From 2db6555139e1e961a4c8847f1a7e8949ba88147d Mon Sep 17 00:00:00 2001 From: Jeff Lance Date: Mon, 21 Aug 2023 16:12:34 +0200 Subject: [PATCH] add two first valid sequences --- .../images/fig1.asy | 32 + .../images/fig1.pag | 23 + .../images/fig1.pdf | Bin 0 -> 34177 bytes .../images/fig1.png | Bin 0 -> 86972 bytes .../images/fig1.svg | 3619 +++++++++++++++++ .../images/fig2.png | Bin 0 -> 43817 bytes .../images/fig3.png | Bin 0 -> 53903 bytes .../images/fig4.asy | 30 + .../images/fig4.pdf | Bin 0 -> 46591 bytes .../images/fig4.png | Bin 0 -> 41778 bytes .../images/fig4.svg | 2180 ++++++++++ .../images/fig5.asy | 35 + .../images/fig5.pdf | Bin 0 -> 44083 bytes .../images/fig5.png | Bin 0 -> 40204 bytes .../images/fig5.svg | 116 + .../images/fig6.asy | 30 + .../images/fig6.pdf | Bin 0 -> 49731 bytes .../images/fig6.png | Bin 0 -> 5286 bytes .../images/fig6.svg | 1468 +++++++ .../prof - etude de fonction - 2023.1/.pdf | Bin 0 -> 46588 bytes .../prof - etude de fonction - 2023.1/images | 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de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig1.asy new file mode 100644 index 0000000..6fa20b2 --- /dev/null +++ b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig1.asy @@ -0,0 +1,32 @@ +// préambule asymptote +usepackage("amsmath,amssymb"); +usepackage("inputenc","utf8"); +usepackage("icomma"); + +import lib_jl; + +unitsize(1cm,1cm); + +limits(-5.5,5.5); +ylimits(-5.5,5.5); + +xaxis(BottomTop,xmin=-5.5,xmax=5.5,Ticks("%",extend=true,Step=1),p=linewidth(1pt)+gray(0)+dotted); +yaxis(LeftRight,ymin=-5.5,ymax=5.5,Ticks("%",extend=true,Step=1),p=linewidth(1pt)+gray(0)+dotted); + +real F(real x) {return 1/10*x^3+1/10*x^2-1/5*x+1;} +draw(graph(F,-6,6,n=400),linewidth(1pt)+red+solid); + +real G(real x) {return 0.3*(x-1)+1;} +draw(graph(G,-6,6,n=400),linewidth(1pt)+darkblue+solid); + +xlimits(-5.5,5.5,Crop); +ylimits(-5.5,5.5,Crop); + 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etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig5.asy new file mode 100644 index 0000000..5947978 --- /dev/null +++ b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig5.asy @@ -0,0 +1,35 @@ +// préambule asymptote +usepackage("amsmath,amssymb"); +usepackage("inputenc","utf8"); +usepackage("icomma"); + +import lib_jl; + +unitsize(0.5cm,0.125cm); +xlimits(-3,6); +ylimits(-20,20); + +xaxis(BottomTop,xmin=-3,xmax=6,Ticks("%",extend=true,Step=1),p=linewidth(1pt)+gray(.8)+dotted); +yaxis(LeftRight,ymin=-20,ymax=20,Ticks("%",extend=true,Step=1),p=linewidth(1pt)+gray(.8)+dotted); + +real F(real x) {return 2/(x-3)+5;} +draw(graph(F,-4,2.999999,n=400),linewidth(1.5pt)+black+solid); +real G(real x) {return 2/(x-3)+5;} +draw(graph(G,3.000001,6,n=400),linewidth(1.5pt)+black+solid); + +point pA=(3,-10), pB=(3,10); +draw(line(pA,pB),linewidth(1.5pt)+red+dashed); +label(rotate(90)*"$\boldsymbol{x=3}$", (3,-10)+.5, fontsize(8pt)+red); +point pA=(-3,5), pB=(3,5); +draw(line(pA,pB),linewidth(1.5pt)+heavygreen+dashed); +label("$\boldsymbol{y=5}$", (-2,6), fontsize(8pt)+heavygreen); + +xlimits(-3,6,Crop); +ylimits(-20,20,Crop); + +xaxis(axis=YEquals(0),xmin=-3,xmax=6,Ticks(scale(.4)*Label(),beginlabel=true,endlabel=true,begin=true,end=true,NoZero,Step=1,Size=1mm),p=linewidth(1pt)+black,Arrow(2mm),true); +yaxis(axis=XEquals(0),ymin=-20,ymax=20,Ticks(scale(.4)*Label(),beginlabel=true,endlabel=true,begin=true,end=true,NoZero,Step=10,Size=1mm),p=linewidth(1pt)+black,Arrow(2mm),true); + + +shipout(bbox(0.1cm,0.1cm,white)); + diff --git a/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig5.pdf b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig5.pdf