## Calculus with CalcChat and CalcView, Twelfth Edition

By Ron Larson and Bruce Edwards

**Contents:**

Preparation for Calculus 1

P.1 Graphs and Models 2

P.2 Linear Models and Rates of Change 10

P.3 Functions and Their Graphs 19

P.4 Review of Trigonometric Functions 31

Review Exercises 41

P.S. Problem Solving 44

1. Limits and Their Properties 45

1.1 A Preview of Calculus 46

1.2 Finding Limits Graphically and Numerically 52

1.3 Evaluating Limits Analytically 63

1.4 Continuity and One-Sided Limits 74

1.5 Infinite Limits 87

Section Project: Graphs and Limits of

Trigonometric Functions 94

Review Exercises 95

P.S. Problem Solving 98

2. Differentiation 99

2.1 The Derivative and the Tangent Line Problem 100

2.2 Basic Differentiation Rules and Rates of Change 110

2.3 Product and Quotient Rules and Higher-Order

Derivatives 122

2.4 The Chain Rule 133

2.5 Implicit Differentiation 144

Section Project: **Optical Illusions** 151

2.6 Related Rates 152

Review Exercises 161

P.S. Problem Solving 164

3. Applications of Differentiation 165

3.1 Extrema on an Interval 166

3.2 Rolle’s Theorem and the Mean Value Theorem 174

3.3 Increasing and Decreasing Functions and

the First Derivative Test 181

Section Project: Even Polynomial Functions of

Fourth Degree 190

3.4 Concavity and the Second Derivative Test 191

3.5 Limits at Infinity 199

3.6 A Summary of Curve Sketching 209

3.7 Optimization Problems 219

Section Project: Minimum Time 228

3.8 Newton’s Method 229

3.9 Differentials 235

Review Exercises 242

P.S. Problem Solving 246

4. Integration 247

4.1 Antiderivatives and Indefinite Integration 248

4.2 Area 258

4.3 Riemann Sums and Definite Integrals 270

4.4 The Fundamental Theorem of Calculus 281

4.5 Integration by Substitution 296

Section Project: Probability 308

Review Exercises 309

P.S. Problem Solving 312

Logarithmic, Exponential, and

5. Other Transcendental Functions 313

5.1 The Natural Logarithmic Function: Differentiation 314

5.2 The Natural Logarithmic Function: Integration 324

5.3 Inverse Functions 333

5.4 Exponential Functions: Differentiation and Integration 342

5.5 Bases Other than e and Applications 352

Section Project: Using a Graphing Utility to

Estimate Slope 361

5.6 Indeterminate Forms and L’Hôpital’s Rule 362

5.7 Inverse Trigonometric Functions: Differentiation 373

5.8 Inverse Trigonometric Functions: Integration 382

5.9 Hyperbolic Functions 390

Section Project: Mercator Map 399

Review Exercises 400

P.S. Problem Solving 404

6. Differential Equations 405

6.1 Slope Fields and Euler’s Method 406

6.2 Growth and Decay 415

6.3 Separation of Variables and the Logistic Equation 423

6.4 First-Order Linear Differential Equations 432

Section Project: Weight Loss 438

Review Exercises 439

P.S. Problem Solving 442

7. Applications of Integration 443

7.1 Area of a Region Between Two Curves 444

7.2 Volume: The Disk Method 454

7.3 Volume: The Shell Method 465

Section Project: Saturn 473

7.4 Arc Length and Surfaces of Revolution 474

7.5 Work 485

Section Project: Pyramid of Khufu 493

7.6 Moments, Centers of Mass, and Centroids 494

7.7 Fluid Pressure and Fluid Force 505

Review Exercises 511

P.S. Problem Solving 514

8. Integration Techniques and Improper Integrals 515

8.1 Basic Integration Rules 516

8.2 Integration by Parts 523

8.3 Trigonometric Integrals 532

Section Project: The Wallis Product 540

8.4 Trigonometric Substitution 541

8.5 Partial Fractions 550

8.6 Numerical Integration 559

8.7 Integration by Tables and Other Integration Techniques 566

8.8 Improper Integrals 572

Review Exercises 583

P.S. Problem Solving 586

9. Infinite Series 587

9.1 Sequences 588

9.2 Series and Convergence 599

Section Project: Cantor’s Disappearing Table 608

9.3 The Integral Test and p-Series 609

Section Project: The Harmonic Series 615

