Question 1 1 pts Calculate the curvature and torsion 7 at time t = 1 of a curve r(t) where v(1) = (0.4722, 2.1426, 0.6216), a(1) = (-1.6432, 2.5850, -0.3864), Then cos(k) + sin(T) is da (1) dt -0.5033 O 1.2716 -0.5244 O-1.8600 -0.5828 O 1.1397 O1.0570 -0.0484
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- Find the unit tangent vector to the curve defined by F(t) = (5 cos(t), 5 sin(t), 2 sin²(t)) 3T 4 at t= 5 7 (³) = (-2/ T r(t) = y(t) = Use the unit tangent vector to write the parametric equations of a tangent line to the curve at t = 3π 4 5 >(t) =| −1 2 5 √2 2√ 13 Submit Question 5 2v 13 5 2√13 26 t 5 2√13 t X X 26 X X D JUN 13 2 ali SA NConsider the parametric equations x(t) = 1 + 3t2, y(t) = 4 + 21³, -1Find parametric equations for the tangent line to the curve x=cos(t), y=sin(t), z=t at the point(cos(5pi/6),sin(5pi/6),5pi/6). Use u as the parameter.x(u) = y(u) = z(u) = (The line should be parametrized so that it passes through the given point at u=0).Consider the parametric equations y = e' +t te [1, 10] Briefly explain how you know that the curve is only traversed one time fromt = 1 to t = 10.3. Consider the parametric curve C given by r = te', y =t + cos t. (a) Find the equation of the tangent line to the curve C when t = 0. (b) Is the curve is concave upward or downward when t = 0?2. The parametric equations of a curve is x = 3 sint, y = 3 cost and z= = 4. (a) Sketch the curve r(t) for 0 ≤ t ≤ 2π. (b) Find the followings when t = 0: (i) The unit tangent vector, T. (ii) The principal unit normal, N.Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. x = V t2 + 15, y = In(t2 + 15), z = t; (4, In(16), 1) (x(0), vto), zte) = (|2. The parametric equations for the graph are x= V1-t, y= arccost %3D Determine the equation of the tangent line to the equation at t =1.ii. Find parametric equations for the Line through (7, 5) and (-5, 7) 7. Calculate dy/dx at the point indicated: f(0) = (7tan 0, cos O), 0=a/41. The parametric equations for the astroid below are x = sin (t), y = – cos* (t) on 0 < t < 2n. a) Find the starting point on the astroid and mark it on the graph as well as the direction of travel as t increases. dy b) Show all work to find the derivative at the point where t 2 -1 dxWhich is a parametric equation for the curve 25 = (x + 7)2 + (y + 5)?? A. c(t) = B. c(t) = (25 cos(t) – 7, 25 sin(t) – 5) C. c(t) = (5 cos(t) – 7,5 sin(t) – 5) D. c(t) = (25 cos(t) – 5, 25 sin(t) – 7) = (5 cos(t) – 5,5 sin(t) – 7) - -Consider the following curve. x = sin(6t), y = -cos(6t), z = 18t Using the given parametric equations, give the corresponding vector equation r(t). r(t) = (sin (6t))i (cos (6t))j + (18t)k ✔ Find r'(t) and Ir'(t). r'(t) = (6 cos (6t))i + (6 sin (6t))j + (18) k |r'(t) = 6√10 Find the equation of the normal plane of the given curve at the point (0, 1, 3). Now consider the osculating plane of the given curve at the point (0, 1, 3). Determine each of the following. T(t) = 1 T'(t) = |T'(t) = N(t) = XSEE MORE QUESTIONS