new file mode 100644 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reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig6.asy new file mode 100644 index 0000000..c8b5ff5 --- /dev/null +++ b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig6.asy @@ -0,0 +1,30 @@ +// préambule asymptote +usepackage("amsmath,amssymb"); +usepackage("inputenc","utf8"); +usepackage("icomma"); + +import lib_jl; + +unitsize(1cm,1cm); +xlimits(-2,7); +ylimits(-3,5); + +xaxis(BottomTop,xmin=-2,xmax=7,Ticks("%",extend=true,Step=1),p=linewidth(1pt)+gray(0)+dotted); +yaxis(LeftRight,ymin=-3,ymax=5,Ticks("%",extend=true,Step=1),p=linewidth(1pt)+gray(0)+dotted); + +real F(real x) {return x-3+1/x;} +draw(graph(F,-0.9999999999,8,n=400),linewidth(1.5pt)+black+solid); +real G(real x) {return x-3;} +draw(graph(G,-0.9999999999,8,n=400),linewidth(1.5pt)+red+solid+dashed); + +label(rotate(45)*"$\boldsymbol{ y = x-3 }$", (5,2)+.5, fontsize(12pt)+red); + +xlimits(-2,7,Crop); +ylimits(-3,5,Crop); + +xaxis(axis=YEquals(0),xmin=-2,xmax=7,Ticks(scale(.7)*Label(),beginlabel=true,endlabel=true,begin=true,end=true,NoZero,Step=10,Size=1mm),p=linewidth(1pt)+black,Arrow(2mm),true); +yaxis(axis=XEquals(0),ymin=-3,ymax=5,Ticks(scale(.7)*Label(),beginlabel=true,endlabel=true,begin=true,end=true,NoZero,Step=1,Size=1mm),p=linewidth(1pt)+black,Arrow(2mm),true); + + +shipout(bbox(0.1cm,0.1cm,white)); + diff --git a/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig6.pdf b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig6.pdf new file mode 100644 index 0000000000000000000000000000000000000000..787167c688bcadd74294a183fd438a2f4f3b283a GIT binary patch literal 49731 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reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/prof - etude de fonction - 2023.1/partieI.tex b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/prof - etude de fonction - 2023.1/partieI.tex new file mode 100644 index 0000000..edd330f --- /dev/null +++ b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/prof - etude de fonction - 2023.1/partieI.tex @@ -0,0 +1,93 @@ +% !TeX root = prof_-_etude_de_fonction_-_2023.1.tex + +\partie{Rappels : dérivation} + +\sspartie{Nombre dérivé, tangente} + +\noindent +\begin{multicols}{2} + \noindent + On considère une fonction $f$ définie sur un intervalle $I$, sa courbe représentative $\ronde{C}_f$ dans un repère orthogonal, et un réel $a$ dans $I$. + + Si $\ronde{C}_f$ admet une tangente $\mathsf{T}$ au point d'abscisse $a$ non parallèle à l'axe des abscisses, on dit que $f$ est dérivable en $a$. + + Le coefficient directeur de la tangente $\mathsf{T}$ au point $a$ est le nombre dérivé $f^{\prime}(a)$.\\\\ + \columnbreak + + \begin{center} + \includegraphics[scale=.6]{fig1.pdf} + \end{center} +\end{multicols} + +\medskip + +\begin{propriete}[équation de la tangente] + La tangente $\mathsf{T}$ à $\ronde{C}_f$ au point de coordonnées $\coord{a, f(a)}$ a pour équation $y = f^{\prime}(a)(x - a) + f(a)$. +\end{propriete} + +\sspartie{Fonction dérivée et dérivées usuelles} + +\noindent +On dit que la fonction \textbf{$\boldsymbol{f}$ est dérivable sur l'intervalle $I$} si elle est dérivable en tout point $a$ de $I$.\\ + +\noindent +La \textbf{fonction dérivée de $\boldsymbol{f}$} est notée $f^{\prime}$ et correspond à la fonction qui à tout réel $x$ de $I$, associe le nombre $f^{\prime}(x)$. + +\medskip + +\ssspartie{Dérivées des fonctions usuelles} + +\begin{center} + \begin{tabularx}{.7\textwidth}{| C{4.5cm} | >{\centering\arraybackslash}X | C{4.5cm} |} + \hline + \textbf{\small$ \boldsymbol{f(x)} $} & \textbf{\footnotesize est dérivable sur ...