Proof of The Corollary
Suppose F is an antiderivative of f, with f continuous on . Let
- .
By the first part of the theorem, we know G is also an antiderivative of f. It follows by the mean value theorem that there is a number c such that G(x) = F(x) + c, for all x in . Letting x = a, we have
which means c = − F(a). In other words G(x) = F(x) − F(a), and so
Read more about this topic: Fundamental Theorem Of Calculus
Famous quotes containing the words proof of the, proof of and/or proof:
“The fact that several men were able to become infatuated with that latrine is truly the proof of the decline of the men of this century.”
—Charles Baudelaire (1821–1867)
“In the reproof of chance
Lies the true proof of men.”
—William Shakespeare (1564–1616)
“If any doubt has arisen as to me, my country [Virginia] will have my political creed in the form of a “Declaration &c.” which I was lately directed to draw. This will give decisive proof that my own sentiment concurred with the vote they instructed us to give.”
—Thomas Jefferson (1743–1826)