This worksheet does not need to be handed in, but you are responsible for the material. The purpose of the lab is to explore the concepts of limits, continuity, and differentiability in multivariable functions.

Geometrically, a function of two variables is differentiable at a point if its graph looks like a plane when you zoom in close enough. Said another way, a differentiable function is well-approximated near that point by a linear function (whose graph is the plane that best fits the graph of the function). Most of the functions we study in calculus are ``nice'', meaning they are continuous, and often differentiable as well. In this lab we examine some functions that are not so nice.

We will study the function

To get started, launch Maple, type
` with(plots): with(linalg):`,
and define **F** as above.
We will use Maple to geometrically test the
differentiability of **F** at the point .

Use your plot of the graph of **F** to estimate and
. Discuss how you can use the
geometric information from the graph of **F** to determine your estimates.
Now symbolically compute the actual value of the two
quantities you estimated above and compare them with your estimates.

Your plot of the graph of

In fact, we will show that **G** is * not*
differentiable at .

What phenomenon do you observe? Explain how this
indicates that **G** can not be differentiable at . (You do
not need to print these plots.)

- Symbolically compute both partial derivatives of
**G**and note that the denominators of these derivatives make it hard to determine whether the derivatives exist at ! - Fortunately, we can think back to earlier in the year and compute the
partial derivatives a different way.
Write down and and compute the
derivatives of these one-variable functions at zero. Why does this approach
verify the existence of and
?
- Use the values of the partial derivatives at to write down
tangent vectors to the graph of
**G**at in the coordinate directions. - Parametrize the plane based at the point and spanned
by the tangent vectors you computed above, and graphically display
this plane together with the graph of
**G**on the domain`x=-1..1, y=-1..1`. Print out a copy of this plot.

- Verify the above statement by
picking a direction through the origin in the -plane
for which the tangent vector does
not seem to lie in the plane spanned by tangent vectors in the coordinate
directions. Parametrize a line in this direction and
evaluate
**G**along this line. (*Hint:*All non-vertical lines passing through the origin are of the form**y=mx**and their parametrizations in the**x,y**-plane are therefore .) - Show that the tangent vector at the origin to the curve parametrized by does not lie in the plane found in the previous activity.
- Illustrate this by sketching in the tangent vector on the plot which includes the graph of the plane.
- Using this activity, can you show
from the definition of differentiability (p. 176)
that
**G**is not differentiable at ?

If you finish this lab with extra time, you may want to think about the following continuation of the lab:

There is a theorem (p. 181) that says that **G**
must be differentiable ** if** the partial derivatives for **G** exist
and are continuous. We've just shown that the partial derivatives for
**G** exist at and that **G** is * not* differentiable there.
So * at least one of the partial derivatives of G must not be
continuous at *. (If you follow this logic, consider
going into mathematics.)

- Make a plot of the graph of on the domain
`x=-1..1, y=-1..1`. Use the plot to show graphically why this new function is not continuous at .(

*Hint:*To be continuous, the limit must be the same from any direction. Be careful of what Maple is showing you---technically the partial derivatives of**G**are not defined at , so Maple tries to ``sew up'' everything there. You should concentrate on the behavior of the graph outside of this messy area.) - Verify analytically (by using the formula for ) the discontinuity you observed above.

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Last modified: Nov 21 1996

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