微积分 第2版 第2卷PDF电子书下载

A Preview of Calculus 2

1 Functions and Models 10

1.1 Four Ways to Represent a Function 11

1.2 Mathematical Models 24

1.3 New Functions from Old Functions 38

1.4 Graphing Calculators and Computers 49

1.5 Exponential Functions 56

1.6 Inverse Functions and Logarithms 64

1.7 Parametric Curves 75

Laboratory Project Running Circles around Circles 83

Review 84

Principles of Problem Solving 88

2 Limits and Derivatives 94

2.1 The Tangent and Velocity Problems 95

2.2 The Limit of a Function 100

2.3 Calculating Limits Using the Limit Laws 110

2.4 Continuity 119

2.5 Limits Involving Infinity 130

2.6 Tangents，Velocities，and Other Rates of Change 142

2.7 Derivatives 150

Writing Project Early Methods for Finding Tangents 157

2.8 The Derivative as a Function 157

2.9 Linear Approximations 171

2.10 What Does f’ Say about f？ 175

Review 181

Focus on Problem Solving 185

3 Differentiation Rules 188

3.1 Derivatives of Polynomials and Exponential Functions 189

Applied Project Building a Better Roller Coaster 198

3.2 The Product and Quotient Rules 199

3.3 Rates of Change in the Natural and Social Sciences 206

3.4 Derivatives of Trigonometric Functions 218

3.5 The Chain Rule 225

Laboratory Project Bezier Curves 236

Applied Project Where Should a Pilot Start Descent？ 237

3.6 Implicit Differentiation 237

3.7 Derivatives of Logarithmic Functions 245

Discovery Project Hyperbolic Functions 251

3.8 Linear Approximations and Differentials 252

Laboratory Project Taylor Polynomials 257

Review 258

Focus on Problem Solving 261

4Applications of Differentiation 264

4.1 Related Rates 265

4.2 Maximum and Minimum Values 271

Applied Project The Calculus of Rainbows 279

4.3 Derivatives and the Shapes of Curves 280

4.4 Graphing with Calculus and Calculators 291

4.5 Indeterminate Forms and l’Hospital’s Rule 298

Writing Project The Origins of l’Hospital’s Rule 307

4.6 Optimization Problems 307

Applied Project The Shape of a Can 318

4.7 Applications to Economics 319

4.8 Newton’s Method 324

4.9 Antiderivatives 329

Review 336

Focus on Problem Solving 340

5 Integrals 344

5.1 Areas and Distances 345

5.2 The Definite Integral 357

5.3 Evaluating Definite Integrals 369

Discovery Project Area Functions 379

5.4 The Fundamental Theorem of Calculus 380

Writing Project Newton，Leibniz，and the Invention of Calculus 388

5.5 The Substitution Rule 389

5.6 Integration by Parts 396

5.7 Additional Techniques of Integration 403

5.8 Integration Using Tables and Computer Algebra Systems 409

Discovery Project Patterns in Integrals 415

5.9 Approximate Integration 416

5.10 Improper Integrals 428

Review 438

Focus on Problem Solving 442

6 Applications of Integration 446

6.1 More about Areas 447

6.2 Volumes 453

Discovery Project Rotating on a Slant 466

6.3 Arc Length 467

Discovery Project Arc Length Contest 472

6.4 Average Value of a Function 473

Applied Project Where to Sit at the Movies 476

6.5 Applications to Physics and Engineering 476

6.6 Applications to Economics and Biology 487

6.7 Probability 492

Review 499

Focus on Problem Solving 502

7 Differential Equations 506

7.1 Modeling with Differential Equations 507

7.2 Direction Fields and Euler’s Method 512

7.3 Separable Equations 522

Applied Project Which Is Faster，Going Up or Coming Down？ 530

7.4 Exponential Growth and Decay 531

Applied Project Calculus and Baseball 540

7.5 The Logistic Equation 541

7.6 Predator-Prey Systems 550

Review 557

Focus on Problem Solving 560

8 Infinite Sequences and Series 562

8.1 Sequences 563

Laboratory Project Logistic Sequences 573

8.2 Series 573

8.3 The Integral and Comparison Tests； Estimating Sums 583

8.4 Other Convergence Tests 592

8.5 Power Series 600

8.6 Representations of Functions as Power Series 605

8.7 Taylor and Maclaurin Series 611

8.8 The Binomial Series 622

Writing Project How Newton Discovered the Binomial Series 626

8.9 Applications of Taylor Polynomials 626

Applied Project Radiation from the Stars 634

8.10 Using Series to Solve Differential Equations 635

Review 640

Focus on Problem Solving 643

9 Vectors and the Geometry of Space 646

9.1 Three-Dimensional Coordinate Systems 647

9.2 Vectors 652

9.3 The Dot Product 661

9.4 The Cross Product 667

Discovery Project The Geometry of a Tetrahedron 675

9.5 Equations of Lines and Planes 676

9.6 Functions and Surfaces 685

9.7 Cylindrical and Spherical Coordinates 694

Laboratory Project Families of Surfaces 699

Review 700

Focus on Problem Solving 703

10 Vector Functions 704

10.1 Vector Functions and Space Curves 705

10.2 Derivatives and Integrals of Vector Functions 711

10.3 Arc Length and Curvature 717

10.4 Motion in Space 725

Applied Project Kepler’s Laws 735

10.5 Parametric Surfaces 736

Review 742

Focus on Problem Solving 745

11 Partial Derivatives 748

11.1 Functions of Several Variables 749

11.2 Limits and Continuity 760

11.3 Partial Derivatives 766

11.4 Tangent Planes and Linear Approximations 779

11.5 The Chain Rule 790

11.6 Directional Derivatives and the Gradient Vector 798

11.7 Maximum and Minimum Values 811

Applied Project Designing a Dumpster 820

Discovery Project Quadratic Approximations and Critical Points 821

11.8 Lagrange Multipliers 822

Applied Project Rocket Science 829

Applied Project Hydro-Turbine Optimization 830

Review 831

Focus on Problem Solving 836

12Multiple Integrals 838

12.1 Double Integrals over Rectangles 839

12.2 Iterated Integrals 849

12.3 Double Integrals over General Regions 854

12.4 Double Integrals in Polar Coordinates 863

12.5 Applications of Double Integrals 868

12.6 Surface Area 878

12.7 Triple Integrals 883

Discovery Project Volumes of Hyperspheres 893

12.8 Triple Integrals in Cylindrical and Spherical Coordinates 893

Applied Project Roller Derby 900

Discovery Project The Intersection of Three Cylinders 901

12.9 Change of Variables in Multiple Integrals 901

Review 910

Focus on Problem Solving 914

13 Vector Calculus 916

13.1 Vector Fields 917

13.2 Line Integrals 924

13.3 The Fundamental Theorem for Line Integrals 936

13.4 Green’s Theorem 945

13.5 Curl and Divergence 952

13.6 Surface Integrals 960

13.7 Stokes’ Theorem 971

Writing Project Three Men and Two Theorems 977

13.8 The Divergence Theorem 978

13.9 Summary 985

Review 986

Focus on Problem Solving 989