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        1 GROWTH 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        12 PROOF THAT THE SECANT GRADIENT IS p + q THE QUADRATICS y = ax2 THE GENERAL QUADRATIC:  y = ax2 + 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    20 USING 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            30 INTRODUCTION SKETCHING PARABOLAS OPTIMISING PRACTICAL EXAMPLES FORMING YOUR OWN EQUATION	Applications of the differentiation learnt in the previous chapter
5    Cubics and Beyond        40 DIFFERENCE OF SQUARES, CUBES ETC. FORMULAE [1]  GRADIENT OF y = x3, x4, x5 etc. [2]  GRADIENT OF y = axn 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        49 THE EQUATION y2 = x ANOTHER CURVE: y2 = x3	The secant-tangent method applied to fractional powers Page 51: square root equivalent to a power of ½
7    Negative Powers            55 THE CURVE y = 1/x ANOTHER CURVE: y = 1/x2 OTHER CURVES WITH NEGATIVE n	The secant-tangent method applied to negative powers Page 55: 1/x = x-1
8    Ratio of Gradients        60 THE CURVE y = x2 THE CURVE y = x3 THE CURVES y = axn RATIO OF GRADIENTS = n A NEGATIVE POWER FINDING GRADIENTS USING THE RATIO	Finding gradients using: Ratio of Gradients = n
9      Area Under a Parabola    66 RATIOS OF AREAS AREA OF A STRIP USING SYMMETRY	Using a simple ratio to get areas under y = ax2
10    Ratio of Areas            76 DIFFERENCE OF SQUARES OF x-LIMITS DIFFERENCE OF x-LIMITS COMBINATIONS CUBICS ETC.	Extending the previous chapter to areas under polynomials
11    Integration                86 THE CONSTANT OF INTEGRATION THE PATTERN INTEGRATION SYMBOL INTEGRATING TERM BY TERM FINDING THE CONSTANT	Integration as the reverse of differentiation
12    Integration and Area        94 INTEGRATION 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                    99 DISPLACEMENT VELOCITY GETTING VELOCITY FROM DISPLACEMENT ACCELERATION DISPLACEMENT, VELOCITY, ACCELERATION USING INTEGRATION	An application of differentiation and integration
14    Differential Equations    107 WHAT 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.