The Gradient Function

The Gradient Function

Introduction

The gradient of any line is the steepness at which it slopes; on

straight lines it can be worked out by drawing a right angled triangle

using the line itself as the hypotenuse to find out the ∆y, and ∆x.

The gradient of a line can then be worked out by dividing ∆y by ∆x.

The following graphic shows an example:

[IMAGE]

However, with a curved graph, the gradient is different at different

points. To work out the gradient at a point of a curved graph, a

tangent would have to be drawn, and the gradient of it measured. The

longer the tangent is, the more accurate the result if done by eye.

The following graphic is an example:

[IMAGE]

Because this method is inherently inaccurate, to improve the accuracy

we could either use a computer program to draw an accurate tangent, or

use the small increment method. The small increment method is where

the gradient of a chord, from the point, to another point of the line

a short distance away, is worked out to find the gradient between the

two points. So, for example, if it was a curve of y=x2, the gradient

at x=3 would be measured using a chord from [3, 9] to [3.0001,

9.00060001] and so the gradient would be 0.00060001 divided by 0.0001

which is 6.0001 or 6 as an integer.

The Small Increment Method

I given a curve, y=x2, and a point, x=2, I can calculate an

approximate gradient by using a chord with a second point a small

distance away from x=2.

[IMAGE]

Using this method, when adding on 0.1, the gradient would be like

this:

G = 4.41 - 4

2.1 - 2

G = 0.41

0.1

G = 4.1

I am expecting for the gradient given to become more like the gradient

of its tangent if I decrease the amount added onto x for the second

point.

The most accurate value would be zero as at that point it would be a