Name:
FINAL EXAM A
MA227
Open Book. No notes, etc. No calculators.
CIRCLE YOUR ANSWER. You must show your work and justify your
answers to receive credit.
1. Suppose a projectile is fired at a 45 degree angle from the top of a building
that is 50 feet tall with an initial speed of v0ft./sec.
(a) Write a position vector that describes the motion of the projectile
for all time t.
(b) If the projectile is 10 feet above the ground 5√2 seconds later, what
was the initial speed?
2. The radius of a right circular cone is increasing at a rate of 3
2in/sec while
its height is decreasing at a rate of 2
3in/sec. At what rate is the volume of
the cone changing when the radius is 9 inches and the height is 12 inches?
3. Find the directional derivative of f(x, y) = √xy at P(3,27) in the direction
of Q(15,36).
4. The dimensions, length, width, and height, of a closed rectangular box1are
measured to be approximately 40 cm, 30 cm, and 50 cm, respectively, with
the same error in each measurement. If you wish to know the surface area
of the box within a maximum error of 120 cm2, what is the maximum error
that you can tolerate in measuring the length, width, and height?
5. Let Dbe the closed triangular region with vertices (1,0), (5,0), and (1,4),
which includes the boundary. Find the absolute maximum and absolute
minimum values of f(x,y ) = 13 + xy −x−2yon the set D.
6. Find the surface area of the part of the paraboloid x=y2+z2that lies
inside the cylinder y2+z2= 25.
7. Evaluate
Z2
−2Z√4−y2
0Z√4−x2
−y2
0
y2px2+y2+z2dzdxdy.
8. (a) Show that the vector field F(x, y) = yi+ (x+ 4y)jis conservative and
find a function f(x, y) such that F(x, y) = ∇f(x, y). (b) Evaluate
ZC
F·dr
where Cis the upper semicircle that starts at (0,5) and ends at (9,5).
9. Use Green’s Theorem to find the work done by the force F(x, y)=2x(2x+
y)i+ 2xy2jin moving a particle from the origin along the x-axis to (1,0),
then along the (straight) line segment to (0,1), and then back to the origin
along the y-axis.
Date: August 25, 2006.
1Being closed it must have a top.
1