Math 209 - Assignment 2 Answer Key - Course Hero

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Math 209
Assignment 2 — Solutions
2
2
2
1. Let R = ln(u
+ v
+ w
), u = x + 2y, v = 2x
y, and w = 2xy. Use the Chain Rule to
∂R
∂R
find
and
when x = y = 1 .
∂x
∂y
Solution:
The Chain Rule gives
∂R
∂R
∂u
∂R
∂v
∂R
∂w
=
+
+
∂x
∂u
∂x
∂v
∂x
∂w
∂x
2u
2v
2w
=
1 +
2 +
(2y).
2
2
2
2
2
2
2
2
2
u
+ v
+ w
u
+ v
+ w
u
+ v
+ w
When x = y = 1, we have u = 3, v = 1, and w = 2, so
∂R
6
2
4
18
9
=
1 +
2 +
2 =
=
.
∂x
14
14
14
14
7
∂R
∂R
∂u
∂R
∂v
∂R
∂w
=
+
+
∂y
∂u
∂y
∂v
∂y
∂w
∂y
2u
2v
2w
=
2 +
( 1) +
(2x).
2
2
2
2
2
2
2
2
2
u
+ v
+ w
u
+ v
+ w
u
+ v
+ w
When x = y = 1, we have u = 3, v = 1, and w = 2, so
∂R
6
2
4
18
9
=
2 +
( 1) +
2 =
=
.
∂x
14
14
14
14
7
∂z
∂z
2. Find
and
if xyz = sin(x + y + z).
∂x
∂y
Solution:
Let F (x, y, z) = xyz
sin(x + y + z) = 0. Then, we have
∂F
∂z
yz
cos(x + y + z)
∂x
=
=
,
∂F
∂x
xy
cos(x + y + z)
∂z
∂F
∂z
xz
cos(x + y + z)
∂y
=
=
.
∂F
∂y
xy
cos(x + y + z)
∂z

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