763 lines
31 KiB
TeX
763 lines
31 KiB
TeX
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% Dérivation
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%-----------------------
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\question
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\begin{center}
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\textbf{Durée : 10min \hspace{3cm} Thème : Dérivation}
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\end{center}
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Calculer la dérivé des fonctions suivantes
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\begin{parts}
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\begin{multicols}{2}
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\Block{set A = Polynom.random(degree = 1, name = 'f')}
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\part $\Var{A.name}:x \mapsto \Var{A}$
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\begin{savedSolution}
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$ \Var{A.derivate().name}(x) = \Var{A.derivate()}$
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\end{savedSolution}
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\Block{set A = Polynom.random(degree = 1, name = 'g')}
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\part $\Var{A.name}:x \mapsto \Var{A}$
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\begin{savedSolution}
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$ \Var{A.derivate().name}(x) = \Var{A.derivate()}$
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\end{savedSolution}
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\Block{set A = Polynom.random(degree = 2, name = 'h')}
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\part $\Var{A.name}:x \mapsto \Var{A}$
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\begin{savedSolution}
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$ \Var{A.derivate().name}(x) = \Var{A.derivate()}$
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\end{savedSolution}
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\Block{set A = Polynom.random(degree = 2, name = 'i')}
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\part $\Var{A.name}:x \mapsto \Var{A}$
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\begin{savedSolution}
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$ \Var{A.derivate().name}(x) = \Var{A.derivate()}$
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\end{savedSolution}
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\Block{set A = Polynom.random(degree = 3, name = 'j')}
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\part $\Var{A.name}:x \mapsto \Var{A}$
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\begin{savedSolution}
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$ \Var{A.derivate().name}(x) = \Var{A.derivate()}$
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\end{savedSolution}
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\end{multicols}
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\end{parts}
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\question
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\begin{center}
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\textbf{Durée : 10min \hspace{3cm} Thème : Dérivation}
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\end{center}
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Calculer la dérivé des fonctions suivantes
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\begin{parts}
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\begin{multicols}{2}
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\Block{set A = Polynom.random(degree = 2, name = 'f')}
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\part $\Var{A.name}:x \mapsto \Var{A}$
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\begin{savedSolution}
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$ \Var{A.derivate().name}(x) = \Var{A.derivate()}$
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\end{savedSolution}
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\Block{set A = Polynom.random(degree = 1, name = 'g')}
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\part $\Var{A.name}:x \mapsto \Var{A}$
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\begin{savedSolution}
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$ \Var{A.derivate().name}(x) = \Var{A.derivate()}$
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\end{savedSolution}
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\Block{set A = Polynom.random(degree = 3, name = 'h')}
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\part $\Var{A.name}:x \mapsto \Var{A}$
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\begin{savedSolution}
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$ \Var{A.derivate().name}(x) = \Var{A.derivate()}$
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\end{savedSolution}
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\Block{set A = Polynom.random(degree = 3, name = 'i')}
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\part $\Var{A.name}:x \mapsto \Var{A}$
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\begin{savedSolution}
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$ \Var{A.derivate().name}(x) = \Var{A.derivate()}$
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\end{savedSolution}
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\Block{set A = Polynom.random(degree = 2, name = 'j')}
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\part $\Var{A.name}:x \mapsto \Var{A}$
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\begin{savedSolution}
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$ \Var{A.derivate().name}(x) = \Var{A.derivate()}$
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\end{savedSolution}
