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Source LaTeX icone Cours_Activite_Tangentes



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Type: Cours
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Description
Activité de mathématiques en 1ère STI2D - notion (graphique) de tangente
Niveau
Première STI2D
Mots clé
tangente, graphique, cercle, courbe, tangente à une courbe, cours de mathématiques, maths, première, 1ère, STI2D
Voir aussi:

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\usepackage{hyperref}
\hypersetup{
    pdfauthor={Yoann Morel},
    pdfsubject={Tangentes  une courbe},
    pdftitle={Activit mathmatique: Tangentes  une courbe},
    pdfkeywords={Mathmatiques, 1STI, 1STI2D, premire, STI, STI2D, 
      tangente, tangentes}
}
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\author{Y. Morel}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
%\thispagestyle{empty}

\vspace*{-0.6cm}


\hfill{\LARGE \bf \TITLE}
\hfill $1^{\mbox{\scriptsize{re}}}STI2D$
\vspace{-0.2cm}

\hspace{-0.8cm}
\bgmp[t]{6.5cm}
\bgex On considre le demi-cercle $\mathcal{C}$ de rayon $1$. 

Tracer les tangentes  $\mathcal{C}$ aux points d'abscisse $-0,5$, 
$0$ et $0,5$. 

\psset{unit=2.6cm}
\begin{pspicture}(-1.1,-1)(1.3,2)
  \psline{->}(-1.2,0)(1.2,0)
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  \rput(-0.1,-0.1){$0$}
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  \psline(-0.5,0.05)(-0.5,-0.05)
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\enex
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La courbe $\Cf$, reprsentative d'une fonction $f$, est donne 
ci-dessous. 
Construire les tangentes  cette courbe aux points d'abscisses 
$-5$; $-4$; $-1$; $0$; $2$, $4$ et $5$. 

\ct{
\psset{xunit=0.9cm,yunit=0.8cm}
\begin{pspicture}(-7,-6.5)(6.5,6.7)
  \psline[linewidth=1.4pt]{->}(-6.4,0)(6.4,0)
  \psline[linewidth=1.4pt]{->}(0,-6.4)(0,6.4)
  \multido{\i=-6+1}{13}{
    \psline[linestyle=dashed](-6.2,\i)(6.2,\i)
    \rput(-0.3,\i){$\i$}
    \psline[linestyle=dashed](\i,-6.2)(\i,6.2)
    \rput(\i,-0.3){$\i$}
  }
  \psplot[linewidth=1.2pt,plotpoints=500]{-6.4}{6.3}{
    x 180 mul 3.1415 div sin 
    x abs 1.1 exp mul 
    0.85 mul
    1 add
  }
  \rput(-6.6,1){$\Cf$}
\end{pspicture}
}
\enex
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\psline(-1,-0.55)(19,-0.55)

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\bgex 
Tracer dans le repre suivant l'allure d'une courbe passant par tous
les points $A$, $B$, $C$, $D$, $E$ et $F$. 

\vspace{0.7cm}


\vspace{0.6cm}
\psset{unit=0.7cm}
\begin{pspicture}(-6,-6.2)(6.5,7.5)

  \psline[linewidth=1.4pt]{->}(-6.4,0)(7.4,0)
  \psline[linewidth=1.4pt]{->}(0,-6.4)(0,7.4)

  \multido{\i=-6+1}{14}{
    \psline[linestyle=dashed](-6.2,\i)(7.2,\i)
    \rput(-0.3,\i){$\i$}
    \psline[linestyle=dashed](\i,-6.2)(\i,7.2)
    \rput(\i,-0.3){$\i$}
  }
  %\psplot[linewidth=1.2pt,plotpoints=500]{-6.4}{7.2}{
  %  x 180 mul 3.1415 div 3.1415 mul 4 div cos 
  %  x 7 add 1 exp mul 
  %  0.5 mul 
  %  1 add
  %}

