| |
| |
| Introduction | |
| |
| |
| About This Book | |
| |
| |
| Conventions Used in This Book | |
| |
| |
| How to Use This Book | |
| |
| |
| Foolish Assumptions | |
| |
| |
| How This Book Is Organized | |
| |
| |
| Icons Used in This Book | |
| |
| |
| Where to Go from Here | |
| |
| |
| |
| An Overview of Calculus | |
| |
| |
| |
| What Is Calculus? | |
| |
| |
| What Calculus Is Not | |
| |
| |
| So What Is Calculus Already? | |
| |
| |
| Real-World Examples of Calculus | |
| |
| |
| |
| The Two Big Ideas of Calculus: Differentiation and Integration | |
| |
| |
| Defining Differentiation | |
| |
| |
| Investigating Integration | |
| |
| |
| Sorting Out Infinite Series | |
| |
| |
| |
| Why Calculus Works | |
| |
| |
| The Limit Concept: A Mathematical Microscope | |
| |
| |
| What Happens When You Zoom In | |
| |
| |
| Two Caveats--or Precision, Preschmidgen | |
| |
| |
| |
| Warming Up with Calculus Prerequisites | |
| |
| |
| |
| Pre-Algebra and Algebra Review | |
| |
| |
| Fine-Tuning Your Fractions | |
| |
| |
| Absolute Value: Absolutely Easy | |
| |
| |
| Empowering Your Powers | |
| |
| |
| Rooting for Roots | |
| |
| |
| Logarithms--This Is Not an Event at a Lumberjack Competition | |
| |
| |
| Factoring Schmactoring, When Am I Ever Going to Need It? | |
| |
| |
| Solving Quadratic Equations | |
| |
| |
| |
| Funky Functions and Their Groovy Graphs | |
| |
| |
| What Is a Function? | |
| |
| |
| What Does a Function Look Like? | |
| |
| |
| Common Functions and Their Graphs | |
| |
| |
| Inverse Functions | |
| |
| |
| Shifts, Reflections, Stretches, and Shrinks | |
| |
| |
| |
| The Trig Tango | |
| |
| |
| Studying Trig at Camp SohCahToa | |
| |
| |
| Two Special Right Triangles | |
| |
| |
| Circling the Enemy with the Unit Circle | |
| |
| |
| Graphing Sine, Cosine, and Tangent | |
| |
| |
| Inverse Trig Functions | |
| |
| |
| Identifying with Trig Identities | |
| |
| |
| |
| Limits | |
| |
| |
| |
| Limits and Continuity | |
| |
| |
| Take It to the Limit--Not | |
| |
| |
| Linking Limits and Continuity | |
| |
| |
| The 33333 Limit Mnemonic | |
| |
| |
| |
| Evaluating Limits | |
| |
| |
| Easy Limits | |
| |
| |
| The "Real Deal" Limit Problems | |
| |
| |
| Evaluating Limits at Infinity | |
| |
| |
| |
| Differentiation | |
| |
| |
| |
| Differentiation Orientation | |
| |
| |
| Differentiating: It's Just Finding the Slope | |
| |
| |
| The Derivative: It's Just a Rate | |
| |
| |
| The Derivative of a Curve | |
| |
| |
| The Difference Quotient | |
| |
| |
| Average Rate and Instantaneous Rate | |
| |
| |
| To Be or Not to Be? Three Cases Where the Derivative Does Not Exist | |
| |
| |
| |
| Differentiation Rules--Yeah, Man, It Rules | |
| |
| |
| Basic Differentiation Rules | |
| |
| |
| Differentiation Rules for Experts--Oh, Yeah, I'm a Calculus Wonk | |
| |
| |
| Differentiating Implicity | |
| |
| |
| Getting into the Rhythm with Logarithmic Differentiation | |
| |
| |
| Differentiating Inverse Functions | |
| |
| |
| Scaling the Heights of Higher Order Derivatives | |
| |
| |
| |
| Differentiation and the Shape of Curves | |
| |
| |
| Taking a Calculus Road Trip | |
| |
| |
| Finding Local Extrema--My Ma, She's Like, Totally Extreme | |
| |
| |
| Finding Absolute Extrema on a Closed Interval | |
| |
| |
| Finding Absolute Extrema over a Function's Entire Domain | |
| |
| |
| Locating Concavity and Inflection Points | |
| |
| |
| Looking at Graphs of Derivatives Till They Derive You Crazy | |
| |
| |
| The Mean Value Theorem--GRRRRR | |
| |
| |
| |
| Your Problems Are Solved: Differentiation to the Rescue! | |
| |
| |
| Getting the Most (or Least) Out of Life: Optimization Problems | |
| |
| |
| Yo-Yo a Go-Go: Position, Velocity, and Acceleration | |
| |
| |
| Related Rates--They Rate, Relatively | |
| |
| |
| Tangents and Normals: Joined at the Hip | |
| |
| |
| Straight Shooting with Linear Approximations | |
