136.275, Assignment No. 5

February 7, 2003

The assignment is due Friday, February 14 , 2002 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 one variable, and let z=f(). 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 (1,1) in the direction of u = (2,-1). [6]

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

3. Prove that if a function f is differentiable at the point (x,y) and if in two nonparallel directions , then in all directions. [9]

4. Find parametric equations for the tangent line to the curve of intersection of the cone

and the plane x+2y +2z = 20 at the point (4,3,5). [8]

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]