How to calculate local quadratic approximation?

A quadratic approximation is a Taylor series with 3 terms. It looks like this, in general:

q(x)= f(x0) + f'(x0)(x-x0) + f''(x0)(x-x0)^2/2

We need to choose a good x0. If this were a linear approximation, 81 would be a good choice since we know both sqrt(81) and the derivative of sqrt(x) at x=81. The second derivative term, however, is of the order of -3/2, so there isnt much we can do about it. So, lets start:

f(81)=9

f'(81)=1/2*x^(-1/2)=1/2*(1/sqrt(81))=1/18

f''(81)=-1/4*x^(-3/2)=-.000349

q(x)=9 + 1/18(x-81) - .000349(x-81)^2

Plug in 81.04 for x to get the local quadratic approximation at x=81.04

Quadratic Approximation

The question asks :- are all third and better partial derivatives of f(x, y) equivalent to 0 whilst x=a, y=b. word that interior the Taylor enhance of f(x, y) those greater derivatives have words like (x-a)^n * (y-b)^m , the place (m+n)>2 and so the third and better derivatives will vanish whilst x=a and y=b.