Answer
$$0.086$$
Work Step by Step
$$f_x=y^2 -y \sin (x-1) \\ f_x(1,2)=2^2 -(2) \sin (1-1)=4 \\ f_y(x,y) =2xy+\cos (x-1) \\ f_{y}(1,2) =2(1)(2)+\cos (1-1)=5 \\f_{xx}(x,y)=-y \cos (x-1) \\f_{yy}(x,y)=2x \\ f_{xy}(x,y) =2y- \sin (x-1)$$
Error:
$|E(x,y)| \leq \dfrac{1}{2} \times (4.3) [ |x-1| +|y-2|)^2$
or, $ E \leq \dfrac{4.3}{2} \times (0.1+0.1)^2 =0.086$
