6 questions

K Use the definitions to compute the unit binormal vector and torsion of the following curve. r(t) = (2 sint- 2t cost,2 cost + 2t sin t,5t) Compute the unit binormal vector. B = 0OO AW Score: 6.25%, 0.5 of 8 points Points: 0 of 1 The formula x(x) =- [1+ (f'(x)) 273/2 expresses the curvature of a twice-differentiable plane curve as a function of x. Use this formula to find the curvature function of the following curve. f(x) = In (3 cos x) for - 7 The curvature function is k(x) = (Simplify your answer.) Find the unit tangent vector T and the principal unit normal vector N for the following parameterized curve. Verify that |T| = [N| = 1 and T . N = 0. r(t) = (8 sin t,8 cost,6t) T= 0OO [vxal Use the alternative curvature formula k = to find the curvature of the following parameterized curve. 1v / 3 r(1) = (5e' cost,5e' sin t,5e') k=[ uit ingin of the following curve. Simplify the arc length integral as much as possible before finding an approx r(t) = (7 cost, 42 sint), for Osts 2x Write the arc length integral and simplify as much as possible. Choose the correct answer below. A. L=7 1+35 cos 2tdt 2 x O B. L= 49 sin 21+ 1764 cos 2tdt 2x O C. L= 245 cost dt OD. L= 245 (sint+ cost) at The length of the curve is approximately units. (Round to two decimal places as needed.) Find the length of the following two-dimensional curve. K r(t) = (19 cost,19 sint), for Osts n Set-up the integral that gives the arc length. L= Sat 0 (Type exact answers.)