Results 1 to 4 of 4

Math Help - Taylor approximation.

  1. #1
    Senior Member Pinkk's Avatar
    Joined
    Mar 2009
    From
    Uptown Manhattan, NY, USA
    Posts
    419

    Taylor approximation.

    Use Taylor approximation on e^{-x^{2}} to compute \int_{0}^{1} e^{-x^{2}}\,dx to three decimal places and prove the accuracy of your answer using the theorem that if f is of class C^{k+1} on an interval I and |f^{(k+1)}(x)|le M for x\in I, then |R_{a, k}(h)|\le \frac{M}{(k+1)!}|h|^{k+1}.

    So I know how to find the taylor expansion of e^{-x^{2}} about zero and then integrate each term, but I do not know how to approximate to ensure that the approximation is good to three decimal places nor do I know how to apply that theorem to prove it. Any help would be appreciated.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    14,973
    Thanks
    1121
    You have that the error in the polynomial value is less than \frac{M}{(k+1)!}|h|^{k+1} (M is an upper bound on e^{-x^2}) so the error in the integral is less than the integral of that from 0 to 1. And since that is a constant, it is just that number times the length of the interval, which is 1.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member Pinkk's Avatar
    Joined
    Mar 2009
    From
    Uptown Manhattan, NY, USA
    Posts
    419
    Ah, okay. And I guess to ensure that my approximation is to three decimal places, I show that that remainder is less than 0.5 \times 10^{-4}, right?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Apr 2005
    Posts
    14,973
    Thanks
    1121
    Yes.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. taylor approximation help
    Posted in the Calculus Forum
    Replies: 1
    Last Post: March 13th 2010, 08:31 AM
  2. Taylor approximation
    Posted in the Calculus Forum
    Replies: 7
    Last Post: March 10th 2010, 06:55 AM
  3. Taylor approximation of cos
    Posted in the Calculus Forum
    Replies: 2
    Last Post: November 5th 2009, 05:38 PM
  4. taylor approximation
    Posted in the Calculus Forum
    Replies: 3
    Last Post: August 14th 2009, 06:15 AM
  5. taylor approximation
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 14th 2006, 08:25 PM

Search Tags


/mathhelpforum @mathhelpforum