The Gradient Function

dy/dx

Increment Method

 However there is another way called increment method. This method gives a more accurate approximation on the gradient.  What you do is on the graph you take part of the curve and 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. The line should follow almost the same path as the curve. On a computer the more you zoom in the more accurate the gradient will be. You then use the same formula as before 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 on a computer you would get the exact same answer. An easy and accurate way to do this 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 y=x² you can work out the y co-ordinate. In this case you would do 3.01 X 3.01, which would be the same as 3.01². This then gives you the value 9.0601 as a y co-ordinate. 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 discovered 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.

 

y=x

I am now going to look at the line y=x.  As you can tell from the equation, the gradient is always going to be one, as they are always going to be the same values as each other.

  

 

 

There is no need to use the small increment method here, as I know that the gradient is accurate as y=x is always going to equal 1. A formula to work out the gradient function for this equation is:

 

y/x

Another formula is of course just simple 1.

y=x³

Now I am going to look at the line y=x³. I predict that this line will look similar to y=x² but it will be steeper.  I have decided to complete both this stage and all the stages after this stage of my investigation with the use of a computer.  I will use the computer to work out and draw the graphs I will be using.  This will save time, make it easier to work out any equations or formulas and the graphs and results will be a lot more accurate as computers don’t make as many mistakes as humans and the graphs will be more detailed.  This will help me to work out and calculate more precise answers, results, equations and formulas.

 

 

Please refer to graph on separate piece of paper

 

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

3 X x2

Below are the calculations that I used to check that the formula works:

 

3 X 12=3
3 X 22=12
3 X 32=27
3 X 42=48

 

As you can see my formula does work, they give almost exactly the same answers for the gradient as my increment results, which means it must be right, also the increment method must be more accurate.

 

y=x4

I am now going to look at the gradient of the line y=x4, I think this could have a very similar formula to the previous ones as looking back on the other equations the rule that I have used for y=x³ works for the two previous ones assuming I adapt it.  For example in the equation y=x³ the formula was 2x, but I can also use my new formula to get the same results except of course instead of multiplying by three you would multiply by 2 as the equation is to the power of 2. So I would expect 4 X x3 will be the formula that works for the equation y=x4.

 

My prediction was right, the formula for this line is 4 X x3

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

 

 

Looking at my previous formulas I have come up with a formula that will work for any equation, although I would need to check that it does work for negative, fractions etc. The formula is as follows:

nx (n-1)

This formula has been developed from the other formulas that I have discovered for the equations, as you can see from the table above, it is fairly obvious what the formula is going to have to be. The n stands for the power in the equation

e.g. y=x², n would equal ². 

y=x5

I am going to just check that my formula does work with the equation y=x5 although I am sure it will work as it has worked for all the previous equations.

5 X 14=5

5 X 24=80
5 X 34=405
5 X 44=1280

Indeed my rule does work, however now that the gradients are such high numbers it is more noticeable that the small increment method is not perfect.