10. The absolute curvature of a smooth curve with parametrization ('!/J, I) at a point Xo = '!/J(to) is the number

. B(t)

I\;(xo) = hm O( ) , t--+to (. t when this limit exists, where B(t) is the angle between '!/J'(t) and '!/J'(to), and let) is the arc length of '!/J(I) from '!/J(t) to '!/J(to). (Thus I\; measures how rapidly B(t) changes with respect to arc length.)

(a) Givena,b ERn, b =f 0, prove that the absolute curvature of the line A = '!/J(I), where '!/J(t) :=a+tband I:= (-00,00), is zero at each pointxo on A.

(b) Prove that the absolute curvature of the circle ofradius r (namely, G = '!/J(I), where '!/J(t) = (rcost,rsint) and I = [0,2rr)) is l/r at each point Xo on G.