The Tangent Plane Approximation

To get a first order approximation to a function f(x,y) we use a similar idea. However, a linear function in two variables is a plane. Thus, we need to relate the constants in the equation of a plane:

z = m(x - x₀) + n(y - y₀) + d

to the function that is being approximated. In order for this plane to touch the graph of f(x,y) at (x₀,y₀,f(x₀,y₀)) the constant d must be equal to f(x₀,y₀). The slope of the plane in the x and y directions must e equal to the slope of the function in the x and y directions. Thus, we have the linear approximation to a function of two variables as

$$f(x,y) \approx f(x_0,y_0) + \frac{\partial f}{\partial x}(x_0,y_0) (x-x_0) + \frac{\partial f}{\partial y}(x_0,y_0) (y-y_0).$$

Of course, the farther the point (x,y) is from (x₀,y₀) the worse the approximation will be.

There is another version of this formula which is commonly seen. By rearranging the terms, we get

$$f(x,y) - f(x_0,y_0) \approx \frac{\partial f}{\partial x}(x_0,y_0) (x-x_0) + \frac{\partial f}{\partial y}(x_0,y_0) (y-y_0)$$

or, in more compact notation,

$$\Delta f \approx f_x(x_0,y_0) \Delta x + f_y(x_0,y_0) \Delta y.$$

This makes it a little easier to write down higher order approximations as well as to see what happens for a function of more than two variables. To get a linear approximation to g(x,y,z) at the point (x₀,y₀,z₀,g(x₀,y₀,z₀)) we use the formula

$$\Delta g \approx g_x \Delta x + g_y \Delta y + g_z \Delta z$$

where are partial derivatives are to be evaluated at the point (x₀,y₀,z₀).