# IMPLICIT FUNCTIONS

An implicit function is one given by
**F: f(x,y,z)=k**, where k is a constant.
Unlike the other two examples, the
tangent plane to an implicitly defined function
is much more difficult to find.
As with graphs and parametric plots, we must use
another device as a tool for finding the plane.
This device is known as the gradient. The
gradient equals
**(derivative of 1st component w/ respect to x,
derivative of 2nd component w/ respect to y,
..., nth component w/ respect to nth
variable)**.

The gradient is the perpendicular or normal
vector to the surface at a certain point.
How does this relate to tangent planes? Well,
for implicit surfaces, the tangent plane is the
set of points (x,y,z) that satisfy the equation
**(grad f(a,b,c))((x,y,z)-(a,b,c)) = 0
**where (a,b,c) is a specific point.
(This means that the gradient is, at all times,
perpendicular to our tangent plane.
So, to get our tangent plane, we simply derive
the plane perpendicular to our gradient
vector.)

**ex.4** Let us define a function f, f:
x^2+ y^2+ z^2=2. The point that we have
interest in is (a,b,c)=(1,1,0).
grad f = (2x, 2y, 2z)

grad f(a,b,c) = (2,2,0)

(x,y,z)-(a,b,c) = (x-1, y-1, z)

The resulting formula of our tangent plane to f
at (a,b,c) is
**(x-1, y-1, z).(2,2,0) =0**
Note that the third component of the
plane is given free range.

### QUESTION #3

**A.** We define our function implicitly as
q:= z^3+y^2+x=4. Our point of interest is
(2,1,1).

**B.** Find the gradient vector to q.

**C.** Plug in your point of interest.

**D.** After you have this gradient, find
the plane perpendicular to this vector, and
voila, you have your tangent plane.

**E.** Use the spiffy *Maple* tool to check
your work. (If you have not already pasted in the maple code that auxilliary
functions, then do so now.) The syntax is as follows:

ImplicitTan(function, x=xmin..xmax,
y=ymin..ymax, z=zmin..zmax, point =[x0,y0,z0]);

**Note:** Remember that when using ImplicitTan on your own
that the point you specify must lie on the surface, otherwize
you will not get tangent planes.

Next: **When Problems Arise**
Back: **Tangent
Planes to Parametric Surfaces**