136.275, Assignment No. 4

January 15, 2003

The assignment is due Wednesday, January 22, 2002 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, 0,1, for f. [5]

2. Let for (x,y,z) (0,0,0).

a)Show that by using the definition of the limit. [6]

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)Find and wherever they exist . [7]

c)Can f be differentiable at (1,0) ? Explain by using b). [2]

5. Let .

a) Show that f is differentiable at (0,1) by using the definition of differentiability. [7]

b) State a theorem by which you know that f is differentiable everywhere. [2].

Total [ 44/42]