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Math Help - Second degree taylor polynomial help

  1. #1
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    Second degree taylor polynomial help

    I need a little bit of help with a question.

    Whats the second degree taylor polynomial (taylor vector quadratic) for the function

    f(x) = x1^3 x2^2 about the point [2, 3] <------- Vector


    Thanks if anyone can help
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by nicolas123 View Post
    f(x) = x1^3 x2^2 about the point [2, 3] <------- Vector
    It is rather difficult to understand what function you are referring to. Perhaps you mean:

    f(x_1,x_2)=x_1^3x_2^2

    At any rate:

    p(x_1,x_2)=f(2,3)+\dfrac{{\partial f}}{{\partial x_1}}(2,3)(x_1-2)+\dfrac{{\partial f}}{{\partial x_2}}(2,3)(x_2-3)+
    \dfrac{{\partial^2 f}}{{\partial x_1}^2}(2,3)(x_1-2)^2+2\dfrac{{\partial^2 f}}{{\partial x_1\partial x_2}}(2,3)(x_1-2)(x_2-3)+\dfrac{{\partial^2 f}}{{\partial x_2}^2}(2,3)(x_2-3)^2<br />
=\ldots

    Regards.

    Fernando Revilla
    Last edited by FernandoRevilla; November 19th 2010 at 11:52 AM.
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  3. #3
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    Another way to do it:

    A polynomial already is a Taylor's polynomial. To get it around (2, 3) (which is NOT a vector), let u= x_1- 2 and v= x_2- 3, so that x_1= u+ 2 and x_2= v+ 3. Then x_1^3x_2^2= (u+ 2)^3(v+ 3)^2= (u^3+ 6u^2+ 12u+ 8)(v^2+ 6v+ 9) = u^3(v^2+ 6v+ 9)+ 6u^2(v^2+ 6v+ 9)+ 12u(v^2+ 6v+ 9)+ 8(v^2+ 6v+ 9) = u^3v^2+ 26u^2+v+ 6u^3v+ 9u^3+ 6u^2v^2+ 36u^2v+ 54u^2v+ 12uv^2+ 72uv+ 108u+ 8v^2+ 48v+ 72.

    To get the second degree polynomial, just drop all terms where the exponents sum to 3 or larger, then replace u and v with x_1- 2 and x_2- 3 again.
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