From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik)
Subject: Re: Mean Value Theorem
Date: 27 Dec 1999 19:10:10 -0500
Newsgroups: sci.math

In article <3863D5C4.1C724FFE@cableol.co.uk>,
neenag  <neenag@cableol.co.uk> wrote:
>What exactly is the mean value theorem?  I've looked it up in calculus
>books but haven't found it, although I've heard references to it in some
>proofs. Is it more straightfoward than I think?
>
>Neenag.

(1) What is it? (2) What is it good for?

Ad (1): The "two-function MVT" says: if the real-valued functions f(x) and
g(x) are continuous on [a, b] and differentiable on (a, b) then

for some c in (a, b),  f'(c) * (g(b) - g(a)) = g'(c) * (f(b) - f(a))

You could see the conclusion, with the extra assumption that g' is never
zero on (a, b), in the form 

   (f(b) - f(a)) / (g(b) - g(a)) = f'(c) / g'(c)

The simple consequence, when g(x)=x for all x, so that g'(x)=1, is easy to
write down, and it is often called the MVT. 

Caveat: MVT is no longer valid for complex-valued or vector-valued
functions. Counterexamples exist. (Consider cos(x) + i * sin(x) for x
between 0 and pi.)

Ad (2): The theorem, as stated, is a tool for a proof of the L'Hopital's
Rule. Its special case, Rolle's Theorem, is used to derive many kinds of
remainder terms in expansions, such as Taylor's.

But: one of practical uses of MVT is a recipe to produce estimates in the
form of inequalities. Namely, the Mean Value Inequality, this time
available for (careful) extensions to vector functions, says:

 Let f be continuous in [a, b], differentiable in (a, b), and let it be
known that m <= f'(x) <= M  for some finite numbers (or even infinite
limits, but that's less useful) m and M. Then

  m <= (f(b) - f(a)) / (b - a) <= M

This is helpful in estimating the accuracy of approximate calculations
(often hand-wavingly presented as "differential approximations").

(Start of a tirade that can be skipped:)
 Many students come to the study of Mathematics with a conviction,
instilled perhaps by previous education, that 

 "equations are what Maths is all about". (???)

 They call an expression an "equation" even if it visibly lacks the
equality sign (=). I've seen and heard students of Linear Programming
point to an inequality and call it an equation. 

 They are in for a shock when they have to handle inequalities, for
example in the traditional definition of a limit and illustrations of
that. And serious Numerical Methods just can't work without reasonable
knowledge and skills concerning inequalities.  And didn't I see a program
called "equation editor" which handles mathematical expressions and
couldn't give a hoot whether they are equations or not?

(End of tirade)

Happy New Year, ZVK(Slavek).