Find the second-order Taylor polynomial for f(x,y)=sin(x+y)f(x,y) = \sin(x + y)f(x,y)=sin(x+y) centered at (0,0)(0, 0)(0,0).
P2(x,y)=x+yP_2(x,y) = x + yP2(x,y)=x+y
P2(x,y)=x+y−12(x2+2xy+y2)P_2(x,y) = x + y - \frac{1}{2}(x^2 + 2xy + y^2)P2(x,y)=x+y−21(x2+2xy+y2)
P2(x,y)=1+x+y+12xyP_2(x,y) = 1 + x + y + \frac{1}{2}xyP2(x,y)=1+x+y+21xy
P2(x,y)=(x+y)2−16(x+y)3P_2(x,y) = (x+y)^2 - \frac{1}{6}(x+y)^3P2(x,y)=(x+y)2−61(x+y)3