Details

<p><b>Introduction 1</b></p> <p>About This Book 1</p> <p>Foolish Assumptions 2</p> <p>Icons Used in This Book 2</p> <p>Beyond the Book 3</p> <p>Where to Go from Here 3</p> <p><b>Part 1: Pre-Calculus Review 5</b></p> <p><b>Chapter 1: Getting Down to Basics: Algebra and Geometry 7</b></p> <p>Fraction Frustration 7</p> <p>Misc. Algebra: You Know, Like Miss South Carolina 9</p> <p>Geometry: When Am I Ever Going to Need It? 11</p> <p>Solutions for This Easy, Elementary Stuff 16</p> <p><b>Chapter 2: Funky Functions and Tricky Trig 25</b></p> <p>Figuring Out Your Functions 25</p> <p>Trigonometric Calisthenics 29</p> <p>Solutions to Functions and Trigonometry 33</p> <p><b>Part 2: Limits and Continuity 41</b></p> <p><b>Chapter 3: A Graph Is Worth a Thousand Words: Limits and Continuity 43</b></p> <p>Digesting the Definitions: Limit and Continuity 44</p> <p>Taking a Closer Look: Limit and Continuity Graphs 46</p> <p>Solutions for Limits and Continuity 50</p> <p><b>Chapter 4: Nitty-Gritty Limit Problems 53</b></p> <p>Solving Limits with Algebra 54</p> <p>Pulling Out Your Calculator: Useful “Cheating” 59</p> <p>Making Yourself a Limit Sandwich 61</p> <p>Into the Great Beyond: Limits at Infinity 63</p> <p>Solutions for Problems with Limits 67</p> <p><b>Part 3: Differentiation 77</b></p> <p><b>Chapter 5: Getting the Big Picture: Differentiation Basics 79</b></p> <p>The Derivative: A Fancy Calculus Word for Slope and Rate 79</p> <p>The Handy-Dandy Difference Quotient 81</p> <p>Solutions for Differentiation Basics 84</p> <p><b>Chapter 6: Rules, Rules, Rules: The Differentiation Handbook 89</b></p> <p>Rules for Beginners 89</p> <p>Giving It Up for the Product and Quotient Rules 92</p> <p>Linking Up with the Chain Rule 94</p> <p>What to Do with Y’s: Implicit Differentiation 98</p> <p>Getting High on Calculus: Higher Order Derivatives 101</p> <p>Solutions for Differentiation Problems 103</p> <p><b>Chapter 7: Analyzing Those Shapely Curves with the Derivative 117</b></p> <p>The First Derivative Test and Local Extrema 117</p> <p>The Second Derivative Test and Local Extrema 120</p> <p>Finding Mount Everest: Absolute Extrema 122</p> <p>Smiles and Frowns: Concavity and Inflection Points 126</p> <p>The Mean Value Theorem: Go Ahead, Make My Day 129</p> <p>Solutions for Derivatives and Shapes of Curves 131</p> <p><b>Chapter 8: Using Differentiation to Solve Practical Problems 147</b></p> <p>Optimization Problems: From Soup to Nuts 147</p> <p>Problematic Relationships: Related Rates 150</p> <p>A Day at the Races: Position, Velocity, and Acceleration 153</p> <p>Solutions to Differentiation Problem Solving 157</p> <p><b>Chapter 9: Even More Practical Applications of Differentiation 173</b></p> <p>Make Sure You Know Your Lines: Tangents and Normals 173</p> <p>Looking Smart with Linear Approximation 177</p> <p>Calculus in the Real World: Business and Economics 179</p> <p>Solutions to Differentiation Problem Solving 183</p> <p><b>Part 4: Integration and Infinite Series 191</b></p> <p><b>Chapter 10: Getting into Integration 193</b></p> <p>Adding Up the Area of Rectangles: Kid Stuff 193</p> <p>Sigma Notation and Riemann Sums: Geek Stuff 196</p> <p>Close Isn’t Good Enough: The Definite Integral and Exact Area 200</p> <p>Finding Area