136.275, Assignment No. 5

February 6, 2004

The assignment is due Friday, February 13 , 2004 in class. Late assignments receive a mark zero.

1. a) Use the chain rule to find the values of and at r=2 and ,

if . [5]

b) Let f be a differentiable function of three variables, and let w=f(x-y,y-z,z-x). Show that . [4]

2. The temperature (in degrees Celsius) at a point (x,y) on a metal plate in the xy plane is: .

a) Find the rate of change of temperature at (0,1) in the direction of u = (2,-1). [6]

b) An ant at (0,1) wants to walk in the direction in which the temperature drops most rapidly. Find a unit vector in that direction. [3]

3. Show that if f is differentiable and z = x f(x/y), then all of the tangent planes to the graph of this equation pass through the origin. [8]

4. Let (|R² , d ) be the usual metric space. Let A=D(0,1) \ {0} , B= |NX {0} and

C = { (r, ) : r = , } . For each of A, B and C determine :

a) interior points and boundary points,

b) are they open, closed or neither,

c) are they bounded or not.

Explain your work. [9]

5. Find the local and the absolute extremes of f(x, y) = xy² on the region R in |R² for

which y 0 and x²+y² 1. [8]

Total [ 43/42]