Gradient Function.

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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 a-g. They go from x=-3 to x=4. Below are the calculations for their gradients. (I am using the formula (y2-y1)/(x2-x1)to calculate the gradient of the line.

a= (12-6)/(-3.5--2.5) = 6/-1 = -6 b= (6-2)/(-2.5--1.5) = 4/-1 = -4

c= (2-0)/(-1.5--0.5) = 2/-1 = -2 d= (2-0)/(1.5-0.5) = 2/1 = 2

e= (6-2)/(2.5-1.5) = 4/-1 = 4 f= (12-6)/(3.5-2.5) = 6/1 = 6

g= (20-12)/(4.5-3.5) = 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. There is always going to be an inaccuracy in a graph, even if that inaccuracy is 0.25 of a millimetre. Therefore, I can only accept these results as estimates. As you can see, they are all twice their x value. Therefore, for the graph of y=x2, the formula for the gradient of a tangent is g=2x.

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I am left with the question of accuracy. I cannot get total accuracy, but there are ways I can get very close to an accurate answer. One such way is to use the method of using a line inside the curve.

(4,16)

(3.5,12.25)

(3.01,9.0601)

(3,9)

Judging by this trend, I can use this method to find more accurate results. If I were to use very short lines, (3,9) to (3.0000001,9.0000006) for example, then use the y step/x step method to find the gradient, I would get an answer very close to the correct gradient. In this case, I would get ...

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