\documentclass{article}
\usepackage{amssymb, amsmath, geometry, multicol, stmaryrd, cancel}
\usepackage{pgfplots, xcolor}


\pgfplotsset{compat=1.18}

\geometry{tmargin=.75in, bmargin=.75in, lmargin=.75in, rmargin = .75in}

\newcommand{\cotan}{\mathrm{cotan}}
\newcommand{\Arcsin}{\mathrm{Arcsin}}
\newcommand{\Arccos}{\mathrm{Arccos}}
\newcommand{\Arctan}{\mathrm{Arctan}}
\newcommand{\ch}{\mathrm{ch}}
\newcommand{\sh}{\mathrm{sh}}

\begin{document}
\setlength{\parindent}{0mm}

\title{Fonctions Usuelles}
\author{Aubin SIONVILLE}
\date{MPI Clemenceau - 2021-2023}
\maketitle

\centering

\section*{Fonctions puissances}

    \begin{tikzpicture}[scale=1.5]
        \begin{axis}[
            axis lines = middle,
            xlabel = {$x$},
            ylabel = {$y$},
            xmin=0,
            xmax=4,
            ymin=0,
            ymax=4,
            samples = 100,
            legend style={at={(1.2,0.5)},anchor=center,legend columns=1}
        ]
        \addplot[red,thick,domain=0:4] {exp(0.2*ln(x))};
        \addlegendentry{$a=0.2$}
        \addplot[orange,thick,domain=0:4] {exp(0.5*ln(x))};
        \addlegendentry{$a=0.5$}
        \addplot[green,thick,domain=0:4] {exp(1*ln(x))};
        \addlegendentry{$a=1$}
        \addplot[blue,thick,domain=0:4] {exp(2*ln(x))};
        \addlegendentry{$a=2$}
        \addplot[purple,thick,domain=0:4] {exp(4*ln(x))};
        \addlegendentry{$a=4$}
        \end{axis}
    \end{tikzpicture}

\[\boxed{\forall a \in \mathbb{R}, \forall x > 0, x^a \triangleq \mathrm{exp}(a \ln x)}\]

Hypothèse : $(a,b) \in (\mathbb{R}^*_+)^2$
    \begin{multicols}{2}
        \[\boxed{\frac{\ln^a(x)}{x^b} \longrightarrow 0}\]
        
        \[\boxed{\frac{\ln^a(x)}{\mathrm{e}^x} \longrightarrow 0}\]
    \end{multicols}

\newpage

\section*{Exponentielle et logarithme de base $a$}
    
    \[\boxed{\forall a > 0, \mathrm{exp}_a \triangleq \mathrm{exp}(x \ln a) = a^x}\]

    \begin{tikzpicture}[scale=1.5]
        \begin{axis}[
            axis lines = middle,
            xlabel = {$x$},
            ylabel = {$y$},
            xmin=-1,
            xmax=4,
            ymin=0,
            ymax=4,
            samples = 100,
            legend style={at={(1.2,0.5)},anchor=center,legend columns=1}
        ]
        \addplot[red,thick,domain=-1:4] {0.2^x};
        \addlegendentry{$a=0.2$}
        \addplot[orange,thick,domain=-1:4] {0.5^x};
        \addlegendentry{$a=0.5$}
        \addplot[green,thick,domain=-1:4] {1^x};
        \addlegendentry{$a=1$}
        \addplot[blue,thick,domain=-1:4] {2^x};
        \addlegendentry{$a=2$}
        \addplot[purple,thick,domain=-1:4] {4^x};
        \addlegendentry{$a=4$}
        \end{axis}
    \end{tikzpicture}

    Pour $a \neq 1$ : Réciproque : $\log_a$
    
    \vspace*{0.5cm}

    Hypothèse : $a \in \mathbb{R}^*_+\setminus\{1\}$
    \[\boxed{\forall x > 0, \log_a(x) \triangleq \frac{\ln(x)}{\ln(a)}}\]

    \begin{tikzpicture}[scale=1.5]
        \begin{axis}[
            axis lines = middle,
            xlabel = {$x$},
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            xmin=-1,
            xmax=4,
            ymin=-4,
            ymax=4,
            samples = 100,
            legend style={at={(1.2,0.5)},anchor=center,legend columns=1}
        ]
        \addplot[red,thick,domain=-1:4] {ln(x)/ln(0.2)};
        \addlegendentry{$a=0.2$}
        \addplot[orange,thick,domain=-1:4] {ln(x)/ln(0.5)};
        \addlegendentry{$a=0.5$}
        \addplot[blue,thick,domain=-1:4] {ln(x)/ln(2)};
        \addlegendentry{$a=2$}
        \addplot[purple,thick,domain=-1:4] {ln(x)/ln(4)};
        \addlegendentry{$a=4$}
        \end{axis}
    \end{tikzpicture}

