moore (jwm2685) – HW12 – gilbert – (55485)
1
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printout
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have
18
questions.
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before answering.
001
10.0points
Find the directional derivative,
f
v
, of
f
(
x, y
) =
radicalbig
6
x
−
2
y
at the point (2
,
−
3) in the direction
v
=
i
+
j
.
1.
f
v
=
1
2
2.
f
v
=
1
3
correct
3.
f
v
=
1
6
4.
f
v
=
2
3
5.
f
v
= 0
Explanation:
For an arbitrary vector
v
,
f
v
=
∇
f
·
parenleftbigg
v

v

parenrightbigg
,
where we have normalized the direction vector
so that it has unit length.
Now the partial derivatives of
f
(
x, y
) =
radicalbig
6
x
−
2
y
are given by
∂f
∂x
=
3
√
6
x
−
2
y
,
and
∂f
∂y
=
−
1
√
6
x
−
2
y
.
Thus
∇
f
(
x, y
) =
∂f
∂x
i
+
∂f
∂y
j
=
parenleftBig
3
√
6
x
−
2
y
parenrightBig
i
−
parenleftBig
1
√
6
x
−
2
y
parenrightBig
j
,
and so
∇
f
(2
,
−
3) =
1
√
2
parenleftBig
i
−
1
3
j
parenrightBig
.
On the other hand,
v
=
i
+
j
=
⇒
v

v

=
1
√
2
(
i
+
j
)
.
But then
∇
f
·
parenleftbigg
v

v

parenrightbigg
=
1
2
parenleftBig
i
−
1
3
j
parenrightBig
·
(
i
+
j
)
.
Consequently,
f
v
=
1
2
parenleftBig
1
−
1
3
parenrightBig
=
1
3
.
keywords:
002
10.0points
Find the gradient of
f
(
x, y
) = 3
xy
2
+
x
3
y .
1.
∇
f
=
(big
3
x
2
y
−
3
y
2
,
6
xy
+
x
3
)big
2.
∇
f
=
(big
3
y
2
+ 3
x
2
y,
6
xy
+
x
3
)big
correct
3.
∇
f
=
(big
3
y
2
+ 3
x
2
y, x
3
−
6
xy
)big
4.
∇
f
=
(big
6
xy
+
x
3
,
3
y
2
+ 3
x
2
y
)big
5.
∇
f
=
(big
x
3
−
6
xy,
3
y
2
+ 3
x
2
y
)big
6.
∇
f
=
(big
6
xy
+
x
3
,
3
x
2
y
−
3
y
2
)big
Explanation:
Since
∇
f
(
x, y
) =
(bigg
∂f
∂x
,
∂f
∂y
)bigg
,