\section*{Exercise 1 -- Part A}
Consider the sequence $\left( u _ { n } \right)$ defined for every natural integer $n$ by:
$$u _ { n } = \int _ { 0 } ^ { n } \mathrm { e } ^ { - x ^ { 2 } } \mathrm {~d} x$$
We will not attempt to calculate $u _ { n }$ as a function of $n$.

\begin{enumerate}
  \item a. Show that the sequence $(u_n)$ is increasing.\\
b. Prove that for every real number $x \geqslant 0$, we have: $- x ^ { 2 } \leqslant - 2 x + 1$, then:
$$\mathrm { e } ^ { - x ^ { 2 } } \leqslant \mathrm { e } ^ { - 2 x + 1 }$$
Deduce that for every natural integer $n$, we have: $u _ { n } < \frac { \mathrm { e } } { 2 }$.\\
c. Prove that the sequence $(u_n)$ is convergent. We will not attempt to calculate its limit.

  \item In this question, we propose to obtain an approximate value of $u _ { 2 }$.

In the orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath})$ below, we have drawn the curve $\mathscr{C}_f$ representing the function $f$ defined on the interval $[0;2]$ by $f(x) = \mathrm{e}^{-x^2}$, and the rectangle OABC where $\mathrm{A}(2;0)$, $\mathrm{B}(2;1)$ and $\mathrm{C}(0;1)$.\\
We have shaded the region $\mathscr{D}$ between the curve $\mathscr{C}_f$, the horizontal axis, the vertical axis and the line with equation $x = 2$.

Consider the random experiment consisting of choosing a point $M$ at random inside the rectangle OABC.\\
We admit that the probability $p$ that this point belongs to the region is: $p = \frac{\text{area of } \mathscr{D}}{\text{area of } \mathrm{OABC}}$.\\
a. Justify that $u_2 = 2p$.\\
b. Consider the following algorithm:

\begin{center}
\begin{tabular}{|l|l|}
\hline
L1 & Variables: $N, C$ integers; $X, Y, F$ real numbers \\
\hline
L2 & Input: Enter $N$ \\
\hline
L3 & Initialization: $C$ takes the value 0 \\
\hline
L4 & Processing: \\
\hline
L5 & For $k$ varying from 1 to $N$ \\
\hline
L6 & $X$ takes the value of a random number between 0 and 2 \\
\hline
L7 & $Y$ takes the value of a random number between 0 and 1 \\
\hline
L8 & If $Y \leqslant \mathrm{e}^{-X^2}$ then \\
\hline
L9 & $C$ takes the value $C + 1$ \\
\hline
L10 & End if \\
\hline
L11 & End for \\
\hline
L12 & Display $C$ \\
\hline
L13 & $F$ takes the value $C/N$ \\
\hline
L14 & Display $F$ \\
\hline
\end{tabular}
\end{center}

i. What does the condition on line $L8$ allow us to test regarding the position of point $M(X;Y)$?\\
ii. Interpret the value $F$ displayed by this algorithm.\\
iii. What can we conjecture about the value of $F$ when $N$ becomes very large?\\
c. By running this algorithm for $N = 10^6$, we obtain $C = 441138$.

We admit in this case that the value $F$ displayed by the algorithm is an approximate value of the probability $p$ to within $10^{-3}$.\\
Deduce an approximate value of $u_2$ to within $10^{-2}$.
\end{enumerate}

\section*{Part B}
The sign, modeled by the region $\mathscr{D}$ defined in Part A, is cut from a rectangular sheet of 2 meters by 1 meter. It is represented in an orthonormal coordinate system $(\mathrm{O}, \vec{\imath}, \vec{\jmath})$; the chosen unit is the meter.

For $x$ a real number belonging to the interval $[0;2]$, we denote:
\begin{itemize}
  \item $M$ the point on the curve $\mathscr{C}_f$ with coordinates $(x; \mathrm{e}^{-x^2})$,
  \item $N$ the point with coordinates $(x; 0)$,
  \item $P$ the point with coordinates $\left(0; \mathrm{e}^{-x^2}\right)$,
  \item $A(x)$ the area of rectangle $ONMP$.
\end{itemize}

\begin{enumerate}
  \item Justify that for every real number $x$ in the interval $[0;2]$, we have: $A(x) = x\mathrm{e}^{-x^2}$.
  \item Determine the position of point $M$ on the curve $\mathscr{C}_f$ for which the area of rectangle $ONMP$ is maximum.
  \item The rectangle $ONMP$ of maximum area obtained in question 2. must be painted blue, and the rest of the sign in white. Determine, in $\mathrm{m}^2$ and to within $10^{-2}$, the measure of the surface to be painted blue and that of the surface to be painted white.
\end{enumerate}