oh (myo92) – HW09 – mostovyi – (54020)
1
This
printout
should
have
20
questions.
Multiplechoice questions may continue on
the next column or page – find all choices
before answering.
001
10.0points
From the contour map of
f
shown below
decide whether
f
x
, f
y
are positive, negative,
or zero at
P
.
0
0
2
2
4
4
6
6
P
x
y
1.
f
x
>
0
,
f
y
>
0
2.
f
x
<
0
,
f
y
>
0
3.
f
x
<
0
,
f
y
= 0
4.
f
x
>
0
,
f
y
<
0
5.
f
x
>
0
,
f
y
= 0
6.
f
x
<
0
,
f
y
<
0
correct
Explanation:
When we walk in the
x
direction from
P
we
are walking downhill, so
f
x
<
0. On the other
hand, when we walk in the
y
direction from
P
we are again walking downhill, so
f
y
<
0 also.
Consequently, at
P
f
x
<
0
,
f
y
<
0
.
keywords: contour map, slope, partial deriva
tive,
002
10.0points
Determine
f
x
when
f
(
x , y
) = (
x
2

y
)(2
y
2

x
)
.
1.
f
x
= 2
y
+ 2
xy
2
+ 3
x
2
2.
f
x
= 2
xy
2

2
y
+ 3
x
2
3.
f
x
= 2
y

2
xy
2
+ 3
x
2
4.
f
x
= 4
xy
2

y

3
x
2
5.
f
x
=
y

4
xy
2

3
x
2
6.
f
x
= 4
xy
2
+
y

3
x
2
correct
Explanation:
From the Product Rule we see that
f
x
= 2
x
(2
y
2

x
)

(
x
2

y
)
.
Consequently,
f
x
= 4
xy
2
+
y

3
x
2
.
003
10.0points
Determine
f
y
when
f
(
x, y
) = sin(2
x

y
)

y
cos(2
x

y
)
.
1.
f
y
= 2 sin(2
x

y
)

y
cos(2
x

y
)
2.
f
y
=
y
sin(2
x

y
)
3.
f
y
=

y
sin(2
x

y
)
4.
f
y
=

y
cos(2
x

y
)
5.
f
y
=
y
cos(2
x

y
)
6.
f
y
=

2 cos(2
x

y
)

y
sin(2
x

y
)
correct
oh (myo92) – HW09 – mostovyi – (54020)
2
7.
f
y
= 2 cos(2
x

y
) +
y
sin(2
x

y
)
8.
f
y
=

2 sin(2
x

y
) +
y
cos(2
x

y
)
Explanation:
From the Product Rule we see that
f
y
=

cos(2
x

y
)

cos(2
x

y
)

y
sin(2
x

y
)
.
Consequently,
f
y
=

2 cos(2
x

y
)

y
sin(2
x

y
)
.
004
10.0points
Find the slope in the
x
direction at the
point
P
(0
,
2
, f
(0
,
2)) on the graph of
f
when
f
(
x, y
) = 3(2
x
+
y
)
e

xy
.
1.
slope =

6
correct
2.
slope =

2
3.
slope =

10
4.
slope =

8
5.
slope =

4
On the other hand,
differentiating
f
(
x, y
)
with respect to
y
holding
x
fixed, we see that
f
y
= 4
y
+
x
2
.
006
10.0points
Find the value of
f
x
at (3
,
2) when
f
(
x, y
) = 4
x
3

4
x
2
y

7
x
+ 5
y.
2.
f
x
= 8
x
+ 2
xy,
f
y
= 4
y
+
x
2
correct
3.
f
x
= 8
x
+
xy,
f
y
= 4
y
+
x
2
4.
f
x
= 4
y
+
x
2
,
f
y
= 8
x
+ 2
xy
5.
f
x
= 8
x
+ 2
x,
f
y
= 4
y
+
x
2
Explanation:
Differentiating
f
(
x, y
) with respect to
x
holding
y
fixed, we see that
f
x
= 8
x
+ 2
xy
.