How Do the Two Theorems of the Fundamental Theorem of Calculus Imply Each Other?

[roam]

There are two theorems:

The fundamental theorem of calculus: \int_{a}^{b}F'(x) = F(b) - F(a)

And the theorem that states if f is continious on [a,b]and g:[a,b]->R is defined by
g(x) = \int_{a}^{x}f(t) dt, then g is differentiable on (a,b) and;

g'(x) = \frac{d}{dx}(\int^{x}_{a}f(t)) = f(x)

"There’s one question that asks if it is possible to readily show that each one implies the other, by a few lines of manipulation."

Does anyone know how to do that?

Let f be cts on [a,b], and let F(x) = \int f(x) dx be called F, the antiderivative of f. Then

\int_{a}^{b}f(x) dx = F(b) - F(a)

By theorem \frac{d}{dx}(\int^{x}_{a}f(t)) = f(x), we know \int_{a}^{x}f(t) dt = F(b) - F(a) & F(x) has the same derivative, f(x). Hence there will be a constant c such that \int_{a}^{x}f(t)dt = F(x)+c When x = a, we get

F(a) + c = \int_{a}^{a}f(t)dt = 0 so c = -F(a)

Hence, \int_{a}^{x}f(t)dt = F(x)-F(a)
When x = b this yields

\int_{a}^{b}f(t)dt = F(b)-F(a)

I'm not sure if this really answers the question though.

 

[mathwonk]

do you know the mean value theorem?

 

[roam]

A little, how can I use it to show that in those two theorems each one implies the other one?