Lab #6

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 .

ACTIVITY: Find a number small enough so that the graph of F over the rectangle x=1-epsilon..1+epsilon, y=1/5-epsilon..1/5+epsilon looks like a plane. (Use the ``Constrained'' projection.) Print out a copy of this plot, and indicate on the plot the value you used for .

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 F near the point gives evidence to the fact that F is differentiable at . For the rest of the lab we will investigate the differentiability of the function G defined by

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

ACTIVITY: The Geometric Approach Define G in Maple (The simplify command my be useful). Use the ideas from the first activity to give geometric evidence to the fact that G is not differentiable at . In other words, zoom in on the graph of G around the point , picking smaller and smaller -rectangles (again, it's important to use the ``Constrained'' option to observe things at the correct scale).

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

ACTIVITY: The Symbolic Approach
• 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.

ACTIVITY: The Limit Approach It should be apparent from the above activity that some tangent vectors to the graph of G at do not lie in the plane you computed above!

• 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.