Results 1 to 2 of 2

Math Help - Taylors theorem question.

  1. #1
    C.E
    C.E is offline
    Junior Member
    Joined
    Mar 2009
    Posts
    34

    Taylors theorem question.

    Hi, I have been stuck on the following for ages, any guidance would be welcome.

    A function f: R --> R is C^3 and,

    f(a+h)= f(a) + f'(a + 0.5h)h

    for real a and h 0

    by applying Taylors theorem to f and f' show that the third

    derivative of f is identically 0.

    I can see that f'(a + 0.5h)h is the Lagrange error term

    and I know that similarly f'(a+05h)=f'(a)+0.5f'' \deltah

    Therefore f(x)= f(a) + (f'(a) + 0.5 f''(a))(x-a).

    Is this even the right way to attempt the question? If so, how does the above imply that the third derivative of f is always zero?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Taylor's theorem for f says f(a+h) = f(a) + hf'(a) + \tfrac12h^2f''(a) + \tfrac16f'''(a+\theta h) ( for some \theta between 0 and 1).

    Taylor's theorem for f' says f'(a+<br />
\tfrac h2) = f'(a) + \tfrac h2f''(a) + \tfrac12\bigl(\tfrac h2\bigr)^2f'''(a+\phi h) ( for some \phi between 0 and 1/2). Multiply that equation by h, subtract it from the previous one, and use the information that f(a+h) = f(a) + hf'(a+\tfrac h2). You'll end up with the equation 4f'''(a+\theta h) = 3f'''(a+\phi h). Now let h\to0 to see that 4f'''(a) = 3f'''(a), from which obviously f'''(a)=0.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Taylors Series Confusion
    Posted in the Calculus Forum
    Replies: 2
    Last Post: April 28th 2011, 11:14 PM
  2. Taylors Theorem
    Posted in the Calculus Forum
    Replies: 6
    Last Post: July 19th 2010, 09:54 AM
  3. Deriving Taylors for 1/(1-X)
    Posted in the Calculus Forum
    Replies: 2
    Last Post: March 13th 2009, 12:57 AM
  4. taylors theorem
    Posted in the Calculus Forum
    Replies: 1
    Last Post: August 11th 2008, 02:38 AM
  5. taylors series
    Posted in the Calculus Forum
    Replies: 4
    Last Post: May 23rd 2007, 01:05 PM

Search Tags


/mathhelpforum @mathhelpforum