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\title{FORMULAIRE DE MATHEMATIQUES}
\date{}
\author{}

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\begin{document}

%\thispagestyle{fancy}

\lhead{FORMULAIRE DE MATHÉMATIQUES, Série STI (toutes spécialités)}
\chead{} \rhead{}
\lfoot{}
\cfoot{}
\rfoot{\begin{footnotesize}\end{footnotesize}}

\maketitle
\thispagestyle{fancy}

\small
\begin{multicols}{2}

\section{PROBABILIT\'ES}
Si $A$ et $B$ sont incompatibles : $P(A\cup B)= P(A) + P(B)$\\

Dans le cas général : $P(A\cup B)= P(A) + P(B) - P(A \cap B)$\\

$P\left(\overline{A}\right) = 1 - P(A)\quad ;\quad P(\Omega)= 1\quad ;\quad P(\emptyset) = 0$\\

Dans le cas équiprobable : 
$ P(A)= \dfrac {\mbox{Nombres d'éléments de }A}{\mbox{Nombres d'éléments de }\Omega}$\\
\bigskip

\textit{Variable aléatoire}\medskip\\
Fonction de répartition : $F(x)=P(X\leq x)$\medskip\\
Espérance mathématiques : $\displaystyle E(X)=\sum_{i=1}^{n} p_i x_i$\medskip\\
Variance : $\displaystyle V(X)=\sum_{i=1}^{n} p_i \left(x_i -E(X)\right)2=\sum_{i=1}^{n} p_i x_i2-\left(E(X)\right)2$\medskip\\
Écart type : $\displaystyle \sigma(X)=\sqrt{V(X)}$

\vspace*{2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{ALG\`EBRE}

\subsection{
\textit{Nombres complexes}
}

Forme algébrique : $z=x+iy$ \medskip\\ 
Forme trigonométrique : $z=\rho(\cos \theta+i\sin \theta)=\rho e^{i\theta}, \rho>0$\medskip\\
\begin{tabular}{cl}
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&
\begin{minipage}{4cm}
\vspace*{-3cm}
$\overrightarrow{OM}=x\vec{u}+y\vec{v}$\\
$\overline{OP}=x=\Re(z)=\rho \cos \theta$\\
$\overline{OQ}=y=\Im(z)=\rho \sin \theta$\\
$OM=\rho=|z|=\sqrt{x2+y2}$
\end{minipage}
\end{tabular}\\



\textit{Opérations algébriques }\medskip\\
$z+z'=(x+iy)+(x'+iy')=(x+x')+i(y+y')$\\
$zz'=(x+iy)(x'+iy')=(xx'-yy')+i(xy'+x'y)$\\

\ \\
\ \\
\textcolor{white}{.}\\

\textit{Conjugué }\smallskip\\
$z=x+iy=\rho e^{i\theta}\quad ;\quad \overline{z}=x-iy=\rho e^{-i\theta}$\bigskip\\
$x=\dfrac{1}{2}(z+\overline{z})\quad ;\quad y=\dfrac{1}{2i}(z-\overline{z})$\bigskip\\
$\overline{z+z'}=\overline{z}+\overline{z'}\quad ;\quad \overline{zz'}=\overline{z}\overline{z'}$\bigskip\\
$z\overline{z'}=x^2+y^2=|z|^2$\bigskip\\
$\dfrac{1}{z}=\dfrac{\overline{z}}{z\overline{z}}=\dfrac{x}{x^2+y^2}+i\dfrac{-y}{x^2+y^2}=\dfrac{1}{\rho}e^{-i\theta}$\bigskip\\
\textit{Module et argument d'un produit, d'un quotient}\bigskip\\
$zz'=\left(\rho e^{i\theta}\right)\left(\rho' e^{i\theta'}\right)=\rho \rho' e^{i(\theta+\theta')}$\bigskip\\
$|zz'|=|z||z'|$\bigskip\\
$\dfrac{z}{z'}=\dfrac{\rho e^{i\theta}}{\rho' e^{i\theta'}}=\dfrac{\rho}{\rho'} e^{i(\theta-\theta')}$\bigskip\\
$\left|\dfrac{z}{z'}\right|=\dfrac{|z|}{|z'|}$\bigskip\\
$z^n=\left(\rho e^{i\theta}\right)^n=\rho^n e^{in\theta}, \, n\in \Z$\bigskip\\
\textit{Inégalité triangulaire}\bigskip\\
$\left||z|-|z'|\right|\leq \left|z+z'\right| \leq |z|+|z'|$\bigskip\\

