\section*{Exercise 1 - Part A}
Here are two curves $\mathcal { C } _ { 1 }$ and $\mathcal { C } _ { 2 }$ which give for two people $P _ { 1 }$ and $P _ { 2 }$ of different body compositions the concentration $C$ of alcohol in the blood (blood alcohol level) as a function of time $t$ after ingestion of the same quantity of alcohol. The instant $t = 0$ corresponds to the moment when the two individuals ingest the alcohol.\\
$C$ is expressed in grams per litre and $t$ in hours.

\begin{enumerate}
  \item The function $C$ is defined on the interval $\left[ 0 ; + \infty \left[ \right. \right.$ and we denote $C ^ { \prime }$ its derivative function. At an instant $t$ positive or zero, the rate of appearance of alcohol in the blood is given by $C ^ { \prime } ( t )$.\\
At what instant is this rate maximal?\\
It is often said that a person of weak body composition experiences the effects of alcohol more quickly.
  \item On the previous graph, identify the curve corresponding to the person with the largest body composition. Justify the choice made.
  \item A person on an empty stomach ingests alcohol. It is admitted that the concentration $C$ of alcohol in their blood can be modelled by the function $f$ defined on $[ 0 ; + \infty [$ by
$$f ( t ) = A t \mathrm { e } ^ { - t }$$
where $A$ is a positive constant that depends on the body composition and the quantity of alcohol ingested.\\
a. We denote $f ^ { \prime }$ the derivative function of the function $f$. Determine $f ^ { \prime } ( 0 )$.\\
b. Is the following statement true?\\
``For equal quantities of alcohol ingested, the larger $A$ is, the more corpulent the person is.''
\end{enumerate}

\section*{Part B - A particular case}
Paul, a 19-year-old student of average body composition and a young driver, drinks two glasses of rum. The concentration $C$ of alcohol in his blood is modelled as a function of time $t$, expressed in hours, by the function $f$ defined on $[ 0 ; + \infty [$ by
$$f ( t ) = 2 t \mathrm { e } ^ { - t } .$$

\begin{enumerate}
  \item Study the variations of the function $f$ on the interval $[ 0 ; + \infty [$.
  \item At what instant is the concentration of alcohol in Paul's blood maximal? What is its value then? Round to $10 ^ { - 2 }$ near.
  \item Recall the limit of $\frac { \mathrm { e } ^ { t } } { t }$ as $t$ tends to $+ \infty$ and deduce from it that of $f ( t )$ at $+ \infty$. Interpret the result in the context of the exercise.
  \item Paul wants to know after how much time he can take his car. We recall that the legislation allows a maximum concentration of alcohol in the blood of $0,2 \mathrm {~g} . \mathrm { L } ^ { - 1 }$ for a young driver.\\
a. Prove that there exist two real numbers $t _ { 1 }$ and $t _ { 2 }$ such that $f \left( t _ { 1 } \right) = f \left( t _ { 2 } \right) = 0,2$.\\
b. What minimum duration must Paul wait before he can take the wheel in full compliance with the law?\\
Give the result rounded to the nearest minute.
  \item The minimum concentration of alcohol detectable in the blood is estimated at $5 \times 10 ^ { - 3 }$ g.L${}^{ - 1 }$.\\
a. Justify that there exists an instant $T$ from which the concentration of alcohol in the blood is no longer detectable.\\
b. The following algorithm is given where $f$ is the function defined by $f ( t ) = 2 t \mathrm { e } ^ { - t }$.
\begin{center}
\begin{tabular}{ | l l | }
\hline
Initialization: & $t$ takes the value 3,5 \\
 & $p$ takes the value 0,25 \\
 & $C$ takes the value 0,21 \\
Processing: & While $C > 5 \times 10 ^ { - 3 }$ do: \\
 & $\quad \mid \quad t$ takes the value $t + p$ \\
 & $\quad C$ takes the value $f ( t )$ \\
Output: & End While \\
Display $t$ & \\
\hline
\end{tabular}
\end{center}
Copy and complete the following table of values by executing this algorithm. Round the values to $10 ^ { - 2 }$ near.
\begin{center}
\begin{tabular}{ | c | c | c | c | }
\hline
 & Initialization & Step 1 & Step 2 \\
\hline
$p$ & 0,25 & & \\
\hline
$t$ & 3,5 & & \\
\hline
$C$ & 0,21 & & \\
\hline
\end{tabular}
\end{center}
What does the value displayed by this algorithm represent?
\end{enumerate}