136.275, Assignment No. 4
January 16, 2004
The assignment is due Friday, January 23, 2004 in class. Late assignments receive a mark zero.
1.
Let 
.
a) Draw the domain of f. Explain . [4]
b) Draw the level curves k= -1, 0,2 for f. [4]
2.
Let 
for
(x,y,z) 
(0,0,0).
a)
Show that 
by using the 
definition of
the limit. [7]
b)
Can f(x,y,z) be redefined at (0,0,0) such that it is
continuous on 
? Explain.[2]
3.
Show that 
does not
exist. [6]
4.
Let 
.
a) Sketch the graph of f. [3]
b)
For which values of 
do
and 
exist? Show your
work. [7]
c) Can f be differentiable at (1,0) ? Explain by using b). [2]
5.
Let 
.
a) Show that f is differentiable at (0,0) by using the definition of differentiability. [6]
b) Look at the partial derivatives fx and fy of f. State a theorem by which you know that f is differentiable everywhere. [2].
Total [ 43/42]