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Linearization Error

  1. Jan 4, 2012 #1
    1. The problem statement, all variables and given/known data

    I've been given the assignment to make a linearization of the function f(x) = Sin(x) e^(x) by using Taylors polynominal of 3. degree.

    Next is to find the approximately error of f(x) when |x| < 1 (expressed as a function of x).

    2. Relevant equations

    So finding the linearization is rather easy to get through, however finding the estimated error is where I just can't figure it out.

    The formula that I've been given from my book is as follows:

    [itex](f''(s)/2)(x-a)^2[/itex]

    for some s lying between a og x. But finding the s value is just something, I have no idea how to figure out. I tried to search some and found out that you could just find the maximum y- value of your function, within that interval, and put it instead of f''(s). Tried that and got a very unexact approximation.

    Hope some of you have an idea, how to solve this problem :)
     
  2. jcsd
  3. Jan 4, 2012 #2

    Ray Vickson

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    Science Advisor
    Homework Helper

    You are using the wrong error formula. The formula above is for the error in a _linear_ approximation (i.e., using a polynomial of degree 1). You have been asked to use a polynomial of degree 3 (a cubic approximation), so the error will involve the 4th derivative.

    Anyway, you can easily estimate the "maximum possible absolute error" as a function of (x-a), just by estimating max_s{|f''''(s)|, a <= s <= x} and using that instead of |f''''(s)|. However, to do this we need to be able to compute exactly the values of exp(x)*sin(x) at various x. This would be OK if we wanted to do this once and then use the approximate formula thereafter (in other words, do a lot of work off-line in order to validate a formula to be used on-line). An alternative might be to use a cruder estimate that does not involve already knowing how to compute exp(x)*sin(x). Again, though, you would be estimating the maximum possible error, not the actual error itself.

    RGV
     
  4. Jan 8, 2012 #3
    Thanks for the answer. And sorry for the late reply.

    I'm still quite unclear of what you mean, to be exact. In words, the assigment says that I have to estimate the error for |x| < 1. So I guess that has to be the maximum error, in form of a function.

    But then you say, then when I have a 3rd degree Taylor's polynominal, I have to use f^(n+1) = f''''(s). But how exactly to find the maximum value of s within that interval numerical? One thing is that I can see it quite clearly on the graph, but if it happens to be a graph, where you almost have to identical points, with just a slightly difference, then there must be some way to solve it numerical.
     
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