# Quadratic/ Cubic/ etc approximations without the Taylor series

It's easy to convince someone that the linear approximation is the best line which approximates a function at a point because everyone learns early that the derivative of a function is just the slope of that function at the point.

However...

Assuming I'd never heard of Taylor series, is there some way you could convince me that $f(a)-f'(a)(x-a) - \frac 12f''(a)(x-a)^2$ is the best quadratic function that describes the function $f$ at the point $x=a$?

Bonus points: How about the best cubic or quartic or quintic or .... approximation?

Write $f(x)\approx a_0+a_1(x-a)+a_1(x-a)^2$. Then, determine coefficients by taking successive derivatives and letting $x=a$.
So, $f(a)=a_0$. $f'(a)=a_1$. $f''(a)=2a_2$.
Well, to be awkward and annoying, you can define the derivative as the number $f'(a)$ so that $$\frac{f(a+h)-f(a)-h f'(a)}{h} \to 0$$ as $h \to 0$, i.e. so that $$f(a+h) = f(a) + h f'(a) + o(h).$$ In a similar way, you can define the second derivative as the number $f''(a)$ such that $$\frac{f(a+h)-f(a)-h f'(a)-h^2 f''(a)/2 }{h^2} \to 0,$$ and so on. That there is only one such number $f^{(k)}(a)$ for each $k$ up to $n$ is exactly what it means for a function to be $n$-times differentiable.