[rsj_hn]

A first order approximation is always less accurate than a zeroth order approximation for bounded functions, as the first order approximation will have unbounded error whereas the zeroth order approximation will not -- unless you are in the degenerate case of the first order itself being exactly zero. The first order approximation is infinitely worse. Most (all?) things in the world are bounded. Hence if you must choose between just a first and zero-th order approximation, the zero-th order is the way to go for long run predictions. Cue the XKCD comic about the expected number of weddings.

On the other hand, if you are not interested in making long run predictions but only short run predictions, then first order approximations will tend to be more accurate in a small region around the base, but that region might be quite small.