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Quiz #3
NAME:
1. Let f ( x, y)  x3 cos( xy)
a. Find f x ( x, y ), f y ( x, y ), and f xy ( x, y )
 
b. Evaluate f 1,  .
 2
2.
Evaluate the following limit or show that it does not exist.
 x2 y 
a.
Lim  2
 x , y    0,0 x  y 2 


b.
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 x2 y 
 x , y    0,0  x 4  y 2 


Lim
 x 3 y 4
if ( x, y )  (0,0)

3.
Let f ( x, y )   x 2  y 2
. Find f x (0,0) and f y (0,0) or show that they do

if ( x, y)  (0,0)
 0
not exist..
4.
Find the directional derivative of F ( x, y, z) 
v
5.
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1
3
Evaluate
i
1
3
j
1
3
x y
z
in the direction of the vector
k , at the point 1,2,5 .
w
at the point (r , s)  (1, 1) if w  x 2 y  y 2 z and x  2r  s, y  r  2s, and z  2rs .
s
6.
Find a tangent plane to the surface z  xy 2  yx 2 at the point (2,1,6) .
7.
Find a point on the paraboloid 2 x  z 2  3 y 2  0 at which the tangent plane is parallel to
the plane 5 x  y  z  1
8. Prove that if z  f ( x, y), with x(s, t )  s 2  t 2 and y(s, t )  t 2  s 2 , then
f
f
t s  0
s
t
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9. For the contour map for z  f ( x, y ) shown below, estimate each of the following quantities.
Explain briefly how you are getting your answer.
(a) f x (2, 1) and f y (2, 1)
(b) f (2, 1)
(c) Du  f (2, 1) , where u is a unit vector in the direction of f (2, 1)
(d) Sketch the vector f (2, 1) on the contour map for f using (2, 1) as the initial point.
(e) Sketch a unit vector v with initial point (2, 1) such that Dv  f (2, 1)  0 .
7
5
3
1
-1
-3
-5
3
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Concepts:
10.
Let f :
2

3
be given by f ( x1 , x2 )  ( x1  x2 , x1 x2 , x12 x2 ) and let x :
by x (t1 , t2 , t3 )  (t1t2 , t ) .
2
1
a) Write an explicit formula for x f


b) Find the derivative, D x f .
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3

2
be given