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Taylor Series with Remainder

  1. Mar 16, 2008 #1
    1. The problem statement, all variables and given/known data

    Find the 3rd-order Maclaurin Polynomial (i.e. P3,o(u)) for the function f(u) = sin u, together with an upper bound on the magnitude of the associated error (as a function of u), if this is to be used as an approximation to f on the interval [0,2].

    I did the question fine except for the upper bound error.

    2. Relevant equations

    |Rn,o(u)| <= (k|u|^4)/(4!)

    Where 0 <= sin(u) <= 2, and |sin(u)| <= k


    3. The attempt at a solution

    Okay, since 0 <= sin(u) <= 2 and |sin(u)| <= k, shouldn't k = sin(2)? The actual answer is k = 1, but if sin(u) is bounded between 0 and 2, it can't have a value greater than sin(2) so saying that it's max is 1 (at sin(pi/2) is an overestimation for this particular problem by my thought process.

    Any enlightening comments as to why it's 1 and not sin(2)? Thanks!
     
  2. jcsd
  3. Mar 17, 2008 #2

    Gib Z

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    The interval is [0,2] for values of u. So [itex]0\leq u \leq 2[/itex]. What is the maximum value of sin u on that interval?
     
  4. Mar 17, 2008 #3
    Right, pi/2 = 1.57 which is less than 2.

    Okay then the max value is 1. But if u was bounded between zero and 1 then the max value would be sin1 right?
     
  5. Mar 18, 2008 #4

    Gib Z

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    Yes, because sin u is a strictly increasing function over that interval.
     
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