In simple terms, what is the meaning of "second-order local approximation"?

Question

In simple terms, what is the meaning of “second-order local approximation”?

I have seen in used in Economics literature, but I presume it has a mathematical basis and meaning.

Answer 1

The goal is to match a quadratic (that is, a polynomial of degree 2, or second order) to the information near a particular (that is, local) point: you approximate the value, its rate of change (first derivative), and its acceleration (second derivative). So, for instance, you can approximate $\sin(x)+\cos(x)$ around $x=0$ as $1 + x - x^2/2$, and the approximation works reasonably well for quite a good distance:

Look at how well the orange line matches the blue over $-0.5<x<0.5$. A first order approximation won't do as well, going well too high at both $x=0.5$ and $x=-0.5$.

Answer 2

Do you know about the Taylor series approximation to $f$ near $x_0$? The second-order Taylor approximation of $f$ near $x_0$ is \begin{equation} f(x) \approx f(x_0) + \nabla f(x_0)^T(x - x_0) + \frac12 (x - x_0)^T \nabla^2 f(x_0) (x - x_0). \end{equation} Here $\nabla^2 f(x_0)$ is the Hessian of $f$ at $x_0$.

This is a standard topic in calculus.