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Identités remarquables}

(valables sur $\C$ et donc sur $\R$)\\
$(a+b)^2 = a^2 + 2ab + b^2 \quad  ;\quad (a-b)^2 = a^2 - 2ab + b^2$ \medskip\\
$(a+b)^3 = a^3 + 3a^2b+3ab^2 + b^3$\medskip\\
$(a-b)^3 = a^3 - 3a^2b+3ab^2 - b^3$\medskip\\
$a^2 - b^2 = (a+b)(a-b)\quad  ;\quad a^2 + b^2 = (a+ib)(a-ib)$
\vspace{3cm}\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{multicols}

\newpage

\begin{multicols}{2}
\subsection{\textit{Trigonométrie}}
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&
\begin{minipage}{4cm}
\vspace*{-5cm}
$\overline{OP}= \cos \theta$\medskip\\
$\overline{OQ}=\sin \theta$\medskip\\
$\cos2 \theta +\sin2\theta=1$\medskip\\
$\tan \theta=\dfrac{\sin \theta}{\cos \theta}\, ,\, \theta \neq \dfrac{\pi}{2}2k\pi$
\end{minipage}
\end{tabular}\\

\textit{Valeurs remarquables} \\

\renewcommand{\arraystretch}{2}
\begin{array}{|c|c|c|c|c|c|c|}
\hline
 & ~~~0~~~ & ~~~\dfrac{\pi}{6}~~~ & ~~~\dfrac{\pi}{4}~~~ & ~~~\dfrac{\pi}{3}~~~ & ~~~\dfrac{\pi}{2}~~~ & ~~~\pi ~~~ \\\hline
~~\sin ~~ & 0 & \dfrac{1}{2} & \dfrac{\sqrt{2}}{2} &  \dfrac{\sqrt{3}}{2} & 1 & 0 \\\hline
\cos & 1 & \dfrac{\sqrt{3}}{2} & \dfrac{\sqrt{2}}{2} & \dfrac{1}{2} & 0 & -1 \\\hline
\tan & 0 & \dfrac{\sqrt{3}}{3} & 1 & \sqrt{3} & & 0 \\\hline
\end{array} \\

\ \\

\textit{Formules d'Euler}\medskip\\
$\cos \theta =\frac{1}{2}\left( e^{i \theta}+e^{-i\theta}\right) \, ,\, \sin \theta =\frac{1}{2i}\left( e^{i\theta}-e^{-i\theta}\right)$\medskip\\

\textit{Formules d'addition}\medskip\\
$e^{i(a+b)}=e^{ia}e^{ib}$\medskip\\
$\cos (a+b)=\cos a \cos b-\sin a \sin b$\medskip\\
$\cos (a-b)=\cos a \cos b+\sin a \sin b$\medskip\\
$\sin (a+b)=\sin a \cos b+\cos a \sin b$\medskip\\
$\sin (a-b)=\sin a \cos b-\cos a \sin b$\medskip\\
$\cos 2a =\cos^2a -\sin2 a=2\cos2 a-1 =1-2\sin^2a$\medskip\\
$\sin 2a =2\sin a \cos a$\medskip\\
$\cos^2a =\dfrac{1}{2}(1+\cos2a)\quad  ;\quad \sin^2a= \dfrac{1}{2}(1-\cos2a)$\medskip\\

\textit{Formules de Moivre}\medskip\\
Pour tout entier naturel non nul $n$, $\left(e^{i \theta}\right)^n=e^{in\theta}$\medskip\\
soit encore $(\cos \theta +i\sin \theta)^n=\cos n\theta +i\sin n\theta$\medskip\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{\textit{Équations du second degré}}