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\end{multicols}
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\end{parts}
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\question
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\begin{center}
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\textbf{Durée : 20min \hspace{3cm} Thème : Dérivation}
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\end{center}
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Calculer la dérivé des fonctions suivantes
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\begin{parts}
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\begin{multicols}{2}
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\Block{set A = Polynom.random(degree = 1, name = 'f')}
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\part $\Var{A.name}:x \mapsto \dfrac{1}{\Var{A}}$
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\begin{savedSolution}
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$ \Var{A.derivate().name}(x) = \dfrac{\Var{(-A.derivate())}}{\Var{Expression([A,2,"^"])}}$
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\end{savedSolution}
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\Block{set A = Polynom.random(degree = 1, name = 'g')}
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\part $\Var{A.name}:x \mapsto \dfrac{1}{\Var{A}}$
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\begin{savedSolution}
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$ \Var{A.derivate().name}(x) = \dfrac{\Var{(-A.derivate())}}{\Var{Expression([A,2,"^"])}}$
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\end{savedSolution}
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\Block{set A = Polynom.random(degree = 2, name = 'h')}
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\part $\Var{A.name}:x \mapsto \dfrac{1}{\Var{A}}$
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\begin{savedSolution}
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$ \Var{A.derivate().name}(x) = \dfrac{\Var{(-A.derivate())}}{\Var{Expression([A,2,"^"])}}$
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\end{savedSolution}
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\Block{set A = Polynom.random(degree = 2, name = 'i')}
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\part $\Var{A.name}:x \mapsto \dfrac{1}{\Var{A}}$
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\begin{savedSolution}
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$ \Var{A.derivate().name}(x) = \dfrac{\Var{(-A.derivate())}}{\Var{Expression([A,2,"^"])}}$
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\end{savedSolution}
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\Block{set A = Polynom.random(degree = 1, name = 'j')}
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\Block{set B = Polynom.random(degree = 1, name = 'f')}
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\part $\Var{A.name}:x \mapsto \dfrac{\Var{A}}{\Var{B}}$
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\begin{savedSolution}
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\Block{set num = Expression([A.derivate(), B, "*", A, B.derivate(), "*", "-"]).simplify()}%
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$ \Var{A.derivate().name}(x) = \dfrac{\Var{num}}{\Var{Expression([B,2,"^"])}}$
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\end{savedSolution}
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\Block{set A = Polynom.random(degree = 1, name = 'k')}
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\Block{set B = Polynom.random(degree = 1, name = 'f')}
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\part $\Var{A.name}:x \mapsto \dfrac{\Var{A}}{\Var{B}}$
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\begin{savedSolution}
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\Block{set num = Expression([A.derivate(), B, "*", A, B.derivate(), "*", "-"]).simplify()}%
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$ \Var{A.derivate().name}(x) = \dfrac{\Var{num}}{\Var{Expression([B,2,"^"])}}$
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\end{savedSolution}
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\end{multicols}
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\end{parts}
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\question
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\begin{center}
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\textbf{Durée : 20min \hspace{3cm} Thème : Dérivation}
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\end{center}
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Calculer la dérivé des fonctions suivantes
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\begin{parts}
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\begin{multicols}{2}
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\Block{set A = Polynom.random(degree = 1, name = 'f')}
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\part $\Var{A.name}:x \mapsto \dfrac{1}{\Var{A}}$
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\begin{savedSolution}
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$ \Var{A.derivate().name}(x) = \dfrac{\Var{(-A.derivate())}}{\Var{Expression([A,2,"^"])}}$
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\end{savedSolution}
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\Block{set A = Polynom.random(degree = 1, name = 'g')}
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\part $\Var{A.name}:x \mapsto \dfrac{1}{\Var{A}}$
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\begin{savedSolution}
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$ \Var{A.derivate().name}(x) = \dfrac{\Var{(-A.derivate())}}{\Var{Expression([A,2,"^"])}}$
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\end{savedSolution}