}\vspace{.2em} & \textbf{\small$ \boldsymbol{f^{\prime}(x)} $} \\ + \hline + \rule[-1ex]{0pt}{2.5ex} $ k $ (constante) & $ \mathbb{R} $ & \rule[-1ex]{0pt}{5ex}$ 0 \vspace{1ex}$ \\ + \hline + \rule[-1ex]{0pt}{2.5ex}$ x $ & $ \mathbb{R} $ & \rule[-1ex]{0pt}{5ex}$ 1 \vspace{1ex}$ \\ + \hline + \rule[-1ex]{0pt}{2.5ex}$ x^n $ où $ n \ge 1 $ & $ \mathbb{R} $& \rule[-1ex]{0pt}{5ex}$ nx^{n-1} $\vspace{1ex} \\ + \hline + \rule[-1ex]{0pt}{2.5ex}$ \dfrac{1}{x} $ & $ \mathbb{R}^{*} $& \rule[-1ex]{0pt}{5ex}$ -\dfrac{1}{x^2} $\vspace{1ex} \\ + \hline + \rule[-1ex]{0pt}{2.5ex}$ \sqrt{x} $ & $ ]0\; ;\; +\infty[ $ & \rule[-1ex]{0pt}{5ex}$ \dfrac{1}{2\sqrt{x}} $\vspace{1ex} \\ + \hline + \rule[-1ex]{0pt}{2.5ex}$ \ln x $ & $]0\ ;\ +\infty[$ & \rule[-1ex]{0pt}{5ex}$\dfrac1{x}$\vspace{1ex} \\ + \hline + \rule[-1ex]{0pt}{2.5ex}$ \e^{x} $ & $\R$ & \rule[-1ex]{0pt}{5ex}$\e^{x}$\vspace{1ex} \\ + \hline + \rule[-1ex]{0pt}{2.5ex}$ \cos x $ & $\R$ & \rule[-1ex]{0pt}{5ex}$-\sin x$\vspace{1ex} \\ + \hline + \rule[-1ex]{0pt}{2.5ex}$ \sin x $ & $\R$ & \rule[-1ex]{0pt}{5ex}$\cos x$\vspace{1ex} \\ + \hline + \end{tabularx} +\end{center} + +\ssspartie{Opérations sur les fonctions dérivées} + +\noindent +On considère $u$ et $v$ deux fonctions dérivables sur un intervalle $I$ de $\R$. +\begin{center} + \begin{tabularx}{.7\textwidth}{| C{4.5cm} | >{\centering\arraybackslash}X | C{4.5cm} |} + \hline + \textbf{$ \boldsymbol{f} $} & \textbf{\footnotesize est dérivable ...}\vspace{.2em} & \textbf{$ \boldsymbol{f^{\prime}} $} \\ + \hline + \rule[-1ex]{0pt}{2.5ex}$ k \times u $ ($k$ constante)& sur $I$ & \rule[-1ex]{0pt}{5ex}$ k \times u' $\vspace{1ex} \\ + \hline + \rule[-1ex]{0pt}{2.5ex}$ u + v $ & sur $I$ & \rule[-1ex]{0pt}{5ex}$ u' + v' $\vspace{1ex} \\ + \hline + \rule[-1ex]{0pt}{2.5ex}$ u \times v $ & sur $I$ & \rule[-1ex]{0pt}{5ex}$ u' \times v + u \times v' $\vspace{1ex} \\ + \hline + \rule[-1ex]{0pt}{2.5ex}$ \dfrac{u}{v} $ & si $v(x) \neq 0$ & \rule[-1ex]{0pt}{5ex}$ \dfrac{u' \times v - u \times v'}{v^2} \vspace{1ex}$ \\ + \hline + \rule[-1ex]{0pt}{2.5ex}$ \dfrac{1}{v} $ & si $v(x) \neq 0$ & \rule[-1ex]{0pt}{5ex}$ \dfrac{-v'}{v^2} \vspace{1ex}$ \\ + \hline + \rule[-1ex]{0pt}{2.5ex}$ u^n $ avec $n \ge 1$ & sur $I$ & \rule[-1ex]{0pt}{5ex}$ n \times u^{\prime} \times u^{n-1} \vspace{1ex}$ \\ + \hline + \rule[-1ex]{0pt}{2.5ex}$ \ln(u) $ & si $u(x) > 0$ & \rule[-1ex]{0pt}{5ex}$ \dfrac{u^{\prime}}{u} $\vspace{1ex} \\ + \hline + \rule[-1ex]{0pt}{2.5ex}$ \e^{u} $ & sur $I$ & \rule[-1ex]{0pt}{5ex}$ u^{\prime} \times \e^{u} $\vspace{1ex} \\ + \hline + \end{tabularx} +\end{center} + diff --git a/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/prof - etude de fonction - 2023.1/partieII.tex b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/prof - etude de fonction - 2023.1/partieII.tex new file mode 100644 index 0000000..0535c88 --- /dev/null +++ b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/prof - etude de fonction - 2023.1/partieII.tex @@ -0,0 +1,153 @@ +% !TeX root = prof_-_etude_de_fonction_-_2023.1.tex + +\partie{Rappels : limites et asymptotes} + +\sspartie{Limites des fonctions de références} + +\ssspartie{Limite en $\boldsymbol{\infty}$} + +\begin{center} + % \resizebox{\textwidth}{!