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Final 21C Winter 2002
1. (9) Compute
f
y where
f
(x,y) = xy e
2 x sin(!y)
3
.
2. a. (8)
Suppose you know at some point that
!
f
!
x
= 2,
!
f
!
y
= -3,
!
x
!
s
= -5,
!
x
!
t
= 5,
!
y
!
s
= 4,
!
y
!
t
= -2.
What is the value of
!
f
!
t
?
b. (8) Suppose in addition that you know
!
s
!
u
= 2,
!
t
!
u
= 3, what is the value of
!
f
!
u
?
3. (13) Change the order of integration for the following integral and evaluate.
2
4
# #
1
1 + y
2
dy dx.
0
2x
4. (12) Write the following integral in spherical coordinates. DO NOT EVALUATE
2
p
1
r
# # #
r dz dr d .
2
q
0
0
r
2
5. (12) Find the critical points of f(x,y) = 6xy
2
- 2x - 3y + 1
3 4
6. (13) Given that (0,0) and (0,4) are critical points for z = 4y + 12yx - 24x - 24 y ,
3 2 ...

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