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:
“When children feel good about themselves, it’s like a snowball rolling downhill. They are continually able to recognize and integrate new proof of their value as they grow and mature.”
—Stephanie Martson (20th century)
“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)
“Sculpture and painting are very justly called liberal arts; a lively and strong imagination, together with a just observation, being absolutely necessary to excel in either; which, in my opinion, is by no means the case of music, though called a liberal art, and now in Italy placed even above the other two—a proof of the decline of that country.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)