Welcome xviii

How to Use This Book to Study for an Exam xix

Two all-purpose study tips xx

Key sections for exam review (by topic) xx

Acknowledgments xxiii

Chapter 1: Functions, Graphs, and Lines 1

1.1 Functions 1

1.1.1 Interval notation 3

1.1.2 Finding the domain 4

1.1.3 Finding the range using the graph 5

1.1.4 The vertical line test 6

1.2 Inverse Functions 7

1.2.1 The horizontal line test 8

1.2.2 Finding the inverse 9

1.2.3 Restricting the domain 9

1.2.4 Inverses of inverse functions 11

1.3 Composition of Functions 11

1.4 Odd and Even Functions 14

1.5 Graphs of Linear Functions 17

1.6 Common Functions and Graphs 19

Chapter 2: Review of Trigonometry 25

2.1 The Basics 25

2.2 Extending the Domain of Trig Functions 28

2.2.1 The ASTC method 31

2.2.2 Trig functions outside [0; 2π] 33

2.3 The Graphs of Trig Functions 35

2.4 Trig Identities 39

Chapter 3: Introduction to Limits 41

3.1 Limits: The Basic Idea 41

3.2 Left-Hand and Right-Hand Limits 43

3.3 When the Limit Does Not Exist 45

3.4 Limits at 1 and —∞ 47

3.4.1 Large numbers and small numbers 48

3.5 Two Common Misconceptions about Asymptotes 50

3.6 The Sandwich Principle 51

3.7 Summary of Basic Types of Limits 54

Chapter 4: How to Solve Limit Problems Involving Polynomials 57

4.1 Limits Involving Rational Functions as χ → αa 57

4.2 Limits Involving Square Roots as χ → α 61

4.3 Limits Involving Rational Functions as χ → ∞ 61

4.3.1 Method and examples 64

4.4 Limits Involving Poly-type Functions as χ → ∞ 66

4.5 Limits Involving Rational Functions as χ → -∞ 70

4.6 Limits Involving Absolute Values 72

Chapter 5: Continuity and Differentiability 75

5.1 Continuity 75

5.1.1 Continuity at a point 76

5.1.2 Continuity on an interval 77

5.1.3 Examples of continuous functions 77

5.1.4 The Intermediate Value Theorem 80

5.1.5 A harder IVT example 82

5.1.6 Maxima and minima of continuous functions 82

5.2 Differentiability 84

5.2.1 Average speed 84

5.2.2 Displacement and velocity 85

5.2.3 Instantaneous velocity 86

5.2.4 The graphical interpretation of velocity 87

5.2.5 Tangent lines 88

5.2.6 The derivative function 90

5.2.7 The derivative as a limiting ratio 91

5.2.8 The derivative of linear functions 93

5.2.9 Second and higher-order derivatives 94

5.2.10 When the derivative does not exist 94

5.2.11 Differentiability and continuity 96

Chapter 6: How to Solve Differentiation Problems 99

6.1 Finding Derivatives Using the Difinition 99

6.2 Finding Derivatives (the Nice Way) 102

6.2.1 Constant multiples of functions 103

6.2.2 Sums and Differences of functions 103

6.2.3 Products of functions via the product rule 104

6.2.4 Quotients of functions via the quotient rule 105

6.2.5 Composition of functions via the chain rule 107

6.2.6 A nasty example 109

6.2.7 Justification of the product rule and the chain rule 111

6.3 Finding the Equation of a Tangent Line 114

6.4 Velocity and Acceleration 114

6.4.1 Constant negative acceleration 115

6.5 Limits Which Are Derivatives in Disguise 117

6.6 Derivatives of Piecewise-Difined Functions 119

6.7 Sketching Derivative Graphs Directly 123

Chapter 7: Trig Limits and Derivatives 127

7.1 Limits Involving Trig Functions 127

7.1.1 The small case 128

7.1.2 Solving problems|the small case 129

7.1.3 The large case 134

7.1.4 The "other" case 137

7.1.5 Proof of an important limit 137

7.2 Derivatives Involving Trig Functions 141

7.2.1 Examples of Differentiating trig functions 143

7.2.2 Simple harmonic motion 145

7.2.3 A curious function 146

