Taylor's Theorem

Taylor's Theorem
Suppose we're working with a function f(x) that is continuous and has n + 1 continuous
derivatives on an interval about x = 0. We can approximate f near 0 by a polynomial Pn(x)
of degree n:

For n = 0, the best constant approximation near 0 is
P0(x) = f(0)
which matches f at 0.

For n = 1, the best linear approximation near 0 is
P1(x) = f(0) + f0(0)x:
Note that P1 matches f at 0 and P01 matches f0 at 0.

For n = 2, the best quadratic approximation near 0 is
P2(x) = f(0) + f0(0)x +
f00(0)
2!
x2: