Problem on approximating a function of two variables
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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:
First-order approximation of functions of several variables
For small $\Delta x$ and $\Delta y$, $$u(x+\Delta x,y + \Delta y) \approx u(x, y) + \partial_x u(x, y) \Delta x + \partial_y u(x, y) \Delta y $$
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 $\Delta y = 0.1$.
Partial derivatives
Computing the partial derivatives, $$\partial_x u(x,y) = y \ \Longrightarrow \ \partial_x u(1,2) = 2 \\\partial_y u(x,y) = x \ \Longrightarrow \ \partial_y u(1,2) = 1.$$
Plugging in, $$w(0.8, 2.1) \approx 2 + 2 \cdot (-0.2) + 1 \cdot (0.1) = 1.7$$
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