## Thomas’ Calculus: Early Transcendentals, Fifteenth Edition

By Joel Hass, Christopher Heil, Przemyslaw Bogacki, Maurice D. Weir and José Luis Zuleta Estrugo

**Contents:**

Preface 9

1 Functions 21

1.1 Functions and Their Graphs 21

1.2 Combining Functions; Shifting and Scaling Graphs 34

1.3 Trigonometric Functions 41

1.4 Exponential Functions 49

1.5 Inverse Functions and Logarithms 54

Questions to Guide Your Review 67

Practice Exercises 67

Additional and Advanced Exercises 69

Technology Application Projects 71

2 Limits and Continuity 72

2.1 Rates of Change and Tangent Lines to Curves 72

2.2 Limit of a Function and Limit Laws 79

2.3 The Precise Definition of a Limit 90

2.4 One-Sided Limits 99

2.5 Limits Involving Infinity; Asymptotes of Graphs 106

2.6 Continuity 120

Questions to Guide Your Review 132

Practice Exercises 133

Additional and Advanced Exercises 134

Technology Application Projects 137

3 Derivatives 138

3.1 Tangent Lines and the Derivative at a Point 138

3.2 The Derivative as a Function 142

3.3 Differentiation Rules 151

3.4 The Derivative as a Rate of Change 161

3.5 Derivatives of Trigonometric Functions 170

3.6 The **Chain Rule** 176

3.7 Implicit Differentiation 184

3.8 Derivatives of Inverse Functions and Logarithms 189

3.9 Inverse Trigonometric Functions 200

3.10 Related Rates 206

3.11 Linearization and Differentials 214

Questions to Guide Your Review 226

Practice Exercises 227

Additional and Advanced Exercises 231

Technology Application Projects 234

4 Applications of Derivatives 235

4.1 Extreme Values of Functions on Closed Intervals 235

4.2 The Mean Value Theorem 243

4.3 Monotonic Functions and the First Derivative Test 250

4.4 Concavity and Curve Sketching 255

4.5 Indeterminate Forms and L’Hôpital’s Rule 268

4.6 Applied Optimization 277

4.7 Newton’s Method 289

4.8 Antiderivatives 294

Questions to Guide Your Review 304

Practice Exercises 305

Additional and Advanced Exercises 308

Technology Application Projects 311

5 Integrals 312

5.1 Area and Estimating with Finite Sums 312

5.2 Sigma Notation and Limits of Finite Sums 322

5.3 The Definite Integral 329

5.4 The Fundamental Theorem of Calculus 342

5.5 Indefinite Integrals and the Substitution Method 354

5.6 Definite Integral Substitutions and the Area Between Curves 361

Questions to Guide Your Review 372

Practice Exercises 372

Additional and Advanced Exercises 375

Technology Application Projects 379

6 Applications of Definite Integrals 380

6.1 Volumes Using Cross-Sections 380

6.2 Volumes Using Cylindrical Shells 391

6.3 Arc Length 399

6.4 Areas of Surfaces of Revolution 405

6.5 Work and Fluid Forces 410

6.6 Moments and Centers of Mass 420

Questions to Guide Your Review 431

Practice Exercises 432

Additional and Advanced Exercises 434

Technology Application Projects 435

7 Integrals and Transcendental Functions 436

7.1 The Logarithm Defined as an Integral 436

7.2 Exponential Change and Separable Differential Equations 447

7.3 Hyperbolic Functions 457

7.4 Relative Rates of Growth 465

Questions to Guide Your Review 470

Practice Exercises 471

Additional and Advanced Exercises 472

8 Techniques of Integration 473

8.1 Using Basic Integration Formulas 473

8.2 Integration by Parts 478

8.3 Trigonometric Integrals 486

8.4 Trigonometric Substitutions 492

8.5 Integration of Rational Functions by Partial Fractions 497

8.6 Integral Tables and Computer Algebra Systems 504

8.7 Numerical Integration 510

8.8 Improper Integrals 520

Questions to Guide Your Review 531

Practice Exercises 532

Additional and Advanced Exercises 534

Technology Application Projects 537

9 Infinite Sequences and Series 538

9.1 Sequences 538

9.2 Infinite Series 551

9.3 The Integral Test 561

9.4 Comparison Tests 567

9.5 Absolute Convergence; The Ratio and Root Tests 572

9.6 Alternating Series and Conditional Convergence 579

9.7 Power Series 586

9.8 Taylor and Maclaurin Series 597

9.9 Convergence of Taylor Series 602

9.10 Applications of Taylor Series 609

Questions to Guide Your Review 618

Practice Exercises 619

Additional and Advanced Exercises 621

Technology Application Projects 623

10 Parametric Equations and Polar Coordinates 624

10.1 Parametrizations of Plane Curves 624

10.2 Calculus with Parametric Curves 633

10.3 Polar Coordinates 642

10.4 Graphing Polar Coordinate Equations 646

10.5 Areas and Lengths in Polar Coordinates 650

10.6 Conic Sections 655

10.7 Conics in Polar Coordinates 663

Questions to Guide Your Review 669