\newpage

\section*{Fonctions trigonométriques}
\begin{tikzpicture}
    \begin{axis}[
        scale only axis = true,
        width = \textwidth,
        height = 0.3*\textheight,
        axis lines = middle,
        xlabel = {$x$},
        ylabel = {$y$},
        domain = -2*pi:2*pi,
        ymin=-1.5,
        ymax=1.5,
        samples = 100,
        xtick = {-2*pi,-1.5*pi,-pi,-pi/2,0,pi/2,pi,1.5*pi,2*pi},
        xticklabels = {$-2\pi$,$-3\pi/2$,$-\pi$,$-\pi/2$,$0$,$\pi/2$,$\pi$,$3\pi/2$,$2\pi$},
        ytick = {-1.5,-1,-0.5,0,0.5,1,1.5},
        legend style={at={(0.5,-0.1)},anchor=north,legend columns=-1}
    ]
    \addplot[red,thick] {sin(x*180/pi)};
    \addlegendentry{$\sin(x)$}
    \addplot[blue,thick] {cos(x*180/pi)};
    \addlegendentry{$\cos(x)$}
    \end{axis}
\end{tikzpicture}

\begin{multicols}{2}
    \[\boxed{\sin'=\cos}\]

    \[\boxed{\cos'=-\sin}\]
\end{multicols}

\vspace{0.1cm}

\begin{multicols}{2}
    Pour $\displaystyle x \not\equiv \frac{\pi}{2}[\pi] :$
        \[\boxed{\tan'(x) = 1+\tan^2(x) = \frac{1}{\cos^2(x)}}\]

    Pour $x \not\equiv 0[\pi] :$
        \[\boxed{\cotan'(x) = -1-\cotan^2(x) = \frac{-1}{\sin^2(x)}}\]
\end{multicols}

\begin{tikzpicture}
    \begin{axis}[
        scale only axis = true,
        width = \textwidth,
        height = 0.3*\textheight,
        axis lines = middle,
        xlabel = {$x$},
        ylabel = {$y$},
        domain = -2*pi:2*pi,
        ymin=-20,
        ymax=20,
        samples = 1000,
        xtick = {-2*pi,-3*pi/2,-pi,-pi/2,0,pi/2,pi,3*pi/2,2*pi},
        xticklabels = {$-2\pi$,$-\frac{3\pi}{2}$,$-\pi$,$-\frac{\pi}{2}$,$0$,$\frac{\pi}{2}$,$\pi$,$\frac{3\pi}{2}$,$2\pi$},
        legend style={at={(0.5,-0.2)},anchor=north,legend columns=-1}
        ]
        \addplot[black!30!green,thick] {tan(x*180/pi)};
        \addlegendentry{$\tan(x)$}
    \end{axis}
    \end{tikzpicture}

\newpage

\section*{Fonctions trigonométriques réciproques}

\subsection*{Arcsin}
\begin{multicols}{2}
    \begin{tikzpicture}
        \begin{axis}[domain=-1:1, samples=500, axis lines*=middle, xtick={-1,1}, ytick={-1.57,1.57}, yticklabels={$-\pi$/2,$\pi$/2}]
        \addplot[color = blue]  {asin(x)/180*pi};
            \end{axis}
    \end{tikzpicture}

    \[\boxed{\Arcsin \triangleq \text{ réciproque de } \begin{array}{ccccc}f & : & \Big[-\frac{\pi}{2}, \frac{\pi}{2}\Big] & \to & [-1,1] \\ & & x & \mapsto & \sin(x) \\\end{array}}\]

    \[\boxed{\begin{array}{c} \text{Continue sur } [-1,1]\\ \\ \text{Dérivable sur } ]-1,1[ \\ \\ \displaystyle\Arcsin'(x) = \frac{1}{\sqrt{1-x^2}}\end{array} }\]
\end{multicols}


