Another thing that I have noticed is that the larger the co-ordinates the smaller the increase in gradient.
As the table above shows there are two methods that I am using for calculating the gradient of line. The first being drawing a tangent at the point, working out the distances on the tangent using the scale on the graph and then using this formula:
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. One of the good things about this method is that it isn’t necessary to sketch the magnified area instead there is another method. All you need to do is take a point e.g. (3,9) and then make up another x co-ordinate which is very close to the first one e.g. 3.01. And then using the lines equation e.g. y=x² you can work out the Y co-ordinate. In this case you would do 3.01*3.01, which would be the same as 3.01². This then gives you the value 9.0601. Now that you have those values you know that the line must cross the point (3,9) and the point (3.01,9.0601). Now you use the formula:
dy/dx
Which gives you 0.0601¸0.01= 6.01
I have discover that the formula for the equation y=x² is 2x. As you can see from the results table the accuracy using the tangent method hasn’t been perfect, but using the small increment method it is possible to get much more accurate set of results.
Now I am going to look at the line y=c³. I predict that this line will look similar to Y=x² but it will be steeper and indeed it is, by entering the two equations into my calculator I can see that the equation y=x³ is steeper than Y=x². Going by the formula for the previous equation I would have thought that the formula for this equation might be 3x.
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*22=12
3*32=27
3*42=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.
