Answer to Question #350677 in Calculus for Dudu

Question

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)

Answer 1

$$f_x=e^x\ln(1+y), f_y=\dfrac{e^x}{1+y}$$

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

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

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

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

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

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

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

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

$$i=0, j=0, 0$$

$$i=0, j=1, y$$

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

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

$$i=1, j=0, 0$$

$$i=1, j=1, xy$$

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

$$i=2, j=0, 0$$

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

$$i=3, j=0, 0$$

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