University Calculus Book

1 Functions

1.1 Functions and Their Graphs 1

1.2 Combining Functions; Shifting and Scaling Graphs 14

1.3 Trigonometric Functions 22

1.4 Exponential Functions 30

1.5 Inverse Functions and Logarithms 36

1.6 Graphing with Calculators and Computers 50

2 Limits and Continuity

2.1 Rates of Change and Tangents to Curves 55

2.2 Limit of a Function and Limit Laws 62

2.3 The Precise Definition of a Limit 74

2.4 One-Sided Limits and Limits at Infinity 84

2.5 Infinite Limits and Vertical Asymptotes 97

2.6 Continuity 103

2.7 Tangents and Derivatives at a Point 115

QUESTIONS TO GUIDE YOUR REVIEW 119

PRACTICE EXERCISES 120

ADDITIONAL AND ADVANCED EXERCISES 122

3 Differentiation

3.1 The Derivative as a Function 125

3.2 Differentiation Rules for Polynomials, Exponentials, Products, and Quotients 134

3.3 The Derivative as a Rate of Change 146

3.4 Derivatives of Trigonometric Functions 157

3.5 The Chain Rule and Parametric Equations 164

3.6 Implicit Differentiation 177

3.7 Derivatives of Inverse Functions and Logarithms 183

3.8 Inverse Trigonometric Functions 194

3.9 Related Rates 201

3.10 Linearization and Differentials 209

3.11 Hyperbolic Functions 221

QUESTIONS TO GUIDE YOUR REVIEW 227

PRACTICE EXERCISES 228

ADDITIONAL AND ADVANCED EXERCISES 234

4 Applications of Derivatives

4.1 Extreme Values of Functions 237

4.2 The Mean Value Theorem 245

4.3 Monotonic Functions and the First Derivative Test 254

4.4 Concavity and Curve Sketching 260

4.5 Applied Optimization 271

4.6 Indeterminate Forms and L’Hôpital’s Rule 283

4.7 Newton’s Method 291

4.8 Antiderivatives 296

QUESTIONS TO GUIDE YOUR REVIEW 306

PRACTICE EXERCISES 307

ADDITIONAL AND ADVANCED EXERCISES 311

5 Integration

5.1 Estimating with Finite Sums 315

5.2 Sigma Notation and Limits of Finite Sums 325

5.3 The Definite Integral 332

5.4 The Fundamental Theorem of Calculus 345

5.5 Indefinite Integrals and the Substitution Rule 354

5.6 Substitution and Area Between Curves 360

5.7 The Logarithm Defined as an Integral 370

QUESTIONS TO GUIDE YOUR REVIEW 381

PRACTICE EXERCISES 382

ADDITIONAL AND ADVANCED EXERCISES 386

6 Applications of Definite Integrals

6.1 Volumes by Slicing and Rotation About an Axis 391

6.2 Volumes by Cylindrical Shells 401

6.3 Lengths of Plane Curves 408

6.4 Areas of Surfaces of Revolution 415

6.5 Exponential Change and Separable Differential Equations 421

6.6 Work 430

6.7 Moments and Centers of Mass 437

QUESTIONS TO GUIDE YOUR REVIEW 444

PRACTICE EXERCISES 444

ADDITIONAL AND ADVANCED EXERCISES 446

7 Techniques of Integration

7.1 Integration by Parts 448

7.2 Trigonometric Integrals 455

7.3 Trigonometric Substitutions 461

7.4 Integration of Rational Functions by Partial Fractions 464

7.5 Integral Tables and Computer Algebra Systems 471

7.6 Numerical Integration 477

7.7 Improper Integrals 487

QUESTIONS TO GUIDE YOUR REVIEW 497

PRACTICE EXERCISES 497

ADDITIONAL AND ADVANCED EXERCISES 500

8 Infinite Sequences and Series

8.1 Sequences 502

8.2 Infinite Series 515

8.3 The Integral Test 523

8.4 Comparison Tests 529

8.5 The Ratio and Root Tests 533

8.6 Alternating Series, Absolute and Conditional Convergence 537

8.7 Power Series 543

8.8 Taylor and Maclaurin Series 553

8.9 Convergence of Taylor Series 559

8.10 The Binomial Series 569

QUESTIONS TO GUIDE YOUR REVIEW 572

PRACTICE EXERCISES 573

ADDITIONAL AND ADVANCED EXERCISES 575

