Name
1-12
/60
MATH 251
Exam 1C
Fall 2015
13
/16
Sections 511/512 (circle one)
Solutions P. Yasskin
14
/12
15
/12
Total
/100
Multiple Choice: (5 points each. No part credit.)
1. If a
and b
2, 6, 2
1, 2, 1 , then
a
2b
a. 1
b. 3
c. 5
d. 6 Correct
e. 13
Solution:
2.
a
2b
| 4, 2, 4 |
16
4
16
6
The plot at the right is the contour plot of which function?
HINT: Where is the level set with value 0?
a. sin x cos y
Correct
b. sin x sin y
c. cos x cos y
d. cos x sin y
e. sin xy
Solution: There is a level set at x 0 but not x
. So it must have a sin x but not cos x
2
factor. There is a level set at y
but not y 0. So it must have a cos y but not sin y or
2
sin xy factor.
1
3. Find the projection of the vector u
onto the vector v
1, 1, 3
between these vectors acute or obtuse?
a.
b.
c.
d.
e.
1,
11
1,
11
2,
3
2,
3
2,
11
1,
11
1,
11
1,
3
1,
3
1,
11
3
11
3
11
2
3
2
3
2
11
, acute
, obtuse Correct
, acute
, obtuse
3
0 So the angle is obtuse.
2, 1, 2
3 3 3
4. Find the asymptotes of the hyperbola 4 x
2
b. y
2
c. y
3
d. y
3
e. y
3
2
2
3
3
2
2
3
3
x
3
x
3
x
2
x
2
2 x
3
3
2
9y
2
2
36.
Correct
2
Solution: In standard form the hyperbola is
x 3 2
y 2 2
cross
0. We solve:
9
4
y 2 2
x 3 2
4 x 3 2
y 2 2
4
9
9
5.
x
3
2
y
9
y
2
2
1. The asymptotes are the
4
2
2 x
3
3
cos 2
b. r
sin 2
c. r
cos 4
d. r
sin 4
Solution: sin 4
Correct
y
2
2 x
3
3
y
The plot at the right is the graph of which polar curve?
a. r
and is the angle
, obtuse
Solution:
u v 2 1 6
u vv
3 2, 1, 2
proj v u
9
v v
a. y
2, 1, 2
0.5
-0.5
0.5
x
-0.5
and cos 4
would have 8 loops. r
0 at
0 which is equation (b).
2
6. Find the volume of the parallelepiped with edge vectors u
w
1, 1, 1 , v
2, 1, 0 , and
0, 1, 2 .
a.
8
b.
4
c. 4 Correct
d. 6
e. 8
Solution: V
|u
v w|
1
1
1
2
1
0
0
1
2
7. Find the line through the points A
| 4|
8, 4, 6
4
and B
10, 5, 9 . It passes through the xy-plane
at:
Correct
a.
4, 2, 0
b.
8, 4, 0
c.
10, 5, 0
d.
8, 4, 0
e.
10, 5, 0
Solution: v
AB
x, y, z
z
6
8. Compute lim
h
a. 4x
3y
b. 8x
6y
4x
2, 1, 3
X A tv
8, 4, 6 t 2, 1, 3
8 2t, 4
3t 0 at t
2 So x, y, z
4h
12y
d. 24x
18y
e. 32x
24y Correct
x
2
4x
3y
2 2, 1, 3
4, 2, 0
2
h
0
c. 16x
Solution:
3y
t, 6 3t
8, 4, 6
4x
3y
2
2 4x
3y 4
32x
24y
3
9. Find the plane tangent to the graph of z
a.
24
b.
16 Correct
c.
8
x 3 e 2y at
2, 0 . The z-intercept is
d. 8
e. 16
f x, y
Solution:
10. If Q 2, 3
x 3 e 2y
f 2, 0
8
z
f 2, 0
f x x, y
3x 2 e 2y
f x 2, 0
12
8
f y x, y
3 2y
f y 2, 0
16
12x
6 and
2x e
Q
2, 3
x
0. 3 and
Q
2, 3
y
f x 2, 0 x
12 x
16y
2
2
f y 2, 0 y
0
16 y
16
c
16
0. 2, estimate Q 2. 2, 2. 7 .
a. 5. 88
b. 5. 9
c. 6. 0
d. 6. 1
e. 6. 12 Correct
Solution: The linear approximation says:
Q x, y
Q tan x, y
Q a, b Q x a, b x a Q y a, b y b
Here x, y
2. 2, 2. 7 and a, b
2, 3 . So
Q 2. 2, 2. 7
Q 2, 3 Q x 2, 3 . 2 Q y 2, 3 . 3
6 0. 3 . 2
0. 2 . 3
6. 12
4
11.
