Question

Find dy/dx ,if x=2 cos t +cos 2t +1 and y=2sint+ sin2t

Answer 1

dx/dt=-2sint+2sin2t

dy/dt=2cost-2cos2t

dy/dx=(dy/dt)/(dx/dt)=(cost-cos2t)/(-sint+sin2t)

dy/dx=2sin3t/2.sint/2/2cos3t/2sint/2

=tan3t/2

d2y/dx2=d/dx(dy/dx)

=d/dt(dy/dx).dt/dx

=(3/2).sec23t/2.1/(-2sint+2sin2t)

=3/4.1/cos2(3t/2).1/(sin2t-sint)

=3/4.1/cos2(3t/2).1/2cos3t/2.sint/2

=(3/8).1/(cos33t/2.sint/2

put t= pi/2

d2y/dx2=3/8. 1/(cos3(3pi/4).sin(pi/4))

=-3/2