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Fundamental theorem of calculus

  1. Mar 2, 2006 #1
    (that's a 3 on the last integral)

    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.
    Last edited: Mar 2, 2006
  2. jcsd
  3. Mar 2, 2006 #2


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    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.
  4. Mar 2, 2006 #3
    Consider what kind of integral we are talking about in I. Consider:
    [tex]\int_0^3x^2 \, dx = (1/3)x^3|_0^3=9 \neq x^2[/tex].

  5. Mar 4, 2006 #4


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    One thing to remember is that:
    [tex]\mathop {\int} \limits_{0} ^ 3 f(x) dx[/tex] is some specific number, whose derivative with respect to x is just a plain 0.
    While this:
    [tex]\mathop {\int} \limits_{0} ^ x f(x) dx[/tex] is different, since the result does depend on what x you choose. And it's a function of x.
    Can you get this? :)
    Last edited: Mar 4, 2006
  6. Mar 4, 2006 #5


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    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".
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