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The Gradient Function.

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Introduction

The Gradient Function

I am trying to find a formula that will work out the gradient of any line (the gradient function)

I am going to start with the simplest cases, e.g. y=x² as they are probably going to be the easiest equations to solve as they are likely to be less complex and hopefully the formulas to the more complex equations will be easier to discover by looking at the previous formulas.

I am going to look at the line y=x² first.

 y=x²

X

1

2

3

4

y

1

4

9

16

One of the most obvious things I notice is that as the co-ordinates increase so does the gradient. Not only can you see that from the results below, but also on the graph you can that the line gets steeper and steeper.

...read more.

Middle

dy/dx

However there is another way called small increment method. This method gives a more accurate approximation on the gradient.  What you do is zoom in on the graph and take part of the curve you take a co-ordinate e.g. (3,9) and (3.01,9.0601).

Now you connect the two points together with a straight line. And because the graph is to a much larger scale the line should follow almost the same path as the curve. Of course the more you zoom in the more accurate the gradient will be. You then use the same formula to work out the gradient.

dy/dx

This should then give you an accurate gradient, it tends to be more accurate than the other method although if you were to draw the tangent and graph perfectly you should get the exact answer.

...read more.

Conclusion

span="1">

Gradient (Small Increment Method)

(1,1)

2

3.0301

(2,8)

8

12.0601

(3,27)

28

27.0901

(4,64)

32

36.855


The formula I predicted doesn’t work for this equation.

I have worked out the gradient function for this line it is more complex than the previous ones, it is:

3*x2


Below are the calculations that I used to check that the formula does work:

3*12=3
3*2
2=12
3*3
2=27
3*4
2=36

As you can see my formula does work, they give almost exactly the same answers for the gradient as my small increment result which means it must be right and also the small increment method must be more accurate, so from now on I am going to just use that method instead of using tangents as well. This will mean I don’t have to draw out the graphs any more.

I am going to put the formulas that I have discovered so far into a table so that they are hopefully easier to interpret.

Equation

y=x

Y =x2

Y =x3

Y = xn

Gradient Function

1

2x

3x2

nx (n-1)

...read more.

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