Mean Value Theorem

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mean value theorem

[′mēn ′val·yü ‚thir·əm]
The proposition that, if a function ƒ (x) is continuous on the closed interval [a,b ] and differentiable on the open interval (a,b), then there exists x0, a <>x0<>b, such that ƒ(b) - ƒ(a) = (b-a)ƒ′(x0). Also known as first law of the mean; Lagrange's formula; law of the mean.

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.

Figure 1

Among the different generalizations of the mean value theorem, note Bonnet’s mean value formula

and its particular case, Cauchy’s mean value formula

References in periodicals archive ?
Then by the mean value theorem and the assumptions, we have
3] Zhang Guangfan, A Note on Mean Value Theorem of Differentials, Mathematics in Practice and Theory, 18(1988), No.
m) and the n-dimensional version of the mean value theorem, we have
N] we used the following interesting mean value theorem, [14].
21 The Generalized Mean Value Theorem (Cauchy's MVT), L' Hospital's Rule, and their Applications
p), and there exit [xi] = q + [theta] (p - q), [theta] [member of] (0,1) by mean value theorem, such that