Math 241 – Exam 2 – 2PM V1
October 20, 2008
55 points possible
1.
Let
F
(
x, y, z
) = (
x
+
e
z
+
y, yx
2
).
(a)
Find the best linear approximation of
f
(
x, y, z
) at
~x
= (1
,
1
,
0).
(b)
Explain what we mean when we say that the linearization in part (a) is the
best
linear
approximation at
~x
= (1
,
1
,
0).
2.
(a)
Let
f
and
g
be two scalarvalued functions. State the Product Rule for
∇
(
fg
).
(b)
Let
f
(
x, y, z
) =
xy
+
e
z
and
g
(
x, y, z
) =
y
2
sin (
z
). Compute
∇
(
fg
)(
x, y, z
).
3.
Let
f
(
x, y, z
) =
xe
y
+
ye
z
+
ze
x
.
(a)
Find the directional derivative of
f
at the point (1
,
1
,
1) in the direction of
v
= (5
,
1
,

2).
(b)
In what direction does
f
have the maximum rate of change? What is this maximum?
4.
Let
F
(
x, y, z
) =
y
2
z
3
i
+ 2
xyz
3
j
+ 3
xy
2
z
2
k
.
(a)
Find
∇ ×
F
.
(b)
Show that
F
is a gradient field by finding a potential function
f
.
5.
Let
F
(
x, y
) = (cos
y
+
x
2
, e
x
+
y
) and
G
(
u, v
) = (
e
u
2
, u

sin
v
).
(a)
Write the formula for
F
◦
G
in terms of (
u, v
).
(b)
Calculate
D
(
F
◦
G
)(0
,
0) using the multivariable Chain Rule. (Hint: Don’t use your
answer from part (a).)
6.
Let
f
(
x, y
) = (
x
4
+
y
4
)
1
/
3
(a)