TITLE: In the mean value theorem, we are guaranteed $c$ such that $f'(c) = (f(b)-f(a))/(b-a)$. Does $c$ have a name?
QUESTION [6 upvotes]: The Mean Value Theorem says approximately that for differentiable $f$, there is a $c \in (a,b)$ such that
$$
f'(c) = \frac{f(b)-f(a)}{b - a}.
$$
I presume that the number $f'(c)$ is the mean value. My question is very short, and very simple. Is there a name for the $c$ guaranteed by the Mean Value Theorem? If I had to name it now, I would call it the mean value abscissa?
Many of these classical theorems were proved before the current language of calculus. This means it is possible, perhaps even likely, that such a $c$ was originally given a name.

REPLY [3 votes]: After asking around for a while, I've decided that perhaps $c$ never did get a name of its own for any particular pair $(a,b)$. However, some authors in the 18th and 19th centuries gave a name to a related object:
$$ f'(\theta_h) = \frac{f(x+h) - f(x)}{h},$$
and called $\theta_h$ Rolle's Function. However I should note that $\theta_h$ is not generically a function; and although it clearly also depends on $x$, it doesn't seem to be customary to include any explicit dependence in the notation.
In a cute paper written with a colleague and to appear in the American Mathematical Monthly, we've decided to keep the name mean value abscissa.