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’Hopital’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
