Name
ID
MATH 253 Honors
1-10
/50
11
/15
12
/10
13
/10
14
/15
Section
FINAL EXAM
Spring 1999
Sections 201-203
P. Yasskin
Multiple Choice: (5 points each)
1. Find the volume of the parallelepiped with
Ÿ3, 2, 0 ,
edges
Ÿ
"1, 1, 2 and
Ÿ0, 4, 1 .
a. "23
b. "19
c. 19
d. 21
e. 23
2. Find the unit tangent vector
T
to the curve
rŸt Ÿ3t, 2t 2 , 4t 3 at the point
rŸ1 Ÿ3, 2, 4 .
a.
b.
c.
d.
e.
3 , 4 , 12
13 13 13
3 , 4 , 12
29
29
29
3 , 4 , 12
169 169 169
3 , 4 , 12
29 29 29
3 , "4 , 12
169 169 169
1
3. If a jet flies around the world from East to West, directly above the equator, in what
B
direction does the unit binormal
a.
b.
c.
d.
e.
North
South
East
West
Down (toward the center of the earth)
4. At the point
plane
a.
b.
c.
d.
e.
point?
where the line
r Ÿt 5,
we have
xyz
Ÿx, y, z 2x " y z
Ÿ2 t, 3 " t, t 2
3
4
5
6
T 4P
>
the pressure and > is the density. At a certain point
Q
5. The temperature in an ideal gas is given by
4
PŸQ Ÿ
>ŸQ 2
>ŸQ Ÿ3,
T ŸQ 24
where 4 is a constant, P is
Ÿ1, 2, 3 ,
we have
"3, 2, 1 PŸQ So at the point Q, the temperature is
a.
b.
c.
d.
e.
intersects the
"1, 2 and its gradient is
TŸQ 4Ÿ"4.5, 0, 2.5 4Ÿ1.5, 0, 2.5 4Ÿ1.5, 2, "4.5 4Ÿ"4.5, 2, "1.5 4Ÿ"1.5, 2, 2.5 2
6. The saddle surface
may be parametrized as
z xy
the plane tangent to the surface at the point
Ÿ1, 2, 2 .
a.
b.
c.
d.
e.
R
Ÿu, v Ÿu, v, uv .
Find
3x y " z 3
2x y " z 2
3x 2y " z 5
2x " y z 2
3x " y z 3
7. Find the minimum value of the function
plane
x 2y 3z
f
x2
y2
z2
on the
14.
a. 0
b. 7
4
c. 7
2
d. 14
e. 28
3
8. Compute
3
9
0
y
; ; 2 y cosŸx 2 dx dy
a. 1 sin 81
4
b. 1 cos 9 " 1
2
2
c. 9 sin 81 cos 9 " 1
2
d.
" 92 sin 81 92 sin y 4
e. 9 sin 81 " cos 9 1
2
9. Compute
;;; z 2 dV
over the solid sphere
x2
y2
z2
t 4.
a. 64=
b.
c.
d.
e.
5
256=
3
48=
5
64=
15
128=
15
10. Compute
0
t x t 1,
a.
b.
c.
d.
e.
3
Ÿx, y , z dS
for
F
over the surface of the cube
;; F
0 t y t 1,
0tzt1
with outward normal.
1
2
3
4
6
4
11. (15 points) Find the area of the diamond shaped region between the curves
1 ex,
y
4
You must use the curvilinear coordinates
y
ex,
y
e "x
u
and
ye "x
y
and
4e "x .
v
ye x .
4
3
y2
1
-1
0
1
x
2
3
-1
5
12. (10 points) Find the mass of a wire in the shape of the curve
0txt =
if the density is
> siny x .
e
4
Note: The wire may be parametrized as
r Ÿt y
lnŸcos x for
Ÿt, lnŸcos t .
6
around the boundary of the triangle with
> x dx z dy " y dz
and
traversed in this order of the
Ÿ0, 0, 0 ,
Ÿ0, 1, 0 Ÿ0, 0, 1 ,
Hint: The yz-plane may be parametrized as
R
Ÿu, v Ÿ0, u, v .
13. (10 points) Compute
vertices
vertices.
7
14.
• F dS
;; (15 points) Compute
S
for
F
sphere
Ÿx
2
x2
y, y, z
y2
2
z2
over the piece of the
25
for
0
tzt4
with normal pointing away from the z-axis.
Hint: Parametrize the upper and lower edges.
8