}{% + \begin{tabular}{|>{\columncolor[HTML]{B5B5B5}}c|c|c|c|c|c|c|c|} + \hline + \rule[-1ex]{0pt}{2.5ex}$f(x)$ & $x^n$ & $\dfrac1{x^n}$ & $\sqrt{x}$ & $\dfrac1{\sqrt{x}}$ & $e^{x}$ & $e^{ax}$ & $\ln(x)$ \\[2ex] \hline + \rule[-1ex]{0pt}{2.5ex}$\lim\limits_{x \to +\infty}f(x)=$ & $+\infty$ & $0$ & $+\infty$ & $0$ & $+\infty$ & + \begin{tabular}{lr} + $+\infty$ & si $a>0$ \\ + $0$ & si $a<0$ + \end{tabular} & $+\infty$ \\ \hline + \rule[-1ex]{0pt}{2.5ex}$\lim\limits_{x \to -\infty}f(x)=$ + & + \begin{tabular}{lr} + $-\infty$ & si $n$ impair \\ + $+\infty$ & si $n$ pair + \end{tabular} + & $0$ & non définie & non définie & $0$ & + \begin{tabular}{lr} + $0$ & si $a>0$ \\ + $+\infty$ & si $a<0$ + \end{tabular} & non définie \\ \hline + \end{tabular}% + % } +\end{center} + +\vspace{-.2cm} + +\ssspartie{Limite en $\boldsymbol{0}$} + +\begin{center} + % \resizebox{\textwidth}{!}{% + \begin{tabular}{|>{\columncolor[HTML]{B5B5B5}} c|c|c|c|} + \hline + \rule[-1ex]{0pt}{2.5ex}$f(x)$ & $\dfrac1{x^n}$ & $\dfrac1{\sqrt{x}}$ & $\ln(x)$ \\[2ex] \hline + \rule[-1ex]{0pt}{2.5ex}$\lim\limits_{\substack{x \to 0 \\ x > 0}}f(x)=$ & $+\infty$ & $+\infty$ & $-\infty$ \\[2ex] \hline + \rule[-1ex]{0pt}{2.5ex}$\lim\limits_{\substack{x \to 0 \\ x < 0}}f(x)=$ + & + \begin{tabular}{lr} + $-\infty$ & si $n$ impair \\ + $+\infty$ & si $n$ pair + \end{tabular} + & non définie & non définie \\ \hline + \end{tabular}% + % } +\end{center} + +\vspace{-.2cm} + +\ssspartie{Opérations sur les limites} + +\begin{tasks}[style=itemize](4) + \task $ f $ et $ g $ sont deux fonctions ; + \task $A$ est soit $+\infty$, soit $-\infty$ ou un réel ; + \task \textsf{\textbf{F. I.}} : Forme indéterminée ; + \task \textup{*} : règle des signes. +\end{tasks} + +\begin{multicols}{2} + ~ + \vspace{-1.3cm} + \begin{center} + % \resizebox{\textwidth}{!}{% + \begin{tabular}{|>{\columncolor[HTML]{FFFFFF}}c |c|c|c|c|c|>{\columncolor[HTML]{EFEFEF}}c|} + \hline + Si $\lim\limits_{x \to A}f(x)=$ & $\ell$ & $\ell$ & $\ell$ & $+\infty$ & $-\infty$ & {$+\infty$} \\ + \hline + Si $\lim\limits_{x \to A}g(x)=$ & $\ell'$ & $+\infty$ & $-\infty$ & $+\infty$ & $-\infty$ & {$-\infty$} \\ + \hline + alors $\lim\limits_{x \to A}f(x)+g(x)=$ & $\ell + \ell'$ & $+\infty$ & $-\infty$ & $+\infty$ & $-\infty$ & {\textsf{\textbf{F. I}}} \\ + \hline + \end{tabular}% + % } + \end{center} + + \columnbreak + + \begin{center} + % \resizebox{\textwidth}{!}{% + \begin{tabular}{|>{\columncolor[HTML]{FFFFFF}}c |c|c|>{\columncolor[HTML]{EFEFEF}}c|c|} + \hline + Si $\lim\limits_{x \to A}f(x)=$ & $\ell$ & $\ell \neq0 $ & {$0$} & $\infty$ \\ + \hline + Si $\lim\limits_{x \to A}g(x)=$ & $\ell'$ & $\infty$ & {$\infty$} & $\infty$ \\ + \hline + alors $\lim\limits_{x \to A}f(x) \times g(x)=$ & $\ell \times \ell'$ & $\infty^{*}$ & {\textsf{\textbf{F. I.}}} & $\infty^{*}$ \\ + \hline + \end{tabular}% + % } + \end{center} +\end{multicols} + +\begin{center} + % \resizebox{\textwidth}{!}{% + \begin{tabular}{|>{\columncolor[HTML]{FFFFFF}}c |c|c|>{\columncolor[HTML]{EFEFEF}}c|c|c|>{\columncolor[HTML]{EFEFEF}}c|} + \hline + Si $\lim\limits_{x \to A}f(x)=$ & $\ell$ & $\ell \neq 0$ & {$0$} & $\ell$ & $\infty$ & {$\infty$} \\ + \hline + Si $\lim\limits_{x \to A}g(x)=$ & $\ell' \neq 0$ & $0$ & {$0$} & $\infty$ & $\ell'$ & {$\infty$} \\ + \hline + alors $\lim\limits_{x \to A}\dfrac{f(x)}{g(x)}=$ & $\dfrac{\ell}{\ell'}$ & $\infty^{*}$ & {\textsf{\textbf{F. I.}}} & $0$ & $\infty^{*}$ & {\textsf{\textbf{F. I.}}} \\ + \hline + \end{tabular}% + % } +\end{center} + +\sspartie{Comportement asymptotique} + +\noindent +On considère une fonction $f$, sa courbe $\ronde{C}_f$, $a$ et $\ell$ des nombres réels. + +\vspace{.4cm} + +\begin{definition}[existence d'asymptote horizontale] + \bi + \item Lorsque $ \lim\limits_{x \to +\infty} f(x) = \ell $, on dit que la courbe $\ronde{C}_f$ admet la droite d'équation $y=\ell$ comme asymptote horizontale en $+\infty$. + \item Lorsque $ \lim\limits_{x \to a} f(x) = +\infty $, on dit que la courbe $\ronde{C}_f$ admet la droite d'équation $x=a$ comme