Mathematical Methods · Unit 3 · Differentiation of exponential and logarithmic functions · Calculus of logarithmic functions
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A biologist is studying the acidity level of a lake affected by industrial runoff. The pH of the lake water can be modelled by the function \(P(t) = 7 - 2\ln(t + 1)\), where \(t\) is the time in weeks since monitoring began and \(0 \leq t \leq 20\). (a) Determine the rate of change of pH with respect to time when \(t = 4\) weeks. [2 marks] (b) Find the time \(t\) when the rate of change of pH is \(-0.1\) units per week. [2 marks]
A newly planted forest has a canopy area, \(A\) (square metres), that can be modelled by the function \(A(t) = 120\ln(3t + 1) + 50\), where \(t\) is the time (years) since planting. (a) Determine the rate of change of the canopy area after 8 years. (2 marks) (b) Find the time when the rate of change of the canopy area is exactly 12 m² per year. Express your answer in years, correct to two decimal places. (2 marks)
The depth, \( D \) (metres), of water in a harbour is modelled by the function \[ D = 8 + 3\ln(t + 1), \quad 0 \leq t \leq 12 \] where \( t \) is the time (hours) since midnight. The rate of change of water depth at 5:00 am is