Gradient Function.
Extracts from this document...
Introduction
Gradient Function Coursework
Gradient Function Coursework
My task is to investigate the relationship between the gradients of tangents on the curves of graphs, such as y=x2. To do this, I will first find the gradient of tangents on the graph y=x2 by drawing the graph.
I have labelled the tangents ag. They go from x=3 to x=4. Below are the calculations for their gradients. (I am using the formula (y2y1)/(x2x1)to calculate the gradient of the line.
a= (126)/(3.52.5) = 6/1 = 6 b= (62)/(2.51.5) = 4/1 = 4
c= (20)/(1.50.5) = 2/1 = 2 d= (20)/(1.50.5) = 2/1 = 2
e= (62)/(2.51.5) = 4/1 = 4 f= (126)/(3.52.5) = 6/1 = 6
g= (2012)/(4.53.5) = 8/1 = 8
As you can see, the results I have obtained are good round numbers. These results however are not accurate to the tangents I drew on the graph.
Middle
12
3
27
3.0000001
27.0000027
27
4
64
4.0000001
64.0000048
48
5
125
5.0000001
125.0000075
75
6
216
6.0000001
216.0000108
108
The first numbers I saw were the first and third numbers. I noticed these because the first number was 3, the power by which I was multiplying the values. The third number was 27, which is 33. This struck me as strange  only this number was cubed, none of the others. To explain this, I tried cross analysing these results with the results I obtained from the y=x2 graph. I saw that the first value was the value of the power by which I multiplied the values. The third value however was not 32, but the second value was 22. Moving back to the results for y=x3, I tried dividing all the results by 3, and was left with the x1 values squared. Therefore, the formula for the gradient of a tangent on the graph of y=x3 was 3x2.
Conclusion
y2
(y2y1)/(x2x1)
4x3
1
1
1.0000001
1.0000004
4.0000006
4
2
16
2.0000001
16.0000032
32.00000238
32
3
81
3.0000001
81.0000108
108.0000054
108
4
256
4.0000001
256.0000256
256.0000097
256
5
625
5.0000001
625.00005
500.0000144
500
6
1296
6.0000001
1296.000086
864.0000213
864

As I thought, the two sets of results are very close. This shows that the formula is correct. I now need to find a formula which link the different formulae together. As the first number in the formulae is always the power being multiplied by x, and the power in the formulae is always 1 less I can determine the linking formula.
This student written piece of work is one of many that can be found in our GCSE Gradient Function section.
Found what you're looking for?
 Start learning 29% faster today
 150,000+ documents available
 Just £6.99 a month