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A Preview of Calculus 2

1 Functions and Models 10

1.1Four Ways to Represent a Function 11

1.2Mathematical Models：A Catalog of Essential Functions 25

1.3New Functions from Old Functions 38

1.4Graphing Calculators and Computers 48

1.5Exponential Functions 55

1.6Inverse Functions and Logarithms 63

Review 77

Principles of Problem Solving 80

2 Limits and Derivatives 86

2.1The Tangent and Velocity Problems 87

2.2The Limit of a Function 92

2.3Calculating Limits Using the Limit Laws 104

2.4The Precise Definition of a Limit 114

2.5Continuity 124

2.6Limits at Infinity；Horizontal Asymptotes 135

2.7Tangents，Velocities，and Other Rates of Change 149

2.8Derivatives 158

Writing Project。Early Methods for Finding Tangents 164

2.9The Derivative as a Function 165

Review 176

Problems Plus 180

3 Differentiation Aules 182

3.1Derivatives of Polynomials and Exponential Functions 183

3.2The Product and Quotient Rules 192

3.3Rates of Change in the Natural and Social Sciences 199

3.4Derivatives of Trigonometric Functions 211

3.5The Chain Rule 217

3.6Implicit Differentiation 227

3.7Higher Derivatives 236

Applied Project。Where Should a Pilot Start Descent? 243

Applied Project。Building a Better Roller Goaster 243

3.8Derivatives of Logarithmic Functions 244

3.9Hyperbolic Functions 250

3.10Related Rates 256

3.11Linear Approximations and Differentials 262

Laboratory Project。Taylor Polynomials 269

Review 270

Problems Plus 274

4 Applications of Differentiation 278

4.1Maximum and Minimum Values 279

Applied Project。The Calculus of Rainbows 288

4.2The Mean Value Theorem 290

4.3How Derivatives Affect the Shape of a Graph 296

4.4Indeterminate Forms and L’Hospital’s Rule 307

Writing Project。The Origins of L’Hospital’s Rule 315

4.5Summary of Curve Sketching 316

4.6Graphing 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

4.10Antiderivatives 353

Review 361

Problems PIus 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 Galculus 413

5.5 The Substitution Rule 414

5.6 The Logarithm Defined as an Integral 422

Review 430

Problems Plus 434

6 Applications of lntegration 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

7.1Integration by Parts 475

7.2Trigonometric Integrals 482

7.3Trigonometric Substitution 489

7.4Integration of Rational Functions by Partial Fractions 496

7.5Strategy for Integration 505

7.6Integration Using Tables and Computer Algebra Systems 511

Discovery Project 。 Patterns in Integrals 517

7.7Approximate Integration 518

7.8Improper Integrals 530

Review 540

Problems Plus 543

8 Further Applications of lntegration 546

8.1Arc Length 547

Discovery Project。Arc Length Contest 554

8.2Area of a Surface of Revolution 554

Discovery Project 。 Rotating on a 51ant 560

8.3Applications to Physics and Engineering 561

8.4Applications to Economics and Biology 571

8.5Probability 575

Review 582

Problems Plus 584

9 Differential Equations 586

9.1Modeling with Differential Equations 587

9.2Direction Fields and Euler’s Method 592

9.3Separable Equations 601

Applied Project。How Fast Does a Tank Drain? 609

Applied Project。Which Is Faster，Going Up or Coming Down? 610

9.4Exponential Growth and Decay 611

Applied Project。Calculus and Baseball 622

9.5The Logistic Equation 623

9.6Linear Equations 632

9.7Predator-Prey Systems 638

Review 644

Problems Plus 648

10 Parametric Equations and Poiar Coordinates 650

10.1Curves Defined by Parametric Equations 651

Laboratory Project。Running Circles around Circles 659

10.2Calculus with Parametric Curves 660

Laboratory Project。5ezier Curves 669

