A function is defined parametrically as x = 4 sin(3t), y = 2 cos(3t). Find and simplify d^2 y/dx^2 in terms of t and y.
We first need to find dy/dx and we use the fact that dy/dx = dy/dt * dt/dx. So we have dy/dt = -6sin(3t) and dx/dt = 12cos(3t). Substituing these in we have dy/dx = -6*sin(3t)1/(12cos(3t) which simplifies to -(1/2)*tan(3t).
We now have dy/dx = -tan(3t)/2.
To find the second derivative we again have to derive w.r.t. t.
d2y/dx2 = d/dt(dy/dx)dt/dx and we have already found dt/dx as 1/(12cos(3t)).
So we have d/dt(-tan(3t)/2) = -3sec2 (3t)/2 and thus
d2y/ dx2 = -3sec2 (3t)/2*(12*cos(3t)) which simplifies to - sec3 (3t) / 8 in terms of t.
In terms of y, we know y = 2 cos(3t) and therefore cos(3t) = y/2, thus cos3(3t) = y3/8 and sec3(3t) = 8/y3
Finally we have d2y/dx2=-1/y3.
BS