  % Tangente en x=0
  %\psplot{-2}{2}{x 2 div 9 2 div add}
  \rput(0,4.5){\Large\bf$\tm$}\rput(0.3,4.8){$C$}
  % Tangente en x=2
  %\psplot{0.2}{4.2}{-9 3.1415 mul 8 div x 2 sub mul 1 add}
  \rput(2,1){\Large\bf$\tm$}\rput(2.3,1.3){$D$}
  % Tangente en x=4
  %\psplot{1}{6}{-0.5 x 4 sub mul -9 2 div add}
  \rput(4,-4.5){\Large\bf$\tm$}\rput(4.3,-4.8){$E$}
  % Tangente en x=-2
  %\psplot{-5}{0.5}{5 3.1415 mul 8 div x 2 add mul 1 add}
  \rput(-2,1){\Large\bf$\tm$}\rput(-2.3,1.3){$B$}
  % Tangente en x=-4
  %\psplot{-6.2}{-1}{-0.5 x 4 add mul -0.5 add}
  \rput(-4,-0.5){\Large\bf$\tm$}\rput(-4.4,-0.7){$A$}
  % Tangente en x=6
  %\psplot{4.2}{7.2}{13 3.1415 mul 8 div x 6 sub mul 1 add}
  \rput(6,1){\Large\bf$\tm$}\rput(6.3,0.6){$F$}
\end{pspicture}
\enex
\enmp\psline(0.45,-0.97)(0.45,12.68)\hspace{0.8cm}
\bgmp[b]{9cm}
\bgex 
Dans le repre ci-dessous, les droites 
$T_{-4}$, $T_{-2}$, $T_0$, $T_2$, $T_4$ et $T_6$ 
sont les tangentes  la courbe $\Cf$, reprsentative d'une fonction
$f$, aux points d'abscisses respectives 
$-4$; $-2$; $0$; $2$; $4$; $6$.

Tracer l'allure de $\Cf$. 

\vspt
\psset{unit=0.7cm}
\begin{pspicture}(-6.,-6.2)(6.5,7.5)

  \psline[linewidth=1.4pt]{->}(-6.4,0)(7.4,0)
  \psline[linewidth=1.4pt]{->}(0,-6.4)(0,7.4)

  \multido{\i=-6+1}{14}{
    \psline[linestyle=dashed](-6.2,\i)(7.2,\i)
    \rput(-0.3,\i){$\i$}
    \psline[linestyle=dashed](\i,-6.2)(\i,7.2)
    \rput(\i,-0.3){$\i$}
  }
  %\psplot[linewidth=1.2pt,plotpoints=500]{-6.4}{7.2}{
  %  x 180 mul 3.1415 div 3.1415 mul 4 div cos 
  %  x 7 add 1 exp mul 
  %  0.5 mul 
  %  1 add
  %}

  % Tangente en x=0
  \psplot{-2.5}{2.5}{x 2 div 9 2 div add}
  \rput(-2.5,3.6){$T_0$}
  % Tangente en x=2
  \psplot{0.2}{3.6}{-9 3.1415 mul 8 div x 2 sub mul 1 add}
  \rput(0.7,6.7){$T_2$}
  % Tangente en x=4
  \psplot{0.8}{6.8}{-0.5 x 4 sub mul -9 2 div add}
  \rput(6.5,-5.4){$T_{4}$}
  % Tangente en x=-2
  \psplot{-5}{0.3}{5 3.1415 mul 8 div x 2 add mul 1 add}
  \rput(-4.6,-3.3){$T_{\mbox{-}2}$}
  % Tangente en x=-4
  \psplot{-6.5}{-0.8}{-0.5 x 4 add mul -0.5 add}
  \rput(-5.5,0.6){$T_{\mbox{-}4}$}
  % Tangente en x=6
  \psplot{4.4}{7.2}{13 3.1415 mul 8 div x 6 sub mul 1 add}
  \rput(4.9,-6.7){$T_6$}
\end{pspicture}
\enex
\enmp

\end{document}

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