Results 1 to 2 of 2

Math Help - Lagrange error bound to estimate sin4 to five decimal places( maclaurin series)

  1. #1
    Junior Member
    Joined
    Jan 2010
    Posts
    26

    Lagrange error bound to estimate sin4 to five decimal places( maclaurin series)

    Lagrange error bound to estimate sin4 to five decimal places( maclaurin series)
    4=pi/45 radians
    |Rn(pi/45)<1*(pi/45)^n+1/(n+1)! < 5*10^-6
    and the answer key says n should be greater than or equal to 3.

    It doesn't make sense .

    Because, if you write out derivatives, the ones with sines will disappear in the polynomial. So, don't we have to ignore sin (since it is maclaurin series)
    So if it is 7rd order polynomial it should be x-(1/3!)x^3 + (x^5)/5!) -(x^7)/7!.
    and therefore we need to look at 9th derivative.

    It seems the answer key just applied the remainder theorem.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by hangainlover View Post
    Lagrange error bound to estimate sin4 to five decimal places( maclaurin series)
    4=pi/45 radians
    |Rn(pi/45)<1*(pi/45)^n+1/(n+1)! < 5*10^-6
    and the answer key says n should be greater than or equal to 3.

    It doesn't make sense .

    Because, if you write out derivatives, the ones with sines will disappear in the polynomial. So, don't we have to ignore sin (since it is maclaurin series)
    So if it is 7rd order polynomial it should be x-(1/3!)x^3 + (x^5)/5!) -(x^7)/7!.
    and therefore we need to look at 9th derivative.

    It seems the answer key just applied the remainder theorem.
    |R_n(x)|= \frac{\max_{u \in [0,x]}(|f^{(n+1)}(u)| |x^{n+1}|}{(n+1)!}

    As f(x)=\sin(x) we may use |f^{(k)}| \le 1 to get:

    |R_n(x)|\le \frac{ |x^{n+1}|}{(n+1)!}.

    So now find n such that:

    \frac{ |(\pi/45)^{n+1}|}{(n+1)!}<5*10^{-6}.

    (you do this last part by checking a few values of n)

    If you don't beleive the book answer compare \sin(\pi/45) with \pi/45 and (\pi/45)-\frac{(\pi/45)^3}{3!}

    CB
    Last edited by CaptainBlack; March 22nd 2010 at 12:17 AM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. decimal places
    Posted in the Math Software Forum
    Replies: 6
    Last Post: July 10th 2010, 04:10 AM
  2. Maclaurin series error of approximation!?!
    Posted in the Calculus Forum
    Replies: 3
    Last Post: March 30th 2010, 05:20 PM
  3. Bound of error(Lagrange Interpolation)
    Posted in the Calculus Forum
    Replies: 3
    Last Post: June 15th 2009, 09:35 AM
  4. Maclaurin series estimate
    Posted in the Calculus Forum
    Replies: 4
    Last Post: April 20th 2009, 09:02 AM
  5. Decimal Places
    Posted in the Algebra Forum
    Replies: 4
    Last Post: April 1st 2008, 02:12 PM

Search Tags


/mathhelpforum @mathhelpforum