# Proof Using Mean Value Theorem

1. Jan 26, 2005

Hello all

Using the Mean Value Theorem, prove that the derivative of the indefinite integral $\int f(x) \ dx$ is $f(x)$

So do I just use the fact that $\int^b_a f(x) \ dx = f(\xi)(b-a)$?

Thanks

2. Jan 27, 2005

is this right?

3. Jan 27, 2005

### HallsofIvy

Staff Emeritus
You CAN'T use the mean value theorem to prove the Fundamental Theorem of Calculus.

4. Jan 27, 2005

not prove but maybe show

5. Jan 27, 2005

### vincentchan

hey... do you forget our rule here... don't give out the answer... delete the link and give him some hints lead to the answer

6. Jan 27, 2005

### learningphysics

7. Jan 27, 2005

### learningphysics

I believe your question should to prove that the derivative of this function:

$$F(x)=\int_a^{x} f(t)dt$$

is f(x). Am I right? The above is a definite integral. The derivative of the indefinite integral is just f(x) by definition. Indefinite integral means anti-derivative.

What is F'(x) from first principles i.e: using the definition of derivative?

8. Jan 27, 2005

$$F'(x) = \frac{F(x+\Delta x) - F(x)}{\Delta x}$$

forgot to put limit

Last edited: Jan 27, 2005
9. Jan 27, 2005

### learningphysics

Careful... we're looking for F'(x) not f'(x).

10. Jan 27, 2005