A Semi-Discrete Analog to The Mean Value Theorem

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The mean value theorem states that if a function is differentiable in an open interval and continuous in its closure
, then there exists a point
such that
. This theorem's semi-discrete analog suggests that if a function is tendable in
, continuous in
, and also that
, then for each
there exists a point
, such that
, where
is the function's tendency, and "tendability of a function" means that the tendency operator can be applied to the function. The definition of the tendency operator is given in the Demonstration "Detachment and Tendency of a Single Variable Function". A comparison between the mean value theorem and its semi-discrete analog is found in the details. When compared to the original theorem, this version depicts a trade-off between the functions for which the theorem's correctness holds and the kind of information that the theorem gives. This version is called semi-discrete due to the decisive role of the
function, whose image is finite. In this Demonstration, you can vary the parameters
,
, and
from the theorem's statement to better understand the theorem. The highlighted yellow points
are those whose existence is assured by the theorem.
Contributed by: Amir Finkelstein (March 2011)
Open content licensed under CC BY-NC-SA
Snapshots
Details
Snapshot 1: setting ,
,
yields only one highlighted point; here the one-sided detachments are omitted
Snapshot 2: setting ,
,
yields two highlighted points; here the one-sided detachments, from which the tendency is calculated, are shown
Snapshot 3: setting ,
,
yields one highlighted point; here the one-sided detachments, from which the tendency is calculated, are shown; although there are two points in the chosen interval whose image is
, only one of them is highlighted—the tendency at the other point is zero
A short and to the point comparison between the mean value theorem and its semi-discrete analog is depicted in this link.
Permanent Citation