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Math Help - Functional Equation

  1. #1
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    Functional Equation

    Greetings,

    I am not quite sure what classification this problem should actually have, but my teacher, who was unable to help me with it, said it was a functional equation.

    The aim is to find the values of a, b, c and d or represent the function on the first line like the one on the second line.
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  2. #2
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    Quote Originally Posted by Logic View Post
    Greetings,

    I am not quite sure what classification this problem should actually have, but my teacher, who was unable to help me with it, said it was a functional equation.

    The aim is to find the values of a, b, c and d or represent the function on the first line like the one on the second line.

    This is just like finding the Taylor series of f center at x=1.

    f(x)=a(x-1)^3+b(x-1)^2+c(x-1)+d


    If we take a few derivatives we will see that

    f'(x)=3a(x-1)^2+2b(x-1)+c

    f''(x)=6a(x-1)+2b

    f'''(x)=6a

    Now compare this with the other functions derivatives

    f(x)=x^3+x^2-x+1
    f'(x)=3x^2+2x-1
    f''(x)=6x+2
    f'''(x)=6

    Now compare the derivatives

    f'''(1)=6 =6a \implies a=1

    f''(1)=6+2=2b \implies b=4

    f'(1)=4=c \implies c=4

    f(1)=2=d

    So we get

    f(x)=(x-1)^3+4(x-1)^2+4(x-1)+2
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