asymptote verticale. + \item Lorsque $ \lim\limits_{x \to +\infty} f(x) - (mx+p)= \ell $, on dit que la courbe $\ronde{C}_f$ admet la droite d'équation $y=mx+p$ comme asymptote oblique en $+\infty$. + \ei +\end{definition} + +\vspace{.2cm} + +\noindent +On retrouve les même situations lorsque $x$ tend vers $-\infty$. + +\vspace{.4cm} + +% exemple +\begin{exemple} + \begin{center} + \begin{tabular}{ccc} + \rule[-1ex]{0pt}{2.5ex} $f(x) = \ln(x+1)$ & \hspace{.2cm} & $g(x) = \dfrac{2}{x-3}+5$ \\ + \rule[-1ex]{0pt}{2.5ex} \includegraphics[scale=.8]{fig4.pdf} & \hspace{.2cm} & \includegraphics[scale=1.25]{fig5.pdf} \\ + \rule[-1ex]{0pt}{2.5ex} la droite $x=-1$ est asymptote verticale & \hspace{.2cm} & les droites $y=5$ et $x=3$ sont asymptotes horizontale et verticale \\ + \end{tabular} + \end{center} + ~\vspace{.4cm} + \begin{center} + \begin{tabular}{ccc} + \rule[-1ex]{0pt}{2.5ex} \hspace{.2cm} & $h(x) = \dfrac{x^2-3x+1}{x}$ & \hspace{.2cm} \\ + \rule[-1ex]{0pt}{2.5ex} \hspace{.2cm} & \includegraphics[scale=.8]{fig6.pdf} & \hspace{.2cm} \\ + \rule[-1ex]{0pt}{2.5ex} \hspace{.2cm} & la droite $y=x-3$ est asymptote oblique & \hspace{.2cm} \\ + \end{tabular} + \end{center} +\end{exemple} diff --git a/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/prof - etude de fonction - 2023.1/partieIII.tex b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/prof - etude de fonction - 2023.1/partieIII.tex new file mode 100644 index 0000000..ea24704 --- /dev/null +++ b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/prof - etude de fonction - 2023.1/partieIII.tex @@ -0,0 +1,45 @@ +% !TeX root = prof_-_etude_de_fonction_-_2023.1.tex + +\newpage + +\partie{Applications} + +\begin{application}[calculer une dérivée] + Déterminer la fonction dérivée de chacune des fonctions sur l'intervalle donné. + + \begin{tasks} (2)[] + \task\ $f_1(x) = x^2 \e^{x}$ \qquad pour $x \in \R$ + \task\ $f_2(x) = 3 \ln(x) - \dfrac5{x} + 2\sqrt{x}$ \qquad pour $ x \in ]0\ ;\ +\infty[ $ + \task\ $f_3 = \dfrac{5x+3}{x-4}$ \qquad pour $x \in ]-\infty\ ;\ 4[ \cup ]4\ ;\ +\infty[$ + \task\ $f_4(x) = \e^{x^2-x}$ \qquad pour $x \in \R$ + \task\ $f_5(x) = \ln(x^3+1)$ \qquad pour $x \in ]0\ ;\ +\infty[$ + \end{tasks} +\end{application} + +\vspace{.4cm} + +\begin{application}[calculer une limite] + Déterminer les limites suivantes: + \begin{tasks} (4)[] + \task\ $ \lim\limits_{x \to +\infty } 5-x^2 \e^{x} $ + \task\ $ \lim\limits_{\substack{ x \to 0 \\ x>0 }} \dfrac{3+x}{\ln x} $ + \task\ $ \lim\limits_{\substack{ x \to -2 \\ x>-2 }} \dfrac{x^2-6x+7}{4+2x} $ + \task\ $ \lim\limits_{x \to -\infty } \e^{x^2-x} $ + \end{tasks} +\end{application} + +\vspace{.4cm} + +\begin{application}[étudier une fonction et ses asymptotes] + Soit $f$ la fonction définie sur $]-\infty\ ;\ -2[\cup]-2\ ;\ +\infty[$ par $f(x)=x+3-\dfrac1{x+2}$ et $\ronde{C}_f$ sa courbe représentative. + \begin{questions} + \item Calculer la dérivée de la fonction $f$ et étudier ses variations. + \item Déterminer les limites de $f$ aux bornes de son ensemble de définition. + \item En déduire l'existence d'une asymptote verticale et préciser son équation. + \item Montrer que la droite $D$ d'équation $y=x+3$ est asymptote oblique à la coiurbe $\ronde{C}_f$ en $-\infty$ et en $+\infty$. + \item Déterminer la position relative de $\ronde{C}_f$ et de la droite $D$. Justifier. + \end{questions} +\end{application} + +\vspace{.4cm} + diff --git a/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/prof - etude de fonction - 2023.1/prof_-_etude_de_fonction_-_2023.log b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/prof - etude de fonction - 