Results 1 to 2 of 2

Math Help - Prove the Fundamental Theorem of Calculus

  1. #1
    Newbie
    Joined
    Mar 2009
    Posts
    24

    Prove the Fundamental Theorem of Calculus

    Hi,

    I would like to know how to prove the following generalization of the Fundamental Theorem of Calculus using the method below.
    Suppose there is a finite set E in [a,b] and functions f,φ : [a,b]> R (reals) such that

    (a) φ is continuous on [a,b]
    (b) φ'(x) = f(x) for all x ∈[a,b]\E
    (c) f ∈R[a,b] (i.e. f is Riemann integrable)

    Then

    ∫f (from a to b) = φ(b) - φ(a)

    Method :Start by assuming E = {a,b} and remember that the Mean Value Theorem for derivatives applied to a (closed) interval [c,d], say, requires continuity on all of [c,d] but only requires differentiability on (c,d).
    Once you have done it for this special case, explain why it is still true if you add another point to E. The general case then follows.

    I am a bit confuse about the method specified by my teacher.
    It would be helpful if someone can guide me through this.
    Thank you very much.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Newbie
    Joined
    Mar 2009
    Posts
    24
    Anyone would have an idea of how my teacher would want the proof to be?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Fundamental Theorem of Calculus
    Posted in the Calculus Forum
    Replies: 1
    Last Post: April 15th 2009, 07:24 PM
  2. fundamental theorem of calculus
    Posted in the Calculus Forum
    Replies: 4
    Last Post: March 5th 2009, 11:17 AM
  3. Fundamental Theorem Of Calculus
    Posted in the Calculus Forum
    Replies: 8
    Last Post: June 19th 2008, 12:22 AM
  4. Replies: 2
    Last Post: June 14th 2007, 06:35 AM
  5. 2nd Fundamental Theorem of Calculus
    Posted in the Calculus Forum
    Replies: 3
    Last Post: November 6th 2006, 07:55 AM

Search Tags


/mathhelpforum @mathhelpforum