What is the Mean Value Theorem in calculus?

Answer 1

There are several generalizations and extensions, but the basic Mean Value Theorem (MVT) by Joseph Louis LaGrange (1797) is this:

Given a function f(x) that is continuous over a closed interval [a,b] and differentiable over the open interval (a,b), then there must exist a place in (a,b) such that the slope of the tangent to f(x) there will equal the slope of the secant over [a,b]. Algebraically stated, there must exist a number t within that interval, i.e., a

f '(t) = (f(b) - f(a)) / (b - a).

Notice that f(x) need not be differentiable at either a or b, as for example in the function for the upper semicircle f(x) = (1 - x)1/2 over domain [-1,1].

A typical restatement of the MVT is: for each x in (a,b) there corresponds at least one t in (a,x) such that f(x) = f(a) + f '(t)(x -a). This is in contrast to the "linear approximation" of f(x) about x=a, which is that f(x) is approximately equal to

f(a) + f '(a)(x -a) as long as x isn't far from a. (In the special case that f(x) is a linear function, then f '(a) = f '(t) = m the slope of the line, and then f(x) = f(a) + m(x -a), and the MVT is identical to the equation of the line, without approximation.)

The best way to appreciate the MVT is to see a graph of f(x) along with its secant line and parallel tangent line.