(1) Find the arc length parametrization for the helix o(t) := (cos(t), sin(t), 2t3/2) at t = 1. (2) Let o(t) be a regular vector function, and let T(t) be the unit tangent vector function of the oriented curve given by o(t). (a) State Dr. Chang's definition of the curvature K(t). (b) State the formula for x(t) in terms of o(t).(8) Let H: x+2y -3z = 1 be the plane in the space. (a) Find a vector function o(t) which generates a circle of radius 2 centered at (2, 1, 1), which is contained in H. (b) Find the parametrization of o(t) by arc length at t = 0. (9) Let o(t) be a regular vector function. Part (a) and (b) are reviews of Calculus I results. (a) Let s = llo'(u)Il du be the arc length function of t, which is one-to-one. Let G(u) be the antiderivative of llo'(u)II, so that G'(u) = llo'(u)Il. Use G(u) to prove that dt s(t) = 11o'(t)11. (b) Let H(t) := G(t) - G(a) where G is the function defined in Part (a), so s = H(t). Let t = t(s) = H-'(s) be the inverse function of H(t), so that H(t(s)) = s. Use the implicit differentiation or the chain rule to prove that d t(s) = ds llo'(t(s))I1 (c) State the definition of the curvature x(t) and the formula of k(t). (d) Use the above results to prove the curvature formula