Answer
See explanation
Work Step by Step
The local linear approximation for a function \(f(x, y)\) of two variables at a point \((x_0, y_0)\) can be expressed as: \[L(x, y) = f(x_0, y_0) + f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0)\] While the tangent plane for \(z = f(x, y)\) at \((x_0, y_0)\) is: \[f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0) - (z - z_0) = 0\] with \(z_0 = f(x_0, y_0)\). So we can express this plane as: \[z = f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0) + f(x_0, y_0)\] The tangent plane gives us an approximation of a surface near a point \((x_0, y_0)\). So for close values to \((x_0, y_0)\), the tangent plane approximates \(f\) at that point. And that's how we define the linear approximation (which has the same expression as the tangent plane).