A right triangle has sides a and b. If a increases
a
from 3 cm to 3. 02 cm, while b decreases from
3
2
4 cm to 3. 98 cm, use the linear approximation to
1
determine whether the hypotenuse increases or
decreases and by how much.
0
0
1
2
3
4
b
a. increases by 0. 0028
b. increases by 0. 004
c. increases by 0. 028
d. decreases by 0. 004 Correct
e. decreases by 0. 028
a2 b2
Solution: c
c da
c db
c dc
a
b
3 . 02
4
5
5
a
a
2
b
. 02
2
b
da
a
0. 004
2
db
b2
decreases
x 2 y 2 15 z . Currently, a fish is at
3, 2, 1 . What is the rate of change of the oxygen density as seen
12. The oxygen density in a fish tank is given by
r
3, 4, 5 and has velocity v
by the fish?
a. 9
b. 115
c. 315 Correct
d. 365
e. 375
Solution:
d
dt
dy
dx
dz
2x 15
x dt
y dt
z dt
2 3 15 5 3 2 4 15 5 2
32
z v1
42
2y 15
1 1
z v2
x2
y2
1 v3
315
5
Work Out: (Points indicated. Part credit possible. Show all work.)
13.
16 points
1 t2, 2 t3, 1 t4
2
3
2
For the parametric curve r t
compute each of the following:
a. velocity v
Solution: v
t, 2t 2 , 2t 3
b. speed |v |
HINT: The quantity inside the square root is a perfect square.
t 2 4t 4 4t 6
t 2t 3
Solution: |v |
2, 16
,8
3
c. arc length L
ds
0,0,0
2
Solution: L
2
|v | dt
0
t2
2
2t 3 dt
t
0
2
t4
2
2
8
10
0
d. acceleration a
Solution: a
1, 4t, 6t 2
e. unit binormal B
î
Solution: v
k
î 12t 4
t 2t 2 2t 3
a
8t 4
6t 3
2t 3
k 4t 2
2t 2
4t 4 , 4t 3 , 2t 2
1 4t 6t 2
a|
16t 8 16t 6 4t 4
4t 4 2t 2 2t 2 2t 2 1
v a
1
2t 2 ,
2t ,
1
B
4t 4 , 4t 3 , 2t 2
2
2
2t 2t 1
2t 2 1 2t 2 1 2t 2 1
|v a |
f. tangential acceleration a T
d|v |
d t 2t 3
Solution: a T
1 6t 2
dt
dt
1 t2, 2 t3, 1 t4
14. 12 points
A wire has the shape of the parametric curve r t
3
2
2
0, 0, 0 and 2, 16 , 8 . Find the mass of the wire if the linear mass density is
3
Don’t simplify the answer.
|v
t, 2t 2 , 2t 3
Solution: v
2, 16
,8
3
M
|v |
2
15.
2t 3
4xz
2
ds
2
4xz |v | dt
0
0,0,0
5
t
2 10
5
t6 t
4 1 t2
2
1 t4
2
2
2t 3 dt
0
t7
between
4xz.
t6
2t 9 dt
0
t8
8
t 10
5
2
0
1184
5
12 points
A mass slides along a wire which has the shape of the parametric curve
1
2 2 3 1 4
rt
t , t , t
between 0, 0, 0 and 2, 16 , 8
under the action of the force
2
3
2
3
F
4z, 3y, 2x . Find the work done by the force.
Solution: F
4z, 3y, 2x
F v 2t 4 t 2t 3 2t 2
W
2, 16
,8
3
0,0,0
2t 4 , 2t 3 , t 2
t 2 2t 3 8t 5
2
F ds
v
2
F v dt
0
0
t, 2t 2 , 2t 3
8t 5 dt
4t 6
3
2
0
28
3
256
3
6