Question 1 to 5

1. Consider the function r(t) = (et cos 2t, 6' sin 2t, 6') which parameterizes a curve C in R3. (a) Find r'(t) and r\" (t) (1)) Give a parametric equation for the tangent line to the curve C at t = 0. (c) Find the length of the curve C over 0 S t g 371'. d 2. Find the derivative assuming r(t) is differentiable: a (r(2t) x tr(t)). 1 1+t 3. Evaluate the integral / (7? + tj + 2(1 + 02/79) dt. 4. Find a vector function that describes the curve of intersection between the surface 1:2 + 3/2 = 1 and the plane y = z using parameter a: = t. Sketch the surface and the curve of intersection. 5. NOTE THIS QUESTION WILL N QT BE GRADED BUT YOU SHOULD DO IT AN Y WAYS. ' , Consider the function r(t) 2 (et cos 2t, 6' sin 225,815) which parameterizes a curve C in R3. (3.) Find a parametric equation for the curve C using arc length 3 as the parameter. That is, nd r(s). (b) Show that the tangent vector to r(s) is a unit tangent vector. \\