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Math Help - differentiating limit statements

  1. #1
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    differentiating limit statements

    if f(x,t) is differentiable on a subset S of the real plane, under what conditions does
    \frac{d}{dx}(lim_{t->a}f(x,t))=lim_{t->a}(\frac{d}{dx}f(x,t))

    e.g: i) \frac{d}{dx}(lim_{t->a}((x-t)^2)=\frac{d}{dx}(x-a)^2=2(x-a)
    and lim_{t->a}(\frac{d}{dx}(x-t)^2)=lim_{t->a}2(x-t)=2(x-a)
    ii) \frac{d}{dx}(lim_{t->0}(t^{-1}*sin(t*x)))=\frac{d}{dx}x=1
    and lim_{t->0}(\frac{d}{dx}t^{-1}*sin(t*x))=lim_{t->0}(cos(t*x))=1
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by JeffN12345 View Post
    if f(x,t) is differentiable on a subset S of the real plane, under what conditions does
    \frac{d}{dx}(lim_{t->a}f(x,t))=lim_{t->a}(\frac{d}{dx}f(x,t))

    e.g: i) \frac{d}{dx}(lim_{t->a}((x-t)^2)=\frac{d}{dx}(x-a)^2=2(x-a)
    and lim_{t->a}(\frac{d}{dx}(x-t)^2)=lim_{t->a}2(x-t)=2(x-a)
    ii) \frac{d}{dx}(lim_{t->0}(t^{-1}*sin(t*x)))=\frac{d}{dx}x=1
    and lim_{t->0}(\frac{d}{dx}t^{-1}*sin(t*x))=lim_{t->0}(cos(t*x))=1
    Let us restate this, since it is the same thing as "When is \frac{d}{dx}\lim_{n\to\infty}f_n(x)=\lim_{n\to\inf  ty}\frac{d}{dx}f_n(x)?". The answer is quite strict. We need that f_n(x) is uniformly convergent, f_n(x) is differentiable for all n, and that f_n'(x) is uniformly convergent.
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  3. #3
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    considering this, under which circumstances is f(x)=\lim_{n->\infty}f_n(x) where  f_n(x)=\Sigma_{k=1}^n(\frac{1}{n}(f(\frac{kx-(n-k)a}{n})+\frac{k(x-a)}{n}f'(\frac{kx+(n-k)a}{n})))
    (I would like if f(x) and f'(x) were uniformly continuous over some subest S of R this would be enough. Is this the case?)
    Last edited by JeffN12345; January 9th 2010 at 02:49 PM.
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by JeffN12345 View Post
    considering this, under which circumstances is  f(x)=\lim_{n->\infty}\Sigma_{k=1}^n(\frac{1}{n}(f(\frac{kx-(n-k)a}{n})+\frac{k(x-a)}{n}f'(\frac{kx+(n-k)a}{n})))
    I remember your last thread Jeff. This gets even more complicated. Write exactly you want to do.
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    im considering the x-derivative of the left-riemann-sum of f(t) over the n-regular partition of the the interval [a,x] and how it should be converge to f(x)
    i would like this to work for any a and x for which the function is riemann-integrable on [a,x]
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  6. #6
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by JeffN12345 View Post
    im considering the x-derivative of the left-riemann-sum of f(t) over the n-regular partition of the the interval [a,x] and how it should be converge to f(x)
    i would like this to work for any a and x for which the function is riemann-integrable on [a,x]
    What is your background? What math do you know? I mean, do you know the concept of uniform convergence? I don't ask this to be condescending but because it will dictate how I answer your question.
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  7. #7
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    im in first year, second term at waterloo and im taking calculus 2 now. We just finished defining an integral and the conditions for riemann-integrability. I think we will learn uniform convergence later this term, but i just read what it is in my textbook after you mentioned it.
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  8. #8
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by JeffN12345 View Post
    im in first year, second term at waterloo and im taking calculus 2 now. We just finished defining an integral and the conditions for riemann-integrability. I think we will learn uniform convergence later this term.
    That amazes me. You are in a course learning about Riemann integrability (I assume that means Riemann-Stieltjes integrability for me), but you haven't learned about uniform convergence?

    Also, don't most Calc II courses do stuff along these lines \int x^n\text{ }dx=\frac{x^{n+1}}{n+1}+C\quad n\in\mathbb{N}
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