Name
ID
MATH 253 Honors
Section
FINAL EXAM
Sections 201-202
1-7
/49
8
/15
9
/10
10
/15
11
/15
Fall 1999
P. Yasskin
Multiple Choice: (7 points each)
P Ÿ4, 4, 4 Its tangent vector is
1. Consider the line through the point
plane
a.
b.
c.
d.
e.
x 2y 3z 7.
which is perpendicular to the
Ÿ3, 2, 1 Ÿ1, 2, 3 Ÿ7, 6, 5 Ÿ5, 6, 7 Ÿ4, 4, 4 x " yz 0
at the point
Which of the following points does not lie on this plane?
2. Find the plane tangent to the hyperbolic paraboloid
P Ÿ6, 3, 2 .
a.
b.
c.
d.
e.
Ÿ
"6, 0, 0 Ÿ0, 3, 0 Ÿ0, 0, 2 Ÿ1,
Ÿ
"1, "1 "1, 1, 1 3. Duke Skywater is flying the Millenium Eagle through a polaron field. His galactic
measured in lightseconds and his velocity is
Ÿ2300, 4200, 1600 measured in lightseconds per second. He measures the strength of
p Ÿ3, 2, 2 the polaron field is
p 274 milliwookies
and its gradient is
milliwookies per lightsecond. Assuming a linear approximation for the polaron field and
that his velocity is constant, how many seconds will Duke need to wait until the polaron
286 milliwookies?
field has grown to
coordinates are
v Ÿ.2, .3, .4 a.
b.
c.
d.
e.
2
3
4
6
12
1
4. Consider the surface S parametrized by
and 0
t v t 4.
Compute
dS
;; F
Ÿu, v Ÿu v, u " v, uv R
where
Ÿy, x, y .
F
for
0
tut2
S
a. "32
b. "16
c. 16
d. 32
e. 64
Ÿu, v Ÿu v, u " v, uv . Find the plane
R
P RŸ1, 2 Ÿ3, "1, 2 .
Which of the following
5. Consider the surface S parametrized by
tangent to this surface at the point
points does not lie on this plane?
a.
b.
c.
d.
e.
Ÿ3, 0, 0 Ÿ0, 4, 0 Ÿ0, 0,
"2 Ÿ1, 1, 0 Ÿ0, 6, 1 6. Compute
area
0
>Ÿ"x 2 y 2 dx 2xy 3 dy t y t 4 " x2
over the complete boundary of the semicircular
traversed counterclockwise.
a. 0
b. 16
c. 4
5
d. 80
5
128
e.
5
7. Compute
3 2
5
y3z2
;; x 3z dy dz 3 dz dx z5 dx dy
over the complete surface of the
S
sphere
x2 y2 z2 4
with outward normal.
a. 512=
21
2
b. 32=
5
128
=
c.
5
d. 16=
3
e. 256=
15
2
8. (15 points) Find the point in the first octant on the surface
z
32
x4y2
which is
closest to the origin.
3
9.
;; x dA
(10 points) Compute
15
R
over the region R in the first quadrant
10
y
bounded by the curves
y x2,
y x4
and
y 16.
5
0
2
x
4
4
10.
(15 points) Find the mass and center of mass
of the solid below the paraboloid
above the xy-plane, if the density is
z 4 " x2
" y2
- x2 y2.
(11 points for setting up the integrals and the
final formula.)
t r
5
11.
(15 points) Find the area and centroid
1
of the right leaf of the rose
r 2 cos 2 2.
-2
-1
0
1
2
(12 points for setting up the integrals
and the final formula.)
-1
6