How to estimate ln(1.1) using quadratic approximation? So the general idea for quadratic approximation is assuming there a function $Q(x)$ we want to estimate near $a$: 
$Q_a(a) = f(a)$
$Q_a'(a) = f '(a)$
$Q_a''(a) = f ''(a)$
But then how do you derive the function $Q_a(x) = f(a) + f '(a)(x-a) + f ''(a) (x-a)^2/2$?
Or can you estimate just using the above one?

" ... the general idea for quadratic approximation is assuming there a function $Q(x)$ such that ... "
This is true, but falls slightly short of how quadratic approximation is normally set up - it's missing the stipulation that $Q(x)$ be a quadratic polynomial (in this context at least).
Deriving the function from here works a few ways - you can posit that $Q(x)=Ax^2+Bx+C$ and solve for $A,B,C$ given the conditions on the derivatives, but this is a bit laborious.
Simpler is to realise that any quadratic can also be written in the form $A(x-a)^2+B(x-a)+C$ and do the same thing. This leads to the formula you quote.
