Earlier we saw how the
two partial derivatives and can be thought of as the slopes of
traces. We want to extend this idea out
a little in this section. The graph of a
function is a surface in (three dimensional space) and so we can
now start thinking of the plane that is “tangent” to the surface as a point.
Let’s start out with a point and let’s let represent the trace to for the plane (i.e.
allowing x to vary with y held fixed) and we’ll let represent the trace to for the plane (i.e.
allowing y to vary with x held fixed). Now, we know that is the slope of the tangent line to the trace and is the slope of the tangent line to the trace . So, let be the tangent line to the trace and let be the tangent line to the trace .
The tangent plane will then be the plane that contains the
two lines and . Geometrically this plane will serve the same
purpose that a tangent line did in Calculus I.
A tangent line to a curve was a line that just touched the curve at that
point and was “parallel” to the curve at the point in question. Well tangent planes to a surface are planes
that just touch the surface at the point and are “parallel” to the surface at
the point. Note that this gives us a
point that is on the plane. Since the
tangent plane and the surface touch at the following point will be on both the
surface and the plane.
What we need to do now is determine the equation of the
tangent plane. We know that the general equation of a plane is given
where is a point that is on the plane, which we know
already. Let’s rewrite this a
little. We’ll move the x terms and y terms to the other side and divide both sides by c.
Doing this gives,
Now, let’s rename the constants to simplify up the notation
a little. Let’s rename them as follows,
With this renaming the equation of the tangent plane
and we need to determine values for A and B.
Let’s first think about what happens if we hold y fixed, i.e. if we assume that . In this case the equation of the tangent
This is the equation of a line and this line must be tangent
to the surface at (since it's part of the tangent plane). In addition, this line assumes that (i.e.
fixed) and A is the slope of this
line. But if we think about it this is
exactly what the tangent to is, a line tangent to the surface at assuming that . In other words,
is the equation for and we know that the slope of is given by . Therefore we have the following,
If we hold x fixed
at the equation of the tangent plane becomes,
However, by a similar argument to the one above we can see
that this is nothing more than the equation for and that it’s slope is B or . So,
The equation of the tangent plane to the surface given by at is then,
Also, if we use the fact that we can rewrite the equation of the tangent
We will see an easier derivation of this formula (actually a
more general formula) in the next section so if you didn’t quite follow this
argument hold off until then to see a better derivation.
Example 1 Find
the equation of the tangent plane to at .
There really isn’t too much to do here other than taking a
couple of derivatives and doing some quick evaluations.
The equation of the plane is then,
One nice use of tangent planes is they give us a way to
approximate a surface near a point. As
long as we are near to the point then the tangent plane should nearly
approximate the function at that point.
Because of this we define the linear
approximation to be,
and as long as we are “near” then we should have that,
Example 2 Find
the linear approximation to at .
So, we’re really asking for the tangent plane so let’s
The tangent plane, or linear approximation, is then,
For reference purposes here is a sketch of the surface and
the tangent plane/linear approximation.