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A Preview of Calculus 2

1 Functions and Models 10

1.1 Four Ways to Represent a Function 11

1.2 Mathematical Models:A Catalog of Essential Functions 25

1.3 New Functions from Old Functions 38

1.4 Graphing Calculators and Computers 48

1.5 Exponential Functions 55

1.6 Inverse Functions and Logarithms 63

Review 77

Principles of Problem Solving 80

2 Limits and Derivatives 86

2.1 The Tangent and Velocity Problems 87

2.2 The Limit of a Function 92

2.3 Calculating Limits Using the Limit Laws 104

2.4 The Precise Definition of a Limit 114

2.5 Continuity 124

2.6 Limits at Infinity;Horizontal Asymptotes 135

2.7 Tangents,Velocities,and Other Rates of Change 149

2.8 Derivatives 158

Writing Project.Early Methods for Finding Tangents 164

2.9 The Derivative as a Function 165

Review 176

Problems Plus 180

3 Differentiation Rules 182

3.1 Derivatives of Polynomials and Exponential Functions 183

3.2 The Product and Quotient Rules 192

3.3 Rates of Change in the Natural and Social Sciences 199

3.4 Derivatives of Trigonometric Functions 211

3.5 The Chain Rule 217

3.6 Implicit Differentiation 227

3.7 Higher Derivatives 236

Applied Project.Where Should a Pilot Start Descent? 243

Applied Project.Building a Better Roller Coaster 243

3.8 Derivatives of Logarithmic Functions 244

3.9 Hyperbolic Functions 250

3.10 Related Rates 256

3.11 Linear Approximations and Differentials 262

Laboratory Project.Taylor Polynomials 269

Review 270

Problems Plus 274

4 Rpplications of Oifferentiation 278

4.1 Maximum and Minimum Values 279

Applied Project.The Calculus of Rainbows 288

4.2 The Mean Value Theorem 290

4.3 How Derivatives Affect the Shape of a Graph 296

4.4 Indeterminate Forms and L'Hospital's Rule 307

Writing Project.The Origins of L'Hospital's Rule 315

4.5 Summary of Curve Sketching 316

4.6 Graphing with Calculus and Calculators 324

4.7 Optimization Problems 331

Applied Project.The Shape of a Can 341

4.8 Applications to Business and Economics 342

4.9 Newton's Method 347

4.10 Antiderivatives 353

Review 361

Problems Plus 365

5 Integrals 368

5.1 Areas and Distances 369

5.2 The Definite Integral 380

Discovery Project.Area Functions 393

5.3 The Fundamental Theorem of Calculus 394

5.4 Indefinite Integrals and the Net Change Theorem 405

Writing Project.Newton,Leibniz,and the Invention of Calculus 413

5.5 The Substitution Rule 414

5.6 The Logarithm Defined as an Integral 422

Review 430

Problems Plus 434

6 Rpplications of Integration 436

6.1 Areas between Curves 437

6.2 Volumes 444

6.3 Volumes by Cylindrical Shells 455

6.4 Work 460

6.5 Average Value of a Function 464

Applied Project.Where to Sit at the Movies 468

Review 468

Problems Plus 470

7 Techniques of Integration 474

7.1 Integration by Parts 475

7.2 Trigonometric Integrals 482

7.3 Trigonometric Substitution 489

7.4 Integration of Rational Functions by Partial Fractions 496

7.5 Strategy for Integration 505

7.6 Integration Using Tables and Computer Algebra Systems 511

Discovery Project.Patterns in Integrals 517

7.7 Approximate Integration 518

7.8 Improper Integrals 530

Review 540

Problems Plus 543

8 Further Rpplications of Integration 546

8.1 Arc Length 547

Discovery Project.Arc Length Contest 554

8.2 Area of a Surface of Revolution 554

Discovery Project.Rotating on a Slant 560

8.3 Applications to Physics and Engineering 561

8.4 Applications to Economics and Biology 571

8.5 Probability 575

Review 582

Problems Plus 584

9 Differential Equations 586

9.1 Modeling with Differential Equations 587

9.2 Direction Fields and Euler's Method 592

9.3 Separable Equations 601

Applied Project.How Fast Does a Tank Drain? 609

Applied Project.Which Is Fasteer,Going Up or Coming Down? 610

9.4 Exponential Growth and Decay 611

Applied Project.Calculus and Baseball 622

9.5 The Logistic Equation 623

9.6 Linear Equations 632

9.7 Predator-Prey Systems 638

Review 644

