Let R3 EXERCISE 1.41. γ : 1 → be a unit-speed space curve with component functions denoted by γ(t) = (x(t),y(t),2(t). The plane curve (t)-(x(t), y(t)) represents the projection of γ onto the xy-plane. Assume that γ, is nowhere parallel to (0,0,1), so that γ is regular. Let K and K denote the curvature functions of γ and γ respectively. Let v, v denote the velocity functions of γ and γ respectively (1) Prove that R 2 RV2. In particular, at a time t E I for which v(t) (t). lies in the xy-plane, we have K(t) 2 (2) Suppose the trace of ry lies on the cylinder {(x, y, z) E R3 2 +y2 1). At a time t E 1 for which y(t) lies in the xy-plane (so that γ is tangent to the "waist" of the cylinder), conclude that K(t) 2 1. Is there any upper bound for K(t) under these conditions? Find an optimal lower bound for K(t) at a time t E 1 when v(t) makes the angle θ with the xy-plane.