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)

1
Expert's answer
2022-06-15T16:40:18-0400
fx=exln(1+y),fy=ex1+yf_x=e^x\ln(1+y), f_y=\dfrac{e^x}{1+y}

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

fxy=fyx=ex1+y,f_{xy}=f_{yx}=\dfrac{e^x}{1+y},

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

fxyy=ex(1+y)2,fxxy=ex1+yf_{xyy}=-\dfrac{e^x}{(1+y)^2}, f_{xxy}=\dfrac{e^x}{1+y}

f(0,0)=0,fx(0,0)=0,fy(0,0)=1,f(0,0)=0, f_x(0,0)=0, f_y(0,0)=1,

fxx(0,0)=0,fyy(0,0)=1,fxy(0,0)=1,f_{xx}(0,0)=0, f_{yy}(0,0)=-1, f_{xy}(0,0)=1,

fxxx(0,0)=0,fyyy(0,0)=2,f_{xxx}(0,0)=0, f_{yyy}(0,0)=2,

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

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

i=0,j=1,yi=0, j=1, y

i=0,j=2,12y2i=0, j=2, -\dfrac{1}{2}y^2

i=0,j=3,26y3i=0, j=3, -\dfrac{2}{6}y^3

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

i=1,j=1,xyi=1, j=1, xy

i=1,j=2,12xy2i=1, j=2, -\dfrac{1}{2}xy^2

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

i=2,j=1,x2yi=2, j=1, x^2y

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

P3(x,y)=y12y213y3+xy12xy2+x2yP_3(x,y)=y-\dfrac{1}{2}y^2-\dfrac{1}{3}y^3+xy-\dfrac{1}{2}xy^2+x^2y


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!
LATEST TUTORIALS
APPROVED BY CLIENTS