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
sin2? + cos2? = 1 176
The Product Rule 176
The Quotient Rule 176
Your Sunglasses 176
Index 177