Soient $a$, $b$, $c$ des nombres réels, $a\neq 0$ et $\Delta=b^2-4ac$.\medskip\\
l'équation $az^2+bz+c=0$ admet :
\begin{itemize}
	\item[-] si $\Delta >0$, deux solutions réelles :\\
	$$z_1=\frac{-b+\sqrt{\Delta}}{2a}\; \mathrm{et} \; z_2=\frac{-b-\sqrt{\Delta}}{2a}$$
  
  \item[-] si $\Delta =0$, une solution réelle double :\\
	$$z_1=z_2=\frac{-b}{2a}$$
	
	\item[-] si $\Delta <0$, deux solutions complexes conjuguées :\\
	$$z_1=\frac{-b+i\sqrt{-\Delta}}{2a}\; \mathrm{et} \;  z_2=\frac{-b-i\sqrt{-\Delta}}{2a}$$
\end{itemize}

Dans tous les cas : $az^2+bz+c=a\left(z-z_1\right)\left(z-z_2\right).$\medskip\\
$z_1+z_2=-\dfrac{b}{a}$ \quad , \quad $z_1z_2=\dfrac{c}{a}$\medskip\\
\ \\
\ \\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Suites arithmétiques, suites géométriques}
\textit{Suites arithmétiques}\\

Premier terme $u_0 \quad;\quad u_{n+1} = u_n + a \quad ;\quad u_n = u_0 + na$\\

$1 + 2 + \cdots + n = \dfrac {n(n+1)}{2}$\medskip\\

\textit{Suites géométriques}\\

Premier terme $u_0\quad ;\quad u_{n+1} = b u_n \quad ;\quad u_n = u_0 b^n$\\

$\mbox{Si }b \not= 1,\quad S_n = 1 + b + b2 +\cdots + b^n = \dfrac {1-b^{n+1}}{1-b}$\medskip\\
$\mbox{Si }b = 1,\quad S_n = n+1 $
\vspace{5cm}\\
\end{multicols}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage

\section{ANALYSE}

\subsection{Propriétés algébriques des fonctions usuelles}

\begin{enumerate}
	\item \textbf{Fonctions logarithme et exponentielle}
	\begin{multicols}{3}
	$\ln 1=0$\medskip \\
	$\ln e=1$ \medskip \\
	$\ln(ab)= \ln a + \ln b$\medskip \\
	$\ln\left(\dfrac{a}{b}\right)=\ln a-\ln b$\\\vspace{1cm}
	
  
  Si $x\in \left]-\infty~;~+\infty\right[$ et $y\in\left]\,0~;~+\infty\right[$,\\
  $y=\exp x=e^x$ équivaut à $x=\ln y$\medskip \\
  $e^0=1$\medskip \\
  $e^{a+b}=e^ae^b$\medskip \\
  $e^{a-b}=\dfrac{e^a}{e^b}$
  
  
  $a^x=e^{x\ln a}\quad (a>0)$\medskip \\
  $\left(e^a\right)^b=e^{ab}$\medskip \\
  $\ln a^x=x\ln a$\medskip \\
  \vspace{0.5cm}
	\end{multicols}
	
	\item  \textbf{Fonctions puissances}
	\begin{multicols}{3}
	$x^\alpha=e^{\alpha\ln x}\quad (x>0)$\medskip \\
	$x^0=1$\medskip \\
	  \vspace{0.5cm}

  $x^{\alpha+\beta}=x^\alpha x^\beta$\medskip \medskip \\
  $x^{\alpha-\beta}=\dfrac{x^\alpha}{x^\beta}$\medskip \\
  \vspace{0.5cm}
  
  $\left(x^\alpha\right)^\beta=x^{\alpha \beta}$\medskip \\
  Si $n\in \N^*$, $x\in[\,0;+\infty[$ et $y\in[\,0;+\infty[$\medskip \\
  $y=\sqrt[n]{x}$ \;équivaut à $x=y^n$\medskip \\
	\end{multicols}
\end{enumerate}
 