Problems Plus 648

10 Parametric Equations and Polar Coordinates 650

10.1 Curves Defined by Parametric Equations 651

Laboratory Project.Running Circles around Circles 659

10.2 Calculus with Parametric Curves 660

Laboratory Project.Bézier Curves 669

10.3 Polar Coordinates 669

10.4 Areas and Lengths in Polar Coordinates 679

10.5 Conic Sections 684

10.6 Conic Sections in Polar Coordinates 692

Review 696

Problems Plus 699

11 Infinite Sequences and Series 700

11.1 Sequences 701

Laboratory Project.Logistic Sequences 713

11.2 Series 713

11.3 The Integral Test and Estimates of Sums 723

11.4 The Comparison Tests 730

11.5 Alternating Series 735

11.6 Absolute Convergence and the Ratio and Root Tests 740

11.7 Strategy for Testing Series 747

11.8 Power Series 749

11.9 Representations of Functions as Power Series 754

11.10 Taylor and Maclaurin Series 760

Laboratory Project.An Elusive Limit 772

11.11 The Binomial Series 772

Writing Project.How Newton Discovered the Binomial Series 776

11.12 Applications of Taylor Polynomials 776

Applied Project.Radiation from the Stars 785

Review 786

Problems Plus 789

12 Vectors and the Geometru of Space 792

12.1 Three-Dimensional Coordinate Systems 793

12.2 Vectors 798

12.3 The Dot Product 807

12.4 The Cross Product 814

Discovery Project.The Geometry of a Tetrahedron 822

12.5 Equations of Lines and Planes 822

Laboratory Project.Putting 3D in Perspective 832

12.6 Cylinders and Quadric Surfaces 832

12.7 Cylindrical and Spherical Coordinates 839

Laboratory Project.Families of Surfaces 844

Review 844

Problems Plus 847

13 Vector Functions 848

13.1 Vector Functions and Space Curves 849

13.2 Derivatives and Integrals of Vector Functions 856

13.3 Arc Length and Curvature 862

13.4 Motion in Space:Velocity and Acceleration 870

Applied Project.Kepler's Laws 880

Review 881

Problems Plus 884

14 Partial Derivatives 886

14.1 Functions of Several Variables 887

14.2 Limits and Continuity 902

14.3 Partial Derivatives 909

14.4 Tangent Planes and Linear Approximations 923

14.5 The Chain Rule 931

14.6 Directional Derivatives and the Gradient Vector 940

14.7 Maximum and Minimum Values 953

Applied Project.Designing a Dumpster 963

Discovery Project.Quadratic Approximations and Critical Points 964

14.8 Lagrange Multipliers 965

Applied Project.Rocket Science 972

Applied Project.Hydro-Turbine Optimization 973

Review 974

Problems Plus 978

15 Multiple Integrals 980

15.1 Double Integrals over Rectangles 981

15.2 Iterated Integrals 989

15.3 Double Integrals over General Regions 995

15.4 Double Integrals in Polar Coordinates 1003

15.5 Applications of Double Integrals 1009

15.6 Surface Area 1019

15.7 Triple Integrals 1023

Discovery Project.Volumes of Hyperspheres 1032

15.8 Triple Integrals in Cylindrical and Spherical Coordinates 1033

Applied Project.Roller Derby 1039

Discovery Project.The Intersection of Three Cylinders 1040

15.9 Change of Variables in Multiple Integrals 1041

Review 1049

Problems Plus 1052

16 Vector Calculus 1054

16.1 Vector Fields 1055

16.2 Line Integrals 1062

16.3 The Fundamental Theorem for Line Integrals 1074

16.4 Green's Theorem 1083

16.5 Curl and Divergence 1090

16.6 Parametric Surfaces and Their Areas 1098

16.7 Surface Integrals 1109

16.8 Stokes'Theorem 1121

Writing Project.Three Men and Two Theorems 1126

16.9 The Divergence Theorem 1127

16.10 Summary 1134

Review 1135

Problems Plus 1138

17 Second-Order Differential Equations 1140

17.1 Second-Order Linear Equations 1141

17.2 Nonhomogeneous Linear Equations 1147

17.3 Applications of Second-Order Differential Equations 1155

17.4 Series Solutions 1163

Review 1167

Appendixes R 1

A Numbers,Inequalities,and Absolute Values A 2

B Coordinate Geometry and Lines A 10

C Graphs of Second-Degree Equations A 16

D Trigonometry A 24

E Sigma Notation A 34

F Proofs of Theorems A 39

G Complex Numbers A 49

H Answers to Odd-Numbered Exercises A 57

Index A 125