Taylor Polynomial - intuition

Question

How do adding higher derivatives of the function on the same point gives a better approximation?

Answer 1

Here's a very rough, intuition-based way to understand it. Suppose we'd like to write a (smooth) function as a power series. In other words,

$$f(x)=a_0+a_1x+a_2x^2+\dotsm$$

Then by evaluating $f(0)$ in the expansion above, we see $f(0)=a_0$. So we have determined the first constant. Then calculating $f'$,

$$f'(x)=a_1+2a_2x+3a_3x^2+\dotsm.$$

And thus, $f'(0)=a_1$ by evaluating the expansion above. So this determines $a_1$. By differentiating again, we find that $f''(0)=2a_2$, and thus $a_2=f''(0)/2$. Continuing this way gives us $a_n$ as functions of nth order derivatives.