136.275, Assignment No. 3

November 25, 2002

The assignment is due Monday, December 2, 2002 in class. Late assignments receive a mark zero.

1. Find an equation for the surface consisting of all points P for which the distance

from P to the x-axis is twice the distance from p to the yz- plane. Identify and sketch

the surface by first drawing the traces in x=1 plane , and z=0 plane. [7]

2. The vector valued function r is given by r(t) = < t, t, sint > , -2 t 2 .

a) Sketch the graph of r (t) and show the direction of increasing t. [5]

b) Find the tangent vector and the tangent line to the graph of r(t) at t=. [5]

c) Guess what is the osculating plane ( TN plane) and what is the binormal vector B(t) for the curve given by r(t) (without calculating them the long way .) [3]

3. Let r(t)= <t, > . Find . [7]

4. a) Find the arc length parametrization of the line segment x= -5+2t, y=-t, z=3-2t

with -1 t 5. [5]

b) Use the parametrization equations obtained in a) to find the point on the line

segment that is 3 units away from the point (-7, 1 , 5) . How long is the line

segment (again, use the arc length parametrization from a) )? [4]

5. At what point(s) does y= have maximum curvature? (Take r(t)= <t, , 0>.) [7]

Total [ 43/42]