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Math Help - Taylor series for function of several variables

  1. #1
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    Taylor series for function of several variables

    Write down the Taylor series up to degree of five for:

    cos(x+y) about the point (0,0)

    For doing questions like these. Should I just remember the formula for the Taylor expansion of f about the point x_0=(x_0,y_0)

    \displaystyle f(x,y)=f(x_0)+\frac{1}{1!}(f_x(x_0)(x-x_0)+f_y(x_0)(y-y_0))+\frac{1}{2!}(f_{xx}(x_0)(x-x_0)^2+2f_{xy}(x_0)(x-x_0)(y-y_0)+f_{yy}(x_0)(y-y_0)^2)+...

    Let P=(0,0)

    f=1 at P
    f_x=-sin(x+y)=0 at P
    f_y=-sin(x+y)=0 at P
    f_xx=-cos(x+y)=-1 at P
    f_yy=-cos(x+y)=-1 at P
    f_xy=-cos(x+y)=-1 at P

    cos(x+y)\approx 1+\frac{1}{1!}(0(x-0)+y(y-0))+\frac{1}{2!}(-1(x-0)^2-2(x-0)(y-0)-1(y-0)^2)

    Is there an easier way of doing these questions?
    Last edited by acevipa; November 5th 2010 at 05:19 PM.
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  2. #2
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    What you are doing is the easiest method.
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  3. #3
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    Quote Originally Posted by Prove It View Post
    What you are doing is the easiest method.
    So should I just remember the formula I wrote, or is there a way to derive this?
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  4. #4
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    The derivation requires about three pages of tricky algebra.
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  5. #5
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    I've just thought of something:

    I know that the expansion of \cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}+...

    So isn't it easier just to do the following:

    1-\frac{(x+y)^2}{2!}+\frac{(x+y)^4}{4!}+...
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