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Thread: 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:

    $\displaystyle cos(x+y)$ about the point $\displaystyle (0,0)$

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

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

    Let $\displaystyle P=(0,0)$

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

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

    Is there an easier way of doing these questions?
    Last edited by acevipa; Nov 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 $\displaystyle \cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}+...$

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

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