Selecting $x_0$ in approximations using Taylor polynomials?

Question

Are there any general rules for how to pick $x_0$ (relative to $x$) in approximations using Taylor polynomials?

What if $x=x_0$?

Answer 1

Pick $x_0$ to be the center of the region in which you want the Taylor approximation to be accurate. Taylor approximations are very good near one point (your $x_0$), and get worse quite rapidly as you move away from this point. Specifically, at $x=x_0$, the approximation is perfect -- the error is zero.

If you want an approximation that's good near two points, instead of one, use Hermite methods.

If you want an approximation that's equally good throughout an entire interval, use Chebyshev approximation.