What is the FTC...exactly?

[Nootropic]

So talking to my calculus teacher, I happened to mention that the Fundamental Theorem of Calculus states that integration and differentiation, the central operations of calculus, are inverses of each other. He then said, not quite, that's just a consequence of the FTC. Though I'm aware of how it's stated in mathematical terms*, I'm not exactly sure if there's a better verbal formulation or I happen to missing some point? Anybody?

 

 

* F'(x) = f(x), where F(x) is the integral of f(t) with the limits of integration from 0 to x. Sorry for being latex-deficient!

[eric l]

If my memory does not fail me (and it's been over 40 years by now that I learned calculus) the statement that integration is the inverse of differentiation is only valid when the primary function is continuous.

[Nootropic]

Oh, well, yes, that is something you have to throw in there . But I've talked to other family members who are mathematicians, and they said my interpretation is correct (if the function is, of course, continuous), but it's not something you can base everything off of. The actual Fundamental Theorem of Calculus is the matematical statement, not a formulation of words. I suppose that has a bit to do with Plato's mathematical world, in a sense.

[Bombadil]

What the Fundamental Theorem of Calculus says is that the derivative of the integral of F(x) is f(x). What this means is that the integral form A to B can be found assuming that it is continuous by finding the value of the anti-derivative at the point B and subtracting the value of the anti-derivative at A.

[Nootropic]

That would actually be the second part of it, or not exactly the part I'm discussing. Interestingly enough, I mentioned something similar (read the first post) and my teacher mentioned it being a consequence, and was not the theorem itself. He was, I'm assuming, alluding to a more realist interpretation of mathematics, rather than giving the theorem a verbal formulation, he was, as I see now, saying that the matematical formulation is the theorem (which is true) and that a verbal formulation is not quite the theorem itself.