Calculus I

Calculus I

by W. Michael Kelley
Calculus I

Calculus I

by W. Michael Kelley

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Overview

Let's face it, most students don't take calculus because they find it intellectually stimulating.

It's not . . . at least for those who come up on the wrong side of the bell curve! There they are, minding their own business, working toward some non-science related degree, when . . . BLAM! They get next semester's course schedule in the mail, and first on the list is the mother of all loathed college courses . . . CALCULUS!

Not to fear—Idiot's Guides®: Calculus I is a curriculum-based companion book created with this audience in mind. This new edition continues the tradition of taking the sting out of calculus by adding more explanatory graphs and illustrations and doubling the number of practice problems!

By the time readers are finished, they will have a solid understanding (maybe even a newfound appreciation) for this useful form of math. And with any luck, they may even be able to make sense of their textbooks and teachers.

Product Details

ISBN-13: 9781465451682
Publisher: DK
Publication date: 07/12/2016
Series: Idiot's Guides
Pages: 352
Sales rank: 653,847
Product dimensions: 7.70(w) x 9.10(h) x 0.80(d)

About the Author

W. Michael Kelley is a former award-winning calculus teacher and the author of six math books, including The Complete Idiot's Guide to Algebra, Second Edition, and The Humongous Book of Calculus Problems. Kelley received an award from the Maryland Council of Teachers of Mathematics recognizing him as an Outstanding High School Mathematics Teacher and four-years-running title of Most Popular Teacher in his home school. Kelley is also the founder and editor of calculus-help.com.

Table of Contents

Part 1 The Roots of Calculus 1

1 What Is Calculus, Anyway? 3

What's the Purpose of Calculus? 4

Finding the Slopes of Curves 4

Calculating the Area of Bizarre Shapes 4

Justifying Old Formulas 5

Calculating Complicated x-Intercepts 5

Visualizing Graphs 5

Finding the Average Value of a Function 6

Calculating Optimal Values 6

Who's Responsible for This? 7

Ancient Influences 7

Newton vs. Leibniz 9

Will I Ever Learn This? 11

2 Polish Up Your Algebra Skills 13

Walk the Line: Linear Equations 14

Common Forms of Linear Equations 14

Calculating Slope 16

Interpreting Linear Graphs 18

You've Got the Power: Exponential Rules 21

Breaking Up Is Hard to Do: Factoring Polynomials 22

Greatest Common Factor 23

Special Factoring Patterns 23

Solving Quadratic Equations 24

Method 1 Factoring 25

Method 2 Completing the Square 25

Method 3 The Quadratic Formula 26

Synthesizing the Quadratic Solution Methods 27

3 Equations, Relations, and Functions 31

What Makes a Function Tick? 31

Working with Graphs of Functions 36

Functional Symmetry 39

Graphs to Know by Heart 43

Constructing an Inverse Function 45

Parametric Equations 47

What's a Parameter? 47

Converting to Rectangular Form 48

4 Trigonometry: Last Stop Before Calculus 51

Getting Repetitive: Periodic Functions 51

Introducing the Trigonometric Functions 53

Sine (Written as y = sin x) 54

Cosine (Written as y = cos x) 54

Tangent (Written as y = tan x) 55

Cotangent (Written as y = cot x) 56

Secant (Written as y = sec x) 57

Cosecant (Written as y = csc x) 57

What's Your Sine: The Unit Circle 59

Incredibly important Identities 61

Pythagorean identities 62

Double-Angle Formulas 63

Solving Trigonometric Equations 64

Part 2 Laying the Foundation for Calculus 67

5 Take It to the Limit 69

What Is a Limit 70

Can Something Be Nothing? 71

One-Sided Limits 74

When Does a Limit Exist? 78

When Does a Limit Not Exist? 79

6 Evaluating Limits Numerically 85

The Major Methods 86

Substitution Method 86

Factoring Method 87

Conjugate Method 88

What If Sorting Works? 90

Limits and Infinity 90

Vertical Asymptotes 90

Horizontal Asymptotes 92

Special Limit Theorems 96

Evaluating Limits Graphically 97

Technology Focus: Calculating Limits 99

7 Continuity 103

What Does Continuity Look Like? 104

The Mathematical Definition of Continuity 104

Types of Discontinuity 109

Jump Discontinuity 109

Point Discontinuity 113

Infinite Essential Discontinuity 117

Removable vs. Nonremovable Discontinuity 117

The Intermediate Value Theorem 118

8 The Difference Quotient 121

When a Secant Becomes a Tangent 122

Honey, I Shrunk the δx 123

Applying the Difference Quotient 127

The Alternate Difference Quotient 129

Part 3 The Derivative 131

9 Laying Down the Law for Derivatives 133

When Docs a Derivative Exist? 134

Discontinuity 134

Sharp Point in the Graph 134

Vertical Tangent Line 135

Basie Derivative Techniques 136

The Power Rule 136

The Product Rule 138

The Quotient Rule 139

The Chain Rule 140

Rates of Change 141

Trigonometric Derivatives 144

Tabular and Graphical Derivatives 145

Technology Focus: Calculating Derivatives 150

10 Common Differentiation Tasks 155

Finding Equations of Tangent Lines 156

Implicit Differentiation 159

Differentiating an Inverse Function 161

Parametric Derivatives 164

Technology Focus: Solving Gross Equations 166

Using the Built-In: Equation Solver 166

The Equation-Function Connection 170

11 Using Derivatives to Graph 173

Relative Extrema 174

Finding Critical Numbers 175

Classifying Extrema 176

The Wiggle Graph 178

The Extreme Value Theorem 180

Determining Concavity 182

Another Wiggle Graph 183

The Second Derivative Test 184

12 Derivatives and Motion 187

The Position Equation 188

Velocity 190

Acceleration 191

Vertical Projectile Motion 193

13 Common Derivative Applications 195

Newton's Method 196

Evaluating Limits: L'Hôpital's Rule 199

More Existence Theorems 200

The Mean Value Theorem 201

Rolle's Theorem 203

Related Rates 204

Optimization 208

Part 4 The Integral 215

14 Approximating Area 217

Riemann Sums 218

Right and Left Sums 219

Midpoint Sums 221

The Trapezoidal Rule 222

Simpson's Rule 225

15 Antiderivatives 227

The Power Rule for integration 228

Integrating Trigonometric Functions 230

Separation 232

The Fundamental Theorem of Calculus 233

Part 1 Areas and Integrals Are Related 233

Part 2 Derivative: and Integrals Are Opposites 235

u-Substiution 236

Tricky u-Substitution and Long Division 237

Technology Focus: Definite and Indefinite Integrals 239

16 Applications of the Fundamental Theorem 245

Calculating Area Between Two Curves 246

The Mean Value Theorem for Integration 249

A Geometric Interpretation 249

The Average Value Theorem 251

Finding Distance Traveled 253

Accumulation Functions 255

Arc Length 256

Rectangular Equations 256

Parametric Equations 257

Part 5 Differential Equations and More 259

17 Differential Equations 261

Separation of Variables 262

Types of Solutions 263

Family of Solutions 264

Specific Solutions 266

Exponential Growth and Decay 267

18 Visualizing Differential Equations 275

Linear Approximation 276

Slope Fields 277

Euler's Method 281

Technology Focus: Slope Fields 285

19 Final Exam 289

Appendixes

A Solutions to "You've Got Problems" 301

B Glossary 317

Index 323

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