d50e3c…: “Besides the great city”

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Compute the degree $2$ Taylor polynomial for $F(x,y) = \sin (x)+ye^x$ centered at $(0,1)$ . Start by making a table of partial derivatives along with their value when evaluated at $(0,1)$ : $\begin {array}{lcl} F(x,y) = \sin (x)+ ye^x & \Rightarrow &F(0,1) = \answer {1}\\ F^{(1,0)}(x,y) = \answer {\cos (x)+ye^x} & \Rightarrow & F^{(1,0)}(0,1) = \answer {2}\\ F^{(0,1)}(x,y) = \answer {e^x} &\Rightarrow & F^{(0,1)}(0,1) = \answer {1}\\ F^{(2,0)}(x,y) = \answer {ye^x-\sin (x)} &\Rightarrow & F^{(2,0)}(0,1) = \answer {1}\\ F^{(0,2)}(x,y) = \answer {0} &\Rightarrow & F^{(0,2)}(0,1) = \answer {0}\\ F^{(1,1)}(x,y) = \answer {e^x} &\Rightarrow & F^{(1,1)}(0,1) = \answer {1} \end {array}$ The degree $2$ Taylor polynomial is: $P_2(x,y) =\answer {x+x^2/2+y+xy}$