Stony Brook Calculus Materials



Calculus I Handouts

Analysis of polynomial functions by differences.

f is a function of x; fix a number h > 0.

First difference function is \Delta f (x) = f(x + h) - f(x).

Second difference function is \Delta^2 f (x) = \Delta f(x + h - \Delta f(x).

Third difference function is \Delta^3 f (x) = \Delta^2 f(x + h) - \Delta^2 f(x).

Etc.

Now suppose f is a polynomial function; f has degree n means that it has the form f(x) = a_n x^n + a_{n-1}x^{n-1}+ ... + a_1 x + a_0 with a_n not 0.

0. If f is of degree 0, it is a constant; \Delta f (x) = 0.

1. If f is of degree 1 (linear) then \Delta f (x) is constant and so \Delta^2 f is zero.

2. If f is of degree 2 (quadratic) then \Delta^2 f (x) is constant and so \Delta^3 f is zero. (Check this for f(x) = x^2; why is this enough?)

3. Fill this in yourselves, and figure out the easiest way to check it.

Etc. State and prove the corresponding fact for polynomials of degree n.