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Differentiability with sinx/x

  1. Dec 6, 2008 #1
    Let f(x)= sinx/x if x [tex]\neq[/tex] 0 and f(0)=1
    Find a polynomial pN of degree N so that
    |f(x)-pN(x)| [tex]\leq[/tex] |x|^(N+1)
    for all x.
    Argue that f is differentiable, f' is differentiable, f" is differentiale .. (all derivatives exist at all points).

    I'm not sure about this one at all. Can you guys help me out?

    Thank You
     
  2. jcsd
  3. Dec 6, 2008 #2

    lurflurf

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    [tex]f(x)=\int_0^1 \cos(x t) dt[/tex]
    so approximate cos first and the integral for f with cos approximated will approximate f.
    The derivatives clearly exist and
    |(D^n)f|<1/n+1
     
  4. Dec 10, 2008 #3
    i still don't understand this, can you elaborate?

    Thank You
     
  5. Dec 12, 2008 #4

    lurflurf

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    Since cos(x t) is smooth the integral will be as well.
    Since
    Cos(x)~1-x^2/2+x^4/24-x^6/720+...
    is a family of approximations of cosine (each member being a sum the first n=1,2,3,... terms) we may repace cosine by an approximation in the integral representation of f to see that
    f~1-x^2/6+x^4/120-x^6/5040+...
    are approximations of f.

    You function f at zero has what is called a removable singularity, a ficticious singularity that is caused by the representation, not by actual properties of the function. By representing the function differently (such as using the integral representation I gave) the singularity and any problems it may cause vanish.
     
  6. Dec 12, 2008 #5

    HallsofIvy

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    Did you consider taking the Taylor's series for sin x, around x= 0, and dividing each term by x? That seems to me to be far simpler than using the integral form.
     
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