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A Preview of Calculus2

1 Functions and Models10

1.1 Four Ways to Represent a Function11

1.2 Mathematical Models24

1.3 New Functions from Old Functions38

1.4 Graphing Calculators and Computers49

1.5 Exponential Functions56

1.6 Inverse Functions and Logarithms64

1.7 Parametric Curves75

Laboratory Project Running Circles around Circles83

Review84

Principles of Problem Solving88

2 Limits and Derivatives94

2.1 The Tangent and Velocity Problems95

2.2 The Limit of a Function100

2.3 Calculating Limits Using the Limit Laws110

2.4 Continuity119

2.5 Limits Involving Infinity130

2.6 Tangents，Velocities，and Other Rates of Change142

2.7 Derivatives150

Writing Project Early Methods for Finding Tangents157

2.8 The Derivative as a Function157

2.9 Linear Approximations171

2.10 What Does f’ Say about f？175

Review181

Focus on Problem Solving185

3 Differentiation Rules188

3.1 Derivatives of Polynomials and Exponential Functions189

Applied Project Building a Better Roller Coaster198

3.2 The Product and Quotient Rules199

3.3 Rates of Change in the Natural and Social Sciences206

3.4 Derivatives of Trigonometric Functions218

3.5 The Chain Rule225

Laboratory Project Bezier Curves236

Applied Project Where Should a Pilot Start Descent？237

3.6 Implicit Differentiation237

3.7 Derivatives of Logarithmic Functions245

Discovery Project Hyperbolic Functions251

3.8 Linear Approximations and Differentials252

Laboratory Project Taylor Polynomials257

Review258

Focus on Problem Solving261

4Applications of Differentiation264

4.1 Related Rates265

4.2 Maximum and Minimum Values271

Applied Project The Calculus of Rainbows279

4.3 Derivatives and the Shapes of Curves280

4.4 Graphing with Calculus and Calculators291

4.5 Indeterminate Forms and l’Hospital’s Rule298

Writing Project The Origins of l’Hospital’s Rule307

4.6 Optimization Problems307

Applied Project The Shape of a Can318

4.7 Applications to Economics319

4.8 Newton’s Method324

4.9 Antiderivatives329

Review336

Focus on Problem Solving340

5 Integrals344

5.1 Areas and Distances345

5.2 The Definite Integral357

5.3 Evaluating Definite Integrals369

Discovery Project Area Functions379

5.4 The Fundamental Theorem of Calculus380

Writing Project Newton，Leibniz，and the Invention of Calculus388

5.5 The Substitution Rule389

5.6 Integration by Parts396

5.7 Additional Techniques of Integration403

5.8 Integration Using Tables and Computer Algebra Systems409

Discovery Project Patterns in Integrals415

5.9 Approximate Integration416

5.10 Improper Integrals428

Review438

Focus on Problem Solving442

6 Applications of Integration446

6.1 More about Areas447

6.2 Volumes453

Discovery Project Rotating on a Slant466

6.3 Arc Length467

Discovery Project Arc Length Contest472

6.4 Average Value of a Function473

Applied Project Where to Sit at the Movies476

6.5 Applications to Physics and Engineering476

6.6 Applications to Economics and Biology487

6.7 Probability492

Review499

Focus on Problem Solving502

7 Differential Equations506

7.1 Modeling with Differential Equations507

7.2 Direction Fields and Euler’s Method512

7.3 Separable Equations522

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

7.4 Exponential Growth and Decay531

Applied Project Calculus and Baseball540

7.5 The Logistic Equation541

7.6 Predator-Prey Systems550

Review557

Focus on Problem Solving560

8 Infinite Sequences and Series562

8.1 Sequences563

Laboratory Project Logistic Sequences573

8.2 Series573

8.3 The Integral and Comparison Tests； Estimating Sums583

8.4 Other Convergence Tests592

8.5 Power Series600

8.6 Representations of Functions as Power Series605

8.7 Taylor and Maclaurin Series611

8.8 The Binomial Series622

Writing Project How Newton Discovered the Binomial Series626

8.9 Applications of Taylor Polynomials626

Applied Project Radiation from the Stars634

8.10 Using Series to Solve Differential Equations635

Review640

Focus on Problem Solving643

9 Vectors and the Geometry of Space646

9.1 Three-Dimensional Coordinate Systems647

9.2 Vectors652

9.3 The Dot Product661

9.4 The Cross Product667

Discovery Project The Geometry of a Tetrahedron675

9.5 Equations of Lines and Planes676

9.6 Functions and Surfaces685

9.7 Cylindrical and Spherical Coordinates694

Laboratory Project Families of Surfaces699

Review700

Focus on Problem Solving703

10 Vector Functions704

10.1 Vector Functions and Space Curves705

10.2 Derivatives and Integrals of Vector Functions711

10.3 Arc Length and Curvature717

10.4 Motion in Space725

Applied Project Kepler’s Laws735

10.5 Parametric Surfaces736

Review742

Focus on Problem Solving745

11 Partial Derivatives748

11.1 Functions of Several Variables749

11.2 Limits and Continuity760

11.3 Partial Derivatives766

11.4 Tangent Planes and Linear Approximations779

11.5 The Chain Rule790

11.6 Directional Derivatives and the Gradient Vector798

11.7 Maximum and Minimum Values811

Applied Project Designing a Dumpster820

Discovery Project Quadratic Approximations and Critical Points821

11.8 Lagrange Multipliers822

Applied Project Rocket Science829

Applied Project Hydro-Turbine Optimization830

Review831

Focus on Problem Solving836

12Multiple Integrals838

12.1 Double Integrals over Rectangles839

12.2 Iterated Integrals849

12.3 Double Integrals over General Regions854

12.4 Double Integrals in Polar Coordinates863

12.5 Applications of Double Integrals868

12.6 Surface Area878

12.7 Triple Integrals883

Discovery Project Volumes of Hyperspheres893

12.8 Triple Integrals in Cylindrical and Spherical Coordinates893

Applied Project Roller Derby900

Discovery Project The Intersection of Three Cylinders901

12.9 Change of Variables in Multiple Integrals901

Review910

Focus on Problem Solving914

13 Vector Calculus916

13.1 Vector Fields917

13.2 Line Integrals924

13.3 The Fundamental Theorem for Line Integrals936

13.4 Green’s Theorem945

13.5 Curl and Divergence952

13.6 Surface Integrals960

13.7 Stokes’ Theorem971

Writing Project Three Men and Two Theorems977

13.8 The Divergence Theorem978

13.9 Summary985

Review986

Focus on Problem Solving989