The sequence of values given by the iterative formula \[ x_{n+1}=\frac{2 x_{n}^{6}+12 x_{n}}{3 x_{n}^{5}+8} \] with initial value \( x_{1}=2 \), converges to \( \alpha \). (i) Use the formula to calculate \( \alpha \) correct to 4 decimal places. Give the result of each iteration to 6 decimal places. (ii) State an equation satisfied by \( \alpha \) and hence find the exact value of \( \alpha \).

Find the average rate of change of \( h(x)=-2 x^{2}-5 x \) from \( x=-4 \) to \( x=1 \). Simplify your answer as much as possible.

2) La ecuación del movimiento de una particula está dada por: \( s(t)=t^{3}-3 t \) Encontrar: a) La velocidad como función de \( t \) b) La aceleración después de 4 segundos. c) la aceleración cuando la velocidad es cero.

b Hence show that \( \int_{2}^{3} \ln y \mathrm{~d} y=\ln \left(\frac{27}{4 \mathrm{e}}\right) \)

A national fast food company has just analyzed the effect of the latest sales promotion. It has determined that the sales decay for its milkshakes is given by \( S=285,000 e^{-0.7 x} \), where \( S \) is the weekly sales and \( x \) is the number of weeks that have passes since the end of the sales promotion. a) Determine the sales 2 weeks after the end of the promotion: Round answer to 2 decimal places. Sales \( =\$ \) b) If no new campaign is started, how many weeks after the end of the promotion will sales drop below 30,000 ? Round answer to 1 decimal place. Num. weeks =

7. c) \( \frac{1}{1+x}-\frac{2}{x^{2}-1} \)