An understanding of this and the PointSlope Formula will make it clear that the equation of a secant (which intersects (a, f(a)) and (b, f(b)) ) is: y = {[f(b)  f(a)] / [b  a]}(x  a)  f(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 point lying between the endpoints of the cord such that the tangent at that point is parallel to 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
By Rolle's Theorem, there is some x in (a , b) for which g '(x) = 0, and it follows
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 lim_{h→0} (f(x+h)f(x))/h exists or is equal to ± infinity.
The first mean value theorem for integration states:
In particular (φ(t) = 1), there exists x in (a , b) with
The second mean value theorem for integration states:
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