2023.1/prof_-_etude_de_fonction_-_2023.log new file mode 100644 index 0000000..f1d39b1 --- /dev/null +++ b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/prof - etude de fonction - 2023.1/prof_-_etude_de_fonction_-_2023.log @@ -0,0 +1,1302 @@ +This is pdfTeX, Version 3.141592653-2.6-1.40.25 (TeX Live 2023/Arch Linux) (preloaded format=pdflatex 2023.8.3) 9 AUG 2023 11:09 +entering extended mode + restricted \write18 enabled. + %&-line parsing enabled. +**\scrollmode + 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+*\fontsize{12}{14.4}\selectfont% + +* +(Please type a command or say `\end') +*\usefont{\ASYencoding}{\ASYfamily}{\ASYseries}{\ASYshape}% + +* +(Please type a command or say `\end') +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-5$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-4$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-3$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-2$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-1$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$1$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(5.87494pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(0.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$2$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(5.87494pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(0.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$3$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(5.87494pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(0.0pt)dim + +*\fontsize{12}{14.4}\selectfont% + +* +(Please type a command or say `\end') +*\usefont{\ASYencoding}{\ASYfamily}{\ASYseries}{\ASYshape}% + +* +(Please type a command or say `\end') +*\setbox\ASYbox=\hbox{$\boldsymbol{ x=-1 }$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(43.55823pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(7.73332pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.59998pt)dim + +*\fontsize{12}{14.4}\selectfont% + +* +(Please type a command or say `\end') +*\usefont{\ASYencoding}{\ASYfamily}{\ASYseries}{\ASYshape}% + +* +(Please type a command or say `\end') +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$0$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(5.87494pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(0.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-5$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-4$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-3$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-2$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-1$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$1$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(5.87494pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(0.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$2$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(5.87494pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(0.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$3$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(5.87494pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(0.0pt)dim + +*\fontsize{12}{14.4}\selectfont% + +* +(Please type a command or say `\end') +*\usefont{\ASYencoding}{\ASYfamily}{\ASYseries}{\ASYshape}% + +* +(Please type a command or say `\end') +*\setbox\ASYbox=\hbox{$\boldsymbol{ x=-1 }$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(43.55823pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(7.73332pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.59998pt)dim + +*\fontsize{12}{14.4}\selectfont% + +* +(Please type a command or say `\end') +*\usefont{\ASYencoding}{\ASYfamily}{\ASYseries}{\ASYshape}% + +* +(Please type a command or say `\end') +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$0$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(5.87494pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(0.