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Linear Approximation of a Function of Two Independent Variables

In a Nut Shell: Recall that the "linear approximation", L(x), of a function of one independent

variable, x, provides a way to find the value of f(x) at a neighboring point of a where

x is a point close to a. This approximation directly relates to the slope of f(x) at x = a.

The figure below depicts the linear approximation, L(x), of the function, f(x).

where f(x) is the value of the function f at x

f(a) is the value of the function f(x) at x = a;
pick x to be slightly larger or slightly smaller than a

x ˗ a is the interval; it is best to have a small interval

f '(a) is the slope of the function f(x) at x = a

L(x) is the linear approximation of f(x) near x = a

L(x) = f(a) + df(a)/dx ( x ˗ a )

One can extend the linear approximation of a function, f(x,y) of two independent variables

say x and y by noting that the "surface", f(x,y), may change its slope in both the x and

y-coordinate directions. In this case the relevant slopes at (x,y) = (a,b) are:

∂f(x,y)/∂x evaluated at (x,y) = (a,b) i.e. change of slope in the x-direction

and

∂f(x,y)/∂y evaluated at (x,y) = (a,b) i.e. change of slope in the y-direction

So the linear approximation of a function of two independent variables is:

L(x,y) = f(a,b) + ∂f(a,b)/∂x [ x ˗ a ] + ∂f(a,b)/∂y [ y ˗ b ]

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