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)
“The thing with Catholicism, the same as all religions, is that it teaches what should be, which seems rather incorrect. This is “what should be.” Now, if you’re taught to live up to a “what should be” that never existed—only an occult superstition, no proof of this “should be”Mthen you can sit on a jury and indict easily, you can cast the first stone, you can burn Adolf Eichmann, like that!”
—Lenny Bruce (1925–1966)
“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)