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Math Help - Showing that derivative tends to infinity (Poisson solution)

  1. #1
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    Showing that derivative tends to infinity (Poisson solution)

    Hi,

    In the plot below I have f(\theta) = sin(\theta).

    The second graph of the plot shows that, for x = 1, the plot of P(\theta, x) = f(\theta - x f(\theta)) has an infinite slope at \theta = 0 (called the Poisson non-linear propagation solution).

    I'm trying to show this analytically but not getting the right answer. I reproduce my steps below.

    We want to show that for x = 1, P(\theta = 0, x) has an infinite slope, i.e. dP/d\theta \rightarrow \inf.

    P(\theta, x) = f(\theta - x f(\theta))

    we have f(\theta) = sin(\theta), so

    P(\theta, x) = sin(\theta - x sin(\theta))

    dP/d\theta = cos(\theta - sin(\theta))(1 - x cos(\theta))

    What I'm trying to show is dP/d\theta \rightarrow \inf, but that's clearly not possible from my result above.

    Keen to hear where I'm going wrong. Thanks
    Attached Thumbnails Attached Thumbnails Showing that derivative tends to infinity (Poisson solution)-poisson.png  
    Last edited by algorithm; June 21st 2013 at 01:58 PM.
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  2. #2
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    Re: Showing that derivative tends to infinity (Poisson solution)

    Use the definition of a derivative at a point x=a. where the derivative of some function at a specified point is lim x->a (f(x)-f(a))/(x-a) to solve your problem
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  3. #3
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    Re: Showing that derivative tends to infinity (Poisson solution)

    This is a double post of this thread. Please direct your comments to the original thread.

    -Dan
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  4. #4
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    Re: Showing that derivative tends to infinity (Poisson solution)

    @ShadowKnight8702: I'm doing a similar sort of thing but not getting the right answer. From post #1 I get an answer that won't tend to infinity under any circumstances.

    @topsquark: The problem is more concisely put in this thread.
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