with the Trapezoid Rule and Simpson’s Rule 202</p> <p>Solutions to Getting into Integration 205</p> <p><b>Chapter 11: Integration: Reverse Differentiation 213</b></p> <p>The Absolutely Atrocious and Annoying Area Function 213</p> <p>Sound the Trumpets: The Fundamental Theorem of Calculus 216</p> <p>Finding Antiderivatives: The Guess-and-Check Method 219</p> <p>The Substitution Method: Pulling the Switcheroo 221</p> <p>Solutions to Reverse Differentiation Problems 225</p> <p><b>Chapter 12: Integration Rules for Calculus Connoisseurs 229</b></p> <p>Integration by Parts: Here’s How u du It 229</p> <p>Transfiguring Trigonometric Integrals 233</p> <p>Trigonometric Substitution: It’s Your Lucky Day! 235</p> <p>Partaking of Partial Fractions 237</p> <p>Solutions for Integration Rules 241</p> <p><b>Chapter 13: Who Needs Freud? Using the Integral to Solve Your Problems</b><b> 255</b></p> <p>Finding a Function’s Average Value 255</p> <p>Finding the Area between Curves 256</p> <p>Volumes of Weird Solids: No, You’re Never Going to Need This 258</p> <p>Arc Length and Surfaces of Revolution 265</p> <p>Solutions to Integration Application Problems 268</p> <p><b>Chapter 14: Infinite (Sort of) Integrals</b><b> 277</b></p> <p>Getting Your Hopes Up with L’Hôpital’s Rule 278</p> <p>Disciplining Those Improper Integrals 280</p> <p>Solutions to Infinite (Sort of) Integrals 283</p> <p><b>Chapter 15: Infinite Series: Welcome to the Outer Limits 287</b></p> <p>The Nifty nth Term Test 287</p> <p>Testing Three Basic Series 289</p> <p>Apples and Oranges . . . and Guavas: Three Comparison Tests 291</p> <p>Ratiocinating the Two “R” Tests 295</p> <p>He Loves Me, He Loves Me Not: Alternating Series 297</p> <p>Solutions to Infinite Series 299</p> <p><b>Part 5: The Part of Tens 309</b></p> <p><b>Chapter 16: Ten Things about Limits, Continuity, and Infinite Series 311</b></p> <p>The 33333 Mnemonic 311</p> <p>First 3 over the “l”: 3 parts to the definition of a limit 312</p> <p>Fifth 3 over the “l”: 3 cases where a limit fails to exist 312</p> <p>Second 3 over the “i”: 3 parts to the definition of continuity 312</p> <p>Fourth 3 over the “i”: 3 cases where continuity fails to exist 312</p> <p>Third 3 over the “m”: 3 cases where a derivative fails to exist 313</p> <p>The 13231 Mnemonic 313</p> <p>First 1: The nth term test of divergence 313</p> <p>Second 1: The nth term test of convergence for alternating series 313</p> <p>First 3: The three tests with names 313</p> <p>Second 3: The three comparison tests 314</p> <p>The 2 in the middle: The two R tests 314</p> <p><b>Chapter 17: Ten Things You Better Remember about Differentiation 315</b></p> <p>The Difference Quotient 315</p> <p>The First Derivative Is a Rate 315</p> <p>The First Derivative Is a Slope 316</p> <p>Extrema, Sign Changes, and the First Derivative 316</p> <p>The Second Derivative and Concavity 316</p> <p>Inflection Points and Sign Changes in the Second Derivative 316</p> <p>The Product Rule 317</p> <p>The Quotient Rule 317</p> <p>Linear Approximation 317</p> <p>“PSST,” Here’s a Good Way to Remember the Derivatives of Trig Functions 317</p> <p>Index 319</p>

<p><b>Mark Ryan</b> 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.

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