97 lines
2.7 KiB
TeX
97 lines
2.7 KiB
TeX
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% vim:ft=tex:
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%
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\documentclass[12pt]{article}
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\usepackage[utf8x]{inputenc}
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\usepackage[francais]{babel}
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\usepackage[T1]{fontenc}
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\usepackage{amssymb}
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\usepackage{amsmath}
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\usepackage{amsfonts}
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\title{
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Snippets pour Opytex \\
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Fonctions
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}
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\author{
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Benjamin Bertrand
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}
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\begin{document}
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\maketitle
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\section{Calculer des images}
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\begin{enumerate}
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%-set f = Expression.random("{a}*x^2 + {b}*x + {c}")
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\item $\forall x \in \mathbb{R} \qquad f(x) = \Var{f}$
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Solution:
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\begin{align*}
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f(0) &= \Var{f(0).explain() | join('=')} \\
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f(1) &= \Var{f(1).explain() | join('=')} \\
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f(2) &= \Var{f(2).explain() | join('=')} \\
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f({10}) &= \Var{f(10).explain() | join('=')} \\
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f({100}) &= \Var{f(100).explain() | join('=')}
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\end{align*}
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\end{enumerate}
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\section{Résolution d'équation du 2nd degré}
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%- macro solveEquation(P)
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On commence par calculer le discriminant de $P(x) = \Var{P}$.
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\begin{eqnarray*}
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\Delta & = & b^2-4ac \\
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\Var{P.delta.explain()|calculus(name="\\Delta")}
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\end{eqnarray*}
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\Block{if P.delta > 0}
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comme $\Delta = \Var{P.delta} > 0$ donc $P$ a deux racines
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\begin{eqnarray*}
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x_1 & = & \frac{-b - \sqrt{\Delta}}{2a} = \frac{\Var{-P.b} - \sqrt{\Var{P.delta}}}{2 \times \Var{P.a}} = \Var{P.roots[0] } \\
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x_2 & = & \frac{-b + \sqrt{\Delta}}{2a} = \frac{\Var{-P.b} + \sqrt{\Var{P.delta}}}{2 \times \Var{P.a}} = \Var{P.roots[1] }
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\end{eqnarray*}
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Les solutions de l'équation $\Var{P} = 0$ sont donc $\mathcal{S} = \left\{ \Var{P.roots[0]}; \Var{P.roots[1]} \right\}$
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\Block{elif P.delta == 0}
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Comme $\Delta = 0$ donc $P$ a une racine
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\begin{eqnarray*}
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x_1 = \frac{-b}{2a} = \frac{-\Var{P.b}}{2\times \Var{P.a}} = \Var{P.roots[0]} \\
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\end{eqnarray*}
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La solution de $\Var{P} = 0$ est donc $\mathcal{S} = \left\{ \Var{P.roots[0]}\right\}$
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\Block{else}
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Alors $\Delta = \Var{P.delta} < 0$ donc $P$ n'a pas de racine donc l'équation $\Var{P} = 0$ n'a pas de solution.
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\Block{endif}
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%- endmacro
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\begin{enumerate}
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%-set P = Expression.random("{a}*x^2 + {b}*x + {c}", ["b**2-4*a*c>0"])
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\item Étude du polynôme $P$, $\forall x \in \mathbb{R} \quad P(x) = \Var{P}$
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Solution:
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\Var{solveEquation(P)}
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%-set P = Expression.random("{a}*x^2 + {b}*x + {c}", ["b**2-4*a*c==0"])
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\item Étude du polynôme $P$, $\forall x \in \mathbb{R} \quad P(x) = \Var{P}$
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Solution:
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\Var{solveEquation(P)}
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%-set P = Expression.random("{a}*x^2 + {b}*x + {c}", ["b**2-4*a*c<0"])
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\item Étude du polynôme $P$, $\forall x \in \mathbb{R} \quad P(x) = \Var{P}$
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Solution:
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\Var{solveEquation(P)}
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\end{enumerate}
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\end{document}
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