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Introduction | |

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About This Book | |

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Conventions Used in This Book | |

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Foolish Assumptions | |

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Icons Used in This Book | |

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Where to Go from Here | |

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Calculus: No Big Deal | |

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So What Is Calculus Already? | |

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Real-World Examples of Calculus | |

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Differentiation | |

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Integration | |

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Why Calculus Works | |

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Limits: Math microscopes | |

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What happens when you zoom in | |

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Limits and Continuity | |

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Taking It to the Limit | |

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Three functions with one limit | |

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One-sided limits | |

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Limits and vertical asymptotes | |

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Limits and horizontal asymptotes | |

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Instantaneous speed | |

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Limits and Continuity | |

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The hole exception | |

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Evaluating Limits | |

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Easy Limits | |

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Limits to memorize | |

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Plug-and-chug limits | |

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"Real" Limit Problems | |

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Factoring | |

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Conjugate multiplication | |

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Miscellaneous algebra | |

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Limits at Infinity | |

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Horizontal asymptotes | |

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Solving limits at infinity | |

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Differentiation Orientation | |

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The Derivative: It's Just Slope | |

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The slope of a line | |

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The derivative of a line | |

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The Derivative: It's Just a Rate | |

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Calculus on the playground | |

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The rate-slope connection | |

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The Derivative of a Curve | |

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The Difference Quotient | |

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Average and Instantaneous Rate | |

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Three Cases Where the Derivative Does Not Exist | |

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Differentiation Rules | |

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Basic Differentiation Rules | |

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The constant rule | |

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The power rule | |

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The constant multiple rule | |

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The sum and difference rules | |

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Differentiating trig functions | |

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Exponential and logarithmic functions | |

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Derivative Rules for Experts | |

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The product and quotient rules | |

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The chain rule | |

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Differentiating Implicitly | |

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Differentiation and the Shape of Curves | |

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A Calculus Road Trip | |

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Local Extrema | |

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Finding the critical numbers | |

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The First Derivative Test | |

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The Second Derivative Test | |

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Finding Absolute Extrema on a Closed Interval | |

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Finding Absolute Extrema over a Function's Entire Domain | |

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Concavity and Inflection Points | |

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Graphs of Derivatives | |

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The Mean Value Theorem | |

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Differentiation Problems | |

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Optimization Problems | |

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The maximum area of a corral | |

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Position, Velocity, and Acceleration | |

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Velocity versus speed | |

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Maximum and minimum height | |

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Velocity and displacement | |

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Speed and distance traveled | |

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Acceleration | |

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Tying it all together | |

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Related Rates | |

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A calculus crossroads | |

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Filling up a trough | |

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Linear Approximation | |

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Introduction to Integration | |

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Integration: Just Fancy Addition | |

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Finding the Area under a Curve | |

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Dealing with negative area | |

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Approximating Area | |

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Approximating area with left sums | |

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Approximating area with right sums | |

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Approximating area with midpoint sums | |

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Summation Notation | |

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Summing up the basics | |

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Writing Riemann sums with sigma notation | |

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Finding Exact Area with the Definite Integral | |

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Integration: Backwards Differentiation | |

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Antidifferentiation: Reverse Differentiation | |

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The Annoying Area Function | |

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The Fundamental Theorem | |

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Fundamental Theorem: Take Two | |

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Antiderivatives: Basic Techniques | |

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Reverse rules | |

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Guess and check | |

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Substitution | |

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Integration for Experts | |

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Integration by Parts | |

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Picking your u | |

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Tricky Trig Integrals | |

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Sines and cosines | |

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Secants and tangents | |

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Cosecants and cotangents | |

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Trigonometric Substitution | |

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Tangents | |

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Sines | |

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Secants | |

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Partial Fractions | |

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The denominator contains only linear factors | |

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The denominator contains unfactorable quadratic factors | |

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The denominator contains repeated factors | |

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Equating coefficients | |

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Using the Integral to Solve Problems | |

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The Mean Value Theorem for Integrals and Average Value | |

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The Area between Two Curves | |

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Volumes of Weird Solids | |

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The meat-slicer method | |

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The disk method | |

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The washer method | |

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The matryoshka doll method | |

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Arc Length | |

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Improper Integrals | |

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Improper integrals with vertical asymptotes | |

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Improper integrals with infinite limits of integration | |

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Eight Things to Remember | |

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a<sup>2</sup> - b<sup>2</sup> = (a - b)(a + b) | |

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0/5 = 0 But 5/0 Is Undefined | |

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SohCahToa | |

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Trig Values to Know | |

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sin<sup>2</sup> ï¿½ + cos<sup>2</sup> ï¿½ = 1 | |

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The Product Rule | |

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The Quotient Rule | |

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Your Sunglasses | |

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Index | |