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# City Emission Levels

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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