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% ##############################################################
\def\monnumero{ 1 }
\def\madate{Novembre 2025}
% ########################### MES PARAMETRES ##############################################################

% en-tete
\lhead{\footnotesize{\textsf{Lycée Schuman Perret\\ \madate}}} %gauche
\chead{\textsf{\textcolor{red}{\textbf{\textsc{Série d'exercices No\monnumero}}}}} %centre
\rhead{\footnotesize{\textsf{CIRA\textsuperscript{1}}}}%droit
%pied de page
\rfoot{\scriptsize{\textsf{Page~\thepage{}~sur~\pageref{LastPage}}}}
\lfoot{\scriptsize{\textsf{\textsc{Stéphane Le Méteil }}}}

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\begin{document}%\everymath{\displaystyle}

\exo{Nombre dérivé}

\bigskip

\fbox{\begin{minipage}{0.99\linewidth}\onehalfspacing

{\bf Rappel } : $f'(a)$ s'appelle nombre dérivé d'une fonction $f$ en un point $A$ d'abscisse $a$ (ou nombre dérivé de $f$ en $a$).

Il correspond graphiquement au coefficient directeur de la tangente à la courbe représentative de $f$ au point d'abscisse $a$, c'est à dire à la droite qui approxime au mieux courbe représentative de la fonction $f$ autour de $a$.\end{minipage}}

\bigskip

{\small \bf Soit ci-dessous, la courbe représentative d'une fonction $f$ définie sur l'intervalle $[-4;4]$ dans le plan muni d'un repère orthonormal.}

\medskip

Les droites $T_1$ et $T_2$ sont les tangentes respectives à la courbe aux points d'abscisse 0 et -2.

\medskip

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\vskip 1cm 

\begin{enumerate}
	 \item Par lecture graphique, donner :
	\begin{enumerate}[a)]
		\item $f(0) = $\\
		\item $f(-2) = $
	\end{enumerate}
	
  \item Lire graphiquement : 

  \begin{enumerate}[a)]
    \item Le coefficient directeur de $T_1$ :\hskip 2cm On écrit alors $f'(0) = $\\
    \item Le coefficient directeur de $T_2$ :\hskip 2cm On écrit alors $f'(-2) = $
  \end{enumerate}
 
 
\end{enumerate}


\eject

\exo {\'Equation de tangente}

\bigskip

\fbox{\begin{minipage}{0.99\linewidth}\onehalfspacing

{\bf Rappel } : L'équation réduite de la tangente à la courbe représentative représentant $f$  au point $A$ d'abscisse $a$
               ( aussi appelée tangente en $A$ ou tangente en $a$ ) est : $y = f'(a)\times (x-a) + f(a)$ .\end{minipage}}

\bigskip


\small \bf On a représenté la fonction $f$ définie sur $[-2,5;3]$ par : $$f(x) = \dfrac13x^3-\frac12x^2-2x+\dfrac76$$

\medskip

Déterminer par le calcul les images des points d'abscisses $-2$, $-1$, $1$ et $2$. Vérifier les réponses sur le graphique.

\bigskip

\doublespacing

\begin{minipage}{0.40\linewidth}

\begin{enumerate}
  \item

  $f(-2) = $

  $f(-1) = $

  $f(1) = $

  $f(2) = $

\item Compléter le tableau suivant :

\begin{tabular}{|*{5}{>{\centering\arraybackslash}p{1cm}|} }
  \hline
  \tvi $a$ & $-2$ & $-1$ & $~1$ & $~2$ \\ \hline
  \tvi $f(a)$ &  &  &  &  \\ \hline
  \tvi $f'(a)$ &  &  &  &  \\ \hline
\end{tabular}

\end{enumerate}

\end{minipage}

\vskip 1cm

\begin{enumerate}\setcounter{enumi}{2}

\item Donner une équation des tangentes en $-2$, en $-1$, en $1$ et en $2$.

\end{enumerate}


\medskip


\vskip -11cm\hfill
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\end{minipage}

\eject

\exo {Courbe représentative.}

\begin{enumerate}

\item Tracer sur l'intervalle $[-4;4]$ une courbe qui remplisse les différents critères et sa/ses tangente(s).

\hskip -2cm $\begin{tabularx}{\linewidth}{cccc}
  a)~ $f(2)=-1$ &b)~ $f(-1) = 2$ & c) ~ $f(-2) = 1$~ ; ~ $f(3) = -1$ & d) ~   $f(-3) = 1$~ ; ~ $f(1) = -1$\\\\
   $f'(2) = 1$& $f'(-1)=1$&$f'(-2) = -2$~ ; ~ $f'(3) = 2$& $f'(-3) = 2$~ ; ~ $f'(1) = 0$\\\\


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\end{tabularx}$



\item La courbe ci-contre représente une fonction $f$.

\begin{minipage}{0.6\linewidth}

$(d_1)$, $(d_2)$ et $(d_3)$ sont les tangentes à cette courbe respectivement aux points
d'abscisses -4 , 1 et 3.

Par lecture graphique, déterminer :

\begin{enumerate}

\item $f(-4) = $\qquad $f(1) = $\qquad $f(3) = $

$f'(-4) = $\qquad $f'(1) = $\qquad $f'(3) = $

\item En déduire les équations réduites des droites :

\qquad $(d_1)$, $(d_2)$ et $(d_3)$

\end{enumerate}

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\vskip 1cm

\item Dessiner la représentation graphique d'une fonction qui vérifie les points suivants simultanément :


\begin{minipage}{0.4\linewidth}\doublespacing

 $f$ est croissante sur $[-4;1]$

 $f(-4) = 2$ et $f'(-4) = 2$

 $f(-1) = 3$ et $f'(-1) = 0$

 $f(4) = 4$ et $f'(4) = 1$

 $f$ admet un minimum en 2

 et $f(2) = -1$

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\end{enumerate}

\end{document}
