Calculus Essentials For Dummies

by Ryan, Mark

• ISBN13: 9780470618356
• ISBN10: 0470618353
• Format: Paperback
• Copyright: 2010-05-17
• Publisher: For Dummies

Summary

Many colleges and universities require students to take at least one math course, and Calculus I is often the chosen option. Calculus Essentials For Dummies provides explanations of key concepts for students who may have taken calculus in high school and want to review the most important concepts as they gear up for a faster-paced college course. Free of review and ramp-up material, Calculus Essentials For Dummies sticks to the point with content focused on key topics only. It provides discrete explanations of critical concepts taught in a typical two-semester high school calculus class or a college level Calculus I course, from limits and differentiation to integration and infinite series. This guide is also a perfect reference for parents who need to review critical calculus concepts as they help high school students with homework assignments, as well as for adult learners headed back into the classroom who just need a refresher of the core concepts.The Essentials For Dummies Series Dummies is proud to present our new series, The Essentials For Dummies. Now students who are prepping for exams, preparing to study new material, or who just need a refresher can have a concise, easy-to-understand review guide that covers an entire course by concentrating solely on the most important concepts. From algebra and chemistry to grammar and Spanish, our expert authors focus on the skills students most need to succeed in a subject.

Author Biography

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.

Table of Contents

Introduction	p. 1
About This Book	p. 1
Conventions Used in This Book	p. 2
Foolish Assumptions	p. 2
Icons Used in This Book	p. 3
Where to Go from Here	p. 3
Calculus: No Big Deal	p. 5
So What Is Calculus Already?	p. 5
Real-World Examples of Calculus	p. 7
Differentiation	p. 8
Integration	p. 9
Why Calculus Works	p. 11
Limits: Math microscopes	p. 11
What happens when you zoom in	p. 12
Limits and Continuity	p. 15
Taking It to the Limit	p. 15
Three functions with one limit	p. 15
One-sided limits	p. 17
Limits and vertical asymptotes	p. 18
Limits and horizontal asymptotes	p. 19
Instantaneous speed	p. 19
Limits and Continuity	p. 22
The hole exception	p. 23
Evaluating Limits	p. 25
Easy Limits	p. 25
Limits to memorize	p. 25
Plug-and-chug limits	p. 26
"Real" Limit Problems	p. 26
Factoring	p. 27
Conjugate multiplication	p. 27
Miscellaneous algebra	p. 28
Limits at Infinity	p. 29
Horizontal asymptotes	p. 30
Solving limits at infinity	p. 31
Differentiation Orientation	p. 33
The Derivative: It's Just Slope	p. 34
The slope of a line	p. 35
The derivative of a line	p. 36
The Derivative: It's Just a Rate	p. 36
Calculus on the playground	p. 36
The rate-slope connection	p. 38
The Derivative of a Curve	p. 39
The Difference Quotient	p. 41
Average and Instantaneous Rate	p. 47
Three Cases Where the Derivative Does Not Exist	p. 48
Differentiation Rules	p. 49
Basic Differentiation Rules	p. 49
The constant rule	p. 49
The power rule	p. 49
The constant multiple rule	p. 50
The sum and difference rules	p. 51
Differentiating trig functions	p. 52
Exponential and logarithmic functions	p. 53
Derivative Rules for Experts	p. 53
The product and quotient rules	p. 53
The chain rule	p. 54
Differentiating Implicitly	p. 59
Differentiation and the Shape of Curves	p. 61
A Calculus Road Trip	p. 61
Local Extrema	p. 63
Finding the critical numbers	p. 63
The First Derivative Test	p. 65
The Second Derivative Test	p. 66
Finding Absolute Extrema on a Closed Interval	p. 69
Finding Absolute Extrema over a Function's Entire Domain	p. 71
Concavity and Inflection Points	p. 73
Graphs of Derivatives	p. 75
The Mean Value Theorem	p. 78
Differentiation Problems	p. 81
Optimization Problems	p. 81
The maximum area of a corral	p. 81
Position, Velocity, and Acceleration	p. 83
Velocity versus speed	p. 84
Maximum and minimum height	p. 86
Velocity and displacement	p. 87
Speed and distance traveled	p. 88
Acceleration	p. 89
Tying it all together	p. 90
Related Rates	p. 91
A calculus crossroads	p. 91
Filling up a trough	p. 94
Linear Approximation	p. 96
Introduction to Integration	p. 101
Integration: Just Fancy Addition	p. 101
Finding the Area under a Curve	p. 103
Dealing with negative area	p. 105
Approximating Area	p. 105
Approximating area with left sums	p. 105
Approximating area with right sums	p. 109
Approximating area with midpoint sums	p. 111
Summation Notation	p. 113
Summing up the basics	p. 113
Writing Riemann sums with sigma notation	p. 114
Finding Exact Area with the Definite Integral	p. 117
Integration: Backwards Differentiation	p. 119
Antidifferentiation: Reverse Differentiation	p. 119
The Annoying Area Function	p. 121
The Fundamental Theorem	p. 124
Fundamental Theorem: Take Two	p. 126
Antiderivatives: Basic Techniques	p. 128
Reverse rules	p. 128
Guess and check	p. 131
Substitution	p. 132
Integration for Experts	p. 137
Integration by Parts	p. 137
Picking your u	p. 139
Tricky Trig Integrals	p. 141
Sines and cosines	p. 142
Secants and tangents	p. 144
Cosecants and cotangents	p. 147
Trigonometric Substitution	p. 147
Tangents	p. 148
Sines	p. 150
Secants	p. 152
Partial Fractions	p. 152
The denominator contains only linear factors	p. 152
The denominator contains unfactorable quadratic factors	p. 153
The denominator contains repeated factors	p. 155
Equating coefficients	p. 156
Using the Integral to Solve Problems	p. 157
The Mean Value Theorem for Integrals and Average Value	p. 158
The Area between Two Curves	p. 160
Volumes of Weird Solids	p. 162
The meat-slicer method	p. 162
The disk method	p. 164
The washer method	p. 166
The matryoshka doll method	p. 168
Arc Length	p. 170
Improper Integrals	p. 172
Improper integrals with vertical asymptotes	p. 173
Improper integrals with infinite limits of integration	p. 175
Eight Things to Remember	p. 177
a2 - b2 = (a - b)(a + b)	p. 177
0/5 = 0 But 5/0 Is Undefined	p. 177
SohCahToa	p. 177
Trig Values to Know	p. 178
sin2 ¿ + cos2 ¿ = 1	p. 178
The Product Rule	p. 178
The Quotient Rule	p. 178
Your Sunglasses	p. 178
Index	p. 179
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