Chapter 8: Implicit Differentiation and Related Rates 149

8.1 Implicit Differentiation 149

8.1.1 Techniques and examples 150

8.1.2 Finding the second derivative implicitly 154

8.2 Related Rates 156

8.2.1 A simple example 157

8.2.2 A slightly harder example 159

8.2.3 A much harder example 160

8.2.4 A really hard example 162

Chapter 9: Exponentials and Logarithms 167

9.1 The Basics 167

9.1.1 Review of exponentials 167

9.1.2 Review of logarithms 168

9.1.3 Logarithms, exponentials, and inverses 169

9.1.4 Log rules 170

9.2 Difinition of *e* 173

9.2.1 A question about compound interest 173

9.2.2 The answer to our question 173

9.2.3 More about e and logs 175

9.3 Differentiation of Logs and Exponentials 177

9.3.1 Examples of Differentiating exponentials and logs 179

9.4 How to Solve Limit Problems Involving Exponentials or Logs 180

9.4.1 Limits involving the Difinition of *e* 181

9.4.2 Behavior of exponentials near 0 182

9.4.3 Behavior of logarithms near 1 183

9.4.4 Behavior of exponentials near ∞ or -∞1 184

9.4.5 Behavior of logs near ∞ 187

9.4.6 Behavior of logs near 0 188

9.5 Logarithmic Differentiation 189

9.5.1 The derivative of χ^{a} 192

9.6 Exponential Growth and Decay 193

9.6.1 Exponential growth 194

9.6.2 Exponential decay 195

9.7 Hyperbolic Functions 198

Chapter 10: Inverse Functions and Inverse Trig Functions 201

10.1 The Derivative and Inverse Functions 201

10.1.1 Using the derivative to show that an inverse exists 201

10.1.2 Derivatives and inverse functions: what can go wrong 203

10.1.3 Finding the derivative of an inverse function 204

10.1.4 A big example 206

10.2 Inverse Trig Functions 208

10.2.1 Inverse sine 208

10.2.2 Inverse cosine 211

10.2.3 Inverse tangent 213

10.2.4 Inverse secant 216

10.2.5 Inverse cosecant and inverse cotangent 217

10.2.6 Computing inverse trig functions 218

10.3 Inverse Hyperbolic Functions 220

10.3.1 The rest of the inverse hyperbolic functions 222

Chapter 11: The Derivative and Graphs 225

11.1 Extrema of Functions 225

11.1.1 Global and local extrema 225

11.1.2 The Extreme Value Theorem 227

11.1.3 How to find global maxima and minima 228

11.2 Rolle's Theorem 230

11.3 The Mean Value Theorem 233

11.3.1 Consequences of the Mean Value Theorem 235

11.4 The Second Derivative and Graphs 237

11.4.1 More about points of inection 238

11.5 Classifying Points Where the Derivative Vanishes 239

11.5.1 Using the first derivative 240

11.5.2 Using the second derivative 242

Chapter 12: Sketching Graphs 245

12.1 How to Construct a Table of Signs 245

12.1.1 Making a table of signs for the derivative 247

12.1.2 Making a table of signs for the second derivative 248

12.2 The Big Method 250

12.3 Examples 252

12.3.1 An example without using derivatives 252

12.3.2 The full method: example 1 254

12.3.3 The full method: example 2 256

12.3.4 The full method: example 3 259

12.3.5 The full method: example 4 262

Chapter 13: Optimization and Linearization 267

13.1 Optimization 267

13.1.1 An easy optimization example 267

13.1.2 Optimization problems: the general method 269

13.1.3 An optimization example 269

13.1.4 Another optimization example 271

13.1.5 Using implicit Differentiation in optimization 274

13.1.6 A difficult optimization example 275

13.2 Linearization 278

13.2.1 Linearization in general 279

13.2.2 The Differential 281

13.2.3 Linearization summary and examples 283

13.2.4 The error in our approximation 285

13.3 Newton's Method 287

Chapter 14: L'Hôpital's Rule and Overview of Limits 293