9.4 Comparisons of Series 616

9.5 Alternating Series 623

9.6 The Ratio and Root Tests 631

9.7 Taylor Polynomials and Approximations 640

9.8 Power Series 651

9.9 Representation of Functions by Power Series 661

9.10 Taylor and Maclaurin Series 668

Review Exercises 680

P.S. Problem Solving 684

Conics, Parametric Equations, and

10. Polar Coordinates 685

10.1 Conics and Calculus 686

10.2 Plane Curves and Parametric Equations 700

Section Project: Cycloids 709

10.3 Parametric Equations and Calculus 710

10.4 Polar Coordinates and Polar Graphs 719

Section Project: Cassini Oval 728

10.5 Area and Arc Length in Polar Coordinates 729

10.6 Polar Equations of Conics and Kepler’s Laws 738

Review Exercises 746

P.S. Problem Solving 750

11. Vectors and the Geometry of Space 751

11.1 Vectors in the Plane 752

11.2 Space Coordinates and Vectors in Space 762

11.3 The Dot Product of Two Vectors 770

11.4 The Cross Product of Two Vectors in Space 779

11.5 Lines and Planes in Space 787

Section Project: Distances in Space 797

11.6 Surfaces in Space 798

11.7 Cylindrical and Spherical Coordinates 808

Review Exercises 815

P.S. Problem Solving 818

12. Vector-Valued Functions 819

12.1 Vector-Valued Functions 820

Section Project: Witch of Agnesi 827

12.2 Differentiation and Integration of Vector-Valued

Functions 828

12.3 Velocity and Acceleration 836

12.4 Tangent Vectors and Normal Vectors 845

12.5 Arc Length and Curvature 855

Review Exercises 867

P.S. Problem Solving 870

13. Functions of Several Variables 871

13.1 Introduction to Functions of Several Variables 872

13.2 Limits and Continuity 884

13.3 Partial Derivatives 894

13.4 Differentials 904

13.5 Chain Rules for Functions of Several Variables 911

13.6 Directional Derivatives and Gradients 919

13.7 Tangent Planes and Normal Lines 931

Section Project: Wildflowers 939

13.8 Extrema of Functions of Two Variables 940

13.9 Applications of Extrema 948

Section Project: Building a Pipeline 955

13.10 Lagrange Multipliers 956

Review Exercises 964

P.S. Problem Solving 968

14. Multiple Integration 969

14.1 Iterated Integrals and Area in the Plane 970

14.2 Double Integrals and Volume 978

14.3 Change of Variables: Polar Coordinates 990

14.4 Center of Mass and Moments of Inertia 998

Section Project: Center of Pressure on a Sail 1005

14.5 Surface Area 1006

Section Project: Surface Area in Polar Coordinates 1012

14.6 Triple Integrals and Applications 1013

14.7 Triple Integrals in Other Coordinates 1024

Section Project: Wrinkled and Bumpy Spheres 1030

14.8 Change of Variables: Jacobians 1031

Review Exercises 1038

P.S. Problem Solving 1042

15. Vector Analysis 1043

15.1 Vector Fields 1044

15.2 Line Integrals 1055

15.3 Conservative Vector Fields and Independence of Path 1069

15.4 Green’s Theorem 1079

Section Project: Hyperbolic and Trigonometric Functions 1087

15.5 Parametric Surfaces 1088

15.6 Surface Integrals 1098

Section Project: Hyperboloid of One Sheet 1109

15.7 Divergence Theorem 1110

15.8 Stokes’s Theorem 1118

Review Exercises 1124

P.S. Problem Solving 1128

16. Additional Topics in Differential Equations (Online)*

16.1 Exact First-Order Equations

16.2 Second-Order Homogeneous Linear Equations

16.3 Second-Order Nonhomogeneous Linear Equations

Section Project: Parachute Jump

16.4 Series Solutions of Differential Equations

Review Exercises

P.S. Problem Solving

Appendices

Appendix A: Proofs of Selected Theorems A2

Appendix B: Integration Tables A3

Appendix C: Precalculus Review (Online)*

Appendix D: Rotation and the General Second-Degree

Equation (Online)*

Appendix E: Complex Numbers (Online)*

Appendix F: Business and Economic Applications (Online)*

Appendix G: Fitting Models to Data (Online)*

Answers to All Odd-Numbered Exercises A7

Index A123

*Available at the text companion website Larson Calculus.com