The Mean Value Theorem of real analysis states the following: Let f:[a , b] -> R be continuous on [a , b] and differentiable on (a , b). Then, for some x in (a , b) f '(x) = ( f(b) - f(a) ) / (b - a) The formula ( f(b) - f(a) ) / (b - a) gives the gradient 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 gradient of the tangent to the curve at the point (x , f(x) ). Thus the MVT 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. Proof of MVT: Define g(x) = f(x) + hx , where h is a constant. Since f is continuous on [a , b] and differentiable on (a , b) the same is true of g. We choose h so that g satisfies the conditions of Rolle's Theorem. Then f(a) + ha = f(b) + hb => h = -( f(b) - f(a) ) / (b - a) By Rolle's Theorem, there is some x in (a , b) for which g ' (x) = 0. i.e. f ' (x) = -h = ( f(b) - f(a) ) / (b - a) , as required |
Mean value theorem of differential calculusIn 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 integrationThe 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. |
In real analysis, the mean value theorem for differentiation states the following:
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
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.
The first mean value theorem for integration states:
The second mean value theorem for integration states: