  We investigate functions with the property that for every interval, the slope
at the midpoint of the interval is the same as the average slope. More
generally, we find functions whose average slopes over intervals are given by
the slope at a weighted average of the endpoints of those intervals. This is
equivalent to finding functions satisfying a weighted mean value property. In
the course of our exploration, we find connections to harmonic functions that
prompt us to explore multivariable analogues and the existence of "weighted
harmonic functions."