\subsection{\textit{Limites usuelles des fonctions}}
	\begin{enumerate}
	\item \textbf{Fonctions}
	\begin{multicols}{2}
	\textit{Comportement à l'infini}\medskip \\
	$\displaystyle \lim_{x \rightarrow +\infty } \ln x = +\infty$ \medskip \\
	$\displaystyle \lim_{x \rightarrow +\infty } e^x =+\infty$    \medskip \\
	$\displaystyle \lim_{x \rightarrow -\infty } e^x =0$    \medskip \\
	Si $\alpha >0$, $\displaystyle \lim_{x \rightarrow +\infty } x^{\alpha} = +\infty$ et si $\alpha <0$, $\displaystyle \lim_{x \rightarrow +\infty } x^{\alpha}=0$\medskip \\
	 
	 \textit{Croissances comparées à l'infini}\medskip \\
	 $\displaystyle \lim_{x \rightarrow +\infty } \frac{e^x}{x} = +\infty$\medskip \\
	 $\displaystyle \lim_{x \rightarrow -\infty } xe^x = 0$\medskip \\
	 $\displaystyle \lim_{x \rightarrow +\infty } \frac{\ln x}{x} = 0$\medskip \\
	 Si $\alpha>0$, $\displaystyle \lim_{x \rightarrow +\infty } \frac{e^x}{x^\alpha} = +\infty$\medskip \\
	 Si $\alpha>0$, $\displaystyle \lim_{x \rightarrow +\infty } x^\alpha e^{-x} = 0$\medskip \\
	 Si $\alpha>0$, $\displaystyle \lim_{x \rightarrow +\infty } \frac{\ln x}{x^\alpha} = 0$\medskip \\
	 
	 
		\textit{Comportement à l'origine}\medskip \\
	$\displaystyle \displaystyle \lim_{x \rightarrow 0 } \ln x = -\infty$ \medskip \\
	Si $\alpha >0$, $\displaystyle \displaystyle \lim_{x \rightarrow 0 } x^{\alpha} =0$ et si $\alpha <0$, $\displaystyle \displaystyle \lim_{x \rightarrow 0} x^{\alpha}=+\infty$\medskip \\
	
	\vspace{0.2cm}
	\textit{Comportement à l'origine de} $\ln (1+x)$, $e^x$, $\sin x$\medskip \\
 	$\displaystyle \lim_{h \rightarrow 0 } \frac{\ln(1+h)}{h} = 1$\medskip \\
 	$\displaystyle \lim_{h \rightarrow 0 } \frac{e^h -1}{h} = 1$\medskip \\
 	$\displaystyle \lim_{h \rightarrow 0 } \frac{\sin h}{h} = 1$ 
 \vspace{3cm}\\
	\end{multicols}
	
	\item \textbf{Suites (S\'ERIES STI, spécialités génie électronique et génie électrotechnique)}
	
	Si $k>1$, $\displaystyle \lim_{n \rightarrow +\infty } k^n=+\infty$ \quad ; \quad Si $0<k<1$, $\displaystyle \lim_{n \rightarrow +\infty } k^n=0$ 
\newpage

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{\textit{Dérivées et primitives \begin{scriptsize}
(Les formules ci-dessous peuvent servir à la fois pour calculer des dérivées et des primitives)
\end{scriptsize}}}

 	\begin{multicols}{2}
%	\begin{minipage}[t]{8cm}
\textbf{1. Dérivées et primitives des fonctions usuelles}


