Calculus Essentials For Dummies

Mark Ryan is the owner of The Math Center in Chicago, Illinois, where he teaches students in all levels of mathematics, from pre-algebra to calculus. He is the author of Calculus For Dummies and Geometry For Dummies.

• Introduction 1About This Book 1Conventions Used in This Book 2Foolish Assumptions 2Icons Used in This Book 3Where to Go from Here 3Chapter 1: Calculus: No Big Deal 5So What is Calculus Already? 5Real-World Examples of Calculus 7Differentiation 8Integration 9Why Calculus Works 11Limits: Math microscopes 11What happens when you zoom in 12Chapter 2: Limits and Continuity 15Taking it to the Limit 15Three functions with one limit 15One-sided limits 17Limits and vertical asymptotes 18Limits and horizontal asymptotes 18Instantaneous speed 19Limits and Continuity 21The hole exception 22Chapter 3: Evaluating Limits 25Easy Limits 25Limits to memorize 25Plug-and-chug limits 26“Real” Limit Problems 26Factoring 27Conjugate multiplication 27Miscellaneous algebra 28Limits at Infinity 29Horizontal asymptotes 30Solving limits at infinity 31Chapter 4: Differentiation Orientation 33The Derivative: It’s Just Slope 34The slope of a line 35The derivative of a line 36The Derivative: It’s Just a Rate 36Calculus on the playground 36The rate-slope connection 38The Derivative of a Curve 39The Difference Quotient 40Average and Instantaneous Rate 46Three Cases Where the Derivative Does Not Exist 47Chapter 5: Differentiation Rules 49Basic Differentiation Rules 49The constant rule 49The power rule 49The constant multiple rule 50The sum and difference rules 51Differentiating trig functions 52Exponential and logarithmic functions 52Derivative Rules for Experts 53The product and quotient rules 53The chain rule 54Differentiating Implicitly 59Chapter 6: Differentiation and the Shape of Curves 61A Calculus Road Trip 61Local Extrema 63Finding the critical numbers 63The First Derivative Test 65The Second Derivative Test 66Finding Absolute Extrema on a Closed Interval 69Finding Absolute Extrema over a Function’s Entire Domain 71Concavity and Inflection Points 73Graphs of Derivatives 75The Mean Value Theorem 78Chapter 7: Differentiation Problems 81Optimization Problems 81The maximum area of a corral 81Position, Velocity, and Acceleration 83Velocity versus speed 84Maximum and minimum height 86Velocity and displacement 87Speed and distance travelled 88Acceleration 89Tying it all together 90Related Rates 91A calculus crossroads 91Filling up a trough 94Linear Approximation 97Chapter 8: Introduction to Integration 101Integration: Just Fancy Addition 101Finding the Area under a Curve 103Dealing with negative area 105Approximating Area 105Approximating area with left sums 105Approximating area with right sums 108Approximating area with midpoint sums 110Summation Notation 112Summing up the basics 112Writing Riemann sums with sigma notation 113Finding Exact Area with the Definite Integral 116Chapter 9: Integration: Backwards Differentiation 119Antidifferentiation: Reverse Differentiation 119The Annoying Area Function 121The Fundamental Theorem 124Fundamental Theorem: Take Two 126Antiderivatives: Basic Techniques 128Reverse rules 128Guess and check 130Substitution 132Chapter 10: Integration for Experts 137Integration by Parts 137Picking your u 139Tricky Trig Integrals 141Sines and cosines 141Secants and tangents 144Cosecants and cotangents 147Trigonometric Substitution 147Case 1: Tangents 148Case 2: Sines 150Case 3: Secants 151Partial Fractions 152Case 1: The denominator contains only linear factors 152Case 2: The denominator contains unfactorable quadratic factors 153Case 3: The denominator contains repeated factors 155Equating coefficients 155Chapter 11: Using the Integral to Solve Problems 157The Mean Value Theorem for Integrals and Average Value 158The Area between Two Curves 160Volumes of Weird Solids 162The meat-slicer method 162The disk method 163The washer method 165The matryoshka doll method 166Arc Length 168Improper Integrals 171Improper integrals with vertical asymptotes 171Improper integrals with infinite limits of integration 173Chapter 12: Eight Things to Remember 175a2- b2 = (a - b)(a + b) 1750/5 = 0 But 5/0 is Undefined 175SohCahToa 175Trig Values to Know 176sin2ϴ + cos2ϴ = 1 176The Product Rule 176The Quotient Rule 176Your Sunglasses 176Index 177