The Manga Guide to Calculus

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Overview

Noriko is just getting started as a junior reporter for the Asagake Times. She wants to cover the hard-hitting issues, like world affairs and politics, but does she have the smarts for it? Thankfully, her overbearing and math-minded boss, Mr. Seki, is here to teach her how to analyze her stories with a mathematical eye.

In The Manga Guide to Calculus, you'll follow along with Noriko as she learns that calculus is more than just a class designed to weed out would-be science majors. You'll see that calculus is a useful way to understand the patterns in physics, economics, and the world around us, with help from real-world examples like probability, supply and demand curves, the economics of pollution, and the density of Shochu (a Japanese liquor).

Mr. Seki teaches Noriko how to:

Use differentiation to understand a function's rate of change
Apply the fundamental theorem of calculus, and grasp the relationship between a function's derivative and its integral
Integrate and differentiate trigonometric and other complicated functions
Use multivariate calculus and partial differentiation to deal with tricky functions
Use Taylor Expansions to accurately imitate difficult functions with polynomials

Whether you're struggling through a calculus course for the first time or you just need a painless refresher, you'll find what you're looking for in The Manga Guide to Calculus.

This EduManga book is a translation from a bestselling series in Japan, co-published with Ohmsha, Ltd. of Tokyo, Japan.

Product Details

ISBN-13: 9781593271947
Publisher: No Starch Press San Francisco, CA
Publication date: 8/28/2009
Pages: 256
Sales rank: 172,353
Series: Manga Guide to Science Series
Product dimensions: 7.00 (w) x 9.20 (h) x 0.80 (d)

Meet the Author

Hiroyuki Kojima received his PhD in Economics from the Graduate School of Economics, Faculty of Economics, at the University of Tokyo. He has worked as a lecturer and is now an associate professor in the Faculty of Economics at Teikyo University in Tokyo, Japan. While well-regarded as an economist, he is also active as an essayist and has published a wide range of books on mathematics and economics at the fundamental, practical, and academic levels.

Preface xi

Prologue: What is a Function? 1

Exercise 14

1 Let's Differentiate a Function! 15

Approximating with Functions 16

Calculating the Relative Error 27

The Derivative in Action! 32

Step 1 34

Step 2 34

Step 3 35

Calculating the Derivative 39

Calculating the Derivative of a Constant, Linear, or Quadratic Function 40

Summary 40

Exercises 41

2 Let's Learn Differentiation Techniques! 43

The Sum Rule of Differentiation 48

The Product Rule of Differentiation 53

Differentiating Polynomials 62

Finding Maxima and Minima 64

Using the Mean Value Theorem 72

Using the Quotient Rule of Differentiation 74

Calculating Derivatives of Composite Functions 75

Calculating Derivatives of Inverse Functions 75

Exercises 76

3 Let's Integrate a Function! 77

Illustrating the Fundamental Theorem of Calculus 82

Step 1 When the Density Is Constant 83

Step 2 When the Density Changes Stepwise 84

Step 3 When the Density Changes Continuously 85

Step 4 Review of the Imitating Linear Function 88

Step 5 Approximation $$ Exact Value 89

Step 6 p(x) Is the Derivative of q(x) 90

Using the Fundamental Theorem of Calculus 91

Summary 93

A Strict Explanation of Step 5 94

Using Integral Formulas 95

Applying the Fundamental Theorem 101

Supply Curve 102

Demand Curve 103

Review of the Fundamental Theorem of Calculus 110

Formula of the Substitution Rule of Integration 111

The Power Rule of Integration 112

Exercises 113

4 Let's Learn Integration Techniques! 115

Using Trigonometric Functions 116

Using Integrals with Trigonometric Functions 125

Using Exponential and Logarithmic Functions 131

Generalizing Exponential and LogarithmicFunctions 135

Summary of Exponential and Logarithmic Functions 140

More Applications of the Fundamental Theorem 142

Integration by Parts 143

Exercises 144

5 Let's Learn About Taylor Expansions! 145

Imitating with Polynomials 147

How to Obtain a Taylor Expansion 155

Taylor Expansion of Various Functions 160

What Does Taylor Expansion Tell Us? 161

Exercises 178

6 Let's Learn About Partial Differentiation! 179

What Are Multivariable Functions? 180

The Basics of Variable Linear Functions 184

Partial Differentiation 191

Definition of Partial Differentiation 196

Total Differentials 197

Conditions for Extrema 199

Applying Partial Differentiation to Economics 202

The Chain Rule 206

Derivatives of Implicit Functions 218

Exercises 218

Epilogue: What Is Mathematics For? 219

A Solutions to Exercises 225

Prologue 225

Chapter 1 225

Chapter 2 225

Chapter 3 226

Chapter 4 227

Chapter 5 228

Chapter 6 229

B Main Formulas, Theorems, and Functions Covered in this Book 231

Linear Equations (Linear Functions) 231

Differentiation 231

Derivatives of Popular Functions 232

Integrals 233

Taylor Expansion 234

Partial Derivatives 234

Index 235

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Anonymous

Posted December 6, 2011

Fun way to learn a tricky subject

I've been trying to teach myself calculus for ages, but the textbooks are difficult to understand. This book is written as if someone is there to breakdown the concepts and explain them.
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Fun way to learn a tricky subject