Name  ________________       Calculus Final Exam       Due Date: Beginning of Exam Thursday

Takehome Problem.  

Note that you must clearly and completely show all your work in order to get full credit.

1.  Show that the following curve is a unit speed curve and compute T, N, B.

alpha(t) = vector([1/3*(t+1)^(3/2), 1/3*(1-t)^(3/2), 1/2*2^(1/2)*t]), ` for `*abs(t) < 1





2.  It can be shown that since the curve in 1. is a unit speed curve, its curvature function kappa(t)and torsion function tau(t)are defined by the equations below. Use your results in 1. to find these two functions. (Note that these are just standard 1-D functions defined on the reals.)

`Rate of change of Tangent vector field:    `*`T '`(t) = kappa(t)*N(t)

`Rate of change of Binormal vector field:  `*`B '`(t) = -tau(t)*N(t)





3.  Give an example of a unit speed curve that has both nonzero constant curvature and nonzero constant torsion, and show that this is the case.