Price $25 (including postage)

## Description

The book gives a very simple and easy method to approach Calculus without the usual notations. Though these notations are introduced in the book the teacher/parent can choose when and how to do that.

## Details

143 + vii pages.

Size: 24.5cm by 17cm.

Paperback. 2015

Author: Kenneth Williams

ISBN 978-1-902517-40-7.

## Preface & Introduction

**PREFACE**

The aim of this book is to introduce Calculus in a simple holistic way that even young children can assimilate. It is a first introduction to Calculus and can be used in different ways.

* Prerequisites are: basic graph work, an understanding of what a gradient is and how to measure it (though this is briefly explained in the text) and a basic knowledge of algebra.

* Calculus is normally taught later in the school career, but here you will see that this does not have to be the case.

* Calculus can be taught and understood without the complex terminology and symbolism that is usually associated with this subject.

* The approach described here allows the possibility of connecting at any point with the conventional approach.

* So the simple Calculus methods shown are also connected in this book to the standard notations.

* The teacher/parent can therefore choose the appropriate material and introduce standard notations as appropriate for their class/child.

**Possible paths through the book.**

Without understanding formal algebra a good introduction to Calculus is still possible because children naturally generalise but may be puzzled by the use of letters to express those generalisations. So taking only the essential ideas from Chapters 1, 2, 3, 9, 10 a grounding in Calculus is entirely possible. (Chapter 5 can also be used by extending the idea of Chapter 3).

Path A Chapters: **1 2 3 - **(5)** - - - 9 10 - - - -**

Path B would be for a child who knows basic algebra as well as gradients but needs a gentle introduction. This child may go through Chapters 1, 2, 3, 4, 5, 9, 10.

Path B Chapters:** 1 2 3 4 5 - - - 9 10 -** - (13) (14)

Path C would go through the entire book.

Path C Chapters: **1 2 3 4 5 6 7 8 9 10 11 12 13 14**

Chapters 6, 7, 8 deal with negative and fractional powers so may be felt a little harder (though the text does explain what is needed), but the beautiful graphical designs here are very attractive and easy to do.

Chapters 11, 12 introduce the integration symbol and are really present to connect the approach of this book to the traditional approach.

Chapters 4 and 13 are applications of Calculus, and Chapter 14 is an easy introduction to simple differential equations.

This book aims to open up the field of Calculus in a natural way, so that even young children can appreciate and apply the techniques in their thinking, in their study of mathematics and in the solution of problems.

**INTRODUCTION**

Calculus is all about change and growth. It is a way of measuring change and growth. And since these are everywhere it is no real surprise that Calculus has applications in many areas: economic, medical, engineering and so on.

All growth is lawful, each has its own way of changing. And these laws of growth can be treated mathematically, with Calculus, so that we can make predictions about future behaviour or find out the laws of growth from growth patterns.

Nature is complex on the surface, and changes lead to more change: one thing affects another, and in particular one thing sets limits on another. So change inevitably leads to more change and limits of change.

The tree is blown by the wind, and if its root system is not strong enough, it may get blown over. A limit was reached. These limits are seen everywhere and are at the basis of Calculus.

## Contents

Preface iv

Introduction v

Detailed Contents vi

1 Growth and Limits 1

2 Gradient of a Secant 12

3 Gradient of a Tangent 20

4 Optimization 30

5 Cubics and Beyond 40

6 Negative Powers 49

7 Fractional Powers 55

8 Ratio of Gradients 60

9 Area Under a Parabola 67

10 Ratio of Areas 77

11 Integration 86

12 Integration and Area 95

13 Motion 100

14 Differential Equations 108

Notes 115

Answers 126

Index 142

## Detailed Contents

1 Growth and Limits 1GROWTH ZERO GROWTH STEADY GROWTH VARIABLE GROWTH LIMITS GRADIENT OF A CURVE LAW OF GROWTH |
Introductory material on growth and gradients and the idea of limits. |

2 Gradient of a Secant 12PROOF THAT THE SECANT GRADIENT IS p + q THE QUADRATICS y = ax ^{2}THE GENERAL QUADRATIC: y = ax ^{2} + bx + c |
Sets things up for the application of limits to get a tangent gradient from a secant gradient in the next chapter. |

