Mean Value Theorem(redirected from Mean value theorems for integration)
mean value theorem[′mēn ′val·yü ‚thir·əm]
Mean Value Theorem
one of the fundamental results of the differential calculus relating an increment of a function f(x) and the values of its derivative. In analytic terms,
f(b) — f(a) = (b — a)f’(c)
where c is some number satisfying the inequality a < c < b. Formula (1) is valid if the function f(x) is continuous on the segment [a, b] and has a derivative at each point of the interval (a, b). In geometric terms (see Figure 1), formula (1) states that the tangent to the curve y = f(x) at a suitable point [c, f(c)] is parallel to the chord passing through the points [a, f(a)] and [b, f(b)]. The mean value theorem was discovered by J. Lagrange in 1797.
Among the different generalizations of the mean value theorem, note Bonnet’s mean value formula
and its particular case, Cauchy’s mean value formula