(i) The altitude of a research drone HHH, measured in decameters, is modeled by the function
H(t)=e−0.4tsec0.5t,0≤t<π H(t) = \text{e}^{-0.4t} \sec 0.5t, \quad 0 \le t < \pi H(t)=e−0.4tsec0.5t,0≤t<πwhere ttt is the time in minutes after deployment.
Find H′(t)H'(t)H′(t).
Hence determine the time ttt at which the altitude of the drone is stationary.
A separate flight path is defined by the implicit relationship
x=ln(2siny),0<y<π2 x = \ln(2\sin y), \quad 0 < y < \frac{\pi}{2} x=ln(2siny),0<y<2πShow that
dydx=exf(x) \frac{\text{d}y}{\text{d}x} = \frac{\text{e}^x}{\text{f}(x)} dxdy=f(x)exwhere f(x)\text{f}(x)f(x) is a function of ex\text{e}^xex that should be found.