9 Polar Coordinates and Conics

9.1 Polar Coordinates 577

9.2 Graphing in Polar Coordinates 582

9.3 Areas and Lengths in Polar Coordinates 586

9.4 Conic Sections 590

9.5 Conics in Polar Coordinates 599

9.6 Conics and Parametric Equations; The Cycloid 606

QUESTIONS TO GUIDE YOUR REVIEW 610

PRACTICE EXERCISES 610

ADDITIONAL AND ADVANCED EXERCISES 612

10 Vectors and the Geometry of Space

10.1 Three-Dimensional Coordinate Systems 614

10.2 Vectors 619

10.3 The Dot Product 628

10.4 The Cross Product 636

10.5 Lines and Planes in Space 642

10.6 Cylinders and Quadric Surfaces 652

QUESTIONS TO GUIDE YOUR REVIEW 657

PRACTICE EXERCISES 658

ADDITIONAL AND ADVANCED EXERCISES 660

11 Vector-Valued Functions and Motion in Space

11.1 Vector Functions and Their Derivatives 663

11.2 Integrals of Vector Functions 672

11.3 Arc Length in Space 678

11.4 Curvature of a Curve 683

11.5 Tangential and Normal Components of Acceleration 689

11.6 Velocity and Acceleration in Polar Coordinates 694

QUESTIONS TO GUIDE YOUR REVIEW 698

PRACTICE EXERCISES 698

ADDITIONAL AND ADVANCED EXERCISES 700

12 Partial Derivatives

12.1 Functions of Several Variables 702

12.2 Limits and Continuity in Higher Dimensions 711

12.3 Partial Derivatives 719

12.4 The Chain Rule 731

12.5 Directional Derivatives and Gradient Vectors 739

12.6 Tangent Planes and Differentials 747

12.7 Extreme Values and Saddle Points 756

12.8 Lagrange Multipliers 765

12.9 Taylor’s Formula for Two Variables 775

QUESTIONS TO GUIDE YOUR REVIEW 779

PRACTICE EXERCISES 780

ADDITIONAL AND ADVANCED EXERCISES 783

13 Multiple Integrals

13.1 Double and Iterated Integrals over Rectangles 785

13.2 Double Integrals over General Regions 790

13.3 Area by Double Integration 799

13.4 Double Integrals in Polar Form 802

13.5 Triple Integrals in Rectangular Coordinates 807

13.6 Moments and Centers of Mass 816

13.7 Triple Integrals in Cylindrical and Spherical Coordinates 825

13.8 Substitutions in Multiple Integrals 837

QUESTIONS TO GUIDE YOUR REVIEW 846

PRACTICE EXERCISES 846

ADDITIONAL AND ADVANCED EXERCISES 848

14 Integration in Vector Fields

14.1 Line Integrals 851

14.2 Vector Fields, Work, Circulation, and Flux 856

14.3 Path Independence, Potential Functions, and Conservative Fields 867

14.4 Green’s Theorem in the Plane 877

14.5 Surfaces and Area 887

14.6 Surface Integrals and Flux 896

14.7 Stokes’Theorem 905

14.8 The Divergence Theorem and a Unified Theory 914

QUESTIONS TO GUIDE YOUR REVIEW 925

PRACTICE EXERCISES 925

ADDITIONAL AND ADVANCED EXERCISES 928

15 First-Order Differential Equations (online)

15.1 Solutions, Slope Fields, and Picard’s Theorem

15.2 First-Order Linear Equations

15.3 Applications

15.4 Euler’s Method

15.5 Graphical Solutions of Autonomous Equations

15.6 Systems of Equations and Phase Planes

16 Second-Order Differential Equations (online)

16.1 Second-Order Linear Equations

16.2 Nonhomogeneous Linear Equations

16.3 Applications

16.4 Euler Equations

16.5 Power Series Solutions

Appendices AP-1

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

A.2 Mathematical Induction AP-7

A.3 Lines, Circles, and Parabolas AP-10

A.4 Trigonometry Formulas AP-19

A.5 Proofs of Limit Theorems AP-21

A.6 Commonly Occurring Limits AP-25

A.7 Theory of the Real Numbers AP-26

A.8 The Distributive Law for Vector Cross Products AP-29

A.9 The Mixed Derivative Theorem and the Increment Theorem AP-30