What are "Finite Differences"?

Can someone explain this mathematical concept to me? MathWorld describes it as the “discrete analog of the derivative”. Of course, MathWorld does a great job of not explaining concepts for the layman or giving any simple examples.

I have a little more than 2 semesters of experience in Calculus. If someone can explain it and give a few simple examples, I’d appreciate it. Thanks!

Splanky

The finite difference of a function f, denoted [symbol]D[/symbol]f, is simply f(x + 1) - f(x). Some authors use [symbol]D[/symbol][sub]h[/sub]f to denote f(x + h) - f(x).

The finite difference of a function has some of the same properties as the derivative–the difference of a constant is 0, the difference of a polynomial of degree n is a polynomial of degree n - 1, it’s a linear operator ([symbol]D[/symbol](af + bg) = a[symbol]D[/symbol]f + b[symbol]D[/symbol]g for real numbers a and b).

You can use the finite difference operator to give an explicit formula for the nth derivative of an infinitely differentiable function, but that’s more of a curiosity than a useful tool.

I’m sure there’s more to it than this, but this is the extent to which I’ve always used it:

Say you have a sequence of numbers. Maybe they’re some data you’re accumulating, and you have reason to believe that they can be modeled by some polynomial function. You can use finite differences to predict future numbers in the sequence.

For example, say you have the numbers:

1
3
6
10
15
21

Take the differences between consecutive numbers:

2
3
4
5
6

Do it again:

1
1
1
1

Notice that now we have a constant sequence. If I have reason to suspect that this pattern will be followed, I can predict the next number in the sequence by working backwards. Sticking a 1 at the end of the final sequence, then the previous sequence would be

2
3
4
5
6
7

and the original sequence would be

1
3
6
10
15
21
28

Another thing: If after taking these finite differences n times, we get a constant sequence, that means the original sequence can be modeled with a polynomial of degree n.

In particular, since our original sequence became constant after taking the difference twice, I know there’s a polynomial p(x) of degree 2 such that p(n)=the nth term of the sequence.

In fact, for this example, p(x)=x(x+1)/2.