Calculus Essentials For Dummies

Title Page......Page 3
Copyright Page......Page 4
Table of Contents
......Page 7
About This Book......Page 11
Foolish Assumptions......Page 12
Where to Go from Here......Page 13
So What Is Calculus Already?......Page 15
Real-World Examples of Calculus......Page 17
Differentiation......Page 18
Integration......Page 19
Limits: Math microscopes......Page 21
What happens when you zoom in......Page 22
Three functions with one limit......Page 25
One-sided limits......Page 27
Limits and horizontal asymptotes......Page 28
Instantaneous speed......Page 29
Limits and Continuity......Page 31
The hole exception......Page 32
Limits to memorize......Page 35
“Real” Limit Problems......Page 36
Conjugate multiplication......Page 37
Miscellaneous algebra......Page 38
Limits at Infinity......Page 39
Horizontal asymptotes......Page 40
Solving limits at infinity......Page 41
Chapter 4 Differentiation Orientation......Page 43
The Derivative: It’s Just Slope......Page 44
The slope of a line......Page 45
Calculus on the playground......Page 46
The rate-slope connection......Page 48
The Derivative of a Curve......Page 49
The Difference Quotient......Page 50
Average and Instantaneous Rate......Page 56
Three Cases Where the Derivative Does Not Exist......Page 57
The power rule......Page 59
The constant multiple rule......Page 60
The sum and difference rules......Page 61
Exponential and logarithmic functions......Page 62
The product and quotient rules......Page 63
The chain rule......Page 64
Differentiating Implicitly......Page 69
A Calculus Road Trip......Page 71
Finding the critical numbers......Page 73
The First Derivative Test......Page 75
The Second Derivative Test......Page 76
Finding Absolute Extrema on a Closed Interval......Page 79
Finding Absolute Extrema over a Function’s Entire Domain......Page 81
Concavity and Inflection Points......Page 83
Graphs of Derivatives......Page 85
The Mean Value Theorem......Page 88
The maximum area of a corral......Page 91
Position, Velocity, and Acceleration......Page 93
Velocity versus speed......Page 94
Maximum and minimum height......Page 96
Total displacement......Page 97
Total distance traveled......Page 98
Acceleration......Page 99
Tying it all together......Page 100
A calculus crossroads......Page 101
Filling up a trough......Page 104
Linear Approximation......Page 107
Integration: Just Fancy Addition......Page 111
Finding the Area under a Curve......Page 113
Approximating area with left sums......Page 115
Approximating area with right sums......Page 118
Approximating area with midpoint sums......Page 120
Summing up the basics......Page 122
Writing Riemann sums with sigma notation......Page 123
Finding Exact Area with the Definite Integral......Page 126
Antidifferentiation: Reverse Differentiation......Page 129
The Annoying Area Function......Page 131
The Fundamental Theorem......Page 134
Fundamental Theorem: Take Two......Page 136
Reverse rules......Page 138
Guess and check......Page 140
Substitution......Page 142
Integration by Parts......Page 147
Picking your u......Page 149
Sines and cosines......Page 151
Secants and tangents......Page 154
Trigonometric Substitution......Page 157
Case 1: Tangents......Page 158
Case 2: Sines......Page 160
Case 3: Secants......Page 161
Case 1: The denominator contains only linear factors......Page 162
Case 2: The denominator contains unfactorable quadratic factors......Page 163
Equating coefficients......Page 165
Chapter 11 Using the Integral to Solve Problems......Page 167
The Mean Value Theorem for Integrals and Average Value......Page 168
The Area between Two Curves......Page 170
The meat-slicer method......Page 172
The disk method......Page 173
The washer method......Page 175
The matryoshka doll method......Page 176
Arc Length......Page 178
Improper integrals with vertical asymptotes......Page 181
Improper integrals with infinite limits of integration......Page 183
SohCahToa......Page 185
Your Sunglasses......Page 186
Index......Page 187
EULA......Page 195