Mean value theorem

In calculus,mean value theorem states, roughly, that givensection ofsmooth curve, there ispoint on that section at whichgradient (slope) ofcurveequal to"average" gradient ofsection.

This theorem was developed by Lagrange. Some mathematicians consider this theorembe the most important theoremcalculus (see also:fundamental theoremcalculus). The theoremnot often usedsolve mathematical problems; rather, itmore commonly usedprove other theorems. The mean value theorem can be usedprove Taylor's theorem,whichisspecial case.

More precisely,theorem states:some continually differentiable curve;every secant, theresome parallel tangent. In addition,tangent runs throughpoint located betweenintersection pointssaid secant.

Sketchmean value theorem

Let f : [a, b] → R be continuous onclosed interval [a, b],differentiable onopen interval (a, b). Then there exists some c(a, b) such that

Generalization: The theoremusually stated inform above, but itactually valid inslightly more general setting: We only needassume that f : [a , b] → Rcontinuous on [a , b],thatevery x(a , b)limit lim_h→0 (f(x+h)-f(x))/h exists orequal± infinity.

Tablecontents

1 Proof 2 Cauchy's mean value theorem 2.1 ProofCauchy's mean value theorem 3 Mean value theoremsintegration

Proof

An understandingthis andPoint-Slope Formula will makeclear thatequation ofsecant (which intersects (a, f(a))(b, f(b)) ) is: y = {[f(b) - f(a)] / [b - a]}(x - a) - f(a).

The formula ( f(b) - f(a) ) / (b - a) givesslope ofline joiningpoints (a , f(a))(b , f(b)), which we callchord ofcurve, while f ' (x) givesslope oftangent tocurve atpoint (x , f(x) ). ThusMean value theorem says that given any chord ofsmooth curve, we can findpoint lying betweenend-points ofchord such thattangent at that pointparallel tochord. The following proof illustrates this idea.

Define g(x) = f(x) + rx , where r isconstant. Since fcontinuous on [a , b]differentiable on (a , b),sametrueg. We choose r so that g satisfiesconditionsRolle's theorem, which means

By Rolle's Theorem, theresome c(a , b)which g '(c) = 0,it follows

as required.

The mean value theorem infollowing formconsidered more useful.

Cauchy's mean value theorem

Cauchy's mean value ismore generalised formmean value theorem. It states: If functions f(t)g(t)both continuous onclosed interval [a,b]differentiable onopen interval (a,b), then there exists some c, such that

Cauchy's mean value theorem can be usedprove l'Hopital's rule.

ProofCauchy's mean value theorem

The proofCauchy's mean value theorembased onsame idea asproofmean value theorem. We aimtransformcurve defined by y = y(t)x = x(t), so thatsatifiesconditionsRolle's theorem.

We definenew function:

where m isconstant, so that

Since FcontinuousF(a) = F(b), by Rolles theorem, there exist some c(a,b) such that F'(c) = 0, i.e.

as required.

Mean value theoremsintegration

The first mean value theoremintegration states:

If f : [a , b] → R iscontinuous functionφ : [a , b] → Ran integrable positive function, then there existsnumber x(a , b) such that

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

The second mean value theoremintegration states:

If f : [a , b] → R ispositivemonotone decreasing functionφ : [a , b] → Ran integrable function, then there existsnumber x(a , b] such that