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Math Help - Taylor's Inequality

  1. #1
    Senior Member mollymcf2009's Avatar
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    Taylor's Inequality

    This is part b. of the problem. I found T_3(x) but I really don't know the process to find the estimate of accuracy of the approximation.
    HEre is the problem:

    a) Approximate f by a Taylor polynomial with degree n at the number a.

    f(x) = e^{4x^2}  a=0 n=3 on 0\leq x \leq 0.3

    My Taylor polynomial:

    T_3 (x) = 1 + 4x^2

    b) Use Taylor's Inequality to estimate the accuracy of the approximation f = T_n(x) when x lies in the given interval. Round to five decimal places.

    The 4th derivative (not sure if this is correct):

    f^4(x) = 192e^{4x^2} + 3072x^2e^{4x^2} + 4096x^3e^{4x^2}

    f^4(0) = 192 *nevermind on this part, can't use zero because a=0

    I really need someone to please take me through each step of how to find the accuracy using Taylor's Inequality, I am not having luck understanding it from my notes. Thanks!!
    Last edited by mollymcf2009; April 19th 2009 at 02:10 PM.
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  2. #2
    MHF Contributor Calculus26's Avatar
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    Rn(x) = Mx^(n+1)/(n+1)!

    Where M is the max of the fourth derivative

    f '''' (x) is increasing on (0,.3) so M is f''''(.3)

    Rn(x)< f''''(.3) (.3)^4/(4!) for all x on (0.3)
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