Chapter 13 - Section 13.6 - Tangent Planes and Differentials - Exercises - Page 729: 37

$0.0222$

$f_x=e^x \cos y \\ \implies f_x(0,0)=e^{0} \cos 0=1$ Next, $f_y(x,y) =-e^x \sin y \\ \implies f_{y}(0,0) =0$ $f_{xx}(x,y)=-e^x \cos y; f_{yy}(x,y)=-e^x \cos y$ and $f_{xy}(x,y) =-e^x \sin y$ The error can be found as: $|E(x,y)| \leq \dfrac{1}{2} \times (1.11) [ |x-0| +|y-0|)^2$ $\implies E \leq \dfrac{1.11}{2} \times (0.1+0.1)^2 =0.0222$