## Problem on approximating a function of two variables

Let $u(x,y) = x y$. Estimate the value of $u(0.8, 2.1)$ using a first-order approximation of $u$ at $(1,2)$.
• ## Solution

We recall the first order approximation of a function of two variables:
To use this formula, we need to find $\Delta x$, $\Delta y$, $u(1,2)$, $\partial_x u(1,2)$, $\partial_y u(1,2)$.
From the problem statement, we observe $u(1,2) = 2$, $\Delta x = -0.2$, and $\Delta y = 0.1$.
Plugging in, $$w(0.8, 2.1) \approx 2 + 2 \cdot (-0.2) + 1 \cdot (0.1) = 1.7$$