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\lhead{\textbf{Spé Maths Term}} \chead{\textbf{CBPM}} \rhead{\textbf{2022-2023}}
\lfoot{\textbf{Function 1}}
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\begin{document}

\begin{tcolorbox}[title= \textbf{Exercise}, colframe=black, colback=white]
\textbf{Part A}

\medskip

Consider the differential equation $(E):y'-y=2\e^{-x}$.

\begin{enumerate}[font=\bfseries]
	\item Determine the real number $\lambda$ so that $y=\lambda \e^{-x}$ is a solution of $(E)$.
	\item 
	\begin{enumerate}[font=\bfseries]
		\item Solve the equation $(E_0):y'-y=0$.
		\item Deduce the general solution of $(E)$.
		\item Verify that the function $g$ defined over $\mathbb{R}$ by $g(x)=\e^x-\e^{-x}$ is a particular solution of $(E)$.
	\end{enumerate}
\end{enumerate}

\vspace{0.5cm}

\textbf{Part B}

\medskip

Consider the function $f$ defined over $\mathbb{R}$ by $f(x)=g(x)-2x$ and designate by $C_f$ its representative curve in an orthonormal system $\left( O; \overrightarrow{i} , \overrightarrow{j} \right)$. $f'$ is the derivative of $f$.

\begin{enumerate}[font=\bfseries]
	\item 
	\begin{enumerate}[font=\bfseries]
		\item Prove that $f$ is odd.
		\item Calculate $\lim\limits_{x \rightarrow +\infty} f(x)$ and $\lim\limits_{x \rightarrow -\infty} f(x)$.
		\item Determine the expression of $f'(x)$, study its sign then set the table of variation of $f$.
	\end{enumerate}
	
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			\clip (-3,-3) rectangle (3,5) ;
			\draw[thick,blue,samples=100,domain=-3:3] plot (\x,{exp(\x)-exp(-\x)-3*\x}) ;
		\end{tikzpicture}
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	\item The adjacent figure shows the representative curve $C_h$ of the function $h$ defined over $\mathbb{R}$ by $h(x)=g(x)-3x$.
	\begin{enumerate}[font=\bfseries]
		\item Write, in Python language, the dichotomy algorithm and give a bounding of $\alpha$ of amplitude $0.001$ knowing that $\alpha$ is between $1$ and $2$.
		\item Using the curve $C_h$, prove that the curve $C_f$ cuts the straight line $(d)$ of equation $y=x$ at three points whose abscissas are to be determined.
	\end{enumerate}
	\item Draw $(d)$ and $C_f$ in the system $\left( O; \overrightarrow{i} , \overrightarrow{j} \right)$.
	\item 
	\begin{enumerate}[font=\bfseries]
		\item Prove that $f$ has an inverse function $f^{-1}$ and determine its domain of definition.
		\item Determine the values of $x$ satisfying $1<f^{-1}(x)<2$.
		\item Draw the representative curve $C_{f^{-1}}$ of $f^{-1}$ in the system $\left( O; \overrightarrow{i} , \overrightarrow{j} \right)$.
	\end{enumerate}
	\item Calculate, in terms of $\alpha$, the area of the region bounded by $C_f$ and $C_{f^{-1}}$.
\end{enumerate}

\vspace{0.5cm}

\textbf{Part C}

\medskip

Consider the sequence $(U_n)$ defined for every $n \in \mathbb{N}$ by $U_0 \in ]0, \alpha[$ and $U_{n+1}=f(U_n)$.

\begin{enumerate}[font=\bfseries]
	\item Prove that for every $n \in \mathbb{N}$, $0<U_n<\alpha$.
	\item Noticing that for every $x \in ]0,\alpha[$, $f(x)<x$, prove that the sequence $(U_n)$ is strictly decreasing and deduce that it is convergent.
\end{enumerate}
\end{tcolorbox}

\end{document}