c3-t3(1) - Name TEST3/MAC2313 Page 1 of 5 Read Me First...

Name: TEST3/MAC2313 Page 1 of 5
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Read Me First: Show all essential work very neatly. Use correct notation when presenting your computations. Write
using complete sentences. Remember this: "=" denotes "equals" , " ⇒ " denotes "implies" , and " ⇔ " denotes "is equivalent to".
Generic vector objects must be denoted by using arrows. Since the answer really consists of all the magic transformations,
do not "box" your final results. Show me all the magic on the page neatly.
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1. (10 pts.)
Obtain an equation for the plane tangent to the graph of
f ( x , y ) = sec( xy ) when ( x
0
, y
0
) = ( π ,1/4). Then obtain a vector equation for the normal line to the surface at the same point.
[ Reminder : z
0
= f ( x
0
, y
0
) ]
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2. (10 pts.)
Let
f ( x , y ) ( y
2
x
2
)
1/3
.
Compute the gradient of f at (-1,3). Then use it to compute the directional derivative
D
u
f (-1,3),
where u is
the unit vector forming an angle of θ = 5 π /6 with respect to the positive x-axis.
∇ f ( x , y )
∇ f ( 1,3)
u
D
u
f ( 1,3)

TEST3/MAC2313 Page 2 of 5
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3. (10 pts.)
Suppose f ( x , y ) = x 2 y . Find all unit vectors u so that
( ) D
u
f ( 1,2) 0
is true for u .