A very quick question about definite integrals

[michonamona]

Homework Statement

F(x) = $$\int^{x}_{0}f(t)dt$$

Then F'(x) = f(x)

what is f'(x)? is this equivalent to f(t)?

Thanks for your help
M

 

[Dick]

F(x) is your integral of f from 0 to x. F'(x) is the derivative of F(x), which is f(x), the value of your integrand f(t) evaluated at t=x. This is just the fundamental theorem of calculus, that the integral is the antiderivative of the integrand.

 

[michonamona]

Thanks for the reply.

so f '(x) IS indeed f(t)? the very same f(t) in F(x)?

 

[Dick]

Thanks for the reply.

so f '(x) IS indeed f(t)? the very same f(t) in F(x)?

No. F'(x) is f(x). But, yes, the derivative of the integral is the function you are integrating, isn't that what the fundamental theorem of calculus is all about?

 

[michonamona]

I'm sorry, I don't think my notations are clear. I understand that big F'(x) = f(x), what I'm concerned with is whether small f '(x) is f(t).

so can we write

F(x) = $$\int^{x}_{0}f'(x)dt$$ = $$\int^{x}_{0}f(t)dt$$

 

[Mark44]

You don't have enough information to determine f'(x). The only information you have is that F is an antiderivative of f. A nearly equivalent way to say this is that f is the derivative of F. IOW, F'(x) = f(x).

For example, if F(x) = x³, F'(x) = f(x) = 3x². To go a step further and find f'(x), you need to know what the function f(x) is.

 

[Dick]

I'm sorry, I don't think my notations are clear. I understand that big F'(x) = f(x), what I'm concerned with is whether small f '(x) is f(t).

so can we write

F(x) = $$\int^{x}_{0}f'(x)dt$$ = $$\int^{x}_{0}f(t)dt$$

How can f'(x) be the same as f(t)? They don't even involve the same variable.

 

[michonamona]

Thanks guys, it now makes sense. I keep getting all the notations mixed up.

I appreciate all of your help

M