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alternative proof of the fundamental theorem of calculus


An alternative proof for the first part involves the use of a formula derived by the method of exhaustionMathworldPlanetmath:

baf(t)𝑑t=(b-a)n=12n-1m=1(-1)m+12-nf(a+m(b-a)/2n).

Given that

F(x)=xaf(t)𝑑t,

and

F(x)=lim

the above formula leads to:

F(x)=limΔx0(x+Δx-x)Δxn=1m=12n-1(-1)m+12-nf(x+mΔx/2n),

or

F(x)=n=1m=12n-1(-1)m+12-nf(x).

Since it can be shown that

n=1m=12n-1(-1)m+12-n=n=12-n=1,

It follows that

F(x)=f(x).

The second part of the proof is identical to the parent.

Title alternative proof of the fundamental theorem of calculusMathworldPlanetmathPlanetmath
Canonical name AlternativeProofOfTheFundamentalTheoremOfCalculus
Date of creation 2013-03-22 15:55:24
Last modified on 2013-03-22 15:55:24
Owner ruffa (7723)
Last modified by ruffa (7723)
Numerical id 4
Author ruffa (7723)
Entry type Proof
Classification msc 26-00