\textbf{Exercise 1 -- Part B}

The proportion of individuals who possess a certain type of equipment in a population is modelled by the function $p$ defined on $[ 0 ; + \infty [$ by
$$p ( x ) = \frac { 1 } { 1 + \mathrm { e } ^ { - 0,2 x } } .$$
The real number $x$ represents the time elapsed, in years, since January 1st, 2000. The number $p ( x )$ models the proportion of equipped individuals after $x$ years. Thus, for this model, $p ( 0 )$ is the proportion of equipped individuals on January 1st, 2000 and $p ( 3.5 )$ is the proportion of equipped individuals in the middle of 2003.

\begin{enumerate}
  \item What is, for this model, the proportion of equipped individuals on January 1st, 2010? Give a value rounded to the nearest hundredth.
  \item \begin{enumerate}
    \item[a.] Determine the direction of variation of the function $p$ on $[ 0 ; + \infty [$.
    \item[b.] Calculate the limit of the function $p$ as $x \to + \infty$.
    \item[c.] Interpret this limit in the context of the exercise.
  \end{enumerate}
  \item It is considered that, when the proportion of equipped individuals exceeds $95\%$, the market is saturated. Determine, by explaining the approach, the year in which this occurs.
  \item The average proportion of equipped individuals between 2008 and 2010 is defined by
$$m = \frac { 1 } { 2 } \int _ { 8 } ^ { 10 } p ( x ) \mathrm { d } x$$
  \begin{enumerate}
    \item[a.] Verify that, for all real $x \geqslant 0$,
$$p ( x ) = \frac { \mathrm { e } ^ { 0,2 x } } { 1 + \mathrm { e } ^ { 0,2 x } }$$
    \item[b.] Deduce an antiderivative of the function $p$ on $[ 0 ; + \infty [$.
    \item[c.] Determine the exact value of $m$ and its approximation to the nearest hundredth.
  \end{enumerate}
\end{enumerate}