graph both the ne screen. Is the .-2 anction k(t). pe of the curve. 0≤t≤8T sa curve y = f(x) tion y = x(x). b a 2t, sin 31). At how bear that the curva- m? rvature function. usion from part (a)? cos t, t) is shown k the curvature is e curvature function. rgest? ture of a plane para- SECTION 13.3 Arc Length and Curvature 47-48 Find the vectors T, N, and B at the given point. 47. r(t) = (1², 1³, 1), (1, 3, 1) A 48. r(t) (cos t, sin t, In cos t), (1, 0, 0) = 869 49-50 Find equations of the normal plane and osculating plane of the curve at the given point. 49. x = sin 2t, y = -cos 2t, z = 4t; (0, 1, 2π) 50. x = ln t, y = 2t, z = ²; (0, 2, 1) 51. Find equations of the osculating circles of the ellipse 9x² + 4y² = 36 at the points (2, 0) and (0, 3). Use a graph- ing calculator or computer to graph the ellipse and both osculating circles on the same screen. 52. Find equations of the osculating circles of the parabola y = x² at the points (0, 0) and (1,2). Graph both oscu- lating circles and the parabola on the same screen. 53. At what point on the curve x = t³, y = 3t, z = t is the normal plane parallel to the plane 6x + 6y - 8z = 1? CAS 54. Is there a point on the curve in Exercise 53 where the osculating plane is parallel to the plane x + y + z = 1? [Note: You will need a CAS for differentiating, for simplify- ing, and for computing a cross product.] 55. Find equations of the normal and osculating planes of the curve of intersection of the parabolic cylinders x = y² and z = x² at the point (1, 1, 1). shn everto 56. Show that the osculating plane at every point on the curve r(t) = (t + 2, 1-t, ²) is the same plane. What can you conclude about the curve? 57. Show that at every point on the curve r(t) = (e' cos t, e' sin t, e') the angle between the unit tangent vector and the z-axis is the same. Then show that the same result holds true for the unit normal and binormal vectors. 58. The rectifying plane of a curve at a point is the plane that T and B at that point. Find the recti-