Gradient Fucntion

Authors Avatar

The Gradient Function

In this coursework I am going to try and find out what the gradient function is for various curved graphs. I will do this by drawing graphs and find the gradient of the tangent of a point.

1.        I am going to draw a graph of y=x². I will use values from 0-5, I will obtain the gradient of the tangent at different points.

This is the table of the results of y=x².

This is the results of the gradient of the tangent:

Gradient of  y=x².

I have noticed that the gradient of the tangent doubles each time. Except for at (4,16) so I fink that this is an anomalious result, which would mean that the tangent must be drawn incorrectly.

I am now going to draw a graph of y=x³. I will use values from 0-5, I will

obtain the gradient of the tangent at different points.

This is the table of the results of y=x³.

This is the results of the gradient of the tangent:

Gradient of  y=x³.

I haven’t noticed a pattern in this set of results, but I think that at (3,27) and (4,64), my gradients are a bit wrong. I think it is that the second difference goes up in 6.

Although I have noticed that for y=x² the first gradient was 2 and for y=x³ the first gradient was 3. So I predict that the first gradient will the number that is the power of x, so in y=x4 the first gradint should be 4.

I am now going to draw a graph of y=x4. I will use values from 0-5, I will

obtain the gradient of the tangent at different points.

This is the table of the results of y=x4.

This is the results of the gradient of the tangent:

Gradient of  y=x4.

I have not noticed a pattern in this  set either, but I think that the gradients are incorrect. This also does not follow my earlier prediction that the first gradient would be 4.

I will now draw a graph for y=x. I will not have to obtain the gradient of the tangent as the answer will always be the same all the way up as it is a linear graph.

Gradient of  y=x.

This is a linear graph so the gradient is the same all the time.

This is a table to compare the results of the gradient of each graph at each point, I will try and see if there is a pattern I can find.

Y=x – the gradient is 1 all the time, this means that the line is linear.

Y=x² - the gradient doubles each time.

Y=x³ - the gradient went wrong on this one and I cannot see a pattern.

Y=x4 - the gradient was also wrong on the one and I cannot see a pattern.

When the x-value is 1 the gradient goes up by 1 each time (that’s how I can tell that the gradients for y=x4 is incorrect)

I have noticed that my results are not relible so they are likey to be incorrect. To try and obtain a more accurate set of results I will use the small increments method. This is more accurate then the gradient method but it is not perfect.

Join now!

Y=x²

This reprents the graph Y=x². Now I  have to use a bit on the curve which is the straightest at the point I want (the circled point). Then I will find the gradient of that straighest bit, but it will not be 100% accurate as that line will have a slight curve.

At (1,1)                         this will be     1.0201 – 1  =    0.0201 = 2.01

                                                1.01 – 1           0.01                

                                        this is correct because to 1sf it is 2

At (2,4)      4 .0401 – 4 = ...

This is a preview of the whole essay