Math 27~ Page 5
10. The directional derivative of f{x,y,z) = xyz at the point (1,2,3) in
the direction from (1 , 2,3) toward (3 , 3,1) is
a) 1/4 b) 11
3 c) 19 d) 13
7
e) 17
6
11. By determining and testing the critical points of f{x,y)
we find that this function has
3 3
x - 3xy + Y ,
a) no critical points
b) a local minimum at (1,1) and a local maximum at (-1, -1)
c) a saddle point at (O,O) and a local minimum at (1,1)
d) a local minimum at (O,O) and a local maximum at (-1 , -1)
e) a saddle point at (1,1) only
12 . Suppose we wish to evaluate the iterated integral
2 1 3
f f x
ye dxdy.
0 y/2
If we reverse the order of integra ti on we have
1 2x XXXXXXXXXX
a) j j x dydx c) J j x
ye ye dydx
a 0 y/2 0
XXXXXXXXXXy/2 3
b) J J x dydx d) J J x
ye ye dydx
a 0 0 0
~, ~
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10. The directional derivative of f{x,y,z) = xyz at the point (1,2,3) in
the direction from (1 ,2,3) toward (3 ,3,1) is
13
11
e) 17
a) b) c) 19 d)
1/4
3 7 6
3 3
x - 3xy + Y ,
11. By determining and testing the critical points of f{x,y)
we find that this function has
a) no critical points
b) a local minimum at (1,1) and a local maximum at (-1, -1)
c) a saddle point at (O,O) and a local minimum at (1,1)
d) a local minimum at (O,O) and a local maximum at (-1 , -1)
e) a saddle point at (1,1) only
integral
evaluate the iterated
Suppose we wish to
12 .
2 1 3
x
ye dxdy.
f f
0 y/2
integra ti on we have
reverse the order of
If we
1 2 3
1 2x 3
x
x
ye dydx
c) j
ye dydx
a) j j
J
y/2
0
a 0
y/2 3
2
2 3
1
x
x
dydx
ye
ye dydx d)
b)
J J
J J
0
0
a 0
~,
~~ XXXXXXXXXX' +-h,,", .." h .... ~yC>.