation A36

F Proofs of Theorems A41

G The Logarithm Defined as an Integral A53

H Answers to Odd-Numbered Exercises A61

Index A143