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\Block{set A = Polynom.random(degree = 2, name = 'h')}
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\part $\Var{A.name}:x \mapsto \dfrac{1}{\Var{A}}$
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\begin{savedSolution}
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$ \Var{A.derivate().name}(x) = \dfrac{\Var{(-A.derivate())}}{\Var{Expression([A,2,"^"])}}$
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\end{savedSolution}
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\Block{set A = Polynom.random(degree = 2, name = 'i')}
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\part $\Var{A.name}:x \mapsto \dfrac{1}{\Var{A}}$
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\begin{savedSolution}
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$ \Var{A.derivate().name}(x) = \dfrac{\Var{(-A.derivate())}}{\Var{Expression([A,2,"^"])}}$
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\end{savedSolution}
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\Block{set A = Polynom.random(degree = 1, name = 'j')}
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\Block{set B = Polynom.random(degree = 1, name = 'f')}
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\part $\Var{A.name}:x \mapsto \dfrac{\Var{A}}{\Var{B}}$
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\begin{savedSolution}
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\Block{set num = Expression([A.derivate(), B, "*", A, B.derivate(), "*", "-"]).simplify()}%
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$ \Var{A.derivate().name}(x) = \dfrac{\Var{num}}{\Var{Expression([B,2,"^"])}}$
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\end{savedSolution}
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\Block{set A = Polynom.random(degree = 1, name = 'k')}
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\Block{set B = Polynom.random(degree = 1, name = 'f')}
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\part $\Var{A.name}:x \mapsto \dfrac{\Var{A}}{\Var{B}}$
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\begin{savedSolution}
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\Block{set num = Expression([A.derivate(), B, "*", A, B.derivate(), "*", "-"]).simplify()}%
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$ \Var{A.derivate().name}(x) = \dfrac{\Var{num}}{\Var{Expression([B,2,"^"])}}$
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\end{savedSolution}
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\end{multicols}
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\end{parts}
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% Polynome 2nd degre
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% ------------------
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\question
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\begin{center}
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\textbf{Durée : 20min \hspace{3cm} Thème : 2nd degré}
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\end{center}
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Tracer le tableau de signe des fonctions suivantes
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\begin{parts}
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\begin{multicols}{2}
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\Block{set f = Polynom.random(degree = 2, name = 'f', conditions = ["{b**2-4*a*c} >0"] )}
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\part $\Var{f.name} : x \mapsto \Var{f}$
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\begin{savedSolution}
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\begin{tikzpicture}
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\tkzTabInit[espcl=2]{$x$/1,Signe de $\Var{f.name}(x)$/2}\Var{f.tbl_sgn_header()}
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\Var{f.tbl_sgn()}
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\end{tikzpicture}
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\end{savedSolution}
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\Block{set f = Polynom.random(degree = 2, name = 'g', conditions = ["{b**2-4*a*c} == 0"])}
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\part $\Var{f.name} : x \mapsto \Var{f}$
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\begin{savedSolution}
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\begin{tikzpicture}
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\tkzTabInit[espcl=2]{$x$/1,Signe de $\Var{f.name}(x)$/2}\Var{f.tbl_sgn_header()}
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\Var{f.tbl_sgn()}
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\end{tikzpicture}
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\end{savedSolution}
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\Block{set f = Polynom.random(degree = 2, name = 'h', conditions = ["{b**2-4*a*c} <0"])}
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\part $\Var{f.name} : x \mapsto \Var{f}$
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\begin{savedSolution}
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\begin{tikzpicture}
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\tkzTabInit[espcl=2]{$x$/1,Signe de $\Var{f.name}(x)$/2}\Var{f.tbl_sgn_header()}
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\Var{f.tbl_sgn()}
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\end{tikzpicture}
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\end{savedSolution}
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\Block{set f = Polynom.random(degree = 2, name = 'i')}
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\part $\Var{f.name} : x \mapsto \Var{f}$
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\begin{savedSolution}
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\begin{tikzpicture}
