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1 Functions and Models10

1.1 Four Ways to Represent a Function11

1.2 Mathematical Models: A Catalog of Essential Functions25

1.3 New Functions from Old Functions38

1.4 Graphing Calculators and Computers48

1.5 Exponential Functions55

1.6 Inverse Functions and Logarithms63

Review77

Principles of Problem Solving80

2 Limits and Derivatives86

2.1 The Tangent and Velocity Problems87

2.2 The Limit of a Function92

2.3 Calculating Limits Using the Limit Laws104

2.4 The Precise Definition of a Limit114

2.5 Continuity124

2.6 Limits at Infinity; Horizontal Asymptotes135

2.7 Tangents, Velocities, and Other Rates of Change149

2.8 Derivatives158

Writing Project · Early Methods for Finding Tangents164

2.9 The Derivative as a Function165

Review176

Problems Plus180

3 Differentiation Rules182

3.1 Derivatives of Polynomials and Exponential Functions183

3.2 The Product and Quotient Rules192

3.3 Rates of Change in the Natural and Social Sciences199

3.4 Derivatives of Trigonometric Functions211

3.5 The Chain Rule217

3.6 Implicit Differentiation227

3.7 Higher Derivatives236

Applied Project · Where Should a Pilot Start Descent?243

Applied Project · Building a Better Roller Coaster243

3.8 Derivatives of Logarithmic Functions244

3.9 Hyperbolic Functions250

3.10 Related Rates256

3.11 Linear Approximations and Differentials262

Laboratory Project · Taylor Polynomials269

Review270

Problems Plus274

4 AppIicatiOnS of Differentiation278

4.1 Maximum and Minimum Values279

Applied Project · The Calculus of Rainbows288

4.2 The Mean Value Theorem290

4.3 How Derivatives Affect the Shape of a Graph296

4.4 Indeterminate Forms and L'Hospital's Rule307

Writing Project · The Origins of L'Hospital's Rule315

4.5 Summary of Curve Sketching316

4.6 Graphing with Calculus and Calculators324

4.7 Optimization Problems331

Applied Project · The Shape of a Can341

4.8 Applications to Business and Economics342

4.9 Newton's Method347

4.10 Antiderivatives353

Review361

Problems Plus365

5 Integrals368

5.1 Areas and Distances369

5.2 The Definite Integral380

Discovery Project · Area Functions393

5.3 The Fundamental Theorem of Calculus394

5.4 Indefinite Integrals and the Net Change Theorem405

Writing Project · Newton, Leibniz, and the Invention of Calculus413

5.5 The Substitution Rule414

5.6 The Logarithm Defined as an Integral422

Review430

Problems Plus434

6 Applications of Integration436

6.1 Areas between Curves437

6.2 Volumes444

6.3 Volumes by Cylindrical Shells455

6.4 Work460

6.5 Average Value of a Function464

Applied Project · Where to Sit at the Movies468

Review468

Problems Plus470

7 Techniques of Integration474

7.1 Integration by Parts475

7.2 Trigonometric Integrals482

7.3 Trigonometric Substitution489

7.4 Integration of Rational Functions by Partial Fractions496

7.5 Strategy for Integration505

7.6 Integration Using Tables and Computer Algebra Systems511

Discovery Project · Patterns in Integrals517

7.7 Approximate Integration518

7.8 Improper Integrals530

Review540

Problems Plus543

8 Further Applications of Integration546

8.1 Arc Length547

Discovery Project · Arc Length Contest554

8.2 Area of a Surface of Revolution554

Discovery Project · Rotating on a slant560

8.3 Applications to Physics and Engineering561

8.4 Applications to Economics and Biology571

8.5 Probability575

Review582

Problems Plus584

9 Differential Equations586

9.1 Modeling with Differential Equations587

9.2 Direction Fields and Euler's Method592

9.3 Separable Equations601

Applied Project How Fast Does a Tank Drain?609

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

9.4 Exponential Growth and Decay611

