\section*{Exercise 2}

\section*{Part A}
We consider the function $f$ defined on the interval $[0; +\infty[$ by:

$$f(x) = \frac{1}{a + \mathrm{e}^{-bx}}$$

where $a$ and $b$ are two strictly positive real constants.\\
We admit that the function $f$ is differentiable on the interval $[0; +\infty[$.\\
The function $f$ has for graphical representation the curve $\mathscr{C}_f$.

We consider the points $\mathrm{A}(0; 0.5)$ and $\mathrm{B}(10; 1)$.\\
We admit that the line (AB) is tangent to the curve $\mathscr{C}_f$ at point A.

\begin{enumerate}
  \item By graphical reading, give an approximate value of $f(10)$.
  \item We admit that $\lim_{x \rightarrow +\infty} f(x) = 1$. Give a graphical interpretation of this result.
  \item Justify that $a = 1$.
  \item Determine the slope of the line (AB).
  \item a. Determine the expression of $f'(x)$ as a function of $x$ and the constant $b$.\\
b. Deduce the value of $b$.
\end{enumerate}

\section*{Part B}
We admit, in the rest of the exercise, that the function $f$ is defined on the interval $[0; +\infty[$ by:

$$f(t) = \frac{1}{1 + \mathrm{e}^{-0.2x}}$$

\begin{enumerate}
  \item Determine $\lim_{x \rightarrow +\infty} f(x)$.
  \item Study the variations of the function $f$ on the interval $[0; +\infty[$.
  \item Show that there exists a unique positive real number $\alpha$ such that $f(\alpha) = 0.97$.
  \item Using a calculator, give a bound for the real number $\alpha$ by two consecutive integers. Interpret this result in the context of the statement.
\end{enumerate}

\section*{Part C}
\begin{enumerate}
  \item Show that, for all $x$ belonging to the interval $[0; +\infty[$, $f(x) = \dfrac{\mathrm{e}^{0.2x}}{1 + \mathrm{e}^{0.2x}}$.
  \item Deduce an antiderivative of the function $f$ on the interval $[0; +\infty[$.
  \item Calculate the average value of the function $f$ on the interval $[0; 40]$, that is:
$$I = \frac{1}{40} \int_0^{40} \frac{1}{1 + \mathrm{e}^{-0.2x}} \,\mathrm{d}x$$
The exact value and an approximate value to the nearest thousandth will be given.
\end{enumerate}