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City Emission Levels

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The emissions of a city from 2000 to 2012 are modelled by the equation


\(p\left( t \right) = \frac{1}{10}\ln\left( t + 1 \right) – \cos\frac{t}{2} + \frac{1}{10}t^{\frac{3}{2}} + 199.3\)

\[0 \leq t \leq 12\]


a) Show that the emissions reach a local maximum in the interval \(8.5 \leq t \leq 8.6\)

[5 marks]


The emissions reach a local minimum between 9 and 11 years after the measurements began.

b) Using the Newton-Raphson procedure once and taking \(t_{0} = 9.9\) as a first approximation, find a second approximation of when the emissions reach a local minimum.

[6 marks]

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