Math 205 |
Problems
Here's a guide for 14.8, what's in this section, what to concentrate on for now, and so forth. But first, a few comments before we start.
Now here's the promised outline or reading guide.
for which the error expression
not only tends to zero as (x,y) --> (a,b).
What more is required is that we can divide this error term by
the (tiny!) distance between (x,y) and (a,b) and the result will still tend
to zero as (x,y) --> (a,b) !
It is in this (explicit) sense that L(x,y) is a good (really good!)
approximation for f(x,y) when (x,y) is close to (a,b). See text or
class notes for precise version of definition.
is not differentiable at (0,0) because it fails
to have partial derivatives there. This function is cone-shaped.
See the sketch in Example 2.
So Example 2 illustrates: nonexistence of partial derivatives implies
nondifferentiability.
does have partial derivatives at (0,0) but still fails to be differentiable at (0,0). So Example 3 illustrates: existence of partial derivatives is not enough to guarantee differentiability.
No proof is given. Problem 14 asks the student to supply a proof. Problem 14 will be on either HWK 13 or HWK 14. After we've worked with the definitions a little longer, this problem will be pretty easy.
f(x,y) = xy/(x^{2} +y^{2}) otherwise
fails to be continuous at (0,0) [ because it has no limit there] and yet it does have partial derivatives at (0,0). So this example illustrates:
is differentiable at every point of its domain. How does it do this? Easy: just find the partial derivatives for f. Observe that they are defined at every point where f is defined. Observe that they are also known to be continuous at all these points. Invoke Thm 14.2. Done.