10.3Polar Coordinates 669

10.4Areas and Lengths in Polar Coordinates 679

10.5Conic Sections 684

10.6Conic Sections in Polar Coordinates 692

Review 696

Problems Plus 699

11 Infinite Sequences and Series 700

11.1Sequences 701

Laboratory Project。Logistic Sequences 713

11.2Series 713

11.3The Integral Test and Estimates of Sums 723

11.4The Comparison Tests 730

11.5Alternating Series 735

11.6Absolute Convergence and the Ratio and Root Tests 740

11.7Strategy for Testing Series 747

11.8Power Series 749

11.9Representations of Functions as Power Series 754

11.10Taylor and Maclaurin Series 760

Laboratory Project。An Elusive Limit 772

11.11The Binomial Series 772

Writing Project。How Newton Discovered the Binomial 5eriee 776

11.12Applications of Taylor Polynomials 776

Applied Project。Radiation from the Stars 785

Review 786

Problems Plue 789

12 Vectors and the Geometrq of Space 792

12.1Three-Dimensional Coordinate Systems 793

12.2Vectors 798

12.3The Dot Product 807

12.4The Cross Product 814

Discovery Project。The Geometry of a Tetrahedron 822

12.5Equations of Lines and Planes 822

Laboratory Project。Putting 3D in Perspective 832

12.6Cylinders and Quadric Surfaces 832

12.7Cylindrical and Spherical Coordinates 839

Laboratory Project。Families of Surfaces 844

Review 844

Problems Plus 847

13 Vector Functions 848

13.1Vector Functions and Space Curves 849

13.2Derivatives and Integrals of Vector Functions 856

13.3Arc Length and Curvature 862

13.4Motion in Space：Velocity and Acceleration 870

Applied Project。Kepler’s Laws 880

Review 881

Problems Plus 884

14 Partial DeriVatiVeS 886

14.1Functions of Several Variables 887

14.2Limits and Continuity 902

14.3Partial Derivatives 909

14.4Tangent Planes and Linear Approximations 923

14.5The Chain Rule 931

14.6Directional Derivatives and the Gradient Vector 940

14.7Maximum and Minimum Values 953

Applied Project。Designing a Dumpster 963

Discovery Project。Quadratic Approximations and Critical Points 964

14.8Lagrange Multipliers 965

Applied ProjectRocket Science 972

Applied ProjectHydro-Turbine Optimization 973

Review 974

Problems Plus 978

15 Multiple Integrals 980

15.1Double Integrals over Rectangles 981

15.2Iterated Integrals 989

15.3Double Integrals over General Regions 995

15.4Double Integrals in Polar Coordinates 1003

15.5Applications of Double Integrals 1009

15.6Surface Area 1019

15.7Triple Integrals 1023

Discovery Project。Volumes of Hyperspheres 1032

15.8Triple Integrals in Cylindrical and Spherical Coordinates 1033

Applied Project。Roller Derby 1039

Discovery Project。The Intersection of Three Cylinders 1040

15.9Change of Variables in Multiple Integrals 1041

Review 1049

Problems Plus 1052

16VeCtor CalCUlUS 1054

16.1Vector Fields 1055

16.2Line Integrals 1062

16.3The Fundamental Theorem for Line Integrals 1074

16.4Green’s Theorem 1083

16.5Curl and Divergence 1090

16.6Parametric Surfaces and Their Areas 1098

16.7Surface Integrals 1109

16.8Stokes’Theorem 1121

Writing Project。Three Men and Two Theorems 1126

16.9The Divergence Theorem 1127

16.10Summary 1134

Review 1135

Problems Plus 1138

17 Second-Drder Differential Equations 1140

17.1Second-Order Linear Equations 1141

17.2Nonhomogeneous Linear Equations 1147

17.3Applications of Second-Order Differential Equations 1155

17.4Series Solutions 1163

Review 1167

AppendixesAl 2

ANumbers，Inequalities，and Absolute ValuesA 2

BCoordinate Geometry and LinesA 10

CGraphs of Second-Degree EquationsA 16

DTrigonometryA 24

ESigma NotationA 34

FProofs of TheoremsA 39

GComplex NumbersA 49

HAnswers to Odd-Numbered ExercisesA 57

IndexA 125