From 70096cf075a9b229a178e855da08f903d792c4d0 Mon Sep 17 00:00:00 2001 From: Lafrite Date: Thu, 14 May 2015 10:59:56 +0200 Subject: [PATCH] need to clean... --- example/1_2ndDeg.tex | 117 ++++++++++++ example/1_corr_DM_0302.tex | 308 ++++++++++++++++++++++++++++++ example/1_play.tex | 0 example/all_2ndDeg.pdf | Bin 0 -> 87917 bytes example/all_corr_DM_0302.pdf | Bin 0 -> 67929 bytes example/poly.tex | 23 +++ example/tpl_2ndDeg.tex | 71 +------ example/tpl_corr_DM_0302.tex | 359 +++++++++++++++++++++++++++++++++++ example/tpl_example.pdf | Bin 0 -> 100735 bytes example/tpl_play.tex | 32 ++++ lib/call_litt.py | 131 ------------- lib/rd_frac.py | 133 ------------- 12 files changed, 847 insertions(+), 327 deletions(-) create mode 100644 example/1_2ndDeg.tex create mode 100644 example/1_corr_DM_0302.tex create mode 100644 example/1_play.tex create mode 100644 example/all_2ndDeg.pdf create mode 100644 example/all_corr_DM_0302.pdf create mode 100644 example/poly.tex create mode 100644 example/tpl_corr_DM_0302.tex create mode 100644 example/tpl_example.pdf create mode 100644 example/tpl_play.tex delete mode 100644 lib/call_litt.py delete mode 100644 lib/rd_frac.py diff --git a/example/1_2ndDeg.tex b/example/1_2ndDeg.tex new file mode 100644 index 0000000..aca36ee --- /dev/null +++ b/example/1_2ndDeg.tex @@ -0,0 +1,117 @@ +\documentclass[a4paper,10pt]{article} +\RequirePackage[utf8x]{inputenc} +\RequirePackage[francais]{babel} +\RequirePackage{amssymb} +\RequirePackage{amsmath} +\RequirePackage{amsfonts} +\RequirePackage{subfig} +\RequirePackage{graphicx} +\RequirePackage{color} + + + +% Title Page +\title{Calcul littéral et statistiques} +\date{\today} + +\begin{document} +\maketitle + + +\section{Polynômes} + + + + + Résoudre l'équation suivante + \begin{eqnarray*} + 3 x^{ 2 } + 6 x + 3 & = & 0 + \end{eqnarray*} + + Solution: + + + + On commence par calculer le discriminant de $P(x) = 3 x^{ 2 } + 6 x + 3$. + \begin{eqnarray*} + \Delta & = & b^2-4ac \\ + \Delta & = & 6^{ 2 } - 4 \times 3 \times 3 \\ +\Delta & = & 36 - 4 \times 9 \\ +\Delta & = & 36 - 36 \\ +\Delta & = & 0 + \end{eqnarray*} + + + Comme $\Delta = 0$ donc $P$ a une racine + + \begin{eqnarray*} + x_1 = \frac{-b}{2a} = \frac{-6}{2\times 3} = -1 \\ + \end{eqnarray*} + + La solution de $3 x^{ 2 } + 6 x + 3 = 0$ est donc $\mathcal{S} = \left\{ -1\right\}$ + + + + + + \bigskip + ~\dotfill + \bigskip + + + + + Résoudre l'équation suivante + \begin{eqnarray*} + x^{ 2 } + 4 x + 2 & = & - 9 x^{ 2 } + 9 x + 5 + \end{eqnarray*} + + Solution: + + On commence par se ramener à une équation de la forme $ax^2+bx+c = 0$. + + + + \begin{align*} + & & x^{ 2 } + 4 x + 2 = - 9 x^{ 2 } + 9 x + 5 \\ + & \Leftrightarrow & x^{ 2 } + 4 x + 2 - ( - 9 x^{ 2 } + 9 x + 5 )= 0 \\ + & \Leftrightarrow & x^{ 2 } + 4 x + 2 + 9 x^{ 2 } - 9 x - 5= 0 \\ + & \Leftrightarrow & ( 1 + 9 ) x^{ 2 } + ( 4 - 9 ) x + 2 - 5= 0 \\ + & \Leftrightarrow & 10 x^{ 2 } - 5 x - 3= 0 + \end{align*} + + On cherche maintenant à résoudre l'équation $10 x^{ 2 } - 5 x - 3 = 0$. + + + + On commence par calculer le discriminant de $P(x) = 10 x^{ 2 } - 5 x - 3$. + \begin{eqnarray*} + \Delta & = & b^2-4ac \\ + \Delta & = & -5^{ 2 } - 4 \times 10 \times ( -3 ) \\ +\Delta & = & 25 - 4 \times ( -30 ) \\ +\Delta & = & 25 - ( -120 ) \\ +\Delta & = & 145 + \end{eqnarray*} + + + comme $\Delta = 145 > 0$ donc $P$ a deux racines + + \begin{eqnarray*} + x_1 & = & \frac{-b - \sqrt{\Delta}}{2a} = \frac{-5 - \sqrt{145}}{2 \times 10} = - \frac{\sqrt{145}}{20} + \frac{1}{4} \\ + x_2 & = & \frac{-b + \sqrt{\Delta}}{2a} = \frac{-5 + \sqrt{145}}{2 \times 10} = \frac{1}{4} + \frac{\sqrt{145}}{20} + \end{eqnarray*} + + Les solutions de l'équation $10 x^{ 2 } - 5 x - 3 = 0$ sont donc $\mathcal{S} = \left\{ - \frac{\sqrt{145}}{20} + \frac{1}{4}; \frac{1}{4} + \frac{\sqrt{145}}{20} \right\}$ + + + + + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: + + \ No newline at end of file diff --git a/example/1_corr_DM_0302.tex b/example/1_corr_DM_0302.tex new file mode 100644 index 0000000..34d355d --- /dev/null +++ b/example/1_corr_DM_0302.tex @@ -0,0 +1,308 @@ +\documentclass[a4paper,10pt, table]{/media/documents/Cours/Prof/Enseignements/tools/style/classDS} +\usepackage{/media/documents/Cours/Prof/Enseignements/2014_2015} + +% Title