Please show all work for 5, 6, and 7, thanks!

5) r(t) = (sin 2t, cos 6t) is the position vector of a particle moving in the plane at time t. Find a) The velocity vector v Co/ = (2cox)6,-6sialbel b) The acceleration vector act/ (-4sin (26), -36 cos(65]) c) The speed at the given time t = " Speed : 12 X r(t ) = ( sin 26, cor60) 8 = 2dy alt)=- 45 in (26) i - Baccar( Gol/j speed : J ( 2 105 26 ) 2 + ( Grin ( boll ? V 4 cas?26 + 36 rich#1 54 ( case ) + 4 sin? (6(3/1 = 12 6) The velocity v(t) of a particle moving in the plane is given along with the position of the particle at time t=0. Find the position of the particle at time t=3 and v(t) = (et + t, et - t), att = 0 position is (1,1) Scottli + let-El;) rit ) = (itj ) 1. ( 3 ) = (1,x 6 t = 0 ( e " + 2 ) ; tle 0 - 2 ) ; + cite; = it; litly + citci= its ci = 0 6j= 0 7) The position of a particle at any time t 2 0 is given by x(t) = t2 - 3, y(t) = 2+3 a) Find the magnitude of the velocity vector at t = 4 :( 12 JZ b) Find the total distance traveled by the particle from t = 0 to t= 4 46.062 units c) Find ~ as a function of x dx = dy * (6/ = t 2 - 3 4 161 : 7 6' J( 26) 2 + (263) " - E 2- 3 / it 3t ? ) 12141 13 + ( 2 141 ' ) ? = 46. 062 1 + 1 = 201 1 267j V ( 32)+ 25% dy - 262 - 1212 solve x= t2-3 8 dx for t & substitute dy