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expand xy2 +cos xy upto fourth degree terms using maclaurin’s series

  • 10 months ago
  • 10 months ago

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  1. oldrin.bataku
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    $$f(x,y)=xy^2+\cos xy\\f_x(x,y)=y^2-y\sin xy\\f_y(x,y)=2xy-x\sin xy\\f_{xx}(x,y)=-y^2\cos xy\\f_{yy}(x,y)=2x-x^2\cos xy\\f_{xy}(x,y)=2y-\sin xy-xy\cos xy\\f_{xxx}(x,y)=y^3\sin xy\\f_{yyy}(x,y)=x^3\sin xy\\f_{xxy}(x,y)=-2y\cos xy+xy^2\sin xy\\f_{xyy}(x,y)=2-2x\cos xy+x^2y\sin xy\\f_{xxxx}(x,y)=y^4\cos xy\\f_{yyyy}(x,y)=x^4\cos xy\\f_{xxyy}(x,y)=x^2y^2\cos xy+4xy\sin xy-2\cos xy\\f_{xxxy}(x,y)=xy^3\cos xy+3y^2\sin xy\\f_{xyyy}(x,y)=3x^2\sin xy+x^3y\cos xy$$ Notice we take advantage of symmetry of mixed partials. Observe so far that at \((0,0)\)$$f=1\\f_x=0\\f_y=0\\f_{xx}=0\\f_{xy}=0\\f_{yy}=0\\f_{xxx}=0\\f_{xxy}=0\\f_{xyy}=2\\f_{yyy}=0\\f_{xxxx}=0\\f_{xxxy}=0\\f_{xxyy}=-2\\f_{xyyy}=0\\f_{yyyy}=0$$Now let's work on our power series expansion: $$\begin{align*}f(x,y)\approx &f\\+&(xf_x+yf_y)\\+&\frac12\left(x^2f_{xx}+2xyf_{xy}+y^2f_{yy}\right)\\+&\frac16\left(x^3f_{xxx}+3x^2yf_{xxy}+3xy^2f_{xyy}+y^3f_{yyy}\right)\\+&\frac1{24}\left(x^4f_{xxxx}+4x^3yf_{xxxy}+6x^2y^2f_{xxyy}+4xy^3f_{xyyyy}+y^4f_{yyyy}\right)\end{align*}$$Plugging in our values, we find:$$f(x,y)\approx1+xy^2-\frac12x^2y^2$$

    • 10 months ago
  2. oldrin.bataku
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    The easy way, however, is just to use our well-known expansions:$$\cos xy=1-\frac12(xy)^2+\frac1{24}(xy)^4-\dots\approx1-\frac12x^2y^2$$Then we just add to our polynomial (which is its own power series expansion):$$f(x,y)\approx1+xy^2-\frac12x^2y^2$$

    • 10 months ago
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