#f(t)=4sin(3t)#
1) #v(t)=(d)/dt (4sin(3t))#
#v(t)=12cos(3t)# (velocity at time t)
Max speed is when #f(t)=0#. (equilibrium)
#0=4sin(3t)#
#0=sin(3t)#
To solve for t, find where #sintheta=0# and divide by 3.
#t=0, pi/3, (2pi)/3#
Substitute any t into #v(t)#.
#v(0)=12cos(0)#
#v=12# (max speed)
2) #f(t)=0#
3) #0=12cos(3t)#
#0=cos(3t)#
#t=pi/6, pi/2# (velocity = 0)
4) #f(pi/6)=4sin(pi/2)#
#f(pi/6)=4#
#f(pi/2)=4sin((3pi)/2)#
#f(pi/2)=-4#
5) Direction changes when #v(t)=0# and the sign of #v(t)# changes.
#v(t)=0# at #t=pi/6, pi/2#
Check sign before and after #t=pi/6, pi/2#.
#v(0)=12cos(0)#
#v(0)=+12#
#v(pi/4)=12cos((3pi)/4)#
#v(pi/4)=-6sqrt(2)#
#v(pi)=12cos(3pi)#
#v(pi)=-12#
Velocity changes sign at #t=pi/6#, therefore the spring changes direction at #t=pi/6#.
