156 lines
4.4 KiB
TeX
156 lines
4.4 KiB
TeX
\documentclass[a5paper,10pt]{article}
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\usepackage{myXsim}
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\usepackage{tasks}
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% Title Page
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\title{DM1 \hfill \Var{Nom}}
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\tribe{TST}
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\date{Toussain 2020}
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\begin{document}
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\maketitle
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\begin{exercise}[subtitle={Fractions}]
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Faire les calculs avec les fraction suivants
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\begin{multicols}{3}
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\begin{enumerate}
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%- set a = Expression.random("{a} / {b} - {c} / {b}", ["b > 1"])
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\item $A = \Var{a}$
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%- set b = Expression.random("{a} / {b} - {c} / {k*b}", ["b > 1", "k>1"])
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\item $B = \Var{b}$
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%- set c = Expression.random("{a} / {b} + {c} / {b-1}", ["b > 1"])
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\item $C = \Var{c}$
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%- set d = Expression.random("{a} / {b} + {c}", ["b > 1"])
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\item $D = \Var{d}$
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%- set e = Expression.random("{a} / {b} * {c} / {b-1}", ["b > 1"])
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\item $E = \Var{e}$
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%- set f = Expression.random("{a} / {b} * {c}", ["b > 1"])
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\item $F = \Var{f}$
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\end{enumerate}
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\end{multicols}
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\end{exercise}
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\begin{solution}
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\begin{enumerate}
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\item
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\[
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\Var{a.simplify().explain() | join('=')}
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\]
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\item
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\[
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\Var{b.simplify().explain() | join('=')}
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\]
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\item
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\[
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\Var{c.simplify().explain() | join('=')}
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\]
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\item
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\[
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\Var{d.simplify().explain() | join('=')}
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\]
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\item
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\[
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\Var{e.simplify().explain() | join('=')}
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\]
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\item
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\[
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\Var{f.simplify().explain() | join('=')}
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\]
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\end{enumerate}
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\end{solution}
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\begin{exercise}[subtitle={Développer réduire}]
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Développer puis réduire les expressions suivantes
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\begin{multicols}{2}
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\begin{enumerate}
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%- set a = Expression.random("({a}x + {b})*({c}x + {b})")
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\item $A = \Var{a}$
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%- set b = Expression.random("({a}x + {b})*({c}x + {b})")
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\item $B = \Var{b}$
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%- set c = Expression.random("({a}x + {b})^2")
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\item $C = \Var{c}$
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%- set d = Expression.random("{c} + x*({a}x + {b})")
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\item $D = \Var{d}$
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%- set e = Expression.random("{c}*x^2 + x*({a}x + {b})")
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\item $E = \Var{e}$
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%- set f = Expression.random("{a}(x+{b})(x+{c})")
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\item $F = \Var{f}$
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\end{enumerate}
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\end{multicols}
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\end{exercise}
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\begin{solution}
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\begin{enumerate}
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\item
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\begin{align*}
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A &= \Var{a.simplify().explain() | join('\\\\&= ')}
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\end{align*}
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\item
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\begin{align*}
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B &= \Var{b.simplify().explain() | join('\\\\&= ')}
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\end{align*}
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\item
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\begin{align*}
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C &= \Var{c.simplify().explain() | join('\\\\&= ')}
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\end{align*}
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\item
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\begin{align*}
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D &= \Var{d.simplify().explain() | join('\\\\&= ')}
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\end{align*}
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\item
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\begin{align*}
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E &= \Var{e.simplify().explain() | join('\\\\&= ')}
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\end{align*}
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\item
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\begin{align*}
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F &= \Var{f.simplify().explain() | join('\\\\&= ')}
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\end{align*}
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\end{enumerate}
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\end{solution}
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\begin{exercise}[subtitle={Étude de fonctions}]
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%- set f = Expression.random("{a}(x-{b})(x-{c})")
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Soit $f(x) = \Var{f.simplify()}$ une fonction définie sur $\R$.
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\begin{enumerate}
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\item Calculer les valeurs suivantes
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\[
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f(1) \qquad f(-2)
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\]
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\item Dériver la fonction $f$
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\item Étudier le signe de $f'$ puis en déduire les variations de $f$.
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\item Est-ce que $f$ admet un maximum? un minimum? Calculer sa valeur.
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\end{enumerate}
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\end{exercise}
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\begin{solution}
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\begin{enumerate}
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\item On remplace $x$ par les valeurs demandées
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\[
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f(1) = \Var{f.simplify()(1).explain() | join('=')}
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\]
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\[
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f(-1) = \Var{f.simplify()(-1).explain() | join('=')}
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\]
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\item Pas de solutions automatiques.
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\item Pas de solutions automatiques.
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\end{enumerate}
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\end{solution}
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%\printsolutionstype{exercise}
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\end{document}
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%%% Local Variables:
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%%% mode: latex
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%%% TeX-master: "master"
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%%% End:
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