Mark Ryan has taught pre-algebra through calculus for more than 25 years. In 1997, he founded The Math Center in Winnetka, Illinois, where he teaches junior high and high school mathematics courses as well as standardized test prep classes. He also does extensive one-on-one tutoring. He is a member of the Authors Guild and the National Council of Teachers of Mathematics.

Introduction 1

About This Book 1

Foolish Assumptions 2

Icons Used in This Book 2

Beyond the Book 3

Where to Go from Here 3

Part 1: Pre-Calculus Review 5

Chapter 1: Getting Down to Basics: Algebra and Geometry 7

Fraction Frustration 7

Misc. Algebra: You Know, Like Miss South Carolina 9

Geometry: When Am I Ever Going to Need It? 11

Solutions for This Easy, Elementary Stuff 16

Chapter 2: Funky Functions and Tricky Trig 25

Figuring Out Your Functions 25

Trigonometric Calisthenics 29

Solutions to Functions and Trigonometry 33

Part 2: Limits and Continuity 41

Chapter 3: A Graph Is Worth a Thousand Words: Limits and Continuity 43

Digesting the Definitions: Limit and Continuity 44

Taking a Closer Look: Limit and Continuity Graphs 46

Solutions for Limits and Continuity 50

Chapter 4: Nitty-Gritty Limit Problems 53

Solving Limits with Algebra 54

Pulling Out Your Calculator: Useful "Cheating" 59

Making Yourself a Limit Sandwich 61

Into the Great Beyond: Limits at Infinity 63

Solutions for Problems with Limits 67

Part 3: Differentiation 77

Chapter 5: Getting the Big Picture: Differentiation Basics 79

The Derivative: A Fancy Calculus Word for Slope and Rate 79

The Handy-Dandy Difference Quotient 81

Solutions for Differentiation Basics 84

Chapter 6: Rules, Rules, Rules: The Differentiation Handbook 89

Rules for Beginners 89

Giving It Up for the Product and Quotient Rules 92

Linking Up with the Chain Rule 94

What to Do with Y's: Implicit Differentiation 98

Getting High on Calculus: Higher Order Derivatives 101

Solutions for Differentiation Problems 103

Chapter 7: Analyzing Those Shapely Curves with the Derivative 117

The First Derivative Test and Local Extrema 117

The Second Derivative Test and Local Extrema 120

Finding Mount Everest: Absolute Extrema 122

Smiles and Frowns: Concavity and Inflection Points 126

The Mean Value Theorem: Go Ahead, Make My Day 129

Solutions for Derivatives and Shapes of Curves 131

Chapter 8: Using Differentiation to Solve Practical Problems 147

Optimization Problems: From Soup to Nuts 147

Problematic Relationships: Related Rates 150

A Day at the Races: Position, Velocity, and Acceleration 153

Solutions to Differentiation Problem Solving 157

Chapter 9: Even More Practical Applications of Differentiation 173

Make Sure You Know Your Lines: Tangents and Normals 173

Looking Smart with Linear Approximation 177

Calculus in the Real World: Business and Economics 179

Solutions to Differentiation Problem Solving 183

Part 4: Integration and Infinite Series 191

Chapter 10: Getting into Integration 193

Adding Up the Area of Rectangles: Kid Stuff 193

Sigma Notation and Riemann Sums: Geek Stuff 196

Close Isn't Good Enough: The Definite Integral and Exact Area 200

Finding Area with the Trapezoid Rule and Simpson's Rule 202

Solutions to Getting into Integration 205

Chapter 11: Integration: Reverse Differentiation 213

The Absolutely Atrocious and Annoying Area Function 213

Sound the Trumpets: The Fundamental Theorem of Calculus 216

Finding Antiderivatives: The Guess-and-Check Method 219

The Substitution Method: Pulling the Switcheroo 221

Solutions to Reverse Differentiation Problems 225

Chapter 12: Integration Rules for Calculus Connoisseurs 229

Integration by Parts: Here's How u du It 229

Transfiguring Trigonometric Integrals 233

Trigonometric Substitution: It's Your Lucky Day! 235

Partaking of Partial Fractions 237

Solutions for Integration Rules 241

Chapter 13: Who Needs Freud? Using the Integral to Solve Your Problems 255

Finding a Function's Average Value 255

Finding the Area between Curves 256

Volumes of Weird Solids: No, You're Never Going to Need This 258

Arc Length and Surfaces of Revolution 265

Solutions to Integration Application Problems 268

Chapter 14: Infinite (Sort of) Integrals 277

Getting Your Hopes Up with L'Hôpital's Rule 278

Disciplining Those Improper Integrals 280

Solutions to Infinite (Sort of) Integrals 283

Chapter 15: Infinite Series: Welcome to the Outer Limits 287

The Nifty nth Term Test 287

Testing Three Basic Series 289

Apples and Oranges . . . and Guavas: Three Comparison Tests 291

Ratiocinating the Two "R" Tests 295

He Loves Me, He Loves Me Not: Alternating Series 297

Solutions to Infinite Series 299

Part 5: The Part of Tens 309

Chapter 16: Ten Things about Limits, Continuity, and Infinite Series 311

The 33333 Mnemonic 311

First 3 over the "l": 3 parts to the definition of a limit 312

Fifth 3 over the "l": 3 cases where a limit fails to exist 312

Second 3 over the "i": 3 parts to the definition of continuity 312

Fourth 3 over the "i": 3 cases where continuity fails to exist 312

Third 3 over the "m": 3 cases where a derivative fails to exist 313

The 13231 Mnemonic 313

First 1: The nth term test of divergence 313

Second 1: The nth term test of convergence for alternating series 313

First 3: The three tests with names 313

Second 3: The three comparison tests 314

The 2 in the middle: The two R tests 314

Chapter 17: Ten Things You Better Remember about Differentiation 315

The Difference Quotient 315

The First Derivative Is a Rate 315

The First Derivative Is a Slope 316

Extrema, Sign Changes, and the First Derivative 316

The Second Derivative and Concavity 316

Inflection Points and Sign Changes in the Second Derivative 316

The Product Rule 317

The Quotient Rule 317

Linear Approximation 317

"PSST," Here's a Good Way to Remember the Derivatives of Trig Functions 317

Index 319