\renewcommand{\arraystretch}{1.8}
\begin{array}{|c|c|c|}
\hline
~~~~~~~~f(x)~~~~~~~~ & ~~~~~~~~f'(x)~~~~~~~~ & \mbox{\textit{Intervalle de validité}} \\ \hline
k & 0 & ]-\infty,+\infty[ \\ \hline
x & 1 & ]-\infty,+\infty[ \\ \hline
x^n,\, n\in \N^* & n x^{n-1} & ]-\infty,+\infty[ \\ \hline
\displaystyle \frac 1x  & \displaystyle -\frac {1}{x2} & 
  ]-\infty,0[ \mbox{ ou }]0,+\infty[ \\[0.3cm] \hline
\displaystyle \frac {1}{x^n},\, n\in \N^* & \displaystyle -\frac {n}{x^{n+1}} & 
  ]-\infty,0[\mbox{ ou }]0,+\infty[ \\[0.3cm] \hline
\sqrt{x} & \displaystyle \frac {1}{2\sqrt x} & ]0,+\infty[ \\[0.3cm] \hline
x^{\alpha}\, ,\, \alpha \in \R& \alpha x^{\alpha -1}& ]0,+\infty[ \\ \hline
\ln x & \displaystyle \frac 1x & ]0,+\infty[ \\[1em] \hline
e^x & e^x & ]-\infty,+\infty[ \\ \hline
\cos x & -\sin x & ]-\infty,+\infty[ \\ \hline
\sin x & \cos x & ]-\infty,+\infty[ \\ \hline
\end{array}

\hspace*{1cm}\textbf{2. Opérations sur les dérivées} \medskip \\
\hspace*{1cm}$(u+v)' = u' + v'$ \medskip \\
\hspace*{1cm}$ \left(ku\right)' = k\, u' $ \medskip \\
\hspace*{1cm}$(uv)' = u'v + uv' $ \medskip \\
\hspace*{1cm}$ \left(\dfrac {1}{u}\right)' = -\dfrac {u'}{u2}$ \bigskip \\
\hspace*{1cm}$\left(\dfrac uv\right)' = \dfrac {u'v - uv'}{v2}$ \bigskip \\
\hspace*{1cm}$(v \circ u)' = (v' \circ u) u' $ \bigskip \\
\hspace*{1cm}$ \left(e^u\right)' = e^u u'  $ \bigskip \\
\hspace*{1cm}$ (\ln u)' = \dfrac {u'}{u},\quad u\mbox{ à valeurs strictement positives}$ \bigskip \\
\hspace*{1cm}$ \left(u^{\alpha}\right)' = n u^{\alpha-1}u'$
	\end{multicols}

\subsection{\textit{Calcul intégral}}
Si $F$ est une primitive de $f$, alors $\displaystyle \int_a^b f(t)\, \mathrm{d}t=F(b)-F(a)$
 	\begin{multicols}{2}
  \textit{Formule de Chasles}\medskip \\
  $\displaystyle \int_a^c f(t)\, \mathrm{d}t=\int_a^b f(t)\, \mathrm{d}t+\int_b^c f(t)\, \mathrm{d}t$  \medskip \\
  $\displaystyle \int_b^a f(t)\, \mathrm{d}t=-\int_a^b f(t)\, \mathrm{d}t$ \medskip \\
  
  \textit{Linéarité} \medskip \\
  $\displaystyle \int_a^b \left(\alpha f(t)+\beta g(t)\right)\, \mathrm{d}t=\alpha \int_a^b f(t)\, \mathrm{d}t +\beta  \int_a^b g(t)\, \mathrm{d}t$  \vspace{1.5cm} \\
  
  \textit{Positivité} \medskip \\
  Si $a\leq b$ et $f\geq 0$, alors $\displaystyle \int_a^b f(t)\, \mathrm{d}t \geq 0$  \medskip \\
  
  \textit{Intégration d'une inégalité} \medskip \\
  Si $a\leq b$ et $f\leq g$, alors $\displaystyle \int_a^b f(t)\, \mathrm{d}t \leq \int_a^b g(t)\, \mathrm{d}t$  \medskip \\
  Si $a\leq b$ et $m\leq f\leq M$, alors $\displaystyle m(b-a)\leq \int_a^b f(t)\, \mathrm{d}t \leq M(b-a)$  \medskip \\

  \textit{Valeur moyenne }de $f$ sur $[a,b]$ : $\displaystyle \dfrac{1}{b-a}\int_a^b f(t)\, \mathrm{d}t $ 
		\end{multicols}

\subsection{\textit{Équations différentielles}}

\renewcommand{\arraystretch}{1.5}
	\begin{tabular}{|l|l|}
	\hline  
	Équations ~~~~~~~~~~~~~~~~~~~~~~~~ & Solutions sur intervalle $]-\infty,+\infty[$ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \\ 
	\hline
	$y'-ay=0$& $f(x)=ke^{ax}$ \\ 
	\hline 
	$y''+\omega^2y=0$ &	$f(x)=A\cos \omega x +B\sin \omega x$ \\ 
	\hline
	\end{tabular} 


\end{document}