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The max value of the fourth derivative

  1. May 8, 2013 #1
    1. The problem statement, all variables and given/known data
    I am trying to write a program to calculate an integral using Simpson's Rule. My program also needs to calculate the error. In the formula for error I need The max value of the fourth derivative. The function is 2.718281828[itex]^{\frac{x^{2}}{2}}[/itex] and the interval is from 0 to 2


    2. Relevant equations
    |E|[itex]\leq[/itex][itex]\frac{(b-a)^{5}}{180n^{4}}[/itex][max|f[itex]^{(4)}[/itex](x)|]


    3. The attempt at a solution
    The fourth derivative of 2.718281828[itex]^{\frac{x^{2}}{2}}[/itex] is approximately
    (x[itex]^{4}[/itex]+6x[itex]^{2}[/itex]+3)*2.718281828[itex]^{\frac{x^{2}}{2}}[/itex]
    When evaluated at 2 this gives 317.729... Since there is a lot of math involved I would appreciate if someone can review this especially since my program is returning strange results. Also, is there anything I can learn by evaluating the fifth derivative?
     
  2. jcsd
  3. May 8, 2013 #2

    Mark44

    Staff: Mentor

    It would be better to write your function exactly, as e(1/2)x2. The fourth derivative is e(1/2)x2(x4 + 6x2 + 3).

    The first factor is always positive and is increasing, so can be ignored for the time being. What about y = x4 + 6x2 + 3? What does the graph of this function look like on the interval [0, 2]? Can you determine where its max. value is?
     
  4. May 9, 2013 #3
    It is certainly helpful to keep it in the form e(1/2)x2 and now I see that the max value of the fourth derivative is 43e2.
    I am not sure why the first factor can be ignored, but for y = x4 + 6x2 + 3 I find a max value of 43 @ x=2.

    So, which is it 43 or 43e2(317.729)?
     
    Last edited: May 9, 2013
  5. May 9, 2013 #4

    Mark44

    Staff: Mentor

    If you'll recall, I said "ignored for the time being."
    For the max value of f(4)(x), use 43 * e2 ≈ 318.

    The reasoning here is that e(1/2)x2 is an increasing function, so the maximum value on an interval is attained at the right endpoint. Presumably your graph of y = x4 + 6x2 + 3 showed that this function also is increasing. As that seems the case, the max value of f(4)(x) occurs when x = 2.
     
  6. May 9, 2013 #5
    OK Now I understand. Thank You.
     
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