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:
“From whichever angle one looks at it, the application of racial theories remains a striking proof of the lowered demands of public opinion upon the purity of critical judgment.”
—Johan Huizinga (1872–1945)
“To cease to admire is a proof of deterioration.”
—Charles Horton Cooley (1864–1929)
“a meek humble Man of modest sense,
Who preaching peace does practice continence;
Whose pious life’s a proof he does believe,
Mysterious truths, which no Man can conceive.”
—John Wilmot, 2d Earl Of Rochester (1647–1680)