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SHOW ALL WORK !!
Calculator Allowed.
Quiz #3
NAME:
1.(8 pts) Let f ( x, y)  x 2 sin( xy)  x
a. Write the equation of the tangent plane to f at the point (1, 2 ) .
b. Find the directional derivative of f at (1, 2 ) in the direction of u  5iˆ  12 ˆj
 x2
if ( x, y )  (0, 0)

2. (8 pts) Let f ( x, y )   x 2  y 2
.

if ( x, y)  (0, 0)
 0
a. Is f continuous at (0,0)? Explain.
b. Find f x (0, 0) or show that it does not exist.
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3(4 pts) a. Sketch the graph of x 2  y 2  4 z 2  1 .
3b(4 pts) Find the equation of the tangent plane to this surface at the point (2,1, 1)
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 xy  y 2 
xy  y 2
w
w
4.(5 pts) If w  f 
is
a
differentiable
function
of
, find :
and
u

2 
2
x
y
xy  y
 xy  y 
5.(4 pts) Evaluate the following limit, or show that the limit does not exist:
Lim ( x 2  y 2 )sin(
(x , y ) (0,0) 

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x2  y 2 )


6.(8 pts) Let f :
2

3
and g :
3

2
and be defined by f ( x, y )   x  y, x 2 , xy  and
g ( x, y, z )   xy, xyz  .
a. Find formulas for the functions f g and g f .
b. Find the matrix of partial derivatives for both f and g :
Df ( x, y ) =
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2

37 23 
7(4 pts) Show that the sphere ( x  10)  ( y  5)   z 
  925 and the ellipsoid
6 



23 
23 
9 x 2  4 y 2  36 z 2  36 are mutually tangent at the point 1,1,
 . The point 1,1,
 is on
6 
6 


both graphs – you do not need to show this.
2
2
8(3 pts) Evaluate the following limit, or show that the limit does not exist.
 x2 y2 
lim 

( x , y )  (0,0) x 3  y 3


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