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

And again...

66 96 126 156 186

And again...

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

And again...

6 6

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.

CB