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
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.
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.)
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.)
( 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.)