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Quiz #3
NAME:
1.(8 pts) Let f ( x, y)  x 2 y  xy 2
a. Find f x ( x, y ), f y ( x, y ), and f yx ( x, y )
b. Evaluate f 1, 1 .
c. Write the equation of the tangent plane to f at the point (1, 1) .
d. Find the directional derivative of f at (1, 1) in the direction of u  3iˆ  1 ˆj
2.(6 pts) If f ( x, y )  sin( xy ) find a vector u such that Du f 1,   is a maximum. Find the value
of Du f 1,   for this vector.
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 xy
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. Is f differentiable at (0,0)? Explain.
f
at ( s, t )  (3, 4) given that x,y, and z are functions of s and t with:
s
f ( x, y, z )  xyz and x(3, 4)  2, y(3, 4)  2, z(3, 4)  3, xs (3, 4)  1, ys (3, 4)  1, zs (3, 4)  2,
4.(5 pts) Evaluate
xt (3, 4)  4, yt (3, 4)  3, and zt (3, 4)  2 .
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 x2  y 2 
x2  y 2
5.(5 pts) If w  f  2
is
a
differentiable
function
of
, show that:
u

2 
x2  y 2
x y 
w
w
x
y
0
x
y
6.(5 pts) Let f :
2

3
be defined by f ( x, y )   x  y, xe xy , xy  .
a. Find the matrix of partial derivatives of f , Df ( x, y ) .
b. Find Df (1, 0)
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7.(8 pts) 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, 0) and f y (2, 0)
(b) f (2, 0)
(c) Du  f (2,0) , where u is a unit vector in the direction of f (2, 0)
(d) Sketch the vector f (2, 0) on the contour map for f using (2,0) as the initial point.
(e) Sketch a unit vector v with initial point (1, 3) such that Dv  f (1, 3)  0 .
-2
-1
1
2
3
4
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0
8.(3 pts) Suppose z  f ( x, y ) and z  g ( x, y ) are differentiable, a
 f  g  Duˆ f  f  Duˆ g
Prove Duˆ   
, provided g ( x, y )  0 .
g2
g
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, and û is a unit vector.