3 Gradient of a Tangent 20USING LIMITS GRADIENT OF ANY PARABOLA QUICK WAY TO GET THE GRADIENT FORMULA THE SPECIAL SYMBOL dy/dx FINDING THE POINT WHICH HAS A GIVEN GRADIENT DIFFERENTIATION THE OPERATOR |
Here we use limits and secant gradients to get tangent gradients. Page 24: dy/dx symbol introduced Page 27: term ‘Differentiation’ introduced Page 28: operator symbol d/dx introduced |

4 Optimisation 30INTRODUCTION SKETCHING PARABOLAS OPTIMISING PRACTICAL EXAMPLES FORMING YOUR OWN EQUATION |
Applications of the differentiation learnt in the previous chapter |

5 Cubics and Beyond 40DIFFERENCE OF SQUARES, CUBES ETC. FORMULAE [1] GRADIENT OF y = x ^{3}, x^{4}, x^{5} etc.[2] GRADIENT OF y = ax ^{n}IN SUMMARY [3] DIFFERENTIATING SEVERAL TERMS SECOND DERIVATIVE |
Extending the secant-tangent method beyond quadratics Page 44: dy/dx = anxn-1 introduced Page 45: differentiating more than one term Page 47: second derivative |

6 Fractional Powers 49THE EQUATION y ^{2} = xANOTHER CURVE: y ^{2} = x^{3} |
The secant-tangent method applied to fractional powers Page 51: square root equivalent to a power of ½ |

7 Negative Powers 55THE CURVE y = 1/x ANOTHER CURVE: y = 1/x ^{2}OTHER CURVES WITH NEGATIVE n |
The secant-tangent method applied to negative powers Page 55: 1/x = x ^{-1} |

8 Ratio of Gradients 60THE CURVE y = x ^{2}THE CURVE y = x ^{3}THE CURVES y = ax ^{n}RATIO OF GRADIENTS = n A NEGATIVE POWER FINDING GRADIENTS USING THE RATIO |
Finding gradients using: Ratio of Gradients = n |

9 Area Under a Parabola 66RATIOS OF AREAS AREA OF A STRIP USING SYMMETRY |
Using a simple ratio to get areas under y = ax^{2} |

10 Ratio of Areas 76DIFFERENCE OF SQUARES OF x-LIMITS DIFFERENCE OF x-LIMITS COMBINATIONS CUBICS ETC. |
Extending the previous chapter to areas under polynomials |

11 Integration 86THE CONSTANT OF INTEGRATION THE PATTERN INTEGRATION SYMBOL INTEGRATING TERM BY TERM FINDING THE CONSTANT |
Integration as the reverse of differentiation |

12 Integration and Area 94INTEGRATION GIVES AREAS NOTATION INTEGRATING SEVERAL TERMS |
Here the ratio of areas method is linked to integration Page 94: notation for definite integrals introduced |

13 Motion 99DISPLACEMENT VELOCITY GETTING VELOCITY FROM DISPLACEMENT ACCELERATION DISPLACEMENT, VELOCITY, ACCELERATION USING INTEGRATION |
An application of differentiation and integration |

14 Differential Equations 107WHAT IS A DIFFERENTIAL EQUATION? CONSTRUCTING A DIFFERENTIAL EQUATION IN REVERSE |
A brief introduction to the concept of Differential Equations (First order only) |

## Back Cover

This book shows a very simple approach to Calculus that can be used as an introduction for even quite young children.

Prerequisites are basic graph work, an understanding of what a gradient is and an elementary knowledge of algebra.

The book shows that the subject can be taught in a very simple way and without the usual use of complex terminology and symbolism.

The standard calculus notations are however introduced here as well (later in the book).

In this way the teacher or parent can connect with the standard approach as appropriate for their class or child.

Kenneth Williams has been studying, researching and teaching Vedic Mathematics for over 40 years. He has published many articles, DVDs and books and has been invited to many countries to give seminars and courses. He gives online courses, including teacher training. Research includes left-to-right calculating, Astronomy, applications of Triples, extension of Tirthaji's 'Crowning Gem', Calculus.