Calculus Essentials for Dummies

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  • Format: Paperback
  • Copyright: 2019-05-14
  • Publisher: For Dummies

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Supplemental Materials

What is included with this book?


Calculus Essentials For Dummies (9781119591207) was previously published as Calculus Essentials For Dummies (9780470618356). While this version features a new Dummies cover and design, the content is the same as the prior release and should not be considered a new or updated product.


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 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

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