This is in response to a post of Sheila King's a few(?) days ago.
She pointed out that her husband often deals with "real world" phenomena that have unusual behaviors as certain of the parameters become very large or very small. What do the functions that describe these behaviors look like?
What I remember from my discussions with physics teachers and my own study of science in college is that in fact, it is NOT that the functions behave oddly at small (or large) values, but rather that assumptions are made about the domains of the functions used to describe these phenomena. Often, scientists will disregard some force because it is so much smaller than another over the domain they're considering. However, if the domain were expanded, the system would need to include other information which may be relevant.
In fact, it may be that the behavior we're describing with a quadratic function (and which fits pretty well with measurements) is really the quotient of a fifth degree polynomial by a third degree polynomial but the parameters are usually so "big" that we don't notice it.
An example like this is to look at logistic growth compared to exponential growth when population is much below the limit. The two functions are not very different, and the exponential growth curve fits the data reasonably well.
I think it may be more important to help students understand the assumptions we make when we model certain behaviors with mathematical functions. They need to learn to evaluate how valid these models are when we either interpolate or extrapolate from measured data.
Elisse Ghitelman Newton North High School Newton, MA, USA