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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
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. 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