Math 1920, Prelim I
September 30, 2010, 7:30 PM to 9:00 PM
You are NOT allowed calculators, the text or any other book or notes. SHOW ALL WORK!
Writing clearly and legibly will improve your chances of receiving the maximum credit that
your solution deserves. Please label the questions in you answer booklet clearly.
Write your name and section number on each booklet you use. You may leave when you
have finished, but if you have not handed in your exam booklet and left the room by 8:45pm,
please remain in your seat so as not to disturb others who are still working.
1. Consider the vectors v=i+2j+akand w=i+j+k.
(a) (10 pts) Find all values of the number a(if any) such that vis perpendicular to
w.
(b) (10 pts) Find all the values of the number a(if any) such that the area of the
parallelogram determined by vand wis equal to √6.
2. (10 pts) Find the plane through the origin perpendicular to the plane 2x+2y+z=1
and perpendicular to the vector v=(1,1,−4).
3. Consider the function g(x, y )=y2−x2.
(a) (9 pts) Sketch the level curves g(x, y)=c,forc=0,1,2.
(b) (4 pts) What is the domain Dof g?
(c) (4 pts) What is the boundary of D?
(d) (4 pts) Is Da closed set, a closed set, both or neither?
4. (9 pts) The wave equation, where a2is constant, is given by
∂2u
∂t2=a2∂2u
∂x2.
It describes the motion of a waveform; examples of such waves could include fluid,
sound, light. Suppose u(x, t) represents the displacement of a vibrating guitar string
at time tat a distance xfrom one end of the string. If u(x, t) = sin(x−at), show that
it satisfies the wave equation.
5. Calculate each of the following limits or show it does not exist.
(a) (10 pts) lim
(x,y)→(4,3)
x=y+1
√x−√y+1
x−y−1.
(b) (10 pts) lim
(x,y)→(0,0)
y
x2+y2.
6. Consider the force vector field given by F=xi+yj+zk.
(a) (10 pts) Calculate the work done on a particle by the force Fwhen the particle
moves along the conical helix r(θ)=(θcos θ)i+(θsin θ)j+θkfrom θ=0to
θ=2π.
(b) (10 pts) Calculate the work done on a particle by Fwhen the particle moves along
a straight line from the origin to the point with co-ordinates (2π, 0,2π). Does the
work done along these two paths depend on the path taken?