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 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 Caveat: MVT is no longer valid for complex-valued or vector-valued functions.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).