Shows the construction of a tangent plane to a surface at a point.

(corresponding notebook for users of Mathematica who want the commands used to generate these graphics)

We start with a surface, in this example the elliptic paraboloid . and we are interested in the tangent plane to the surface S=f(x,y) at a particular point. In this example the point is (2,-1,-5).

Note that many planes go through this point, but we want the one that emulates the surface most closely at this point.

To begin constructing the tangent plane to the surface at the point, we first intersect the surface with the plane y=-1. This intersection results in a curve called the trace of S in the plane y=-1.

We also intersect S with the plane x=2. This intersection also results in a trace. Here we show both curves and emphasize that they intersect at (2,-1,-5).

Each of the traces above has a tangent line at the point (2,-1,-5).

These tangent lines uniquely determine a plane. As it turns out, this is the plane we were seeking.

In general the equation for a plane containing the point (a,b,c) is

A(x - a) + B(y - b) + C(z - c) = 0

where A, B, and C are the components of the normal vector to the plane. In order to calculate these coefficients for our tangent plane, we note that the plane contains the two tangent lines from above. One of those tangent lines has slope and the other tangent line has slope . Some algebra gives us that the equation for the tangent plane to f(x,y) at (a,b,f(a,b)) is

.

This work is part of the Multivariable Calculus with Mathematica Project at the University of Oklahoma.