1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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


    User Avatar
    Homework Helper

    [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
  4. Dec 10, 2008 #3
    i still don't understand this, can you elaborate?

    Thank You
  5. Dec 12, 2008 #4


    User Avatar
    Homework Helper

    Since cos(x t) is smooth the integral will be as well.
    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
    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


    User Avatar
    Science Advisor

    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook