Calculus (3rd Edition)

$$5.07$$
Given $$f(2,4)=5, \quad f_{x}(2,4)=0.3, \quad f_{y}(2,4)=-0.2$$ The linear approximation is given by \begin{aligned} f(a+h, b+k) & \approx f(a, b)+hf_{x}(a, b) +kf_{y}(a, b) \end{aligned} We find $f( 2.1,3.8)$. Here, $h=0.1, k = -0.2$ and $(a,b)=(2,4)$, then \begin{aligned} f(2,4) & \approx f(2, 4)+0.1f_{x}(2, 4) -0.2f_{y}(2, 4) \\ & \approx 5+(0.1)(0.3)+(0.2)(0.2)\\ &\approx 5.07 \end{aligned}