MVC
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1.
Quiz #3
NAME:
(10 pts) Let f ( x, y )  xy sin( x)
a. Find f x ( x, y ), f y ( x, y ), and f yx ( x, y )
 
b. Evaluate f  , 0  .
2 
 
c. Write the equation of the tangent plane to f at the point  , 0  .
2 
 
d. Find the directional derivative of f at  , 0  in the direction of u  3iˆ  4 ˆj
2 
2.
MVC
(5 pts) Evaluate the following limit or show that it does not exist.
 xy  y 3 
Lim  2
 x , y    0,0 x  y 2 


 x2
if ( x, y)  (0, 0)

3. (8 pts) Let f ( x, y )   x 2  y 2
.
 0
if ( x, y)  (0, 0)

a. Is f continuous at (0,0)? Explain.
b. Find f x (0, 0) or show that it does not exist.
c. Find f y (0, 0) or show that it does not exist
4.
MVC
dw
at t  2 given that x,y, and z are functions of t with:
dt
w  x 2 y  y 2 z and x(2)  3, y(2)  1, z (2)  2, x(2)  1, y(2)  0, and z(2)  2. .
(5 pts) Evaluate
5.
(5 pts) Suppose Molly Fane is located on the hyperbolic paraboloid z  y 2  x 2 at the point
(1,1,0). In what direction should she walk for the steepest climb and at what slope is she walking as
she starts out?
6.
(5 pts) Find a tangent plane to the surface y 2  yx 2  xz  yz 3  6 at the point (2,1,1) .
7.
(5 pts) Assuming all appropriate derivatives exist, prove that if z  f ( x, y ), with
x(r , )  r  cos and y (r ,  )  r  sin , then
 f  1  f   f   f 
   2
     
 r  r     x   y 
2
MVC
2
2
2
 x2  y 2
if ( x, y )  (0, 0)
 xy
8. (3 pts) Let f ( x, y )   x 2  y 2
. Show that f xy (0, 0)  f yx (0, 0) .

0
if ( x, y )  (0, 0)

Hint: First find f x (0, y ) and f y ( x, 0) .
MVC