determine the intervals for which the curve r(t)=4sin^3(t)i+7cos^3(t)j is smooth on [0,2π
to me this says:
x = 4sin^3 t
y = 7cos^3 t
dx/dt= 12(sin^2 t)(cost) = 12cost(sint)^2
dy/dt= 21(cos^2 t)(-sint) or -21sint(cost)^2
both dx/dt and dy/dt must be zero
for dx/dt =0 , dy/dt=0
12cost(sint)^2 = 0 , -21sint(cost)^2 = 0
in other words, wherever the sine or cosines are 0
t = 0, π, 2π, π/2 , 3π/2