Practice Exercises 670

Additional and Advanced Exercises 672

Technology Application Projects 674

11 Vectors and the Geometry of Space 675

11.1 Three-Dimensional Coordinate Systems 675

11.2 Vectors 680

11.3 The Dot Product 691

11.4 The Cross Product 699

11.5 Lines and Planes in Space 705

11.6 Cylinders and Quadric Surfaces 714

Questions to Guide Your Review 720

Practice Exercises 720

Additional and Advanced Exercises 722

Technology Application Projects 724

12 Vector-Valued Functions and Motion in Space 725

12.1 Curves in Space and Their Tangents 725

12.2 Integrals of Vector Functions; Projectile Motion 734

12.3 Arc Length in Space 743

12.4 Curvature and Normal Vectors of a Curve 747

12.5 Tangential and Normal Components of Acceleration 753

12.6 Velocity and Acceleration in Polar Coordinates 759

Questions to Guide Your Review 762

Practice Exercises 763

Additional and Advanced Exercises 765

Technology Application Projects 766

13 Partial Derivatives 767

13.1 Functions of Several Variables 767

13.2 Limits and Continuity in Higher Dimensions 775

13.3 Partial Derivatives 784

13.4 The Chain Rule 796

13.5 Directional Derivatives and Gradient Vectors 806

13.6 Tangent Planes and Differentials 815

13.7 Extreme Values and Saddle Points 825

13.8 Lagrange Multipliers 834

13.9 Taylor’s Formula for Two Variables 844

13.10 Partial Derivatives with Constrained Variables 848

Questions to Guide Your Review 852

Practice Exercises 853

Additional and Advanced Exercises 856

Technology Application Projects 858

14 Multiple Integrals 859

14.1 Double and Iterated Integrals over Rectangles 859

14.2 Double Integrals over General Regions 864

14.3 Area by Double Integration 873

14.4 Double Integrals in Polar Form 876

14.5 Triple Integrals in Rectangular Coordinates 883

14.6 Applications 893

14.7 Triple Integrals in Cylindrical and Spherical Coordinates 903

14.8 Substitutions in Multiple Integrals 915

Questions to Guide Your Review 924

Practice Exercises 925

Additional and Advanced Exercises 927

Technology Application Projects 929

15 Integrals and Vector Fields 930

15.1 Line Integrals of Scalar Functions 930

15.2 Vector Fields and Line Integrals: Work, Circulation, and Flux 937

15.3 Path Independence, Conservative Fields, and Potential Functions 950

15.4 Green’s Theorem in the Plane 961

15.5 Surfaces and Area 973

15.6 Surface Integrals 983

15.7 Stokes’ Theorem 993

15.8 The Divergence Theorem and a Unified Theory 1006

Questions to Guide Your Review 1018

Practice Exercises 1019

Additional and Advanced Exercises 1021

Technology Application Projects 1023

16 First-Order Differential Equations 1024

16.1 Solutions, Slope Fields, and Euler’s Method 1024

16.2 First-Order Linear Equations 1032

16.3 Applications 1038

16.4 Graphical Solutions of Autonomous Equations 1044

16.5 Systems of Equations and Phase Planes 1053

17

Questions to Guide Your Review 1059

Practice Exercises 1059

Additional and Advanced Exercises 1061

Technology Application Projects 1061

Second-Order Differential Equations (O nline)

17.1 Second-Order Linear Equations

17.2 Nonhomogeneous Linear Equations

17.3 Applications

17.4 Euler Equations

17.5 Power-Series Solutions

18 Complex Functions (Online)

18.1 Complex Numbers

18.2 Functions of a Complex Variable

18.3 Derivatives

18.4 The Cauchy-Riemann Equations

18.5 Complex Power Series

18.6 Some Complex Functions

18.7 Conformal Maps

Questions to Guide Your Review

Additional and Advanced Exercises

19 Fourier Series and Wavelets (Online)

19.1 Periodic Functions

19.2 Summing Sines and Cosines

19.3 Vectors and Approximation in Three and More Dimensions

19.4 Approximation of Functions

19.5 Advanced Topic: The Haar System and Wavelets

Questions to Guide Your Review

Additional and Advanced Exercises

Appendix A AP-1

A.1 Real Numbers and the Real Line AP-1

A.2 Graphing with Software AP-6

A.3 Mathematical Induction AP-10

A.4 Lines, Circles, and Parabolas AP-13

A.5 Proofs of Limit Theorems AP-23

A.6 Commonly Occurring Limits AP-26

A.7 Theory of the Real Numbers AP-27

A.8 Probability AP-30

A.9 The Distributive Law for Vector Cross Products AP-43

A.10 The Mixed Derivative Theorem and the Increment Theorem AP-44

Appendix B (Online)

B.1 Determinants

B.2 Extreme Values and Saddle Points for Functions of More than Two Variables

B.3 The Method of Gradient Descent

Answers to Odd-Numbered Exercises AN-1

Applications Index AI-1

Subject Index I-1

Credits C-1

A Brief Table of Integrals T-1