Fundamental theorem of calculus

[jesuslovesu]

(that's a 3 on the last integral)
http://img131.imageshack.us/img131/2549/jesus1cj.png

I need to find which of those are true, now I thought I and III were true
for sure. But when I do II with an example f(x) = x^2 I get x^2 - 9, so it's not true right? (I and III are not choices given for the correct answer)I know the fundamental theorem of calculus states that the derivative
of an integral is just the function.

 

[0rthodontist]

Yes, II is not true in general. Why not make a test function for the others also?

You might want to look again at the precise statement of the fundamental theorem of calculus.

 

[topsquark]

(that's a 3 on the last integral)
http://img131.imageshack.us/img131/2549/jesus1cj.png

I need to find which of those are true, now I thought I and III were true
for sure. But when I do II with an example f(x) = x^2 I get x^2 - 9, so it's not true right? (I and III are not choices given for the correct answer)


I know the fundamental theorem of calculus states that the derivative
of an integral is just the function.

Consider what kind of integral we are talking about in I. Consider:
$$\int_0^3x^2 \, dx = (1/3)x^3|_0^3=9 \neq x^2$$.

-Dan

 

[VietDao29]

One thing to remember is that:
$$\mathop {\int} \limits_{0} ^ 3 f(x) dx$$ is some specific number, whose derivative with respect to x is just a plain 0.
While this:
$$\mathop {\int} \limits_{0} ^ x f(x) dx$$ is different, since the result does depend on what x you choose. And it's a function of x.
Can you get this? :)

 

[HallsofIvy]

Okay, you see that if you try some simple function in 2, you get different results on left and right (differ by a constant) so that is not correct.

It has been pointed out that 1 is obviously untrue (the derivative of a constant is 0).

What about 3? Choose some simple functions and see what happens. Of course, examples won't prove a general statement is true but think about the "Fundamental Theorem of Calculus".