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-5$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-4$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-3$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-2$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-1$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$1$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(5.87494pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(0.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$2$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(5.87494pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(0.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$3$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(5.87494pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(0.0pt)dim + +*\fontsize{12}{14.4}\selectfont% + +* +(Please type a command or say `\end') +*\usefont{\ASYencoding}{\ASYfamily}{\ASYseries}{\ASYshape}% + +* +(Please type a command or say `\end') +*\setbox\ASYbox=\hbox{$\boldsymbol{ x=-1 }$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(43.55823pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(7.73332pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.59998pt)dim + +*\fontsize{12}{14.4}\selectfont% + +* +(Please type a command or say `\end') +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$0$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(5.87494pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(0.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-5$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-4$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-3$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-2$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-1$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$1$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(5.87494pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(0.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$2$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(5.87494pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(0.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$3$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(5.87494pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(0.0pt)dim + +*\fontsize{12}{14.4}\selectfont% + +* +(Please type a command or say `\end') +*\usefont{\ASYencoding}{\ASYfamily}{\ASYseries}{\ASYshape}% + +* +(Please type a command or say `\end') +*\setbox\ASYbox=\hbox{$\boldsymbol{ x=-1 }$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(43.55823pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(7.73332pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.59998pt)dim + +*\fontsize{12}{14.4}\selectfont% + +* +(Please type a command or say `\end') +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$0$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(5.87494pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(0.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-5$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-4$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-3$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-2$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-1$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$1$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(5.87494pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(0.