Page +\titre{DM5} +% \seconde \premiereS \PSTMG \TSTMG +\classe{\premiereS} +\date{02 mars 2015} +%\duree{1 heure} +\sujet{1} +% DS DSCorr DM DMCorr Corr +\typedoc{DM} + +\printanswers + +\begin{document} + +\maketitle + +Le barème est donné à titre indicatif, il pourra être modifié. Vous rendrez le sujet avec la copie. + +\begin{questions} + + \question + Résoudre les équations suivantes + + + \begin{eqnarray*} + 8 x^{ 2 } + 5 x - 2 & > &0 \\ + \end{eqnarray*} + + \begin{solution} + On commence par calculer le discriminant de $P(x) = 8 x^{ 2 } + 5 x - 2$. + \begin{eqnarray*} + \Delta & = & b^2-4ac \\ + \Delta & = & 5^{ 2 } - 4 \times 8 ( -2 ) \\ +\Delta & = & 25 - 4 ( -16 ) \\ +\Delta & = & 25 - ( -64 ) \\ +\Delta & = & 89 + \end{eqnarray*} + + + comme $\Delta = 89 > 0$ donc $P$ a deux racines + + \begin{eqnarray*} + x_1 & = & \frac{-b - \sqrt{\Delta}}{2a} = \frac{5 - \sqrt{89}}{2 \times 8} = -0.9 \\ + x_2 & = & \frac{-b + \sqrt{\Delta}}{2a} = \frac{5 + \sqrt{89}}{2 \times 8} = 0.28 + \end{eqnarray*} + + + + Comme $a = 8$, on en déduit le tableau de signe de $P$ + \begin{center} + \begin{tikzpicture} + \tkzTabInit[espcl=2]% + {$x$/1, $P$/2}% + {$-\infty$, -0.9 , 0.28 , $+\infty$} + \tkzTabLine{, +, z, -, z , +,} + \end{tikzpicture} + \end{center} + On regarde maintenant où sont les $+$ dans le tableau de signe pour résoudre l'inéquation. + \end{solution} + + \begin{eqnarray*} + - 3 x^{ 2 } + 2 x + 4 & \leq &0 \\ + \end{eqnarray*} + \begin{solution} + On commence par calculer le discriminant de $Q(x) = - 3 x^{ 2 } + 2 x + 4$. + \begin{eqnarray*} + \Delta & = & b^2-4ac \\ + \Delta & = & 2^{ 2 } - 4 ( -3 ) \times 4 \\ +\Delta & = & 4 - 4 ( -12 ) \\ +\Delta & = & 4 - ( -48 ) \\ +\Delta & = & 52 + \end{eqnarray*} + + + comme $\Delta = 52 > 0$ donc $Q$ a deux racines + + \begin{eqnarray*} + x_1 & = & \frac{-b - \sqrt{\Delta}}{2a} = \frac{2 - \sqrt{52}}{2 \times -3} = 1.54 \\ + x_2 & = & \frac{-b + \sqrt{\Delta}}{2a} = \frac{2 + \sqrt{52}}{2 \times -3} = -0.87 + \end{eqnarray*} + + + + Comme $a = -3$, on en déduit le tableau de signe de $Q$ + \begin{center} + \begin{tikzpicture} + \tkzTabInit[espcl=2]% + {$x$/1, $Q$/2}% + {$-\infty$, -0.87 , 1.54 , $+\infty$} + \tkzTabLine{, -, z, +, z , -,} + \end{tikzpicture} + \end{center} + On regarde maintenant où sont les $-$ dans le tableau de signe pour résoudre l'inéquation. + \end{solution} + + \begin{eqnarray*} + 8 x^{ 2 } + 5 x - 2 & \geq & - 3 x^{ 2 } + 2 x + 4 + \end{eqnarray*} + + + + \begin{solution} + On commence par se ramener à une équation de la forme $ax^2 + bx + c \geq 0$. + \begin{eqnarray*} + 8 x^{ 2 } + 5 x - 2 \geq - 3 x^{ 2 } + 2 x + 4 & \Leftrightarrow & 8 x^{ 2 } + 5 x - 2 - (- 3 x^{ 2 } + 2 x + 4) \geq 0 \\ + & \Leftrightarrow & 8 x^{ 2 } + 5 x - 2 - ( - 3 x^{ 2 } + 2 x + 4 )\geq 0 \\ + & \Leftrightarrow & 8 x^{ 2 } + 5 x - 2 + 3 x^{ 2 } - 2 x - 4\geq 0 \\ + & \Leftrightarrow & ( 8 + 3 ) x^{ 2 } + ( 5 + ( -2 ) ) x + ( -2 ) + ( -4 )\geq 0 \\ + & \Leftrightarrow & 11 x^{ 2 } + 3 x - 6\geq 0 + \end{eqnarray*} + + + + Ensuite on étudie le signe de $R(X) = 11 x^{ 2 } + 3 x - 6$. + \begin{eqnarray*} + \Delta & = & b^2-4ac \\ + \Delta & = & 3^{ 2 } - 4 \times 11 ( -6 ) \\ +\Delta & = & 9 - 4 ( -66 ) \\ +\Delta & = & 9 - ( -264 ) \\ +\Delta & = & 273 + \end{eqnarray*} + + + comme $\Delta = 273 > 0$ donc $R$ a deux racines + + \begin{eqnarray*} + x_1 & = & \frac{-b - \sqrt{\Delta}}{2a} = \frac{3 - \sqrt{273}}{2 \times 11} = -0.89 \\ + x_2 & = & \frac{-b + \sqrt{\Delta}}{2a} = \frac{3 + \sqrt{273}}{2 \times 11} = 0.61 + \end{eqnarray*} + + + + Comme $a = 11$, on en déduit le tableau de signe de $R$ + \begin{center} + \begin{tikzpicture} + \tkzTabInit[espcl=2]% + {$x$/1, $R$/2}% + {$-\infty$, -0.89 , 0.61 , $+\infty$} + \tkzTabLine{, +, z, -, z , +,} + \end{tikzpicture} + \end{center} + On regarde maintenant où sont les $+$ dans le tableau de signe pour résoudre l'inéquation. + + + \end{solution} + + + \question + Tracer le tableau de variation des fonctions suivantes \textit{(Vous pouvez utiliser les nombres à virgules)} + + + \begin{parts} + \part $f:x\mapsto - 10 x^{ 3 } + x^{ 2 } - 7 x + 5$ + \begin{solution} + Pour avoir les