Gavin's Calc II Class Clarification: Oct 11

Let's do an example. Take f(x) = tan(x), and let's find the Taylor polynomial of degree 2 at x=0. We know this is P₂(x) = a₀ + a₁ x + a₂ x², where a₀ = f(0),
a₁ = f'(0), and
a₂ = f''(0) / 2! Calculating these, f(0) = tan(0) = 0, f'(0) = 1 / cos²(0) = 1, and f''(0) = -2 sin(0) / cos³(0) = 0. Therefore P₂(x) = x.

What does this mean? We said that the nth degree Taylor polynomial matches the function and its first n derivatives at the indicated point. Thus we know that

P₂(0) = f(0),
P₂'(0) = f'(0), and
P₂''(0) = f''(0). This is what setting the different values for a_0,1,2 above does. It also means that we can read the values for the different derivatives from the coefficients of the Taylor polynomial -- that is, if I tell you that P₂(x) = x, then you know that f(0) = 0 (because a₀ = 0), f'(0) = 1 (because a₁ = 1) and f''(0) = 0 (because a₂ = 0).