14.1 L'Hôpital's Rule 293

14.1.1 Type A: 0/0 case 294

14.1.2 Type A: ±∞ / ±∞ case 296

14.1.3 Type B1 (∞ - ∞) 298

14.1.4 Type B2 (0 x ±∞) 299

14.1.5 Type C (1^{±∞}, 0^{0}, or ∞^{0}) 301

14.1.6 Summary of L'Hôpital's Rule types 302

14.2 Overview of Limits 303

Chapter 15: Introduction to Integration 307

15.1 Sigma Notation 307

15.1.1 A nice sum 310

15.1.2 Telescoping series 311

15.2 Displacement and Area 314

15.2.1 Three simple cases 314

15.2.2 A more general journey 317

15.2.3 Signed area 319

15.2.4 Continuous velocity 320

15.2.5 Two special approximations 323

Chapter 16: Difinite Integrals 325

16.1 The Basic Idea 325

16.1.1 Some easy examples 327

16.2 Difinition of the Difinite Integral 330

16.2.1 An example of using the Difinition 331

16.3 Properties of Difinite Integrals 334

16.4 Finding Areas 339

16.4.1 Finding the unsigned area 339

16.4.2 Finding the area between two curves 342

16.4.3 Finding the area between a curve and the *y*-axis 344

16.5 Estimating Integrals 346

16.5.1 A simple type of estimation 347

16.6 Averages and the Mean Value Theorem for Integrals 350

16.6.1 The Mean Value Theorem for integrals 351

16.7 A Nonintegrable Function 353

Chapter 17: The Fundamental Theorems of Calculus 355

17.1 Functions Based on Integrals of Other Functions 355

17.2 The First Fundamental Theorem 358

17.2.1 Introduction to antiderivatives 361

17.3 The Second Fundamental Theorem 362

17.4 InDifinite Integrals 364

17.5 How to Solve Problems: The First Fundamental Theorem 366

17.5.1 Variation 1: variable left-hand limit of integration 367

17.5.2 Variation 2: one tricky limit of integration 367

17.5.3 Variation 3: two tricky limits of integration 369

17.5.4 Variation 4: limit is a derivative in disguise 370

17.6 How to Solve Problems: The Second Fundamental Theorem 371

17.6.1 Finding inDifinite integrals 371

17.6.2 Finding Difinite integrals 374

17.6.3 Unsigned areas and absolute values 376

17.7 A Technical Point 380

17.8 Proof of the First Fundamental Theorem 381

Chapter 18: Techniques of Integration, Part One 383

18.1 Substitution 383

18.1.1 Substitution and Difinite integrals 386

18.1.2 How to decide what to substitute 389

18.1.3 Theoretical justification of the substitution method 392

18.2 Integration by Parts 393

18.2.1 Some variations 394

18.3 Partial Fractions 397

18.3.1 The algebra of partial fractions 398

18.3.2 Integrating the pieces 401

18.3.3 The method and a big example 404

Chapter 19: Techniques of Integration, Part Two 409

19.1 Integrals Involving Trig Identities 409

19.2 Integrals Involving Powers of Trig Functions 413

19.2.1 Powers of sin and/or cos 413

19.2.2 Powers of tan 415

19.2.3 Powers of sec 416

19.2.4 Powers of cot 418

19.2.5 Powers of csc 418

19.2.6 Reduction formulas 419

19.3 Integrals Involving Trig Substitutions 421

19.3.1 Type 1: 421

19.3.2 Type 2: 423

19.3.3 Type 3: 424

19.3.4 Completing the square and trig substitutions 426

19.3.5 Summary of trig substitutions 426

19.3.6 Technicalities of square roots and trig substitutions 427

19.4 Overview of Techniques of Integration 429

Chapter 20: Improper Integrals: Basic Concepts 431

20.1 Convergence and Divergence 431

20.1.1 Some examples of improper integrals 433

20.1.2 Other blow-up points 435

20.2 Integrals over Unbounded Regions 437

20.3 The Comparison Test (Theory) 439

20.4 The Limit Comparison Test (Theory) 441

20.4.1 Functions asymptotic to each other 441

20.4.2 The statement of the test 443

20.5 The p-test (Theory) 444

