How to estimate ln(1.1) using quadratic approximation?

Question

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?

Answer 1

What you wrote is the formula for quadratic approximation, which is derived from Taylor series.

In your case, you need to set $f(x)=\ln x$ and $a=1$, then use the formula.

Answer 2

" ... 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.