Exercice 1: On considère la suite \( \left(u_{n}\right)_{n} \) déf par: \( \left\{\begin{array}{l}u_{0}=1 \\ u_{n+1}=\frac{u_{n}^{2}+3}{u_{n}+1}: \forall n \in I N\end{array}\right. \) 1-montrer que : \( \forall n \in I N: 0 \leq u_{n} \prec 3 \). 2-montrer que la suite \( \left(u_{n}\right)_{n} \) est croissante.

Sketch the graph of the exponential function \( g(x) = 3^{x-1} + 2 \), indicating its asymptote.

Find the rectangular coordinates of the point that has polar coordinates \( \left(4,-\frac{2 \pi}{3}\right) \) Rectangular coordinates: (1).D)

The average daily high temperature (in \( \left.{ }^{\circ} \mathrm{C}\right) \) in Karachi, Pakistan, on the \( t^{\text {th }} \) day of the year is approximately \( T(t)=29-5 \cos \left(\frac{2 \pi(t-12)}{365}\right) \). What are the highest and lowest average daily highs in Karachi? Give exact answers. Highest: \( \square{ }^{\circ} \mathrm{C} \) Lowest: \( \square{ }^{\circ} \mathrm{C} \)

On Your Own 9. A bacteria population \( y \) after \( x \) days can be represented by an exponential function whose graph passes through \( (0,100) \) and \( (1,200) \). a. Write a function that represents the population. b. Find the population after 6 days. Does this bacteria population grow faster than the bacteria population in Example 5 ? Explain.

Sketch the graph of the exponential function \( g(x) = 3^{x-1} + 2 \), indicating its asymptote.