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[Calc II] quadratic Chebyshev approximation

  1. Nov 7, 2011 #1
    1. The problem statement, all variables and given/known data
    (a) The quadratic Chebyshev approximation of a function on [-1, 1] can be obtained by finding the coefficients of an arbitrary quadratic y = ax^2 + bx + c which fit the function exactly at the points (-sqrt(3)/2), 0, (sqrt(3)/2). Find the quadratic Chebyshev approximation of e^x on [-1, 1]. Approximate a, b, c to 5 decimal places.

    (b) Find the quadratic truncated Taylor series centered at 0 for e^x.

    2. Relevant equations
    e^x = ax^2 + bx + c
    Maclaurin for e^x


    3. The attempt at a solution
    For part (a), I did not know how to find the quadratic Chebyshev approximation of e^x, but I plugged in the different values of x that were given into e^x = ax^2 + bx + c in order to get 3 equations with 3 unknowns. This gives me a ≈ 0.53204, b ≈ 1.12977, and c = 1.

    As for part (b), since it asks for a truncated Taylor series, I assume I need a value to stop at. I really don't feel confident doing this. Do I need to simply write the Maclaurin series for e^x?
     
  2. jcsd
  3. Nov 7, 2011 #2

    rude man

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    Just a guess, but maybe what's appropriate here is the Maclaurin series for your found quadratic. You could then compare with the series for exp(x).
     
  4. Nov 7, 2011 #3

    lanedance

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    maclaurin series is just a taylor series about zero, and it asks for quadratic so just find terms up to x^2
     
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