泰勒级数
main purpose: taking non-polynomial functions and finding polynomials that
approximate them near some input
$P(x) = f(0) + \frac{df}{dx}(0) \frac{x^1}{1!} + \frac{d^2f}{dx^2}(0)
\frac{x^2}{2!} + \frac{d^3f}{dx^3}(0) \frac{x^3}{3!} + \dots$
=> $P(x) = f(a) + \frac{df}{dx}(a) \frac{x^1}{1!} + \frac{d^2f}{dx^2}(a)
\frac{x^2}{2!} + \frac{d^3f}{dx^3}(a) \frac{x^3}{3!} + \dots$
taylor series is that they translate derivative information at a single point
to approximatioon information around that point