Introduction to the Tangent Plane

The following diagram contains the graph of the function y = x² and the graph of its tangent line at the point (1, 1)

With these graphs, it can be seen that there is a small region near the point (1, 1) where the graph of the tangent line appears identical to the graph of y = x². Correspondingly, the tangent line can frequently be used to approximate the behavior of the graph near the point to which the line is tangent.

In three dimensions, the tangent plane plays a similar role. We have seen that it is an easily obtainable tool. When discussing derivatives of functions of two variables. It is important to note that we can obtain the tangent plane to a surface very easily.

With this graph, we can see that, as with the tangent line in 2-D, there is a small region near the point
(a, b, f (a, b)) where the behavior of the tangent plane and the behavior of the surface z = f (x, y) appear identical.