Calculus Essentials For Dummies

Introduction 1 About This Book 1 Conventions Used in This Book 2 Foolish Assumptions 2 Icons Used in This Book 3 Where to Go from Here 3 Chapter 1: Calculus: No Big Deal 5 So What is Calculus Already? 5 Real-World Examples of Calculus 7 Differentiation 8 Integration 9 Why Calculus Works 11 Limits: Math microscopes 11 What happens when you zoom in 12 Chapter 2: Limits and Continuity 15 Taking it to the Limit 15 Three functions with one limit 15 One-sided limits 17 Limits and vertical asymptotes 18 Limits and horizontal asymptotes 18 Instantaneous speed 19 Limits and Continuity 21 The hole exception 22 Chapter 3: Evaluating Limits 25 Easy Limits 25 Limits to memorize 25 Plug-and-chug limits 26 “Real” Limit Problems 26 Factoring 27 Conjugate multiplication 27 Miscellaneous algebra 28 Limits at Infinity 29 Horizontal asymptotes 30 Solving limits at infinity 31 Chapter 4: Differentiation Orientation 33 The Derivative: It’s Just Slope 34 The slope of a line 35 The derivative of a line 36 The Derivative: It’s Just a Rate 36 Calculus on the playground 36 The rate-slope connection 38 The Derivative of a Curve 39 The Difference Quotient 40 Average and Instantaneous Rate 46 Three Cases Where the Derivative Does Not Exist 47 Chapter 5: Differentiation Rules 49 Basic Differentiation Rules 49 The constant rule 49 The power rule 49 The constant multiple rule 50 The sum and difference rules 51 Differentiating trig functions 52 Exponential and logarithmic functions 52 Derivative Rules for Experts 53 The product and quotient rules 53 The chain rule 54 Differentiating Implicitly 59 Chapter 6: Differentiation and the Shape of Curves 61 A Calculus Road Trip 61 Local Extrema 63 Finding the critical numbers 63 The First Derivative Test 65 The Second Derivative Test 66 Finding Absolute Extrema on a Closed Interval 69 Finding Absolute Extrema over a Function’s Entire Domain 71 Concavity and Inflection Points 73 Graphs of Derivatives 75 The Mean Value Theorem 78 Chapter 7: Differentiation Problems 81 Optimization Problems 81 The maximum area of a corral 81 Position, Velocity, and Acceleration 83 Velocity versus speed 84 Maximum and minimum height 86 Velocity and displacement 87 Speed and distance travelled 88 Acceleration 89 Tying it all together 90 Related Rates 91 A calculus crossroads 91 Filling up a trough 94 Linear Approximation 97 Chapter 8: Introduction to Integration 101 Integration: Just Fancy Addition 101 Finding the Area under a Curve 103 Dealing with negative area 105 Approximating Area 105 Approximating area with left sums 105 Approximating area with right sums 108 Approximating area with midpoint sums 110 Summation Notation 112 Summing up the basics 112 Writing Riemann sums with sigma notation 113 Finding Exact Area with the Definite Integral 116 Chapter 9: Integration: Backwards Differentiation 119 Antidifferentiation: Reverse Differentiation 119 The Annoying Area Function 121 The Fundamental Theorem 124 Fundamental Theorem: Take Two 126 Antiderivatives: Basic Techniques 128 Reverse rules 128 Guess and check 130 Substitution 132 Chapter 10: Integration for Experts 137 Integration by Parts 137 Picking your u 139 Tricky Trig Integrals 141 Sines and cosines 141 Secants and tangents 144 Cosecants and cotangents 147 Trigonometric Substitution 147 Case 1: Tangents 148 Case 2: Sines 150 Case 3: Secants 151 Partial Fractions 152 Case 1: The denominator contains only linear factors 152 Case 2: The denominator contains unfactorable quadratic factors 153 Case 3: The denominator contains repeated factors 155 Equating coefficients 155 Chapter 11: Using the Integral to Solve Problems 157 The Mean Value Theorem for Integrals and Average Value 158 The Area between Two Curves 160 Volumes of Weird Solids 162 The meat-slicer method 162 The disk method 163 The washer method 165 The matryoshka doll method 166 Arc Length 168 Improper Integrals 171 Improper integrals with vertical asymptotes 171 Improper integrals with infinite limits of integration 173 Chapter 12: Eight Things to Remember 175 a2- b2 = (a - b)(a + b) 175 0/5 = 0 But 5/0 is Undefined 175 SohCahToa 175 Trig Values to Know 176 sin2ϴ + cos2ϴ = 1 176 The Product Rule 176 The Quotient Rule 176 Your Sunglasses 176 Index 177