The angle of inclination of a plane curve with parametrization r(t) is defined as the angle θ(t) between the unit tangent vector T(t) and the x-axis (see figure).


(a) Show that ||T '(t)|| = |θ '(t)| and conclude that if r(s) is a parametrization by arc length, then the following is true. (Do this on paper. Your instructor may ask you to turn in this work.)

Hint: Observe that T(t) = < cos(θ(t)), sin(θ(t)) >.

(b) If a particle moves along the path y = x3 with unit speed, how fast is the tangent turning (i.e., how fast is the angle of inclination changing) when the particle passes through the point (1,1)?


i need to figure out b, I've tried a ton of stuff

suggestions?