Determine the intervals for which the curve r(t)=4sin^3(t)i + 7cos^3(t)j is smooth on [0,2pai]

Answer 1

pi, not pai

dx/dt = 12sin^2(t)cos(t)
dy/dt = -21cos^2(t)sin(t)

dx/dt and dy/dt are both zero when sin(2t)=0, or at multiples of pi/2.

So, the open intervals are (0,pi/2),(pi/2,pi),(pi/3pi/2),(3pi/2,2pi).

The graph bears this out:

enter

parametric plot x=4sin^3(t), y=7cos^3(t)

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