Problem 1. (a) Does the limit hm t>0 752 8t t exist? ls yes, find its value. (b) Find the limit 2 . hm - t>1 t 1 ' ln(t) Problem 2. Consider the curve C given by r(t) = (t + 1,tsin(t),tcos(t) 2) and the sphere S with equation (a: 1)2 + 312 + (z + 2)2 = 8. What is the intersection between C and S? Problem 3. Let r(t) = (1 154,1 + t4,68t). (a) Find an equation for the tangent line to this curve at t = 1. (b) Find an antiderivative R(t) of Mt) such that R(O) = (1,1, 2). Problem 4. True or false? Please motivate your answers. (a) The curve with parametric equations m = tcos(t), y = tsin(t), z = t lies on the surface z2 = $2 + 3/2. (b) The curve r(t) = (153, 2133, 3t3) is a line. (e) Let r(t) be a parametrized curve such that r'(t) exists and is contin uous. Then lrf = |r'(t)|. Problem 5. An airplane's position vector at time t is described by the vector function r1(t) 2 (3+ 2t, 4+t, 3t+6). A ying saucer's position vector at time t is described by the vector function r2(t) = (4t + 1. 3 2t, 5t + 3). The plane and the ying saucer started ying at time t = 5 and stop ying at t = 5. (a) Do the plane and the ying saucer collide? That is, are they ever in the same spot at the same time? (b) Do the paths of the plane and the ying saucer intersect? That is, are they ever in the same spot at possibly different times