Tangent Planes and Linear Approximations: Learn It 2

Linear Approximations

You’ve seen linear approximations before in single-variable calculus. The idea is straightforward: use a tangent line to estimate function values near a known point.

Linear Approximations and Differentials

The formula for the linear approximation of a function [latex]f(x)[/latex] at the point [latex]x=a[/latex] is:

[latex]y\approx f(a)+f'(a)(x-a)[/latex]

Here’s what this looks like graphically:

A curve in the xy plane with a point and a tangent to that point. The figure is marked tangent line approximation.
Figure 4. Linear approximation of a function in one variable.

The tangent line gives you a good approximation to [latex]f(x)[/latex] for values of [latex]x[/latex] reasonably close to [latex]x=a[/latex]. When working with functions of two variables, we use the same idea—just replace the tangent line with a tangent plane.

linear approximation of a function of two variables

Given a function [latex]z=f(x, y)[/latex] with continuous partial derivatives at the point [latex](x_0, y_0)[/latex], the linear approximation of [latex]f[/latex] at [latex](x_0, y_0)[/latex] is:

[latex]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).[/latex]

Notice this is exactly the equation of the tangent plane to the surface [latex]z=f(x, y)[/latex] at [latex](x_0, y_0)[/latex]. The linear approximation works because if you know the exact value of [latex]f(x, y)[/latex] at [latex](x_0, y_0)[/latex], then for points [latex](x, y)[/latex] close to [latex](x_0, y_0)[/latex], the tangent plane gives you a value close to the actual value of [latex]f(x, y)[/latex] (Figure 5).

A paraboloid with surface z = f(x, y). There is a point given on the paraboloid P (x0, y0) with a tangent plane at that point. There is a point on the plane which is marked as the linear approximation L(x, y) to f(x0, y0), which is close to the corresponding point on the paraboloid.
Figure 5. Using a tangent plane for linear approximation at a point.
Given the function [latex]f(x, y)=\sqrt{41-4x^{2}-y^{2}}[/latex], approximate [latex]f(2.1, 2.9)[/latex] using point [latex](2, 3)[/latex] for [latex](x_0, y_0)[/latex]. What is the approximate value of [latex]f(2.1, 2.9)[/latex] to four decimal places?