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\tkzTabInit[espcl=2]{$x$/1,Signe de $\Var{f.name}(x)$/2}\Var{f.tbl_sgn_header()}
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\Var{f.tbl_sgn()}
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\end{tikzpicture}
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\end{savedSolution}
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\end{multicols}
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\end{parts}
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\question
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\begin{center}
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\textbf{Durée : 20min \hspace{3cm} Thème : 2nd degré}
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\end{center}
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Tracer le tableau de signe des fonctions suivantes
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\begin{parts}
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\begin{multicols}{2}
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\Block{set f = Polynom.random(degree = 2, name = 'f', conditions = ["{b**2-4*a*c} >0"] )}
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\part $\Var{f.name} : x \mapsto \Var{f}$
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\begin{savedSolution}
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\begin{tikzpicture}
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\tkzTabInit[espcl=2]{$x$/1,Signe de $\Var{f.name}(x)$/2}\Var{f.tbl_sgn_header()}
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\Var{f.tbl_sgn()}
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\end{tikzpicture}
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\end{savedSolution}
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\Block{set f = Polynom.random(degree = 2, name = 'g', conditions = ["{b**2-4*a*c} == 0"])}
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\part $\Var{f.name} : x \mapsto \Var{f}$
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\begin{savedSolution}
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\begin{tikzpicture}
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\tkzTabInit[espcl=2]{$x$/1,Signe de $\Var{f.name}(x)$/2}\Var{f.tbl_sgn_header()}
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\Var{f.tbl_sgn()}
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\end{tikzpicture}
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\end{savedSolution}
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\Block{set f = Polynom.random(degree = 2, name = 'h', conditions = ["{b**2-4*a*c} <0"])}
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\part $\Var{f.name} : x \mapsto \Var{f}$
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\begin{savedSolution}
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\begin{tikzpicture}
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\tkzTabInit[espcl=2]{$x$/1,Signe de $\Var{f.name}(x)$/2}\Var{f.tbl_sgn_header()}
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\Var{f.tbl_sgn()}
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\end{tikzpicture}
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\end{savedSolution}
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\Block{set f = Polynom.random(degree = 2, name = 'i')}
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\part $\Var{f.name} : x \mapsto \Var{f}$
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\begin{savedSolution}
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\begin{tikzpicture}
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\tkzTabInit[espcl=2]{$x$/1,Signe de $\Var{f.name}(x)$/2}\Var{f.tbl_sgn_header()}
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\Var{f.tbl_sgn()}
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\end{tikzpicture}
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\end{savedSolution}
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\end{multicols}
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\end{parts}
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\question
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\begin{center}
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\textbf{Durée : 20min \hspace{3cm} Thème : 2nd degré}
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\end{center}
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Résoudre les équations suivantes
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\begin{parts}
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\begin{multicols}{2}
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\Block{set f = Polynom.random(degree = 2, name = 'f', conditions = ["{b**2-4*a*c} >0"] )}
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\part $\Var{f} = 0$
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\begin{savedSolution}
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$\mathcal{S} = \left\{ \Var{f.roots() | join(";")} \right\}$
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\end{savedSolution}
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\Block{set f = Polynom.random(degree = 2, name = 'g', conditions = ["{b**2-4*a*c} == 0"])}
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\part $\Var{f} = 0$
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\begin{savedSolution}
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$\mathcal{S} = \left\{ \Var{f.roots()[0]} \right\}$
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\end{savedSolution}
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\Block{set f = Polynom.random(degree = 2, name = 'h', conditions = ["{b**2-4*a*c} <0"])}
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\part $\Var{f} = 0$
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\begin{savedSolution}
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Il n'y a pas de solution.