variations de $f$, il faut connaître le signe de sa dérivé. On dérive $P$ + + \begin{eqnarray*} + f'(x) & = & 3 ( -10 ) x^{ 2 } + 2 \times 1 x + 1 ( -7 ) \\ +f'(x) & = & - 30 x^{ 2 } + 2 x - 7 + \end{eqnarray*} + + On étudie le signe de $P'$ + + Ensuite on étudie le signe de $f'(x) = - 30 x^{ 2 } + 2 x - 7$. + \begin{eqnarray*} + \Delta & = & b^2-4ac \\ + \Delta & = & 2^{ 2 } - 4 ( -30 ) ( -7 ) \\ +\Delta & = & 4 - 4 \times 210 \\ +\Delta & = & 4 - 840 \\ +\Delta & = & -836 + \end{eqnarray*} + + + Alors $\Delta = -836 < 0$ donc $f'$ n'a pas de racine. + + + Comme $a = -30$, on en déduit le tableau de signe de $f'$ + \begin{center} + \begin{tikzpicture} + \tkzTabInit[espcl=2]% + {$x$/1, Signe de $f' $/2}% + {$-\infty$, $+\infty$} + \tkzTabLine{, -,} + \end{tikzpicture} + \end{center} + + \end{solution} + + + + \part $g:x\mapsto - 9 x^{ 3 } - 8 x^{ 2 } - 5 x - 2$ + + \begin{solution} + Pour avoir les variations de $g$, il faut connaître le signe de sa dérivé. On dérive $P$ + + \begin{eqnarray*} + g'(x) & = & 3 ( -9 ) x^{ 2 } + 2 ( -8 ) x + 1 ( -5 ) \\ +g'(x) & = & - 27 x^{ 2 } - 16 x - 5 + \end{eqnarray*} + + On étudie le signe de $P'$ + + Ensuite on étudie le signe de $g'(x) = - 27 x^{ 2 } - 16 x - 5$. + \begin{eqnarray*} + \Delta & = & b^2-4ac \\ + \Delta & = & ( -16 )^{ 2 } - 4 ( -27 ) ( -5 ) \\ +\Delta & = & 256 - 4 \times 135 \\ +\Delta & = & 256 - 540 \\ +\Delta & = & -284 + \end{eqnarray*} + + + Alors $\Delta = -284 < 0$ donc $g'$ n'a pas de racine. + + + Comme $a = -27$, on en déduit le tableau de signe de $g'$ + \begin{center} + \begin{tikzpicture} + \tkzTabInit[espcl=2]% + {$x$/1, Signe de $g' $/2}% + {$-\infty$, $+\infty$} + \tkzTabLine{, -,} + \end{tikzpicture} + \end{center} + + \end{solution} + + + \part $h:x\mapsto - 7 x^{ 2 } - 9 x + 3 - f(x)$ + + + + + \begin{solution} + On commence par simplifier l'expression de $h$ + \begin{eqnarray*} + h(x) & = & - 7 x^{ 2 } - 9 x + 3 - f(x) \\ + h(x) & = & - 7 x^{ 2 } - 9 x + 3 - ( - 10 x^{ 3 } + x^{ 2 } - 7 x + 5 ) \\ +h(x) & = & - 7 x^{ 2 } - 9 x + 3 + 10 x^{ 3 } - x^{ 2 } + 7 x - 5 \\ +h(x) & = & 10 x^{ 3 } + ( ( -7 ) + ( -1 ) ) x^{ 2 } + ( ( -9 ) + 7 ) x + 3 + ( -5 ) \\ +h(x) & = & 10 x^{ 3 } - 8 x^{ 2 } - 2 x - 2 + \end{eqnarray*} + + + Pour avoir les variations de $h$, il faut connaître le signe de sa dérivé. On dérive $P$ + + \begin{eqnarray*} + h'(x) & = & 3 \times 10 x^{ 2 } + 2 ( -8 ) x + 1 ( -2 ) \\ +h'(x) & = & 30 x^{ 2 } - 16 x - 2 + \end{eqnarray*} + + On étudie le signe de $P'$ + + Ensuite on étudie le signe de $h'(x) = 30 x^{ 2 } - 16 x - 2$. + \begin{eqnarray*} + \Delta & = & b^2-4ac \\ + \Delta & = & ( -16 )^{ 2 } - 4 \times 30 ( -2 ) \\ +\Delta & = & 256 - 4 ( -60 ) \\ +\Delta & = & 256 - ( -240 ) \\ +\Delta & = & 496 + \end{eqnarray*} + + + comme $\Delta = 496 > 0$ donc $h'$ a deux racines + + \begin{eqnarray*} + x_1 & = & \frac{-b - \sqrt{\Delta}}{2a} = \frac{-16 - \sqrt{496}}{2 \times 30} = -0.1 \\ + x_2 & = & \frac{-b + \sqrt{\Delta}}{2a} = \frac{-16 + \sqrt{496}}{2 \times 30} = 0.64 + \end{eqnarray*} + + + + Comme $a = 30$, on en déduit le tableau de signe de $h'$ + \begin{center} + \begin{tikzpicture} + \tkzTabInit[espcl=2]% + {$x$/1, Signe de $h' $/2}% + {$-\infty$, -0.1 , 0.64 , $+\infty$} + \tkzTabLine{, +, z, -, z , +,} + \end{tikzpicture} + \end{center} + + \end{solution} + \end{parts} + + \question + Appliquer l'algorithme de tri vu en cours à la suite suivante + \begin{center} + \begin{tabular}{|*{6}{c|}} + \hline + 6914 & 6851 & 6532 & 6884 & 6164 & 6495 \\ + \hline + \end{tabular} + + \end{center} + + +\end{questions} + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/example/1_play.tex b/example/1_play.tex new file mode 100644 index 0000000..e69de29 diff --git a/example/all_2ndDeg.pdf b/example/all_2ndDeg.pdf new file mode 100644 index 0000000000000000000000000000000000000000..c97ad52966fcadddca3ae98baec19aeb24c03f2a GIT binary patch literal 87917 zcmce71B@=wvhF{gJ+^Jzwr$(CZDa4Ty=V5=wr$(C{r0)%oSU1xo80%3m)GfZzOG)Y zQfpOGUn>1AQUwt)T1GlnDAJjQ!4)WW0tNzmBP%E#9w>SlQ#*4P3j#JKf`465^kSAa zE~ZWd^kO!KE~X-;#`Y$rP<(t)&Mr=-hPF^1Ys@MP03l&8T@YRnVPi@ozH=0W;%&;J+JA zui#{FtZeE+piMykH-QB7DyAMT1iJso#lO?W$jtJ05&u)d4mH=~*4YqzpVdz)k+fWyyu3i-o#;kSjHF~$S`%tZxvUI*WbFO9R5`1%Cuv{On=}d 