136.275, Assignment No. 3

November 24, 2003

The assignment is due Monday, December 1, 2003 in class. Late assignments receive a mark zero.

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

to the point Q(-1,0,0) and to the plane x=1 . Identify and sketch

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

2. a) For u(t) = r(t)r’(t)x r’’(t)] and r(t) three times differentiable, show that

u’(t) = r(t)r’(t)x r’’’(t)] . [5]

b) For r (t) = < t²,e^-2t+2t, cos2t> find r(t)r’(t)x r’’(t)] . Determine the domain

over which r (t) is a smooth curve. [6]

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

a) Sketch the graph of r(t).(The curve is the intersection of a sphere and a plane). [4]

b) Find the tangent vector and the tangent line to the graph of r(t) at t=. [3]

c) What is the osculating plane ( TN plane) and what is the binormal vector B(t) for

the curve given by r(t). (Do not calculate T and N .) [2]

4. a) Find the arc length of the curve r(t) = < 2cos(t²), 2sin(t²) ,t² > , 0 t 2. [6]

b) Find the arc-length parametrization of r(t) and find the coordinates of the point

that is in the middle of the curve. [5]

5. Find the value of x, x>0, where y=x³ has maximum curvature? (Take r(t)= <t,t³, 0>.)

[6]

Total [ 44/42]