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 analogs and the existence of "weighted harmonic functions".
|Original language||English (US)|
|Number of pages||8|
|Journal||American Mathematical Monthly|
|State||Published - Jun 1 2017|
Bibliographical noteFunding Information:
ACKNOWLEDGMENTS. D. L.-D. was supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE 0228243. P. C. gratefully acknowledges support by the National Science Foundation through grant DMS-1148284.
© The Mathematical Association of America.