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Math Help - showing function differentiable

  1. #1
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    showing function differentiable

    Let [0,1] --->R be continuous. (little f above arrow)
    Define [0,1] --->R (capital F above arrow) by F(x) = \displaystyle\int^{x^2}_0 f \,

    Show that F is differentiable with  F'(x) = 2x f(x^2).


    I seriously have worked on this question for hours and have got nowhere! Can anyone give me any help at all?!

    Thanks in advance!
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  2. #2
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    Quote Originally Posted by hunkydory19 View Post
    Let [0,1] --->R be continuous. (little f above arrow)
    Define [0,1] --->R (capital F above arrow) by F(x) = \displaystyle\int^{x^2}_0 f \,

    Show that F is differentiable with  F'(x) = 2x f(x^2).


    I seriously have worked on this question for hours and have got nowhere! Can anyone give me any help at all?!

    Thanks in advance!
    Let G(x) = \displaystyle\int^{x^2}_0 f \,

    G(x)=F(x^2)-F(0)

    G'(x)=f(x^2)2x-f(0), assume f(0)=0 to get: G'(x)=f(x^2)2x

    I used G instead of F for the left side so you wouldn't confuse the two!
    Last edited by colby2152; February 1st 2008 at 06:22 AM.
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    am i missing something here or did you assume that F(0) = 0
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    Quote Originally Posted by bobak View Post
    am i missing something here or did you assume that F(0) = 0
    Yes, I added that assumption in.
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    Quote Originally Posted by colby2152 View Post
    Yes, I added that assumption in.
    surely F(0) is constant so its differentiates to give 0
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    Quote Originally Posted by bobak View Post
    surely F(0) is constant so its differentiates to give 0
    Good point, because F(0)=\int_0^0 fdx \Rightarrow 0
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    Quote Originally Posted by colby2152 View Post
    Good point, because F(0)=\int_0^0 fdx \Rightarrow 0

    isnt that G(0) ?
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    Quote Originally Posted by bobak View Post
    isnt that G(0) ?
    Yes, in my example, that is G(0)

    Cite back to my original reply.
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    Quote Originally Posted by hunkydory19 View Post
    Let [0,1] --->R be continuous. (little f above arrow)
    Define [0,1] --->R (capital F above arrow) by F(x) = \displaystyle\int^{x^2}_0 f \,

    Show that F is differentiable with  F'(x) = 2x f(x^2).


    I seriously have worked on this question for hours and have got nowhere! Can anyone give me any help at all?!

    Thanks in advance!
    Let  \chi (x) = \int_0^x f(x) dx for x\in [0,1].
    Let \mu (x) = \int_0^{x^2} f(x) dx for x\in [0,1].
    We wish to show \mu is differenciable and find its derivative.

    Note, \mu (x) = \chi (x^2). Now \chi and x^2 are differenciable on (0,1) by the fundamental theorem. This means, \mu ' (x) = 2x \chi ' (x^2) = 2x f(x^2).
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