1. Consider the curve defined by r (t) = ti+5j - 8tk. (a) Find a unit tangent vector for r (t) at the point where t = 3 (b) Find an equation of the line tangent to this curve at the point where t = 3. 2. Consider the curve r (t) = (3t + 7, sin 5at, 2t + 1) . Find an integral that gives the length of this curve between endpoints (13, 0,5) and (22, 0, 11) . Simplify the integral enough so a Calculus 2 student could try it, but do NOT actually evaluate the integral. 3. Define 9 (x, y) = 12 - 3,/y xvy - 4x (a) Find the domain of g and sketch it on the grid below, labeled appropriately. y . (b) Use algebra and a limit law to find: An object moves along trajectory described by M (t) = (x (t), 5t', sint) , where x (t) is some function 4. of t. In a few sentences, describe a calculus-based method to determine whether the object's path is on a sphere centered at the origin. Your answer should include an equation involving a (t). NOTE: You can't actually decide whether this path is on a sphere since we don't have a formula for x (t) . Just explain how you would figure it out if a formula for x (t) were given. 5. An object on Planet Y is initially located at point (3, 4) and shot vertically straight up at speeds, with constant acceleration vector a =. (a) Use calculus to derive the ball's velocity v (t) and position vector r (t) at time t, in terms of s. (b) The object passes through point (165, 526) at time T. Use your work in (a) to find time T and speed s