\vspace{0.5cm}

\subsection*{Arccos}
\begin{multicols}{2}
    \begin{tikzpicture}
        \begin{axis}[domain=-1:1, samples=500, axis lines*=middle, xtick={-1,1}, ytick={0,3.14}, yticklabels={$0$,$\pi$}]
            \addplot[color = blue]  {acos(x)/180*pi};
        \end{axis}
    \end{tikzpicture}

    \[\boxed{\Arccos \triangleq \text{ réciproque de }\begin{array}{ccccc}f & : & [0,\pi] & \to & [-1,1] \\ & & x & \mapsto & \cos(x) \\\end{array}}\]
    
    \[\boxed{\begin{array}{c} \text{Continue sur } [-1,1]\\ \\ \text{Dérivable sur } ]-1,1[ \\ \\ \displaystyle\Arccos'(x) = \frac{-1}{\sqrt{1-x^2}}\end{array} }\]
\end{multicols}

\vspace{0.5cm}

\subsection*{Arctan}
\begin{multicols}{2}

    \begin{tikzpicture}
        \begin{axis}[
            domain=-10:10, samples=500, axis lines*=middle, 
            xtick={-10,-5,0,5,10}, ytick={-1.57,0,1.57}, yticklabels={$-\pi$/2,$0$,$\pi$/2}
        ]
        \addplot[color = blue]  {atan(x)/180*pi};
        \end{axis}
    \end{tikzpicture}
    
    \[\boxed{\Arctan \triangleq \text{ réciproque de } \begin{array}{ccccc}f & : & \Big[-\frac{\pi}{2}, \frac{\pi}{2}\Big] & \to & \mathrm{R} \\ & & x & \mapsto & \tan(x) \\\end{array}}\]

    \[\boxed{\begin{array}{c} \text{Impaire}\\ \\ \text{Continue et dérivable sur } \mathrm{R}\\ \\ \displaystyle\Arctan'(x) = \frac{1}{1+x^2} \\ \\ \Arctan(x) + \Arctan\bigg(\frac{1}{x}\bigg) = \begin{cases} \frac{\pi}{2} & \text{si } x > 0 \\ -\frac{\pi}{2} & \text{si } x < 0 \end{cases}\end{array} }\]
\end{multicols}

\section*{Trigonométrie hyperbolique}

\begin{multicols}{3}
    \[\boxed{\ch(x) \triangleq \frac{e^x + e^{-x}}{2}}\]

    \[\boxed{\sh(x) \triangleq \frac{e^x - e^{-x}}{2}}\]

    \[\boxed{\mathrm{th}(x) \triangleq \frac{\sh(x)}{\ch(x)}}\]
\end{multicols}

\vspace{0.5cm}

\begin{tikzpicture}[scale=1.5]
    \begin{axis}[
        axis lines = middle,
        xlabel = {$x$},
        ylabel = {$y$},
        domain = -4:4,
        ymin=-3,
        ymax=3,
        samples = 100,
        ytick = {-1,0,1},
        legend style={at={(0.8,0.2)},anchor=center,legend columns=1},
    ]
    \addplot[red,thick] {sinh(x)};
    \addlegendentry{$\sh(x)$}
    \addplot[green,thick] {cosh(x)};
    \addlegendentry{$\ch(x)$}
    \addplot[blue,thick] {tanh(x)};
    \addlegendentry{$\mathrm{th}(x)$}
    \end{axis}
\end{tikzpicture}

\vspace{0.5cm}

\[\boxed{\ch^2(x) - \sh^2(x) = 1}\]

\begin{multicols}{2}
    \[\boxed{\ch(a+b) = \ch(a)\ch(b) + \sh(a)\sh(b)}\]

    \[\boxed{\sh(a+b) = \sh(a)\ch(b) + \ch(a)\sh(b)}\]
\end{multicols}

\vspace{0.1cm}

\begin{multicols}{4}

    \[\boxed{\ch \text{ est paire}}\]

    \[\boxed{\ch' = \sh}\]
    
    \[\boxed{\sh \text{ est impaire}}\]

    \[\boxed{\sh' = \ch}\]
\end{multicols}

\vspace{0.5cm}

\begin{multicols}{2}
    \[\boxed{\mathrm{th} \text{ est impaire}}\]

    \[\boxed{\mathrm{th}'(x) = 1-\mathrm{th}^2(x) = \frac{1}{\ch^2(x)}}\]
    
\end{multicols}

\end{document}