2017-2018/3e/DM/DM noel/exo_calculs.tex

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2017-12-06 08:34:44 +00:00
\begin{enumerate}
\item Compléter les pointillés pour qu'il y est bien égalité.
\hspace{-1cm}
\begin{center}
\Block{set a,b,c = random_str("{a},{b},{b*c}", conditions = ["{a} != {b}", "{a} > 1", "{b}>1","{c}>1"]).split(',')}%
$\dfrac{\Var{a}}{\Var{b}} = \dfrac{\ldots}{\Var{c}}$
\hfill
\Block{set a,b,c = random_str("{a},{b},{b*c}", conditions = ["{a} != {b}", "{a} > 1", "{b}>1","{c}>1"]).split(',')}%
$\dfrac{\Var{a}}{\Var{b}} = \dfrac{\ldots}{\Var{c}}$
\hfill
\Block{set a,b,c = random_str("{a},{b},{b*c}", conditions = ["{a} != {b}", "{a} > 1", "{b}>1","{c}>1"]).split(',')}%
$\dfrac{\cdots}{\Var{c}} = \dfrac{\Var{a}}{\Var{b}}$
\hfill
\Block{set a,b,c = random_str("{a},{b},{a*c}", conditions = ["{a} != {b}", "{a} > 1", "{b}>1","{c}>1"]).split(',')}%
$\dfrac{\Var{a}}{\Var{b}} = \dfrac{\Var{c}}{\cdots}$
\end{center}
\vfill
\item Faire les calculs suivants en détaillant les étapes (penser à simplifier les fractions quand c'est possible).
\begin{multicols}{2}
\begin{enumerate}
\Block{set e = Expression.random("{a} / {b} + {c} / {b}", ["{b} > 1", "{a} > 0", "{c} > 0"])}
\item $A = \Var{e}$
\begin{solution}
\begin{eqnarray*}
\Var{e.simplify().explain() | calculus(name = "A")}
\end{eqnarray*}
\end{solution}
\Block{set e = Expression.random("{a} / {b} + {c} / {b}", ["{b} > 1"])}
\item $B = \Var{e}$
\begin{solution}
\begin{eqnarray*}
\Var{e.simplify().explain() | calculus(name = "B")}
\end{eqnarray*}
\end{solution}
\Block{set e = Expression.random("{a} / {b} + {c} / {d*b}", ["{b} > 1", "{c} > 1", "{d} > 1"])}
\item $C = \Var{e}$
\begin{solution}
\begin{eqnarray*}
\Var{e.simplify().explain() | calculus(name = "C")}
\end{eqnarray*}
\end{solution}
\Block{set e = Expression.random("{a} / {b} + {c} / {d*b}", ["{b} > 1", "{d} > 1"])}
\item $D = \Var{e}$
\begin{solution}
\begin{eqnarray*}
\Var{e.simplify().explain() | calculus(name = "D")}
\end{eqnarray*}
\end{solution}
\Block{set e = Expression.random("{a} / {b} * {c}", ["{b} > 1", "{a} > 0", "{c} > 1", "{c} != {b}"])}
\item $E = \Var{e}$
\begin{solution}
\begin{eqnarray*}
\Var{e.simplify().explain() | calculus(name = "E")}
\end{eqnarray*}
\end{solution}
\Block{set e = Expression.random("{a} / {b} * {c} / {d}", ["{b} > 1", "{a} > 0", "{c} > 0", "{d} > 1"])}
\item $F = \Var{e}$
\begin{solution}
\begin{eqnarray*}
\Var{e.simplify().explain() | calculus(name = "F")}
\end{eqnarray*}
\end{solution}
\end{enumerate}
\end{multicols}
\item Évaluer les expressions suivantes
\begin{multicols}{2}
\begin{enumerate}
\Block{set P = Polynom.random(degree = 1)}
\Block{set x = Expression.random("{a}")}
\item $\Var{P}$ en $x = \Var{x}$
\begin{solution}
\begin{eqnarray*}
\Var{P(x).explain()| calculus()}
\end{eqnarray*}
\end{solution}
\Block{set P = Polynom.random(degree = 2)}
\Block{set x = Expression.random("{a}")}
\item $\Var{P}$ en $x = \Var{x}$
\begin{solution}
\begin{eqnarray*}
\Var{P(x).explain()| calculus()}
\end{eqnarray*}
\end{solution}
\Block{set P = Expression.random("{a} x * (x - {b})")}
\Block{set x = Expression.random("{a}")}
\item $\Var{P}$ en $x = \Var{x}$
\Block{set P = Expression.random("({a}x + {b})({c} + {d}x)")}
\Block{set x = Expression.random("{a}")}
\item $\Var{P}$ en $x = \Var{x}$
\end{enumerate}
\end{multicols}
\item Réduire les expressions suivantes
\begin{multicols}{2}
\begin{enumerate}
\Block{set A = Expression.random("{a}x + {c} + {d}x", conditions = ["{a} not in [0,1]", "{d} not in [0,1]"])}
\item A = $\Var{A}$
\begin{solution}
\begin{eqnarray*}
\Var{A.simplify().explain() | calculus()}
\end{eqnarray*}
\end{solution}
\Block{set B = Expression.random("({b} - {a}) + {a}x + {c} + {d}x", conditions = ["{a} not in [0,1]", "{d} not in [0,1]"])}
\item B = $\Var{B}$
\begin{solution}
\begin{eqnarray*}
\Var{B.simplify().explain() | calculus()}
\end{eqnarray*}
\end{solution}
\Block{set C = Expression.random("{a}x^2 + {c} + {d}x + {d} + {a}x", conditions = ["{a} not in [0,1]", "{d} not in [0,1]"])}
\item C = $\Var{C}$
\begin{solution}
\begin{eqnarray*}
\Var{C.simplify().explain() | calculus()}
\end{eqnarray*}
\end{solution}
\Block{set D = Expression.random("{a}x^2 + {c}x^2 + {d}x + {d} + {a}x", conditions = ["{a} not in [0,1]", "{d} not in [0,1]"])}
\item D = $\Var{D}$
\begin{solution}
\begin{eqnarray*}
\Var{D.simplify().explain() | calculus()}
\end{eqnarray*}
\end{solution}
\end{enumerate}
\end{multicols}
\end{enumerate}