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

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!

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.)

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Last modified: Nov 21 1996
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