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Math Help - Proof that G is differentiable on \R & compute G'

  1. #1
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    Proof that G is differentiable on \R & compute G'

    Hullo,

    One more question for the night. I need to prove the following:

    Let  f be a continuous function on  \mathbb{R}
    and define  G(x)=\int_0^{sinx}\; f(t)dt \backepsilon x \in \mathbb{R}.

    Show that  G is differentiable on  \mathbb{R} and compute  F'.

    Thanks in advance for the help.

    -the Doctor
    Last edited by thedoctor818; April 19th 2010 at 08:09 PM. Reason: needed to add 'thanks'
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  2. #2
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    Quote Originally Posted by thedoctor818 View Post
    Hullo,

    One more question for the night. I need to prove the following:

    Let  f be a continuous function on  \mathbb{R}
    and define  G(x)=\int_0^{sinx}\; f(t)dt \backepsilon x \in \mathbb{R}.

    Show that  G is differentiable on  \mathbb{R} and compute  F'.

    Thanks in advance for the help.

    -the Doctor
    I assume you mean G'

    Again, the idea is to make the problem more familiar:

    Let us define the function  H(x) = \int_0^{x}\; f(t)dt

    This function I assume you know how to differentiate.

    Now let's define the function I(x) = sinx, which we also know hoe to differentiate. Now, notice that

     G(x) = H(I(x))

    so  G'(x) = H'(I(x))*I'(x) ...
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  3. #3
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    I appreciate your help, but I still am a bit confused. So, should the derivative be: -sin(x^2)cos(x)?

    I must be having a brain fart, but I am having trouble tonight with the derivative of the basic integral you proposed: \int_0^{x^2}\;f(t)dt

    I do know that H'(x_0)=h(x_0) for this example, but am not sure how this helps me. Sorry, I have been up to long today & am missing the obvious.

    -Michael
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  4. #4
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    Quote Originally Posted by thedoctor818 View Post
    I appreciate your help, but I still am a bit confused. So, should the derivative be: -sin(x^2)cos(x)?

    I must be having a brain fart, but I am having trouble tonight with the derivative of the basic integral you proposed: \int_0^{x^2}\;f(t)dt

    I do know that H'(x_0)=h(x_0) for this example, but am not sure how this helps me. Sorry, I have been up to long today & am missing the obvious.

    -Michael
    H'(x_0)=f(x_0), so H'(I(x))=f(I(x))=f(sinx)
     I'(x) = cosx

    also, I'm not sure where you see an  x^2 , I never squared anything.
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  5. #5
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    Thanks, I believe that has me straightened out.

    -Michael
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  6. #6
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    Quote Originally Posted by thedoctor818 View Post
    Hullo,

    One more question for the night. I need to prove the following:

    Let  f be a continuous function on  \mathbb{R}
    and define  G(x)=\int_0^{sinx}\; f(t)dt \backepsilon x \in \mathbb{R}.

    Show that  G is differentiable on  \mathbb{R} and compute  F'.

    Thanks in advance for the help.

    -the Doctor
    As f is continuous it is Riemann integrable on any (finite) interval, and thus put \int f(x)\,dx=F(x) , i.e. F(x) is a primitive function of f, so:

    G(x)=\int\limits^{\sin x}_0 f(t)\,dt=F(\sin x)-F(0)\Longrightarrow G'(x)=\cos xF'(\sin x)=\cos xf(\sin x) I'm assuming you meant G" and not F'...)

    Tonio
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