TEXAS A&M UNIVERSITY
DEPARTMENT OF MATHEMATICS
MATH 251-509
Final exam version A, 7 May 2008
Name:
In all questions, no analytical work — no points.
1.
Find the distance from the origin to the line


 x = 1 + t,
y = 2 − t,


z = −1 + 2t
Points:
/30
2.
The length x of a side of a triangle is increasing at a rate of 3m/s, the length y of another
side is decreasing at a rate of 2m/s, and the contained angle θ is increasing at a rate of
0.05 radian/s. How fast is the the area of the triangle changing when x = 40m, y = 50m,
and θ = π/6?
3.
Find the center of mass of a right circular cone with height 2 and base radius 1. (Place
the cone so that its base is in the xy-plane with center the origin and its axis along the
positive z-axis).
4.
Evaluate
Z
2
2
x2 y dx − xy 2 dy,
C
where C is the circle x + y = 1 with counterclockwise orientation.
5.
Evaluate
Z
F · dr,
C
where F = h2xy, x2 + 2yi and C is the curve y = sin x from (0, 0) to (π/2, 1).
6.
Evaluate
ZZ
F · dS,
S
where F = hxz, z 2 , 3z − xi and S is the (surface of the) sphere x2 + y 2 + z 2 = 4 with
outward orientation.
7.
Bonus question +10% (no partial credit): Cheburashka and Krokodil Gena are playing calculus riddles to determine who is going to wash the dishes for the next week. Gena
asks Cheburashka to find a vector field G such that
curl G = h2x, 3yz, −xz 2 i.
Help Cheburashka to prove that Gena is playing unfairly.