Results 1 to 5 of 5

Math Help - estimating error Taylor polynomials

  1. #1
    Junior Member
    Joined
    Jun 2011
    From
    Colorado, United States
    Posts
    56

    estimating error Taylor polynomials

    I have been asked to do the following:
    Suppose that f is a function with the property that | f^n(x)| \leq3 for all n and for all x on [-1,1].
    a. Estimate the error if the Taylor polynomial p_{n}(x) (in powers of x) with n=4 is used to approximate f( \frac{1}{2}).
    b. Find the least positive integer n such that p_{n}(x) (in powers of x) will approximate f on [-1,1] to within .001.


    I don't know where to start on this problem. I have no f(x) to work with so what do I do?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor FernandoRevilla's Avatar
    Joined
    Nov 2010
    From
    Madrid, Spain
    Posts
    2,162
    Thanks
    45

    Re: estimating error Taylor polynomials

    Use f(x)=p_4(x)+\dfrac{f^{(5)}(\xi)}{5!}x^5 where \xi is between 0 and x .
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Jun 2011
    From
    Colorado, United States
    Posts
    56

    Re: estimating error Taylor polynomials

    Quote Originally Posted by FernandoRevilla View Post
    Use f(x)=p_4(x)+\dfrac{f^{(5)}(\xi)}{5!}x^5 where \xi is between 0 and x .
    Do I substitute \dfrac{1}{2} in for x? If so, I end up with f(x)=p_4(\dfrac{1}{2})+\dfrac{f^{(5)}(\xi)}{5!} *(\dfrac{1}{2})^5 .
    Or do I substitute \dfrac{1}{2} in for \xi? Which gives me f(x)=p_4(x)+\dfrac{f^{(5)}(\dfrac{1}{2})}{5!} *x^5 .

    In either case, is it possible to get a number as a value for the error with the information I've been given?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor FernandoRevilla's Avatar
    Joined
    Nov 2010
    From
    Madrid, Spain
    Posts
    2,162
    Thanks
    45

    Re: estimating error Taylor polynomials

    f(1/2)-p_4(1/2)=\dfrac{f^{(5)}(\xi)}{5!}(1/2)^5\Rightarrow |f(1/2)-p_4(1/2)|\leq (3/2^5)(1/5!) .
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Jun 2011
    From
    Colorado, United States
    Posts
    56

    Re: estimating error Taylor polynomials

    Quote Originally Posted by FernandoRevilla View Post
    f(1/2)-p_4(1/2)=\dfrac{f^{(5)}(\xi)}{5!}(1/2)^5\Rightarrow |f(1/2)-p_4(1/2)|\leq (3/2^5)(1/5!) .
    The error for \p_{4} is .0007813 and the error for \p_{3} is .007813, so n must be equal to 4 to remain within .001.

    Thank you.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Taylor Polynomials Approx. Error
    Posted in the Calculus Forum
    Replies: 1
    Last Post: October 24th 2009, 01:40 AM
  2. estimating error please help
    Posted in the Calculus Forum
    Replies: 3
    Last Post: November 8th 2008, 07:54 AM
  3. Taylor Polynomials error bounds
    Posted in the Calculus Forum
    Replies: 1
    Last Post: October 8th 2008, 10:46 AM
  4. Replies: 1
    Last Post: March 31st 2007, 10:58 PM
  5. Estimating error using differentials
    Posted in the Calculus Forum
    Replies: 3
    Last Post: March 20th 2007, 09:08 PM

Search Tags


/mathhelpforum @mathhelpforum