Math 222 - 2	First Midquarter Review
Treibergs	Tuesday, Jan. 23, 1996

Let


f(t) = (t, t^2, sin t),

                              x
g(x, y) = [ 2x - 3y, xy,  --------- ],
                          x^2 + y^2
 
h(u, v) = (u^2 - v^2, 2uv),


               /    x^2 y
               |  --------- ,       if  (x, y) ≠ (0, 0);
k(x, y) =     <   x^2 + y^2
               |      
               \      0,            if  (x, y) = (0, 0).



1. Find the velocity vector, acceleration vector and speed of  f(t)  at  t = 1.


2. Sketch the graph and three level curves of  z = x^2 - y^2.


3. Find  dg.


4.  Show that  ∂h/∂u  and  ∂h/∂v   are perpendicular at all  (u,v).


5.  Let  W = { (x,y) : x = 0 } U { (x,y):  4 < x^2 + y^2 < 9 }

    Find the boundary of  W  and the set of interior points of  W.  Is  W  open? 
    
    closed?  both open and closed? neither open nor closed? 


6.  Find an affine function  a(u, v) which best approximates  h(u, v)  at the point  
    
    (u, v) = (3, 1).  Estimate  |h(u, v) - a(u, v)|  when  |(u, v) - (3, 1)| < d.


7.  Let  u = (3. 4). Find  p  and  D_u p where  p(x, y) = x sin (x^2 + y^2).


8.  Find the normal vector and tangent plane to the level surface of the function
	
    q(x, y, z) =  xy + exp(y)(1 + z^2)   at the point  (x,y,z) = (1,2,3).


9.  Show that      lim      k(x, y)  =  0.   
              (x,y)->(0,0)


10. Show that  k(x, y)  is not differentiable at  (0, 0).