APPM 2350 EXAM 1 FALL 2006
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. (20 points) Mark the following statements as True or False. Note that for a statement to be true, it
must be always true, whereas for a statement to be false, it only need be false for a single case. Here
A,Band Care nonzero vectors (thus A6=0, etc.), cis a scalar, and q(t) and r(t) are differentiable
vector-valued functions in three dimensional space. No supporting work need be shown for this
problem.
(a) (A+B)×(A−B) = −2(A×B)
(b) (A×B)·A=B·(B×A)
(c) c(A×B) = (cA)×(cB)
(d) d
dtq(t)·r(t)=dq(t)
dt ·r(t)−dr(t)
dt ·q(t)
(e) If d
dt|r(t)|= 0 for all t, then the points on the curve defined by r(t) all lie on the surface of a
sphere of fixed radius.
2. (20 points) Consider the plane Mgiven by x+y+z= 1 and the points A(2,−1,0), B(−2,3,0), and
C(0,0,−1).
(a) Which of the points A,B, and Care in the plane M?
(b) Find the distance from plane Mto each of the points not on M.
(c) Determine the standard equation of the plane AB C that contains all three points.
(d) Determine a parametric equation for the line of intersection between planes ABC and M.
3. (20 points) Josh, the applied math ski racer, has been blindfolded by his backwards goggles, once
again. Being used to skiing this way, he still breaks some ski race records. The path Josh takes
down the mountain is given by r(t) = cosh(t)i+3
5tj+20 −4
5tk. Josh starts at the top of
the mountain when t= 0 and reaches the bottom of the mountain where the kcomponent z(t) = 0.
You should find the following formulae useful.
cosh(t) = et+e−t
2sinh(t) = et−e−t
2sinh2(t) + 1 = cosh2(t)
(a) How long will it take Josh to reach the bottom of the mountain?
(b) Give Josh’s speed as a function of t, in its most simplified form. What is Josh’s maximum speed
as he skis from the top of the mountain to the bottom?
(c) As he travels down the mountain, is Josh’s acceleration ever in the same direction as his velocity?
Is his acceleration ever in the same direction as his position vector?
(d) Calculate the arc length of Josh’s path as he travels from the top of the mountain to the bottom.
(e) What is his average speed as he travels from the top of the mountain to the bottom?
OVER
