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Math Help - Taylor Polynomial and Remainder

  1. #1
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    Taylor Polynomial and Remainder

    Hi,

    I'm not sure how to present an answer to this kind of question. It states:

    Present the 3rd-Degree Taylor Polynomial P_{a,2}(h) as well as the Lagrange remainder for the function f(x,y) = x^3y + sin(xy).

    Then let a = (1, \pi) and write this polynomial in matrix form.

    Thanks a lot!
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    What kind of difficulties have you had?.

    Regards.

    Fernando Revilla
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  3. #3
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    Well, I don't know how to apply the definitions to find an answer and I haven't read anything on the matrix forms for the taylor polynomial.
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  4. #4
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    The third degree Taylor polynomial for f(x, y), at (1, \pi) is
    f(1, \pi)+ \frac{\partial f(1, \pi)}{\partial x}(x- 1)+ \frac{\partial f(1, \pi)}{\partial y}(y- \pi)+ \frac{1}{2}\frac{\partial^2 f(1, \pi)}{\partial x^2}(x- a)^2+  \frac{1}{2}\frac{\partial^2 f(1, \pi)}{\partial x\partial y}(x- a)(y-\pi) + \frac{1}{2}\frac{\partial^2 f(1,\pi)}{\partial y^2}(y- \pi)^2+ \frac{1}{6}\frac{\partial^3 f(1,\pi)}{\partial x^3}(x- 1)^3+ \frac{1}{6}\frac{\partial^3 f(1, \pi)}{\partial x^2\partial y}(x-1)^2)(y- \pi)+ \frac{1}{6}\frac{\partial^3 f(1,\pi)}{\partial x\partial y^2}(x-1)(y-\pi)^2+ \frac{1}{6}\frac{\partial f(1,\pi)}{\partial y^3}(y- \pi)^3.
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