linear approximation (multivariable)

Using a surface's tangent plane, instead of the true function, to cheaply estimate nearby values.

The linear approximation $L(x,y)$ is literally the tangent plane's equation, only renamed and used for estimation instead of geometry. Near the point of tangency, evaluating the PLANE is much cheaper than evaluating $f$ itself, and often close enough: $f(x,y)\approx L(x,y) = f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)$ , with error that shrinks the closer $(x,y)$ is to $(a,b)$ .

Examples

Estimating $f(4.03,8.96)$ for $f(x,y)=\sqrt{xy}$ using the tangent plane at $(4,9)$ gives $6.0092$ , matching the true value $\approx6.00907$ to four decimal places.