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$2$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(5.87494pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(0.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$3$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(5.87494pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(0.0pt)dim + +*\fontsize{12}{14.4}\selectfont% + +* +(Please type a command or say `\end') +*\usefont{\ASYencoding}{\ASYfamily}{\ASYseries}{\ASYshape}% + +* +(Please type a command or say `\end') +*\setbox\ASYbox=\hbox{$\boldsymbol{ x=-1 }$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(43.55823pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(7.73332pt)dim + 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+\newcolumntype{M}[1]{>{\centering}m{#1}} + +\begin{document} +\pagenumbering{arabic} + +\titre{\Jd Étude de fonctions}{\Jd Étude de fonctions} +\lohead*{BTS2E} +\rofoot*{\anneescolaire} + +\cofoot[\thepage]{\thepage} +\cefoot[\thepage]{\thepage} + +\nskip{3} + +\import{.}{partieI.tex} +\import{.}{partieII.tex} +\import{.}{partieIII.tex} + +\end{document} diff --git a/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/images/fig1.png b/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/images/fig1.png new file mode 100644 index 0000000000000000000000000000000000000000..760d21cc2f49c0714e4cf0e6cf4d927d45c3e24d GIT binary patch literal 33515 zcmXtA2Rzp8_byo>D;}GNgk-WC;j{83MIoG+a>pa2AijQvKP~u#!D_4+LWF^H_o!_rDs=wAzcRcq{T%;bnZZ=OHB*w*jFN8!&RyCDE z_UZF__qijqCdZ?%PQ~Q~?)RQKju*dsH`EtCNaZqMTZ1H{j^2VW2 zFLQysxWC2 zZq3J!Z@AnzlS5sPSv<)f=|&oF^17C~w_J#pGocp)&8Mua%-`St!-o$$J9Zh$*coht 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reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/images/fig5.asy new file mode 100644 index 0000000..7f3f2eb --- /dev/null +++ b/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/images/fig5.asy @@ -0,0 +1,56 @@ +// préambule asymptote +usepackage("amsmath,amssymb"); +usepackage("inputenc","utf8"); +usepackage("icomma"); + +import lib_jl; + +unitsize(1cm,1cm); + +real xmin=-5.5, ymin=-1.5, xmax=5.5, ymax=5.5; + +xlimits(xmin,xmax); +ylimits(ymin,ymax); + +xaxis(BottomTop,xmin=xmin,xmax=xmax,Ticks("%",extend=true,Step=1),p=linewidth(0.5pt)+gray(0)+dotted); +yaxis(LeftRight,ymin=ymin,ymax=ymax,Ticks("%",extend=true,Step=1),p=linewidth(0.5pt)+gray(0)+dotted); + +real F(real x) {return .5*(x+5)*(.5x^2+.25x+.5);} + +draw(graph(F,-6,4,n=400),linewidth(1pt)+deepred+solid); + +xlimits(xmin,xmax,Crop); +ylimits(ymin,ymax,Crop); + +xaxis(axis=YEquals(0),xmin=xmin,xmax=xmax,Ticks("%",beginlabel=true,endlabel=true,begin=true,end=true,NoZero,Step=1,Size=1mm),p=linewidth(1pt)+black,Arrow(2mm),true); +yaxis(axis=XEquals(0),ymin=ymin,ymax=ymax,Ticks("%",beginlabel=true,endlabel=true,begin=true,end=true,NoZero,Step=1,Size=1mm),p=linewidth(1pt)+black,Arrow(2mm),true); + +pair O=(0,0), I=(1,0), J=(0,1), A=(-4,0), B=(1,0), C=(-1.5,0); + +labelx(Label("$O$",NoFill), 0, SW); +label("$I$",I,SE); +label("$J$",J,NW); + +draw(line(A-(0,2),A+(0,2)),deepblue+pen1); +label("$a$",A,SE,deepblue); + +path integrale1=buildcycle((A.x, F(A.x))--A--C--(C.x, F(C.x)),graph(F,A.x,C.x)); +add("integrale1",hatch(H=.5mm,dir=NE,lightred)); +fill(integrale1,pattern("integrale1")); + +label(scale(.9)*"$\int_{a}^{c} f(x)\; dx$",((A.x+C.x)/2,2),deepred); + +path integrale2=buildcycle((C.x, F(C.x))--C--B--(B.x, F(B.x)),graph(F,C.x,B.x)); +add("integrale2",hatch(H=.5mm,dir=NW,lightgreen)); +fill(integrale2,pattern("integrale2")); + 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