Question #350677

Consider the R²-R function f defined by f(x,y)=e^xIn(1+y).

Find the third order Taylor polynomial of f about the point (0,0)

Expert's answer

"f_{xx}=e^x\\ln(1+y), f_{yy}=-\\dfrac{e^x}{(1+y)^2},"

"f_{xy}=f_{yx}=\\dfrac{e^x}{1+y},"

"f_{xxx}=e^x\\ln(1+y), f_{yyy}=\\dfrac{2e^x}{(1+y)^3},"

"f_{xyy}=-\\dfrac{e^x}{(1+y)^2}, f_{xxy}=\\dfrac{e^x}{1+y}"

"f(0,0)=0, f_x(0,0)=0, f_y(0,0)=1,"

"f_{xx}(0,0)=0, f_{yy}(0,0)=-1, f_{xy}(0,0)=1,"

"f_{xxx}(0,0)=0, f_{yyy}(0,0)=2,"

"f_{xyy}(0,0)=-1, f_{xxy}(0,0)=1"

"i=0, j=0, 0"

"i=0, j=1, y"

"i=0, j=2, -\\dfrac{1}{2}y^2"

"i=0, j=3, -\\dfrac{2}{6}y^3"

"i=1, j=0, 0"

"i=1, j=1, xy"

"i=1, j=2, -\\dfrac{1}{2}xy^2"

"i=2, j=0, 0"

"i=2, j=1, x^2y"

"i=3, j=0, 0"

"P_3(x,y)=y-\\dfrac{1}{2}y^2-\\dfrac{1}{3}y^3+xy-\\dfrac{1}{2}xy^2+x^2y"

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