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1.7.17 Product and quotient rules (A-level only)

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Question 39

(i) The altitude of a research drone HHH, measured in decameters, is modeled by the function

H(t)=e−0.4tsec⁡0.5t,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<π

where ttt is the time in minutes after deployment.

ia.

Find H′(t)H'(t)H′(t).

[4]
ib.

Hence determine the time ttt at which the altitude of the drone is stationary.

[3]
ii.

A separate flight path is defined by the implicit relationship

x=ln⁡(2sin⁡y),0<y<π2 x = \ln(2\sin y), \quad 0 < y < \frac{\pi}{2} x=ln(2siny),0<y<2π​

Show that

dydx=exf(x) \frac{\text{d}y}{\text{d}x} = \frac{\text{e}^x}{\text{f}(x)} dxdy​=f(x)ex​

where f(x)\text{f}(x)f(x) is a function of ex\text{e}^xex that should be found.

[5]

1.7.17 Product and quotient rules (A-level only) Questions

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