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zHwsV;99Z=LP0XT`-sHRG-5o;Nn`LJJ<<`{HvP=tcAxKVI!nt}@1H8RJof3_XcEpi_ z-}{SP>*U~P%+Pyf!xyg9RpA)cxe5)>n&3Y$>=vytv3TM<0^Tj}Z-v_K!ExN#oyrEC zlYA7e5P=w_3{0(_T?0j}-*~rbUqQqg@%^={NSRFy0>jkje zhYW}4YXuazA}RO#_8v-2Ny$6ck*!Nb5q)by-&aUVrwI`NrXDF>SQxQ{U`WyhQBWBJ zz{m=~Jvjq-M*4~Gw|^+dhY_0^!fPE5qBIkSQR9g$qdN>H$ZuwTc{3Og~-mrarpeargll2VMbuz++fHK7dUFz?V8GdN@ 0} - comme $\Delta = \Var{P.delta} > 0$ donc $P$ a deux racines - - \begin{eqnarray*} - x_1 & = & \frac{-b - \sqrt{\Delta}}{2a} = \frac{\Var{-P.b} - \sqrt{\Var{P.delta}}}{2 \times \Var{P.a}} = \Var{P.roots()[0] } \\ - x_2 & = & \frac{-b + \sqrt{\Delta}}{2a} = \frac{\Var{-P.b} + \sqrt{\Var{P.delta}}}{2 \times \Var{P.a}} = \Var{P.roots()[1] } - \end{eqnarray*} - - Les solutions de l'équation $\Var{P} = 0$ sont donc $\mathcal{S} = \left\{ \Var{min(P.roots())}; \Var{max(P.roots())} \right\}$ - - \Block{elif P.delta == 0} - Comme $\Delta = 0$ donc $P$ a deux racines - - \begin{eqnarray*} - x_1 = \frac{-b}{2a} = \Var{P.roots()[0]} \\ - \end{eqnarray*} - - La solution de $\Var{P} = 0$ est donc $\mathcal{S} = \left\{ \Var{P.roots()[0]}\right\}$ - - \Block{else} - Alors $\Delta = \Var{P.delta} < 0$ donc $P$ n'a pas de racine donc l'équation $\var{P} = 0$ n'a pas de solution. - - \Block{endif} + \Var{solveEquation(P)} \bigskip ~\dotfill @@ -74,45 +48,16 @@ On commence par se ramener à une équation de la forme $ax^2+bx+c = 0$. - \Block{set R = Polynom_deg2((P-Q)._coef)} + \Block{set R = P - Q} - \begin{eqnarray*} - \Var{P} = \Var{Q} & \Leftrightarrow & \Var{P} - (\Var{Q}) = 0 \\ + \begin{align*} + & & \Var{P} = \Var{Q} \\ \Var{R.explain() | calculus(name = "", sep = "\\Leftrightarrow", end = "= 0")} - \end{eqnarray*} + \end{align*} On cherche maintenant à résoudre l'équation $\Var{R} = 0$. - On commence par calculer le discriminant de $R(x) = \Var{R}$. - \begin{eqnarray*} - \Delta & = & b^2-4ac \\ - \Var{R.delta.explain()|calculus(name="\\Delta")} - \end{eqnarray*} - \Block{set Delta = R.delta} - - \Block{if R.delta > 0} - comme $\Delta = \Var{R.delta} > 0$ donc $R$ a deux racines - - \begin{eqnarray*} - x_1 & = & \frac{-b - \sqrt{\Delta}}{2a} = \frac{\Var{-R.b} - \sqrt{\Var{Delta}}}{2 \times \Var{R.a}} = \Var{R.roots()[0] } \\ - x_2 & = & \frac{-b + \sqrt{\Delta}}{2a} = \frac{\Var{-R.b} + \sqrt{\Var{Delta}}}{2 \times \Var{R.a}} = \Var{R.roots()[1] } - \end{eqnarray*} - - Les solutions de l'équation $\Var{R} = 0$ sont donc $\mathcal{S} = \left\{ \Var{min(R.roots())}; \Var{max(R.roots())} \right\}$ - - \Block{elif R.delta == 0} - Comme $\Delta = 0$ donc $R$ a deux racines - - \begin{eqnarray*} - x_1 = \frac{-b}{2a} = \Var{R.roots()[0]} \\ - \end{eqnarray*} - - La solution de $\Var{R} = 0$ est donc $\mathcal{S} = \left\{ \Var{R.roots()[0]}\right\}$ - - \Block{else} - Alors $\Delta = \Var{R.delta} < 0$ donc $R$ n'a pas de racine donc l'équation $\Var{R} = 0$ n'a pas de solution. - - \Block{endif} + \Var{solveEquation(R)} \end{document} diff --git a/example/tpl_corr_DM_0302.tex b/example/tpl_corr_DM_0302.tex new file mode 100644 index 0000000..1a3cc32 --- /dev/null +++ b/example/tpl_corr_DM_0302.tex @@ -0,0 +1,359 @@ +\documentclass[a4paper,10pt, table]{/media/documents/Cours/Prof/Enseignements/tools/style/classDS} +\usepackage{/media/documents/Cours/Prof/Enseignements/2014_2015} + +% Title Page +\titre{DM5} +% \seconde \premiereS \PSTMG \TSTMG +\classe{\premiereS} +\date{02 mars 2015} +%\duree{1 heure} +\sujet{\Var{infos.num}} +% DS DSCorr DM DMCorr Corr +\typedoc{DM} + +\printanswers + +\begin{document} + +\maketitle + +Le barème est donné à titre indicatif, il pourra être modifié. Vous rendrez le sujet avec la copie. + +\begin{questions} + + \question + Résoudre les équations suivantes + \Block{set P = Polynom_deg2.random(["{a}", "{b}", "{c}"], name = 'P')} + \Block{set Q = Polynom_deg2.random(["{a}", "{b}", "{c}"], name = 'Q')} + \begin{eqnarray*} + \Var{P} & > &0 \\ + \end{eqnarray*} + + \begin{solution} + On commence par calculer le discriminant de $\Var{P.name}(x) = \Var{P}$. + \begin{eqnarray*} + \Delta & = & b^2-4ac \\ + \Var{P.delta.explain()|calculus(name="\\Delta")} + \end{eqnarray*} + + \Block{if P.delta > 0} + comme $\Delta = \Var{P.delta} > 