| |
| |
| Business and Economics Problems | |
| |
| |
| |
| Integration and Infinite Series | |
| |
| |
| |
| Intro to Integration and Approximating Area | |
| |
| |
| Integration: Just Fancy Addition | |
| |
| |
| Finding the Area under a Curve | |
| |
| |
| Dealing with Negative Area | |
| |
| |
| Approximating Area | |
| |
| |
| Getting Fancy with Summation Notation | |
| |
| |
| Finding Exact Area with the Definite Integral | |
| |
| |
| Approximating Area with the Trapezoid Rule and Simpson's Rule | |
| |
| |
| |
| Integration: It's Backwards Differentiation | |
| |
| |
| Antidifferentiation--That's Differentiation in Reverse | |
| |
| |
| Vocabulary, Voshmabulary: What Difference Does It Make? | |
| |
| |
| The Annoying Area Function | |
| |
| |
| The Power and the Glory of the Fundamental Theorem of Calculus | |
| |
| |
| The Fundamental Theorem of Calculus: Take Two | |
| |
| |
| Finding Antiderivatives: Three Basic Techniques | |
| |
| |
| Finding Area with Substitution Problems | |
| |
| |
| |
| Integration Techniques for Experts | |
| |
| |
| Integration by Parts: Divide and Conquer | |
| |
| |
| Tricky Trig Integrals | |
| |
| |
| Your Worst Nightmare: Trigonometric Substitution | |
| |
| |
| The As, Bs, and Cxs of Partial Fractions | |
| |
| |
| |
| Forget Dr. Phil: Use the Integral to Solve Problems | |
| |
| |
| The Mean Value Theorem for Integrals and Average Value | |
| |
| |
| The Area between Two Curves--Double the Fun | |
| |
| |
| Finding the Volumes of Weird Solids | |
| |
| |
| Analyzing Arc Length | |
| |
| |
| Surfaces of Revolution--Pass the Bottle 'Round | |
| |
| |
| L'Hopital's Rule: Calculus for the Sick | |
| |
| |
| Improper Integrals: Just Look at the Way That Integral Is Holding Its Fork! | |
| |
| |
| |
| Infinite Series | |
| |
| |
| Sequences and Series: What They're All About | |
| |
| |
| Convergence or Divergence? That Is the Question | |
| |
| |
| Alternating Series | |
| |
| |
| Keeping All the Tests Straight | |
| |
| |
| |
| The Part of Tens | |
| |
| |
| |
| Ten Things to Remember | |
| |
| |
| Your Sunglasses | |
| |
| |
| a[superscript 2] - b[superscript 2] = (a - b)(a + b) | |
| |
| |
| 0/5 = 0, But 5/0 Is Undefined | |
| |
| |
| Anything[superscript 0] = 1 | |
| |
| |
| SohCahToa | |
| |
| |
| Trigonometric Values for 30, 45, and 60 Degree Angles | |
| |
| |
| sin[superscript 2 theta] + cos[superscript 2 theta] = 1 | |
| |
| |
| The Product Rule | |
| |
| |
| The Quotient Rule | |
| |
| |
| Where You Put Your Keys | |
| |
| |
| |
| Ten Things to Forget | |
| |
| |
| (a + b)[superscript 2] = a[superscript 2] + b[superscript 2]--Wrong! | |
| |
| |
| [radical]a[superscript 2] + b[superscript 2] = a + b--Wrong! | |
| |
| |
| Slope = x[subscript 2] - x[subscript 1]/y[subscript 2] - y[subscript 1]--Wrong! | |
| |
| |
| 3a + b/3a + c = b/c--Wrong! | |
| |
| |
| d/dx[pi superscript 3] = 3[pi superscript 2]--Wrong! | |
| |
| |
| If k Is a Constant, d/dx kx = k'x + kx'--Wrong! | |
| |
| |
| The Quotient Rule Is d/dx (u/v) = v'u - vu'/v[superscript 2]--Wrong! | |
| |
| |
| [function of] x[superscript 2] dx = 1/3x[superscript 3]--Wrong! | |
| |
| |
| [function of] (sinx) dx = cosx + C--Wrong! | |
| |
| |
| Green's Theorem | |
| |
| |
| |
| Ten Things You Can't Get Away With | |
| |
| |
| Give Two Answers on Exam Questions | |
| |
| |
| Write Illegibly on Exams | |
| |
| |
| Don't Show Your Work on Exams | |
| |
| |
| Don't Do All of the Exam Problems | |
| |
| |
| Blame Your Study Partner for Your Low Exam Grade | |
| |
| |
| Tell Your Teacher That You Need an "A" in Calculus to Impress Your Significant Other | |
| |
| |
| Complain That Early-Morning Exams Are Unfair Because You're Not a "Morning Person" | |
| |
| |
| Protest the Whole Idea of Grades | |
| |
| |
| Pull the Fire Alarm During an Exam | |
| |
| |
| Use This Book as an Excuse | |
| |
| |
| Index | |