Contents

Preface xi

To the Student xxiii

Diagnostic Tests xxiv

A Preview of Calculus 1

• Functions and Limits  9
  • Four Ways to Represent a Function 10
  • Mathematical Models: A Catalog of Essential Functions 23
  • New Functions from Old Functions 36
  • The Tangent and Velocity Problems 44
  • The Limit of a Function 50
  • Calculating Limits Using the Limit Laws 62
  • The Precise Definition of a Limit 72
  • Continuity 81

Review 93

Principles of Problem Solving 97

• Derivatives  103

2.1 Derivatives and Rates of Change 104

Writing Project – Early Methods for Finding Tangents114

2.2 The Derivative as a Function 114

2.3 Differentiation Formulas 126

Applied Project – Building a Better Roller Coaster140

2.4 Derivatives of Trigonometric Functions 140

2.5 The Chain Rule 148

Applied Project – Where Should a Pilot Start Descent? 156

2.6 Implicit Differentiation 157

Laboratory Project – Families of Implicit Curves 163

2.7 Rates of Change in the Natural and Social Sciences 164

2.8 Related Rates 176

2.9 Linear Approximations and Differentials 183

Laboratory Project – Taylor Polynomials 189

Review 190

Problems Plus 194

• Applications of Differentiation  197

3.1 Maximum and Minimum Values 198

Applied Project- The Calculus of Rainbows 206

3.2 The Mean Value Theorem 208

3.3 How Derivatives Affect the Shape of a Graph 213

3.4 Limits at Infinity; Horizontal Asymptotes 223

3.5 Summary of Curve Sketching 237

3.6 Graphing with Calculus and Calculators 244

3.7 Optimization Problems 250

Applied Project – The Shape of a Can 262

3.8 Newton’s Method 263

3.9 Antiderivatives 269

Review 275

Problems Plus 279

• Integrals 283

4.1  Areas and Distances 284

4.2 The Definite Integral 295

Discovery Project – Area Functions 309

4.3 The Fundamental Theorem of Calculus 310

4.4 Indefinite Integrals and the Net Change Theorem 321

Writing Project –  Newton, Leibniz, and the Invention of Calculus 329

4.5 The Substitution Rule 330

Review 337

Problems Plus 341

• Applications of Integration       343

5.1 Areas Between Curves 344

Applied Project – The Gini Index 351

5.2 Volumes 352

5.3 Volumes by Cylindrical Shells 363

5.4 Work 368

5.5 Average Value of a Function  373

Applied Project – Calculus and Baseball 376

Review 378

Problems Plus 380

• Inverse Functions:     383

Exponential, Logarithmic, and Inverse Trigonometric Functions

6.1 Inverse Functions 384

Instructors may cover either Sections 6.2 –6.4 or Sections 6.2*–6.4*. See the Preface.

