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Math Help - find the quadratic polynomial.....

  1. #1
    Junior Member
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    find the quadratic polynomial.....

    Find the quadratic polynomial g ( x ) = ax^2 + bx + c which best fits the function f ( x ) = e^x at x = 0 , in the sense that

    g (0) = f (0), and g ′(0) = f ′(0), and g ′′(0) = f ′′(0) .

    I know that e^0 = 1 ,and ax^2 +bx is zero when x is zero , so c has to give me a value of one , but how can I represent that? any help would appreciated!!
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  2. #2
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    Nov 2009
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    Well, this is just the Taylor expansion of the exponential exp(x) around x=0:
    f(x)=f'(x)=f''(x)=exp(x), so f(0)=f'(0)=f''(0)=1.
    Now g(x)=ax+bx+c, g'(x)=2ax+b, g''(x)=2a,
    so g(0)=f(0) =>c=1
    g'(0)=f'(0)=> b=1
    g''(0)=f''(0)=> a=1/2,

    and g(x)= 1+x+x/2.
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  3. #3
    Junior Member
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    thanks umgambit !!!! I didn't know about that theorem.
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