Mean value theorem

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Mean value theorem of differential calculus

In real analysis, the mean value theorem for differentiation states the following:

Let f : [a , b] -> R be continuous on the interval [a , b] and differentiable on (a , b). Then there exists some x in (a , b) with
f '(x) = ( f(b) - f(a) ) / (b - a)

The formula ( f(b) - f(a) ) / (b - a) gives the slope of the line joining the points (a , f(a)) and (b , f(b)), which we call a chord of the curve, while f ' (x) gives the slope of the tangent to the curve at the point (x , f(x) ). Thus the Mean value theorem says that given any chord of a smooth curve, we can find a tangent parallel to that chord and moreover we can take the tangent to some point lying between the end-points of the chord.

The mean value theorem can be used to prove Taylor's theorem, of which it is a special case.

Proof of the theorem: Define g(x) = f(x) + rx , where r is a constant. Since f is continuous on [a , b] and differentiable on (a , b), the same is true of g. We choose r so that g satisfies the conditions of Rolle's theorem, which means

f(a) + ra = f(b) + rb
=> r = -( f(b) - f(a) ) / (b - a)

By Rolle's Theorem, there is some x in (a , b) for which g '(x) = 0, and it follows

f '(x) = g '(x) - r = 0 - r = ( f(b) - f(a) ) / (b - a)

as required.

Generalization: The theorem is usually stated in the form above, but it is actually valid in a slightly more general setting: We only need to assume that f : [a , b] -> R is continuous on [a , b], and that for every x in (a , b) the limit limh->0 (f(x+h)-f(x))/h exists or is equal to +/- infinity.

Mean value theorems for integration

The first mean value theorem for integration states:

If f : [a , b] -> R is a continuous function and φ : [a , b] -> R is an integrable positive function, then there exists a number x in (a , b) such that
abf(t) φ(t) dt    =    f(x) ∫abφ(t) dt.

In particular (φ(t) = 1), there exists x in (a , b) with

abf(t) dt    =    f(x) (b - a).

The second mean value theorem for integration states:

If f : [a , b] -> R is a positive and monotone decreasing function and φ : [a , b] -> R is an integrable function, then there exists a number x in (a , b] such that
abf(t) φ(t) dt    =    (limt->a f(t)) · ∫axφ(t) dt.