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Homework Help: Simpson approximation method

  1. Aug 8, 2010 #1
    1. The problem statement, all variables and given/known data
    Use the Simpson method to estimate [tex]\displaystyle\int_{0}^{1}\cos(x^2)dx[/tex] with an approximation error less than 0.001.

    Well, I have a problem. Actually I'm looking for a bound for the error of approximate method integration by using Simpson's method.

    I have to bring [tex]\displaystyle\int_{0}^{1}\cos(x^2)dx[/tex] with an error less than 0.001.

    I started looking for the fourth order derivative, and got:


    Now, I have to find a bound K for this derivative in the interval [0,1]. What I did do was watch for if they had maximum and minimum on the interval, then calculate the derivative of order five:

    From here I did was ask:

    If [tex]f^5(x)\in{[0,1]}\Rightarrow{f^5(x)=0\Longleftrightarrow{x=0}}[/tex]

    So here I know is that zero is a maximum or minimum. Then I wanted to look for on the concavity of the curve, and I thought the easiest thing would be to look at the sixth derivative, to know how it would behave the fourth derivative of the original function.


    The problem is that when evaluated


    So, I get zero the derivative sixth, and I do not say whether it is concave upwards or downwards is concave.

    What should I do? there may be a less cumbersome to work this, if so I would know.

    Last edited: Aug 8, 2010
  2. jcsd
  3. Aug 8, 2010 #2


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    You could just plot the function and see it hits a maximum on that interval at x=1. Alternately, you can use arguments like x2sin x2≤x2 to find an upper bound for the function. In these types of problems, it's usually not terribly important to find the actual maximum; an upper bound for the maximum is good enough.
  4. Aug 8, 2010 #3
    I complicated it too much. Just by evaluating the 6th derivative in any other point of the interval I could get the concavity of the curve. An other way was just seeing for the value of the 5th derivative, as the concavity changes only in zero then I would know if it creases or decreases.

    Bye, and thanks.
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