Q1) Solve the wave equation of the circular membrane of radius 4 assuming the axisymmetry subjected to the following conditions u(r,0) = u(4,t) =0, t>0 and u,(r,0)=1.

; (x.v) +(0,0) 0; (x.y) = (0,0) Given the hunclion aboue, fund the directional denvatue along

Sketch the curve whose vector equation is Solution r(t) = 6 cos(t) i + 6 sin(t) j + 3tk. The parametric equations for this curve are X = I y = 6 sin(t), z = Since x² + y² = + 36. sin²(t) = The point (x, y, z) lies directly above the point (x, y, 0), which moves counterclockwise around the circle x² + y2 = in the xy-plane. (The projection of the curve onto the xy-plane has vector equation r(t) = (6 cos(t), 6 sin(t), 0). See this example.) Since z = 3t, the curve spirals upward around the cylinder as t increases. The curve, shown in the figure below, is called a helix. ZA (6, 0, 0) (0, 6, 37) I the curve must lie on the circular cylinder x² + y² =

17. If y = tx, where t is the parameter, what are the parametric equations of the curve x' +y' = 8? 2 2t 2 C) x = 2t A) x = y = y = 3 t + 1-1 2 2t 2 , y= 1+1 2t B) x = , y= D) x =- t+1 '