Mathematical Methods · Unit 3 · Differentiation of trigonometric functions and differentiation rules · Calculus of trigonometric functions
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A particle moves along a straight line such that its displacement (metres) from a fixed point at time \(t\) seconds is given by \(s(t) = 3\sin(2t) + 4\cos(t)\), where \(0 \leq t \leq \pi\). Determine the exact time when the particle first changes direction.
A coastal monitoring station tracks the height of water on a slipway due to tidal movement. The height, h metres, of the water above the slipway base at time t hours after midnight is modelled by h(t) = 3.2 + 1.8sin(πt/6), where 0 ≤ t ≤ 24. (a) Calculate the rate of change of water height at t = 2 hours. Give your answer in metres per hour, correct to two decimal places. (b) Calculate the total length of time during the 24-hour period when the rate of change of water height exceeds 0.4 metres per hour.
The height, \( h \) (metres), of the tide above mean sea level at a harbour is modelled by the function \[ h = 2.5 + 1.8 \sin\left(\frac{\pi t}{6}\right), \quad 0 \leq t \leq 24, \] where \( t \) is the time (hours) since midnight. The rate of change of the tide height at 9:00 am is