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Help w/First Fundamental Theorem of Calculus

  1. Dec 10, 2007 #1
    For the first fundamental theorem of calculus, I don't quite understand it...

    I think I got the integral part, but not the interval [a,x]...

    can you guys help me? thx
  2. jcsd
  3. Dec 10, 2007 #2
    Well your question is very vague but I will try and address any misunderstanding you may have.

    The first fundamental theorem of calculus says that given a real valued function defined on [a,b] and if we define a new function F (note that little f and big F are two different functions here) by

    F(x)=integral (lower limit is a, upper limit is x) of f(x) dx

    then F(x) is continuous on all x in the closed interval [a,b]. Furthermore, if the function f is continuous at a point u that is inside the closed interval [a,b] then F is differentiable at u and it's derivative is f(u).

    the first part of the theorem says that if you change the upper limit of integration a LITTLE bit then the change in integrals is also LITTLE. The second part is just saying that if f is continuous at a point then F is SMOOTH (ie differentiable) at that point and that the derivative operation and integral undo one another.

    [a,x] is a subset of [a,b]. It is a fact that if f is integrable on [a,b] and x is a point in the interval [a,b] then f is integrable on [a,x].

    hope this helps.
  4. Dec 10, 2007 #3
    ah i still dun understand...

    can we work on an example from the textbook?
    The example is:
    Find the derivative of
    [tex]\intcost dt[/tex] from 0 to [tex]\sqrt{x}[/tex]

    a. evaluating the integral and differentiating the result
    b. by differentiating the integral directly

    for part a.
    The integral of cos(t) is sin(t). Thus the integral from 0 to [tex]\sqrt{x}[/tex] is
    sin[tex]\sqrt{x}[/tex] - sin(0)
    = sin[tex]\sqrt{x}[/tex]
    Differentiating that, I get [tex]\frac{cos\sqrt{x}}{2\sqrt{x}}[/tex]

    I guess my problem is differentiating an integral directly...

    What would happen if its from sin(x) to x? or from 5 to x?
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