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

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

Select Guide Rating

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

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