MATH 275 – Section 001 – Quiz 4
You may work with other class members on this quiz, but you may not receive
assistance from people not in MATH 275 (Section 001). You must show all of
your work to receive full credit. Do all your work on other sheets of paper and
be sure to staple all the pieces of paper together or YOU WILL GET A ‘ZERO’
ON THE QUIZ. Do not use decimal approximations unless asked to do so. Your
work on this quiz must be handed in by Monday, 23 September 2002 at 9:40
a.m. GOOD LUCK!
1) The point −√3,−1,2is given in rectangular coordinates. Convert this
point to cylindrical and sperical corrdinates.
2) The point 2,−
π
6,−3is given in cylindrical coordinates. Convert this point
to rectangular and spherical coordinates.
3) The point 4√3,5π
4,2π
3is given in spherical coordinates. Convert this
point to rectangular and cylindrical coordinates.
4) Prove:
d
dt [x(t)·y(t)] = x(t)·y0(t) + y(t)·x0(t)
where x(t) and y(t) are vectors of three components each.
5) Let r(t) =
4t
cos 3t
sin 3t
. Find the length of the curve traced out by rfrom
t= 0 to t=π
2.
6) Let r(t) =
1
t
t2
. Find the curvature when t= 3.
7) Let r(t) =
sin 8t
6t
cos 8t
.
a) Find the vectors T,N, and Bwhen t= 0.
b) Find equations for the normal plane and osculating plane for r(t) at
t= 0.