9 - trevino ochoa (dt23653) HW09 mostovyi (54020) This...

trevino ochoa (dt23653) – HW09 – mostovyi – (54020) 1
This print-out should have 20 questions.
Multiple-choice 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
our elevation doesn’t change because we are
walking along a contour, so f
y
= 0.
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
= y - 4 xy
2
- 3 x
2
2. f
x
= 2 y - 2 xy
2
+ 3 x
2
3. f
x
= 4 xy
2
+ y - 3 x
2
4. f
x
= 2 xy
2
- 2 y + 3 x
2
5. f
x
= 2 y + 2 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(4 x - y ) - y cos(4 x - y ) .
1. f
y
= - 2 cos(4 x - y ) - y sin(4 x - y )
correct
2. f
y
= y cos(4 x - y )
3. f
y
= 2 sin(4 x - y ) - y cos(4 x - y )
4. f
y
= 2 cos(4 x - y ) + y sin(4 x - y )
5. f
y
= y sin(4 x - y )
6. f
y
= - y cos(4 x - y )

trevino ochoa (dt23653) – HW09 – mostovyi – (54020) 2
7. f
y
= - y sin(4 x - y )
8. f
y
= - 2 sin(4 x - y ) + y cos(4 x - y )
f
x
= 8 x + xy, f
y
= 8 y + x
2
3. f
x
= 8 x + 2 x, f
y
= 8 y + x
2
004 10.0points
Find the slope in the x -direction at the
2. f
x
= 8 x + xy, f
y
= 8 y + x
2
3. f
x
= 8 x + 2 x, f
y
= 8 y + x
2
point P (0 , 2 , f (0 , 2)) on the graph of f when
f ( x, y ) = 2(2 x + y ) e
- xy
.
1. slope = - 6
2. slope = - 4 correct
3. slope = - 8
4. slope = - 10
5. slope = - 2
On the other hand, differentiating f ( x, y )
with respect to y holding x fixed, we see that
f
y
= 8 y + x
2
.
006 10.0points
Find the value of f
x
at (2 , 1) when
f ( x, y ) = 5 x
3
- 2 x
2
y - 7 x + 2 y.