Applied Project · Calculus and Baseball622

9.5 The Logistic Equation623

9.6 Linear Equations632

9.7 Predator-Prey Systems638

Review644

Problems Plus648

10 Parametric Equations and Polar Coordinates650

10.1 Curves Defined by Parametric Equations651

Laboratory Project · Running Circlee around Circles659

10.2 Calculus with Parametric Curves660

Laboratory Project · Bezler Curves669

10.3 Polar Coordinates669

10.4 Areas and Lengths in Polar Coordinates679

10.5 Conic Sections684

10.6 Conic Sections in Polar Coordinates692

Review696

Problems Plus699

11 Infinite Sequences and Series700

11.1 Sequences701

Laboratory Project · Logietic sequences713

11.2 Series713

11.3 The Integral Test and Estimates of Sums723

11.4 The Comparison Tests730

11.5 Alternating Series735

11.6 Absolute Convergence and the Ratio and Root Tests740

11.7 Strategy for Testing Series747

11.8 Power Series749

11.9 Representations of Functions as Power Series754

11.10 Taylor and Maclaurin Series760

Laboratory Project · An Elusive Limit772

11.11 The Binomial Series772

Writing Project · How Newton Discovered the Binomial Series776

11.12 Applications of Taylor Polynomials776

Applied Project · Radiation from the Stars785

Review786

Problems Plus789

12 Vectors and the Geometry of Space792

12.1 Three-Dimensional Coordinate Systems793

12.2 Vectors798

12.3 The Dot Product807

12.4 The Cross Product814

Discovery Project · The Geometry of a Tetrahedron822

12.5 Equations of Lines and Planes822

Laboratory Project · Putting 3D in Perspective832

12.6 Cylinders and Quadric Surfaces832

12.7 Cylindrical and Spherical Coordinates839

Laboratory Project · Families of Surfaces844

Review844

Problems Plus847

13 Vector Functions848

13.1 Vector Functions and Space Curves849

13.2 Derivatives and Integrals of Vector Functions856

13.3 Arc Length and Curvature862

13.4 Motion in Space: Velocity and Acceleration870

Applied Project · Kepler's Laws880

Review881

Problems Plus884

14 Partial Derivatives886

14.1 Functions of Several Variables887

14.2 Limits and Continuity902

14.3 Partial Derivatives909

14.4 Tangent Planes and Linear Approximations923

14.5 The Chain Rule931

14.6 Directional Derivatives and the Gradient Vector940

14.7 Maximum and Minimum Values953

Applied Project · Designing a Dumpster963

Discovery Project · Quadratic Approximations and Critical Points964

14.8 Lagrange Multipliers965

Applied Project Rocket Science972

Applied Project Hydro-Turbine Optimization973

Review974

Problems Plus978

15 Multiple Integrals980

15.1 Double Integrals over Rectangles981

15.2 Iterated Integrals989

15.3 Double Integrals over General Regions995

15.4 Double Integrals in Polar Coordinates1003

15.5 Applications of Double Integrals1009

15.6 Surface Area1019

15.7 Triple Integrals1023

Discovery Project · Volumes of Hyperspheres1032

15.8 Triple Integrals in Cylindrical and Spherical Coordinates1033

Applied Project · Roller Derby1039

Discovery Project · The Intersection of Three Cylinders1040

15.9 Change of Variables in Multiple Integrals1041

Review1049

Problems Plus1052

16 Vector Calculus1054

16.1 Vector Fields1055

16.2 Line Integrals1062

16.3 The Fundamental Theorem for Line Integrals1074

16.4 Green's Theorem1083

16.5 Curl and Divergence1090

16.6 Parametric Surfaces and Their Areas1098

16.7 Surface Integrals1109

16.8 Stokes' Theorem1121

Writing Project · Three Men and Two Theorems1126

16.9 The Divergence Theorem1127

16.10 Summary1134

Review1135

Problems Plus1138

17 Second-Order Differential Equations1140

17.1 Second-Order Linear Equations1141

17.2 Nonhomogeneous Linear Equations1147

17.3 Applications of Second-Order Differential Equations1155

17.4 Series Solutions1163

Review1167