Good proofs of the Fundamental Theorem of Calculus?
Jeremy
theorem and another one that goes like, g(x) = int( f(t) dt from a to x),
then F = g + C, and g(a) = 0, so F(b) - F(a) = [g(b)+c]-[g(a)+c] = g(b) =
int[f(t) dt from a to b]. But are there any other proofs for this? I am
most interested in proofs that are interesting / clever in some way.
Thanks,
Jeremy
G. A. Edgar
<cfg...@pants.attbi.com> wrote:
How about the one using the gauge integral (Henstock, Kurzweil,
etc...)? It is quite simple to prove, and includes the FTC in some
cases where Riemann and/or Lebesgue integrals fail.
--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/
Proginoskes
> I've seen the proofs for the FTC on integration involving the mean value
> theorem and another one that goes like, g(x) = int( f(t) dt from a to x),
> then F = g + C,
I use this definition and then show that the derivative of g(x) is f(x).
> and g(a) = 0, so F(b) - F(a) = [g(b)+c]-[g(a)+c] = g(b) =
> int[f(t) dt from a to b]. But are there any other proofs for this? I am
> most interested in proofs that are interesting / clever in some way.
I think the proof which I use, which consequently uses the defnition of
the derivative, the intermediate value theorem, the definition of
continuity, the squeeze theorem, and the mean value theorem, summarizes
the first semester of a calculus course nicely. And after I've led the
class through this derivation -- I don't require that they take notes --
I tell them they've seen more real mathematics (as opposed to following
recipes) in the last 20-30 minutes than they've seen in their life
previously. It also shows that these theorems actually work together and
aren't just single entities.
-- Christopher Heckman