Mean value theorems for di erences J. Marshall Ash Summary: Mean value theorems for di erences J. Marshall Ash Abstract. Let M = f(b) f(a) b a be the average slope of the real-valued con- tinuous function f on the closed interval [a; b]. Let 0 < p < b a: A secant line segment connecting (c; f (c)) and (c + p; f (c + p)) of slope M for some c 2 [a; b p] always exists when (b a) =p is an integer. But if p 2 (0; b a) does not have the form (b a) =n for some integer n 2, then an example is constructed for which every secant line segment lying above a subinterval of length p does not have slope M. Applications include two counterintuitive facts involving running certain distances at certain rates. For periodic func- tions the situation is di erent. A generalization for multivariate functions is given. 1. One dimension The mean value theorem says that if f (x) has a derivative at every point x 2 (a; b) and is continuous at x = a and x = b, then there is a c 2 (a; b) such that f0 (c) = Collections: Mathematics