APPM 2350 EXAM 1 FALL 2007
INSTRUCTIONS: Computers, calculators, books, and crib sheets are not permitted. Write
your (1) name, (2) instructor’s name, and (3) lecture number (010, 020 or 030) on the front of your
bluebook. Work all problems. Start each problem on a new page. Show your work clearly and
box your final answer. A correct answer with incorrect or no supporting work may receive no
credit, while an incorrect answer with relevant work may receive partial credit.
1. (25 points) Consider the plane Pdefined by 2x+ 2y+z= 2 and another plane Mthat is
parallel to P. Let Abe a point on the plane Mwith the coordinates (2,2,2).
(a) Determine the standard equation of plane M.
(b) How far is it from plane Mto plane P?
(c) Determine the parameterization of a line from point Ato the closest point on plane P.
(d) Determine the coordinates of the point on plane Pclosest to point A.
2. (25 points) Consider the movement of a particle traveling on the elliptic path defined by
r(t) = cos(t)i+ (cos(t)−1) j+ sin(t)kfor 0 ≤t≤2π.
(a) Set up, but do not evaluate, the integral to determine the length of the path around the
ellipse.
(b) Calculate the acceleration, a(t), of a particle moving on the path described by r(t).
(c) The path r(t) is actually the intersection of a cylinder and a plane. Determine the
standard equation of the cylinder and the plane.
(d) Show that the acceleration, a(t), is always in the plane described in part (c).
(e) What is the torsion, τ, of the path r(t).
3. (25 points) Assume a particle moves along a path in three dimensional space defined by
r(t) = ti−ln(cos(t)) j+ 10 k, for 0 ≤t≤π.
(a) The plane determined by the unit normal and binormal vectors, Nand B, at a point
on the curve is called the local normal plane of the curve. At what point in space, P, is
the local normal plane of r(t) parallel to the plane x+y= 5?
(b) At the point Pfrom part (a), compute the particle’s velocity vand the unit tangent
vector T.
(c) At the point Pfrom part (a), determine the local unit normal vector N.
(d) At the point Pfrom part (a), determine the local unit binormal vector B.
(e) Again, at the point P, determine the curvature κ.
OVER