6.2 Exponential Functions and Their Derivatives 391 6.2* The Natural Logarithmic Function 429

6.3 Logarithmic Functions 404 6.3* The Natural Exponential Function 429

6.4 Derivatives of Logarithmic Functions 410 6.4* General Logarithmic and Exponential Functions 437

6.5 Exponential Growth and Decay 446

6.6 Inverse Trigonometric Functions 453

Applied Project –  Where to Sit at the Movies 461

6.7 Hyperbolic Functions 462

6.8 Indeterminate Forms and l’Hospital’s Rule 469

Writing Project – The Origins of l’Hospital’s Rule 480

Review 480

Problems Plus 485

• Techniques of Integration:       487

7.1Integration by Parts 488

7.2 Trigonometric Integrals 495

7.3 Trigonometric Substitution 502

7.4 Integration of Rational Functions by Partial Fractions 508

7.5 Strategy for Integration 518

7.6 Integration Using Tables and Computer Algebra Systems 524

Discovery Project –  Patterns in Integrals 529

7.7 Approximate Integration 530

7.8 Improper Integrals 543

Review 553

Problems Plus 557

• Further Applications of Integration        561

8.1 Arc Length 562

Discovery Project – Arc Length Contest 569

8.2 Area of a Surface of Revolution 569

Discovery Project – Rotating on a Slant 575

8.3 Applications to Physics and Engineering 576

Discovery Project –  Complementary Coffee Cups 586

8.4  Applications to Economics and Biology 587

8.5 Probability 592

Review 599

Problems Plus 601

• Differential Equations        603

9.1 Modeling with Differential Equations 604

9.2 Direction Fields and Euler’s Method 609

9.3 Separable Equations 618

Applied Project – How Fast Does a Tank Drain? 627

Applied Project – Which Is Faster, Going Up or Coming Down? 628

9.4 Models for Population Growth 629

9.5 Linear Equations 640

9.6 Predator-Prey Systems 646

Review 653

Problems Plus 657

• Parametric Equations and Polar Coordinates        659

10.1 Curves Defined by Parametric Equations 660

Laboratory Project – Running Circles around Circles 668

10.2 Calculus with Parametric Curves 669

Laboratory Project – Bézier Curves 677

10.3 Polar Coordinates 678

Laboratory Project – Families of Polar Curves 688

10.4 Areas and Lengths in Polar Coordinates 689

10.5  Conic Sections 694

10.6 Conic Sections in Polar Coordinates 702

Review 709

Problems Plus 712

11- Infinite Sequences and Series      713

11.1 Sequences 714

Laboratory Project – Logistic Sequences 727

11.2 Series 727

11.3 The Integral Test and Estimates of Sums 738

11.4 The Comparison Tests 746

11.5 Alternating Series 751

11.6 Absolute Convergence and the Ratio and Root Tests 756

11.7 Strategy for Testing Series 763

11.8 Power Series 765

11.9 Representations of Functions as Power Series 770

11.10 Taylor and Maclaurin Series 777

Laboratory Project – An Elusive Limit 791

Writing Project – How Newton Discovered the Binomial Series 791

11.11 Applications of Taylor Polynomials 792

Applied Project – Radiation from the Stars801

Review 802

Problems Plus 805

12- Vectors and the Geometry of Space        809

12.1 Three-Dimensional Coordinate Systems 810

12.2 Vectors 815

12.3 The Dot Product 824

12.4 The Cross Product 832

Discovery Project – The Geometry of a Tetrahedron 840

12.5 Equations of Lines and Planes 840

Laboratory Project -Putting 3D in Perspective 850

12.6 Cylinders and Quadric Surfaces 851

Review 858

Problems Plus 861

13-  Vector Functions      863

13.1 Vector Functions and Space Curves 864

13.2 Derivatives and Integrals of Vector Functions 871

13.3 Arc Length and Curvature 877

13.4 Motion in Space: Velocity and Acceleration 886

Applied Project – Kepler’s Laws 896

Review 897

Problems Plus 900

14- Partial Derivatives        901

14.1 Functions of Several Variables 902

14.2 Limits and Continuity 916

14.3 Partial Derivatives 924

14.4 Tangent Planes and Linear Approximations 939

14.5 The Chain Rule 948

14.6 Directional Derivatives and the Gradient Vector 957

14.7 Maximum and Minimum Values 970

Applied Project – Designing a Dumpster 980

Discovery Project – Quadratic Approximations and Critical Points 980

14.8 Lagrange Multipliers 981

Applied Project – Rocket Science 988

Applied Project – Hydro-Turbine Optimization 990

Review 991

Problems Plus 995

15- Multiple Integrals  997

15.1 Double Integrals over Rectangles 998

15.2 Iterated Integrals 1006

15.3 Double Integrals over General Regions 1012

15.4 Double Integrals in Polar Coordinates 1021

15.5 Applications of Double Integrals 1027

15.6 Surface Area 1037

15.7 Triple Integrals 1041

Discovery Project – Volumes of Hyperspheres 1051

15.8 Triple Integrals in Cylindrical Coordinates 1051

Discovery Project – The Intersection of Three Cylinders 1056

15.9 Triple Integrals in Spherical Coordinates 1057

Applied Project- Roller Derby 1063

15.10 Change of Variables in Multiple Integrals 1064

Review 1073

Problems Plus 1077

16 Vector Calculus 1079

16.1 Vector Fields 1080

16.2 Line Integrals 1087

16.3 The Fundamental Theorem for Line Integrals 1099

16.4 Green’s Theorem 1108

16.5 Curl and Divergence 1115

16.6 Parametric Surfaces and Their Areas 1123

16.7 Surface Integrals 1134

16.8 Stokes’ Theorem 1146

Writing Project – Three Men and Two Theorems 1152

16.9 The Divergence Theorem 1152

16.10 Summary 1159

Review 1160

Problems Plus 1163

17- Second-Order Differential Equations        1165

17.1 Second-Order Linear Equations 1166

17.2 Nonhomogeneous Linear Equations 1172

17.3 Applications of Second-Order Differential Equations 1180

17.4 Series Solutions 1188

Review 1193

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  A34

F Proofs of Theorems A39

G Graphing Calculators and Computers A48

H Complex Numbers A55

I Answers to Odd-Numbered Exercises A63

Index        A135