Results 1 to 2 of 2

Thread: Finding the Taylor polynomial and remainder

  1. #1
    Junior Member
    Joined
    Jul 2011
    Posts
    28

    Finding the Taylor polynomial and remainder

    I want to make sure I'm doing this right.

    Find the third degree Taylor polynomial of $\displaystyle f(x) = \cos x$ centered at c $\displaystyle = $ $\displaystyle \frac {\pi}{6}$. Write the remainder in Lagrange form and find an upper bound for the remainder.

    So $\displaystyle n = 3$ and $\displaystyle c = \frac{\pi}{6}$ right?

    $\displaystyle f(x) = \cos x$; $\displaystyle f{(\frac{\pi}{6}}) = \frac{\sqrt{3}}{2}$
    $\displaystyle f'(x) = -\sin x$; $\displaystyle f'{(\frac{\pi}{6}}) = -\frac{1}{2}$
    $\displaystyle f''(x) = -\cos x$; $\displaystyle f''{(\frac{\pi}{6}}) = -\frac{\sqrt{3}}{2}$
    $\displaystyle f'''(x) = \sin x$; $\displaystyle f'''{(\frac{\pi}{6}}) = \frac{1}{2}$
    (error) $\displaystyle f^{(4)}(x) = \cos x$;

    $\displaystyle P_3(\frac{\pi}{6}) = (\frac{\sqrt{3}}{2}) - (\frac{1}{2 * 1!} (x - \frac{\pi}{6})) - (\frac{\sqrt{3}}{2 * 2!}(x - \frac{\pi}{6})^2) + (\frac{1}{2 * 3!}(x - \frac{\pi}{6})^3) $

    $\displaystyle R_3(x) = \frac{cos (c)}{4!}(x - \frac{\pi}{6})^4$, where $\displaystyle c \exists {[\frac{\pi}{6}, x]}$

    I was confused about the part where it says to find an upper bound for the remainder. Does $\displaystyle c \exists {(\frac{\pi}{6}, x)}$ show the upper bound or do I need more?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    skeeter's Avatar
    Joined
    Jun 2008
    From
    North Texas
    Posts
    16,216
    Thanks
    3702

    Re: Finding the Taylor polynomial and remainder

    $\displaystyle |Error| \le \frac{\left(x - \frac{\pi}{6}\right)^4}{4!}$

    max possible value of $\displaystyle \cos(c) = 1$
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Taylor Polynomial and Remainder
    Posted in the Calculus Forum
    Replies: 3
    Last Post: Dec 10th 2010, 03:05 AM
  2. Replies: 1
    Last Post: Oct 3rd 2009, 10:26 PM
  3. Finding remainder of polynomial division
    Posted in the Algebra Forum
    Replies: 7
    Last Post: Oct 29th 2008, 02:51 PM
  4. Remainder of Taylor Polynomial?
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Jul 16th 2008, 07:47 AM
  5. Taylor Polynomial & Remainder
    Posted in the Calculus Forum
    Replies: 1
    Last Post: May 7th 2008, 12:43 PM

Search Tags


/mathhelpforum @mathhelpforum