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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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How to Use This Book | |

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Foolish Assumptions | |

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How This Book Is Organized | |

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Icons Used in This Book | |

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Where to Go from Here | |

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An Overview of Calculus | |

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What Is Calculus? | |

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What Calculus Is Not | |

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So What Is Calculus Already? | |

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Real-World Examples of Calculus | |

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The Two Big Ideas of Calculus: Differentiation and Integration | |

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Defining Differentiation | |

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Investigating Integration | |

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Sorting Out Infinite Series | |

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Why Calculus Works | |

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The Limit Concept: A Mathematical Microscope | |

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What Happens When You Zoom In | |

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Two Caveats--or Precision, Preschmidgen | |

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Warming Up with Calculus Prerequisites | |

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Pre-Algebra and Algebra Review | |

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Fine-Tuning Your Fractions | |

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Absolute Value: Absolutely Easy | |

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Empowering Your Powers | |

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Rooting for Roots | |

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Logarithms--This Is Not an Event at a Lumberjack Competition | |

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Factoring Schmactoring, When Am I Ever Going to Need It? | |

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Solving Quadratic Equations | |

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Funky Functions and Their Groovy Graphs | |

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What Is a Function? | |

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What Does a Function Look Like? | |

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Common Functions and Their Graphs | |

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Inverse Functions | |

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Shifts, Reflections, Stretches, and Shrinks | |

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The Trig Tango | |

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Studying Trig at Camp SohCahToa | |

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Two Special Right Triangles | |

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Circling the Enemy with the Unit Circle | |

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Graphing Sine, Cosine, and Tangent | |

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Inverse Trig Functions | |

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Identifying with Trig Identities | |

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Limits | |

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Limits and Continuity | |

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Take It to the Limit--Not | |

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Linking Limits and Continuity | |

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The 33333 Limit Mnemonic | |

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Evaluating Limits | |

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Easy Limits | |

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The "Real Deal" Limit Problems | |

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Evaluating Limits at Infinity | |

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Differentiation | |

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Differentiation Orientation | |

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Differentiating: It's Just Finding the Slope | |

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The Derivative: It's Just a Rate | |

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The Derivative of a Curve | |

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The Difference Quotient | |

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Average Rate and Instantaneous Rate | |

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To Be or Not to Be? Three Cases Where the Derivative Does Not Exist | |

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Differentiation Rules--Yeah, Man, It Rules | |

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Basic Differentiation Rules | |

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Differentiation Rules for Experts--Oh, Yeah, I'm a Calculus Wonk | |

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Differentiating Implicity | |

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Getting into the Rhythm with Logarithmic Differentiation | |

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Differentiating Inverse Functions | |

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Scaling the Heights of Higher Order Derivatives | |

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Differentiation and the Shape of Curves | |

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Taking a Calculus Road Trip | |

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Finding Local Extrema--My Ma, She's Like, Totally Extreme | |

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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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Locating Concavity and Inflection Points | |

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Looking at Graphs of Derivatives Till They Derive You Crazy | |

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The Mean Value Theorem--GRRRRR | |

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Your Problems Are Solved: Differentiation to the Rescue! | |

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Getting the Most (or Least) Out of Life: Optimization Problems | |

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Yo-Yo a Go-Go: Position, Velocity, and Acceleration | |

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Related Rates--They Rate, Relatively | |

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Tangents and Normals: Joined at the Hip | |

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Straight Shooting with Linear Approximations | |

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Business and Economics Problems | |

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Integration and Infinite Series | |

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Intro to Integration and Approximating Area | |

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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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Getting Fancy with Summation Notation | |

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Finding Exact Area with the Definite Integral | |

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Approximating Area with the Trapezoid Rule and Simpson's Rule | |

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Integration: It's Backwards Differentiation | |

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Antidifferentiation--That's Differentiation in Reverse | |

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Vocabulary, Voshmabulary: What Difference Does It Make? | |

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The Annoying Area Function | |

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The Power and the Glory of the Fundamental Theorem of Calculus | |

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The Fundamental Theorem of Calculus: Take Two | |

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Finding Antiderivatives: Three Basic Techniques | |

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Finding Area with Substitution Problems | |

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Integration Techniques for Experts | |

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Integration by Parts: Divide and Conquer | |

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Tricky Trig Integrals | |

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Your Worst Nightmare: Trigonometric Substitution | |

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The As, Bs, and Cxs of Partial Fractions | |

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Forget Dr. Phil: Use 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--Double the Fun | |

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Finding the Volumes of Weird Solids | |

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Analyzing Arc Length | |

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Surfaces of Revolution--Pass the Bottle 'Round | |

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L'Hopital's Rule: Calculus for the Sick | |

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Improper Integrals: Just Look at the Way That Integral Is Holding Its Fork! | |

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Infinite Series | |

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Sequences and Series: What They're All About | |

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Convergence or Divergence? That Is the Question | |

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Alternating Series | |

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Keeping All the Tests Straight | |

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The Part of Tens | |

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Ten Things to Remember | |

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Your Sunglasses | |

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a[superscript 2] - b[superscript 2] = (a - b)(a + b) | |

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

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Anything[superscript 0] = 1 | |

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SohCahToa | |

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Trigonometric Values for 30, 45, and 60 Degree Angles | |

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sin[superscript 2 theta] + cos[superscript 2 theta] = 1 | |

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The Product Rule | |

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The Quotient Rule | |

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Where You Put Your Keys | |

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Ten Things to Forget | |

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(a + b)[superscript 2] = a[superscript 2] + b[superscript 2]--Wrong! | |

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[radical]a[superscript 2] + b[superscript 2] = a + b--Wrong! | |

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Slope = x[subscript 2] - x[subscript 1]/y[subscript 2] - y[subscript 1]--Wrong! | |

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3a + b/3a + c = b/c--Wrong! | |

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d/dx[pi superscript 3] = 3[pi superscript 2]--Wrong! | |

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If k Is a Constant, d/dx kx = k'x + kx'--Wrong! | |

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The Quotient Rule Is d/dx (u/v) = v'u - vu'/v[superscript 2]--Wrong! | |

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[function of] x[superscript 2] dx = 1/3x[superscript 3]--Wrong! | |

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[function of] (sinx) dx = cosx + C--Wrong! | |

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Green's Theorem | |

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Ten Things You Can't Get Away With | |

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Give Two Answers on Exam Questions | |

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Write Illegibly on Exams | |

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Don't Show Your Work on Exams | |

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Don't Do All of the Exam Problems | |

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Blame Your Study Partner for Your Low Exam Grade | |

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Tell Your Teacher That You Need an "A" in Calculus to Impress Your Significant Other | |

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Complain That Early-Morning Exams Are Unfair Because You're Not a "Morning Person" | |

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Protest the Whole Idea of Grades | |

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Pull the Fire Alarm During an Exam | |

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Use This Book as an Excuse | |

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Index | |