APPM 2350 EXAM 1 SPRING 2007
INSTRUCTIONS: Computers, calculators, books, and crib sheets are not permitted. Write your
(1) name, (2) instructor’s name, and (3) lecture number (010 or 020) 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) Pete the ant lives somewhere on the plane Pdefined by 2x+ 2y+z= 2. His very
good friend Mary the ant lives on a different plane, M, that is parallel to Pete’s plane. The
coordinates of Mary’s home are (2,2,2).
(a) Determine the standard equation of Mary’s home plane, M.
(b) How far is it from Mary’s plane to Pete’s plane?
(c) Determine the parameterization of a line from Mary’s home to the closest point on Pete’s
plane.
(d) It’s Valentine’s Day and Pete would like to be as close as possible to Mary’s home so
he can serenade her with traditional ant love songs. Assuming that Mary is actually at
home on Valentine’s Day to hear Pete, determine the coordinates of the point on his
plane where he should stand while singing.
2. (25 points) Consider a particle traveling on the elliptic path in space defined by
r(t) = cos(t)i+ sin(t)j+ (1 −cos(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) Consider the position vector of a particle moving along a path in three dimensional
space defined by r(t) = t2i+2
3t3j+tk, for 0 ≤t.
(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 8x+ 8y+ 4z= 10?
(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
