Finite differences.The other day I was messing around with a simple data set. The values of f(x) when f(x) = x^2 with the integers one through five. I found some startling patterns.
X = 1 2 3 4 5
Y = 1 4 9 16 25
I then took the differences of each of the Y-values
3 5 7 9
And then the differences of these values
2 2 2
And found that they all were equal to two. Simple enough right.
After enough trials with different polynomials, I found that this number was always equal to n!*a, where n is equal to the number of times I had to take the difference before I was left with one number or all the numbers where the same, and a is equal to the coefficient. N is also equal to the power of the first part of the equation.
Then, by subtracting x^2 from my original y values I get the set (0,0,0,0,0)
This is an indicator that I have found the correct function.
Now for a slightly more complex problem. 5x^3 + 3x^2 - 2x + 4
Again we will assume that I do not know anything about this function.
but I simply have the data set for the x values 1 to 7
X = 1 2 3 4 5 6 7
Y = 10 52 160 364 694 1180 1852
Now to take the differences
42 108 204 330 486 672
66 96 126 156 186
30 30 30 30
Since I had to take the differences 3 times, we know that the highest power of the simplest function that describes this graph is 3. Simple enough. Plug 3 in for n to find that 3!*x = 30, 6*x=30, x = 5. Now we have the first part of the equation. 5x^3.
The next step is to subtract 5x^3 from the original data to find the y-values...
I'm only going to use the first four for purposes of simplicity.
5 12 25 44
We begin taking differences.
7 13 19
Going through this procedure once again, we find that N=2, and b = 3
Therefore, 3x^2 is the second part of the equation.
so we subtract 3X^2 from the second data set to get...
2 0 -2 -4
Interesting. The values decrease now. This is an indicator that the next coefficient, c, will be negative.
Start taking differences.
-2 -2 -2
That was easy. n=1, c=-2
The next section of the equation is -2x
So far we have 5X^3 +3X^2 -2x. The next bit is easy. Just add 2x to the y-values of the previous data set. You get...
4 4 4 4
The final part of the equation, d.
That means that the simplest equation that passes through each of those values is
X^3 +3X^2 -2x + 4.
I have no idea of what this is, or what this means, or even what form of mathematics this is. I was previously told that this is a rudimentary form of calculus, but I would like a more complete explanation. Sorry if this seems elementary to you, I am only a high school student. Thank you for any help you can provide. Any and every response is appreciated.
The n-th differences of an n-th degree polynomial are constant.