Name:
APPM 2350 Exam #1 Summer 2009
Be sure to include your name and a grading table on the front of your blue book. Also,
include your name on this exam and submit with your blue book. You must work all
of the problems on this exam. Show ALL of your work and BOX IN YOUR FINAL
ANSWERS. A correct answer with no relevant work may receive no credit, a wrong answer
with no work will receive no credit, and an incorrect answer accompanied by some correct
work may receive partial credit. Text books, class notes, crib sheets, cell phones, calculators,
or electronic devices of any kind are NOT permitted. Please clearly indicate the start of
each new problem. Good luck!
1. (20 points) Work the following problems. Not all of them are related to one another.
(a) Determine if the line x= 1 + 3t, y =−2t, z =−1 + tis parallel to the plane
x+ 2y+z=−4.
(b) Find the distance between the line and the plane defined in part (a).
(c) For a given vector v=v1i+v2j+v3k, find all vectors wsuch that v×w=w.
(d) Show that if θis the angle between vand w, then
tan(θ) = |v×w|
v·wwhere 0 ≤θ < π
2.
2. (25 points) A Romulan Warbird just uncloaked and fired directly at the Enterprise.
You and Spock are thrown from your terminals in the explosion. When you get back
to your post, you see that the Enterprise’s primary computer has been badly damaged
and all memory has been lost. You know that you were moving at a constant speed
along a curve in space, however the specific trajectory has been lost. The backup
computer quickly calculates that at the current time r= 2i+ 0j+ 4k,v= 0i+ 4j+ 3k,
a= 4i+ 0j+ 0k. Captain Kirk is attempting to execute evasive maneuvers and needs
information. Using the above information, calculate the following values for him if you
can. If you cannot, clearly state why there is insufficient information.
(a) The unit tangent T.
(b) The unit normal N.
(c) The curvature κ.
(d) The torsion τ.
(e) The value of v·afive seconds before impact with the Romulan plasma torpedo.
3. (30 points)
(a) Complete this problem directly on the exam sheet. Each column in the
figure on page 2 shows a surface and its associated family of level curves. A darker
color corresponds to a smaller value for the level curve. Match each set of surfaces
and associated level curves with one of the four possible functions (below each
column) by circling the function. No justification is needed.
