1. Consider the curve defined by c(t) = (t - 3sin(t/3), 3[1 - cos(t/3)]), where 0 S t

1. Consider the curve defined by c(t) = (t 3sin(t/3), 311 where O 3m (n) (3 points) Calculate the are-length of this path. (b) (2 points) Determine the tangent line to this path at, the point t = 3m. 2. Consider thc vector field F(.T:, y, z) = (x, 1/2, y). Let, +(t) = (01 be a flow line of F. (n) (1 points) Determine the three ordinary differential equations satisfied by 02(t) and 03(t). (b) (3 points) Solve the first two ordinary differential equations in part (a) for (t) ancl +2(t). Then solve the remaining ordinary differential equation for +3(t). (c) (1 points) Determine the flow line that passes through (1, 1, 1) at t = 0. 3. Let F' = (:r2 cosy, (n) (2 points) Show that F is not a gmadient vector field. (b) (3 points) Calculate the line integral LF ds where c(t) = t [0, II.