20.6 The Absolute Convergence Test 447

Chapter 21: Improper Integrals: How to Solve Problems 451

21.1 How to Get Started 451

21.1.1 Splitting up the integral 452

21.1.2 How to deal with negative function values 453

21.2 Summary of Integral Tests 454

21.3 Behavior of Common Functions near ∞ and -∞ 456

21.3.1 Polynomials and poly-type functions near ∞ and -∞ 456

21.3.2 Trig functions near ∞ and -∞ 459

21.3.3 Exponentials near ∞ and -∞ 461

21.3.4 Logarithms near ∞ 465

21.4 Behavior of Common Functions near 0 469

21.4.1 Polynomials and poly-type functions near 0 469

21.4.2 Trig functions near 0 470

21.4.3 Exponentials near 0 472

21.4.4 Logarithms near 0 473

21.4.5 The behavior of more general functions near 0 474

21.5 How to Deal with Problem Spots Not at 0 or ∞ 475

Chapter 22: Sequences and Series: Basic Concepts 477

22.1 Convergence and Divergence of Sequences 477

22.1.1 The connection between sequences and functions 478

22.1.2 Two important sequences 480

22.2 Convergence and Divergence of Series 481

22.2.1 Geometric series (theory) 484

22.3 The nth Term Test (Theory) 486

22.4 Properties of Both Infinite Series and Improper Integrals 487

22.4.1 The comparison test (theory) 487

22.4.2 The limit comparison test (theory) 488

22.4.3 The *p*-test (theory) 489

22.4.4 The absolute convergence test 490

22.5 New Tests for Series 491

22.5.1 The ratio test (theory) 492

22.5.2 The root test (theory) 493

22.5.3 The integral test (theory) 494

22.5.4 The alternating series test (theory) 497

Chapter 23: How to Solve Series Problems 501

23.1 How to Evaluate Geometric Series 502

23.2 How to Use the nth Term Test 503

23.3 How to Use the Ratio Test 504

23.4 How to Use the Root Test 508

23.5 How to Use the Integral Test 509

23.6 Comparison Test, Limit Comparison Test, and *p*-test 510

23.7 How to Deal with Series with Negative Terms 515

Chapter 24: Taylor Polynomials, Taylor Series, and Power Series 519

24.1 Approximations and Taylor Polynomials 519

24.1.1 Linearization revisited 520

24.1.2 Quadratic approximations 521

24.1.3 Higher-degree approximations 522

24.1.4 Taylor's Theorem 523

24.2 Power Series and Taylor Series 526

24.2.1 Power series in general 527

24.2.2 Taylor series and Maclaurin series 529

24.2.3 Convergence of Taylor series 530

24.3 A Useful Limit 534

Chapter 25: How to Solve Estimation Problems 535

25.1 Summary of Taylor Polynomials and Series 535

25.2 Finding Taylor Polynomials and Series 537

25.3 Estimation Problems Using the Error Term 540

25.3.1 First example 541

25.3.2 Second example 543

25.3.3 Third example 544

25.3.4 Fourth example 546

25.3.5 Fifth example 547

25.3.6 General techniques for estimating the error term 548

25.4 Another Technique for Estimating the Error 548

Chapter 26: Taylor and Power Series: How to Solve Problems 551

26.1 Convergence of Power Series 551

26.1.1 Radius of convergence 551

26.1.2 How to find the radius and region of convergence 554

26.2 Getting New Taylor Series from Old Ones 558

26.2.1 Substitution and Taylor series 560

26.2.2 Differentiating Taylor series 562

26.2.3 Integrating Taylor series 563

26.2.4 Adding and subtracting Taylor series 565

26.2.5 Multiplying Taylor series 566

26.2.6 Dividing Taylor series 567

26.3 Using Power and Taylor Series to Find Derivatives 568

26.4 Using Maclaurin Series to Find Limits 570

Chapter 27: Parametric Equations and Polar Coordinates 575

27.1 Parametric Equations 575

27.1.1 Derivatives of parametric equations 578

27.2 Polar Coordinates 581