0$ donc $\Var{P.name}$ a deux racines + + \begin{eqnarray*} + x_1 & = & \frac{-b - \sqrt{\Delta}}{2a} = \frac{\Var{-P.b} - \sqrt{\Var{P.delta}}}{2 \times \Var{P.a}} = \Var{P.roots()[0] } \\ + x_2 & = & \frac{-b + \sqrt{\Delta}}{2a} = \frac{\Var{-P.b} + \sqrt{\Var{P.delta}}}{2 \times \Var{P.a}} = \Var{P.roots()[1] } + \end{eqnarray*} + + + \Block{elif P.delta == 0} + Comme $\Delta = 0$ donc $\Var{P.name}$ a deux racines + + \begin{eqnarray*} + x_1 = \frac{-b}{2a} = \Var{P.roots()[0]} \\ + \end{eqnarray*} + + + \Block{else} + Alors $\Delta = \Var{P.delta} < 0$ donc $\Var{P.name}$ n'a pas de racine. + + \Block{endif} + Comme $a = \Var{P.a}$, on en déduit le tableau de signe de $\Var{P.name}$ + \begin{center} + \begin{tikzpicture} + \tkzTabInit[espcl=2]% + {$x$/1, $P$/2}% + \Var{P.tbl_sgn_header()} + \Var{P.tbl_sgn()} + \end{tikzpicture} + \end{center} + On regarde maintenant où sont les $+$ dans le tableau de signe pour résoudre l'inéquation. + \end{solution} + + \begin{eqnarray*} + \Var{Q} & \leq &0 \\ + \end{eqnarray*} + \begin{solution} + On commence par calculer le discriminant de $Q(x) = \Var{Q}$. + \begin{eqnarray*} + \Delta & = & b^2-4ac \\ + \Var{Q.delta.explain()|calculus(name="\\Delta")} + \end{eqnarray*} + + \Block{if Q.delta > 0} + comme $\Delta = \Var{Q.delta} > 0$ donc $Q$ a deux racines + + \begin{eqnarray*} + x_1 & = & \frac{-b - \sqrt{\Delta}}{2a} = \frac{\Var{-Q.b} - \sqrt{\Var{Q.delta}}}{2 \times \Var{Q.a}} = \Var{Q.roots()[0] } \\ + x_2 & = & \frac{-b + \sqrt{\Delta}}{2a} = \frac{\Var{-Q.b} + \sqrt{\Var{Q.delta}}}{2 \times \Var{Q.a}} = \Var{Q.roots()[1] } + \end{eqnarray*} + + + \Block{elif Q.delta == 0} + Comme $\Delta = 0$ donc $Q$ a une racine + + \begin{eqnarray*} + x_1 = \frac{-b}{2a} = \Var{Q.roots()[0]} \\ + \end{eqnarray*} + + \Block{else} + Alors $\Delta = \Var{Q.delta} < 0$ donc $Q$ n'a pas de racine. + + \Block{endif} + Comme $a = \Var{Q.a}$, on en déduit le tableau de signe de $Q$ + \begin{center} + \begin{tikzpicture} + \tkzTabInit[espcl=2]% + {$x$/1, $Q$/2}% + \Var{Q.tbl_sgn_header()} + \Var{Q.tbl_sgn()} + \end{tikzpicture} + \end{center} + On regarde maintenant où sont les $-$ dans le tableau de signe pour résoudre l'inéquation. + \end{solution} + + \begin{eqnarray*} + \Var{P} & \geq & \Var{Q} + \end{eqnarray*} + + \Block{set R = P-Q} + + \begin{solution} + On commence par se ramener à une équation de la forme $ax^2 + bx + c \geq 0$. + \begin{eqnarray*} + \Var{P} \geq \Var{Q} & \Leftrightarrow & \Var{P} - (\Var{Q}) \geq 0 \\ + \Var{R.explain() | calculus(name = "", sep = "\\Leftrightarrow", end = "\\geq 0")} + \end{eqnarray*} + + \Block{set R = Polynom_deg2(R._coef)} + + Ensuite on étudie le signe de $R(X) = \Var{R}$. + \begin{eqnarray*} + \Delta & = & b^2-4ac \\ + \Var{R.delta.explain()|calculus(name="\\Delta")} + \end{eqnarray*} + + \Block{if R.delta > 0} + comme $\Delta = \Var{R.delta} > 0$ donc $R$ a deux racines + + \begin{eqnarray*} + x_1 & = & \frac{-b - \sqrt{\Delta}}{2a} = \frac{\Var{-R.b} - \sqrt{\Var{R.delta}}}{2 \times \Var{R.a}} = \Var{R.roots()[0] } \\ + x_2 & = & \frac{-b + \sqrt{\Delta}}{2a} = \frac{\Var{-R.b} + \sqrt{\Var{R.delta}}}{2 \times \Var{R.a}} = \Var{R.roots()[1] } + \end{eqnarray*} + + + \Block{elif R.delta == 0} + Comme $\Delta = 0$ donc $R$ a une racine + + \begin{eqnarray*} + x_1 = \frac{-b}{2a} = \Var{R.roots()[0]} \\ + \end{eqnarray*} + + + \Block{else} + Alors $\Delta = \Var{R.delta} < 0$ donc $R$ n'a pas de racine. + + \Block{endif} + Comme $a = \Var{R.a}$, on en déduit le tableau de signe de $R$ + \begin{center} + \begin{tikzpicture} + \tkzTabInit[espcl=2]% + {$x$/1, $R$/2}% + \Var{R.tbl_sgn_header()} + \Var{R.tbl_sgn()} + \end{tikzpicture} + \end{center} + On regarde maintenant où sont les $+$ dans le tableau de signe pour résoudre l'inéquation. + + + \end{solution} + + + \question + Tracer le tableau de variation des fonctions suivantes \textit{(Vous pouvez utiliser les nombres à virgules)} + \Block{set f = Polynom.random(["{a}", "{b}", "{c}", "{d}"], name = 'f')} + \Block{set P = f} + \begin{parts} + \part $f:x\mapsto \Var{P}$ + \begin{solution} + Pour avoir les variations de $\Var{P.name}$, il faut connaître le signe de sa dérivé. On dérive $P$ + \Block{set P1 = P.derivate()} + \begin{eqnarray*} + \Var{P1.explain() | calculus(name = P1.name + "(x)", sep = "=", end = "")} + \end{eqnarray*} + \Block{set P1 = Polynom_deg2(P1._coef, name = P1.name)} + On étudie le signe de $P'$ + + Ensuite on étudie le signe de $\Var{P1.name}(x) = \Var{P1}$. + \begin{eqnarray*} + \Delta & = & b^2-4ac \\ + \Var{P1.delta.explain()|calculus(name="\\Delta")} + \end{eqnarray*} + + \Block{if P1.delta > 0} + comme $\Delta = \Var{P1.delta} > 0$ donc $\Var{P1.name}$ a deux racines + + \begin{eqnarray*} + x_1 & = & \frac{-b - \sqrt{\Delta}}{2a} = \frac{\Var{-P1.b} - \sqrt{\Var{P1.delta}}}{2 \times \Var{P1.a}} = \Var{P1.roots()[0] } \\ + x_2 & = & \frac{-b + \sqrt{\Delta}}{2a} = \frac{\Var{-P1.b} + \sqrt{\Var{P1.delta}}}{2 \times \Var{P1.a}} = \Var{P1.roots()[1] } + \end{eqnarray*} + + + \Block{elif P1.delta == 0} + Comme $\Delta = 0$ donc $\Var{P1.name}$ a une racine + + \begin{eqnarray*} + x_1 = \frac{-b}{2a} = \Var{P1.roots()[0]} \\ + \end{eqnarray*} + + + \Block{else} + Alors $\Delta = \Var{P1.delta} < 0$ donc $\Var{P1.name}$ n'a pas de racine. + + \Block{endif} + Comme $a = \Var{P1.a}$, on en déduit le tableau de signe de $\Var{P1.name}$ + \begin{center} + \begin{tikzpicture} + \tkzTabInit[espcl=2]% + {$x$/1, Signe de $\Var{P1.name} $/2}% + \Var{P1.tbl_sgn_header()} + \Var{P1.tbl_sgn()} + \end{tikzpicture} + \end{center} + + \end{solution} + + \Block{set g = Polynom.random(["{a}", "{b}", "{c}", "{d}"], name = 'g')} + \Block{set P = g} + \part $g:x\mapsto \Var{P}$ + + \begin{solution} + Pour avoir les variations de $\Var{P.name}$, il faut connaître le signe de sa dérivé. On dérive $P$ + \Block{set P1 = P.derivate()} + \begin{eqnarray*} + \Var{P1.explain() | calculus(name = P1.name + "(x)", sep = "=", end = "")} + \end{eqnarray*} + \Block{set P1 = Polynom_deg2(P1._coef, name = P1.name)} + On étudie le signe de $P'$ + + Ensuite on étudie le signe de $\Var{P1.name}(x) = \Var{P1}$. + \begin{eqnarray*} + \Delta & = & b^2-4ac \\ + \Var{P1.delta.explain()|calculus(name="\\Delta")} + \end{eqnarray*} + + \Block{if P1.delta > 0} + comme $\Delta = \Var{P1.delta} > 0$ donc $\Var{P1.name}$ a deux racines + + \begin{eqnarray*} + x_1 & = & \frac{-b - \sqrt{\Delta}}{2a} = \frac{\Var{-P1.b} - \sqrt{\Var{P1.delta}}}{2 \times \Var{P1.a}} = \Var{P1.roots()[0] } \\ + x_2 & = & \frac{-b + \sqrt{\Delta}}{2a} = \frac{\Var{-P1.b} + \sqrt{\Var{P1.delta}}}{2 \times \Var{P1.a}} = \Var{P1.roots()[1] } + \end{eqnarray*} + + + \Block{elif P1.delta == 0} + Comme $\Delta = 0$ donc $\Var{P1.name}$ a une racine + + \begin{eqnarray*} + x_1 = \frac{-b}{2a} = \Var{P1.roots()[0]} \\ + \end{eqnarray*} + + + \Block{else} + Alors $\Delta = \Var{P1.delta} < 0$ donc $\Var{P1.name}$ n'a pas de racine. + + \Block{endif} + Comme $a = \Var{P1.a}$, on en déduit le tableau de signe de $\Var{P1.name}$ + \begin{center} + \begin{tikzpicture} + \tkzTabInit[espcl=2]% + {$x$/1, Signe de $\Var{P1.name} $/2}% + \Var{P1.tbl_sgn_header()} + \Var{P1.tbl_sgn()} + \end{tikzpicture} + \end{center} + + \end{solution} + + \Block{set R = Polynom.random(["{a}", "{b}", "{c}"])} + \part $h:x\mapsto \Var{R} - f(x)$ + + \Block{set h = R - f} + \Block{do h.give_name('h')} + + \begin{solution} + On commence par simplifier l'expression de $h$ + \begin{eqnarray*} + h(x) & = & \Var{R} - f(x) \\ + \Var{h.explain() | calculus(name = h.name + "(x)", sep = "=", end = "")} + \end{eqnarray*} + + \Block{set P = h} + Pour avoir les variations de $\Var{P.name}$, il faut connaître le signe de sa dérivé. On dérive $P$ + \Block{set P1 = P.derivate()} + \begin{eqnarray*} + \Var{P1.explain() | calculus(name = P1.name + "(x)", sep = "=", end = "")} + \end{eqnarray*} + \Block{set P1 = Polynom_deg2(P1._coef, name = P1.name)} + On étudie le signe de $P'$ + + Ensuite on étudie le signe de $\Var{P1.name}(x) = \Var{P1}$. + \begin{eqnarray*} + \Delta & = & b^2-4ac \\ + \Var{P1.delta.explain()|calculus(name="\\Delta")} + \end{eqnarray*} + + \Block{if P1.delta > 0} + comme $\Delta = \Var{P1.delta} > 0$ donc $\Var{P1.name}$ a deux racines + + \begin{eqnarray*} + x_1 & = & \frac{-b - \sqrt{\Delta}}{2a} = \frac{\Var{-P1.b} - \sqrt{\Var{P1.delta}}}{2 \times \Var{P1.a}} = \Var{P1.roots()[0] } \\ + x_2 & = & \frac{-b + \sqrt{\Delta}}{2a} = \frac{\Var{-P1.b} + \sqrt{\Var{P1.delta}}}{2 \times \Var{P1.a}} = \Var{P1.roots()[1] } + \end{eqnarray*} + + + \Block{elif P1.delta == 0} + Comme $\Delta = 0$ donc $\Var{P1.name}$ a une racine + + \begin{eqnarray*} + x_1 = \frac{-b}{2a} = \Var{P1.roots()[0]} \\ + \end{eqnarray*} + + + \Block{else} + Alors $\Delta = \Var{P1.delta} < 0$ donc $\Var{P1.name}$ n'a pas de racine. + + \Block{endif} + Comme $a = \Var{P1.a}$, on en déduit le tableau de signe de $\Var{P1.name}$ + \begin{center} + \begin{tikzpicture} + \tkzTabInit[espcl=2]% + {$x$/1, Signe de $\Var{P1.name} $/2}% + \Var{P1.tbl_sgn_header()} + \Var{P1.tbl_sgn()} + \end{tikzpicture} + \end{center} + + \end{solution} + \end{parts} + + \question + Appliquer l'algorithme de tri vu en cours à la suite suivante + \begin{center} + \begin{tabular}{|*{6}{c|}} + \hline + 6914 & 6851 & 6532 & 6884 & 6164 & 6495 \\ + \hline + \end{tabular} + + \end{center} + + +\end{questions} + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: + diff --git a/example/tpl_example.pdf b/example/tpl_example.pdf new file mode 100644 index 0000000000000000000000000000000000000000..49876de6042d8f56f391a78eb4d2a7cb7ccc12f7 GIT binary patch literal 100735 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zgih=Ng@ueJjQ+wNRtGLJnkKAhWK%E(?giy%sAKHVbWmu_QfXVH%(h7TSd!AOB(16^ zPao#0HJ#{KbE0#PM#rKK7tbl3FJ5kU(wVkSYoqm!q(@)MRFCAoT_rew@oK-esjD5A zMTbwmvnNKkX#TESNq+O*E3A+I!)Uls-~wi-7ndX!l~fb~!`aZlz}VP=OI6j?-;E0Z DX<_$* literal 0 HcmV?d00001 diff --git a/example/tpl_play.tex b/example/tpl_play.tex new file mode 100644 index 0000000..fc76474 --- /dev/null +++ b/example/tpl_play.tex @@ -0,0 +1,32 @@ +\documentclass[a4paper,10pt]{article} +\RequirePackage[utf8x]{inputenc} +\RequirePackage[francais]{babel} +\RequirePackage{amssymb} +\RequirePackage{amsmath} +\RequirePackage{amsfonts} +\RequirePackage{subfig} +\RequirePackage{graphicx} +\RequirePackage{color} + +% Title Page +\title{Calcul littéral et statistiques} +\date{\today} + +\begin{document} +\maketitle + +\Block{set L = [1, 4, 5, 6]} + +\Block{for i in L | shuffle} + \Var{i} + +\Block{endfor} + + + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/lib/call_litt.py b/lib/call_litt.py deleted file mode 100644 index 9437ac3..0000000 --- a/lib/call_litt.py +++ /dev/null @@ -1,131 +0,0 @@ -#!/usr/bin/env python -# encoding: utf-8 - -from random import randint, uniform -from math import sqrt -from jinja2 import Template - -""" -Generate expression for litteral calculous - -3 types of expression (a, b, c != 0, 1) - 1 -> ax + b and eval for x != -b / a - 2 -> ax(bx + c) and eval for x != 0 and x != -c / b - 3 -> ax^2 + b and eval for x != +-sqrt(b/a) (if a and b have same sign) -""" - -def gene_type1(min_ = -10, max_ = 10): - """@todo: Docstring for gene_type1 - - :param min_: @todo - :param max_: @todo - :returns: @todo - - """ - a, b = 0, 0 - while (b in [0]) or (a in [0, 1]): - a = randint(min_, max_) - b = randint(min_, max_) - - return "{}x + {}".format(a,b), [-b/a] - -def gene_type2(min_, max_): - """@todo: Docstring for gene_type2 - - :param min_: @todo - :param max_: @todo - :returns: @todo - - """ - a, b, c = 0, 0, 0 - while (a in [0, 1]) or (b in [0, 1]) or c in [0]: - a = randint(min_, max_) - b = randint(min_, max_) - c = randint(min_, max_) - - return "{}x({}x + {})".format(a,b,c), [0, -c/b] - -def gene_type3(min_ = -10, max_ = 10): - """@todo: Docstring for gene_type3 - - :param min_: @todo - :param max_: @todo - :returns: @todo - - """ - a, b = 0, 0 - while (b in [0]) or (a in [0, 1]): - a = randint(min_, max_) - b = randint(min_, max_) - - if a*(-b) > 0: - VI = [-sqrt(-b/a), sqrt(-b/a)] - else: - VI = [] - - return "{}x^2 + {}".format(a,b), VI - -def get_goodX(VI, approx = 0, min_ = -10, max_ = 10): - """@todo: Docstring for get_goodX - - :param VI: @todo - :returns: @todo - - """ - x = uniform(min_, max_) - if approx == 0: - x = int(x) - else: - x = round(x,approx) - while x in VI: - x = uniform(min_, max_) - if approx == 0: - x = int(x) - else: - x = round(x,approx) - - return x - - - -def fullExo(min_ = -10 , max_ = 10): - """Generate the whole exo - - :param min_: @todo - :param max_: @todo - :returns: @todo - - """ - template = Template(""" -\\begin{equation*} -$A = {{type1}}$ \\qquad $B = {{type2}}$ \\qquad $C = {{type3}} -\\end{equation*} - -Évaluer $A$, $B$ et $C$ pour $x = {{x1}}$ puis $x = {{x2}}$""") - - type1, VI1 = gene_type1(min_, max_) - type2, VI2 = gene_type2(min_, max_) - type3, VI3 = gene_type3(min_, max_) - - VI = VI1 + VI2 + VI3 - - x1, x2 = get_goodX(VI), get_goodX(VI, approx = 1) - - info = {"type1": type1, "type2": type2, "type3": type3, "x1":x1, "x2":x2} - - exo = template.render(**info) - - return exo - -if __name__ == '__main__': - print(fullExo()) - - - - - - -# ----------------------------- -# Reglages pour 'vim' -# vim:set autoindent expandtab tabstop=4 shiftwidth=4: -# cursor: 16 del diff --git a/lib/rd_frac.py b/lib/rd_frac.py deleted file mode 100644 index 9dd3f1b..0000000 --- a/lib/rd_frac.py +++ /dev/null @@ -1,133 +0,0 @@ -#!/usr/bin/env python -# encoding: utf-8 - -from random import randint, random -from random_expression import RdExpression -from arithmetic import gcd -from renders import tex_render - - - -"""Classe which generate randomly fractions calculus - -Types of sums - -add1 -> b / a + c / a -add2 -> b / a + c / ka -add3 -> b / a + e / d -add4 -> f + b / a -add5 -> b / a + f - -where: - a integer > 2 - b integer different from 0 (could be coprime with a) - c integer different from 0 (could be coprime with a or ka) - e integer different from 0 (could be coprime with d) - d integer > 2 ( a not divisible by d and d not divisible by a) - k integer > 2 - f integer different from 0 - - -Types of multiplications - -mult1 -> a x b / c -mult2 -> a x b / c + d / c -mult3 -> a x b / c + d / e -mult4 -> e / f x g / h >>> TODO -mult5 -> i / j x k / l - -where: - a integer different from -1, 0, 1 - b integer different from 0 - c integer different from 0 and 1 (could be coprime with b) - d integer different from 0 - e, g integer different from 0 - f, g integer different from 0 and 1 such that e*g is coprime with f*h - i, k integer different from 0 - j, l integer different from 0 and 1 such that i*k and j*l have divisor in common - - -Types of divisions - -div1 -> a / b : c / d - -where: - a integer different from 0 - b integer different from 0, 1 - c integer different from 0 - d integer different from 0 - - - -#Signs can be mod - - -""" - -add1 = RdExpression("{b} / {a} + {c} / {a}", \ - conditions = ["{a} > 2", "{b} != 0","{c} != 0"]) - -add2 = RdExpression("{b} / {a} + {c} / {k*a}", \ - conditions = ["{a} > 2","{k} > 2", "{b} != 0","{c} != 0"]) - -add3 = RdExpression("{b} / {a} + {e} / {d}", \ - conditions = ["{a} not in [0,1]", "{e} not in [0,1]", "{b} != 0","{d} not in [0,1]"]) - -add4 = RdExpression("{b} / {a} + {f}", \ - conditions = ["{a} > 2", "{b} != 0", "{f} != 0"]) - -add5 = RdExpression("{f} + {b} / {a}", \ - conditions = ["{a} > 2", "{b} != 0", "{f} != 0"]) - - -mult1 = RdExpression("{a} * {b} / {c}",\ - conditions = ["{a} not in [-1, 0, 1]", "{b} != 0", "{c} not in [0,1]"]) - -mult2 = RdExpression("{a} * {b} / {c} + {d} / {c}",\ - conditions = ["{a} not in [-1, 0, 1]", "{b} != 0", "{c} not in [0,1]", \ - "{d} != 0"]) - -mult3 = RdExpression("{a} * {b} / {c} + {d} / {e}",\ - conditions = ["{a} not in [-1, 0, 1]", "{b} != 0", "{c} not in [0,1]", \ - "{d} != 0", "{e} not in [0,1]", "{c} != {e}"]) - -#mult4 = RdExpression("{e} / {f} * {g} / {h}", \ -# conditions = ["{e} != 0", "{g} != 0", "{f} != 0", "{g} != 0", \ -# "gcd({e*g}, {f*h}) == 1"]) - -mult5 = RdExpression("{e} / {f} * {g} / {h}", \ - conditions = ["{e} != 0", "{g} != 0", "{f} not in [0, 1]", "{h} not in [0, 1]"]) - #"gcd({e*g}, {f*h}) != 1"]) - -div1 = RdExpression("{a} / {b} : {c} / {d}", \ - conditions = ["{a} not in [0]", "{b} not in [0,1]", "{c} != 0","{d} not in [0]"]) - -frac = {"add1": add1,\ - "add2": add2,\ - "add3": add3,\ - "add4": add4,\ - "add5": add5, \ - "mult1": mult1,\ - "mult2": mult2,\ - "mult3": mult2,\ - #"mult4": mult2,\ - "mult5": mult5, \ - "div1": div1 } - -if __name__ == '__main__': - print(add1()) - print(add2()) - print(add3()) - print(add4()) - print(add5()) - - print(mult1()) - print(mult2()) - print(mult3()) - #print(mult4()) - print(mult5()) - - print(div1()) -# Reglages pour 'vim' -# vim:set autoindent expandtab tabstop=4 shiftwidth=4: -# cursor: 16 del