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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> � + cos<sup>2</sup> � = 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 | |