27.2.1 Converting to and from polar coordinates 582

27.2.2 Sketching curves in polar coordinates 585

27.2.3 Finding tangents to polar curves 590

27.2.4 Finding areas enclosed by polar curves 591

Chapter 28: Complex Numbers 595

28.1 The Basics 595

28.1.1 Complex exponentials 598

28.2 The Complex Plane 599

28.2.1 Converting to and from polar form 601

28.3 Taking Large Powers of Complex Numbers 603

28.4 Solving *z*^{n} = *w* 604

28.4.1 Some variations 608

28.5 Solving *e*^{z} = *w* 610

28.6 Some Trigonometric Series 612

28.7 Euler's Identity and Power Series 615

Chapter 29: Volumes, Arc Lengths, and Surface Areas 617

29.1 Volumes of Solids of Revolution 617

29.1.1 The disc method 619

29.1.2 The shell method 620

29.1.3 Summary . . . and variations 622

29.1.4 Variation 1: regions between a curve and the *y*-axis 623

29.1.5 Variation 2: regions between two curves 625

29.1.6 Variation 3: axes parallel to the coordinate axes 628

29.2 Volumes of General Solids 631

29.3 Arc Lengths 637

29.3.1 Parametrization and speed 639

29.4 Surface Areas of Solids of Revolution 640

Chapter 30: Differential Equations 645

30.1 Introduction to Differential Equations 645

30.2 Separable First-order Differential Equations 646

30.3 First-order Linear Equations 648

30.3.1 Why the integrating factor works 652

30.4 Constant-coefficient Differential Equations 653

30.4.1 Solving first-order homogeneous equations 654

30.4.2 Solving second-order homogeneous equations 654

30.4.3 Why the characteristic quadratic method works 655

30.4.4 Nonhomogeneous equations and particular solutions 656

30.4.5 Finding a particular solution 658

30.4.6 Examples of finding particular solutions 660

30.4.7 Resolving conicts between *yP* and *yH* 662

30.4.8 Initial value problems (constant-coefficient linear) 663

30.5 Modeling Using Differential Equations 665

Appendix A Limits and Proofs 669

A.1 Formal Difinition of a Limit 669

A.1.1 A little game 670

A.1.2 The actual Difinition 672

A.1.3 Examples of using the Difinition 672

A.2 Making New Limits from Old Ones 674

A.2.1 Sums and Differences of limits|proofs 674

A.2.2 Products of limits|proof 675

A.2.3 Quotients of limits|proof 676

A.2.4 The sandwich principle|proof 678

A.3 Other Varieties of Limits 678

A.3.1 Inffinite limits 679

A.3.2 Left-hand and right-hand limits 680

A.3.3 Limits at ∞ and -∞ 680

A.3.4 Two examples involving trig 682

A.4 Continuity and Limits 684

A.4.1 Composition of continuous functions 684

A.4.2 Proof of the Intermediate Value Theorem 686

A.4.3 Proof of the Max-Min Theorem 687

A.5 Exponentials and Logarithms Revisited 689

A.6 Differentiation and Limits 691

A.6.1 Constant multiples of functions 691

A.6.2 Sums and Differences of functions 691

A.6.3 Proof of the product rule 692

A.6.4 Proof of the quotient rule 693

A.6.5 Proof of the chain rule 693

A.6.6 Proof of the Extreme Value Theorem 694

A.6.7 Proof of Rolle's Theorem 695

A.6.8 Proof of the Mean Value Theorem 695

A.6.9 The error in linearization 696

A.6.10 Derivatives of piecewise-Difined functions 697

A.6.11 Proof of L'Hôpital's Rule 698

A.7 Proof of the Taylor Approximation Theorem 700

Appendix B Estimating Integrals 703

B.1 Estimating Integrals Using Strips 703

B.1.1 Evenly spaced partitions 705

B.2 The Trapezoidal Rule 706

B.3 Simpson's Rule 709

B.3.1 Proof of Simpson's rule 710

B.4 The Error in Our Approximations 711

B.4.1 Examples of estimating the error 712

B.4.2 Proof of an error term inequality 714

List of Symbols 717

Index 719