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\fancyhead[L]{\textsc{Mathématiques} Lycée Schuman-Perret}
\fancyhead[R]{2018-2019}
\fancyfoot[R]{\tiny Stéphane Le Méteil}

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\begin{document}

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{\huge \textbf{\black Compléments sur la fonction exponentielle.}}
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\ovalbox{\tvi Méthode d'Euler}

\bigskip

On part des données : $f(0) = 1$ et $f'(x) = f(x)$ pour tout $x\in\R$\bigskip

L'équation de la tangente en 0 est : $y = f'(0)(x-0)+f(0)$ or $f'(0) = f(0) = 1$ donc $y=x+1$\bigskip

\begin{minipage}{0.6\linewidth}\doublespacing
Lorsque $x$ est proche de 0, la différence graphique entre le point de la courbe et le point de la tangente en 0 est minime. On décide d'approximer le point d'abscisse 0,1 de la courbe avec celui l'abscisse 0,1 de la tangente. Son ordonnée est alors $y=0,1+1 = 1,1$. La courbe passe alors par le point de coordonnées $(0,1 ; 1,1)$\bigskip
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On recommence avec le point suivant, d'abscisse 0,2\bigskip

L'équation de la tangente en 0,1 est : $y = f'(0,1)(x-0,1)+f(0,1)$ or $f'(0,1) = f(0,1) = 1,1$ donc $y=1,1x+0.99$\bigskip

Lorsque $x$ est proche de 0,1, la différence graphique entre le point de la courbe et le point de la tangente en 0,1 est minime. On décide d'approximer le point d'abscisse 0,2 de la courbe avec celui l'abscisse 0,2 de la tangente. Son ordonnée est alors $y=1,1\times 0,1+0,99 = 1,21$. La courbe passe alors par le point de coordonnées $(0,2 ; 1,21)$\bigskip
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Et ainsi de suite.\bigskip

Si $(x_n ; y_n)$ sont les coordonnées d'un point de la courbe alors :

$\diamond$\quad $x_{n+1} = x_n+h$ est l'abscisse suivante

$\diamond$\quad l'équation de la tangente en $x_n$ est alors

\qquad $y=f'(x_n)\times(x-x_n)+f(x_n)\quad = \quad f(x_n)\times(x-x_n)+y_n \quad = \quad y_n\times (x-x_n)+y_n$

$\diamond$\quad l'ordonnée suivante est alors $y_{n+1} = y_n\times (x_{n+1}-x_n)+y_n\quad = \quad y_n\times h + y_n\quad = \quad y_n\times (1+h)$

\qquad La suite $(y_n)$ est géométrique de raison $(1+h)$ et de premier terme 1 donc $y_n = (1+h)^n$\bigskip


On fabrique ainsi la suite de points de coordonnées $(n\times h ;  (1+h)^n)$\bigskip

On fait alors croître $n$ chez les positifs puis chez les négatifs,

On obtient des points qui approximent ceux de la courbe représentant la fonction exponentielle :

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