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\end{savedSolution}
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\Block{set f = Polynom.random(degree = 2, name = 'f', conditions = ["{b**2-4*a*c} >0"] )}
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\part $\Var{f} = 0$
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\begin{savedSolution}
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$\mathcal{S} = \left\{ \Var{f.roots() | join(";")} \right\}$
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\end{savedSolution}
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|
|
\end{multicols}
|
||
|
|
\end{parts}
|
||
|
|
|
||
|
|
\question
|
||
|
|
\begin{center}
|
||
|
|
\textbf{Durée : 20min \hspace{3cm} Thème : 2nd degré}
|
||
|
|
\end{center}
|
||
|
|
Résoudre les équations suivantes
|
||
|
|
\begin{parts}
|
||
|
|
\begin{multicols}{2}
|
||
|
|
|
||
|
|
\Block{set f = Polynom.random(degree = 2, name = 'f', conditions = ["{b**2-4*a*c} >0"] )}
|
||
|
|
\part $\Var{f} = 0$
|
||
|
|
\begin{savedSolution}
|
||
|
|
$\mathcal{S} = \left\{ \Var{f.roots() | join(";")} \right\}$
|
||
|
|
\end{savedSolution}
|
||
|
|
|
||
|
|
\Block{set f = Polynom.random(degree = 2, name = 'g', conditions = ["{b**2-4*a*c} == 0"])}
|
||
|
|
\part $\Var{f} = 0$
|
||
|
|
\begin{savedSolution}
|
||
|
|
$\mathcal{S} = \left\{ \Var{f.roots()[0]} \right\}$
|
||
|
|
\end{savedSolution}
|
||
|
|
|
||
|
|
\Block{set f = Polynom.random(degree = 2, name = 'h', conditions = ["{b**2-4*a*c} <0"])}
|
||
|
|
\part $\Var{f} = 0$
|
||
|
|
\begin{savedSolution}
|
||
|
|
Il n'y a pas de solution.
|
||
|
|
\end{savedSolution}
|
||
|
|
|
||
|
|
\Block{set f = Polynom.random(degree = 2, name = 'f', conditions = ["{b**2-4*a*c} >0"] )}
|
||
|
|
\part $\Var{f} = 0$
|
||
|
|
\begin{savedSolution}
|
||
|
|
$\mathcal{S} = \left\{ \Var{f.roots() | join(";")} \right\}$
|
||
|
|
\end{savedSolution}
|
||
|
|
|
||
|
|
\end{multicols}
|
||
|
|
\end{parts}
|
||
|
|
|
||
|
|
% Variations
|
||
|
|
% ----------
|
||
|
|
|
||
|
|
\question
|
||
|
|
\begin{center}
|
||
|
|
\textbf{Durée : 20min \hspace{3cm} Thème : Variations}
|
||
|
|
\end{center}
|
||
|
|
\Block{set f = Polynom.random(degree = 2, name = 'f')}
|
||
|
|
Soit $\Var{f.name}$ la fonction définie par
|
||
|
|
\begin{eqnarray*}
|
||
|
|
\Var{f.name} : x \mapsto \Var{f}
|
||
|
|
\end{eqnarray*}
|
||
|
|
|
||
|
|
\begin{parts}
|
||
|
|
\part Quel est le domaine de définition de $\Var{f.name}$? Quel est son domaine de dérivation?
|
||
|
|
\begin{savedSolution}
|
||
|
|
Domaine de définition: $\R$ \quad Domaine de dérivation: $\R$
|
||
|
|
\end{savedSolution}
|
||
|
|
\part Calculer la dérivée de $\Var{f.name}$.
|
||
|
|
\begin{savedSolution}
|
||
|
|
$\Var{f.derivate().name} (x) = \Var{f.derivate()}$
|
||
|
|
\end{savedSolution}
|
||
|
|
\part Déterminer le tableau de variations de $\Var{f.name}$.
|
||
|
|
\begin{savedSolution}
|
||
|
|
\begin{tikzpicture}
|
||
|
|
\tkzTabInit[espcl=2]{$x$/1,Variations de $\Var{f.name}(x)$/2}{$-\infty$, $\Var{f.alpha}$, $+\infty$}
|
||
|
|
\Var{f.tbl_variation()}
|
||
|
|
\end{tikzpicture}
|
||
|
|
\end{savedSolution}
|
||
|
|
\end{parts}
|
||
|
|
|
||
|
|
\question
|
||
|
|
\begin{center}
|
||
|
|
\textbf{Durée : 20min \hspace{3cm} Thème : Variations}
|
||
|
|
\end{center}
|
||
|
|
\Block{set f = Polynom.random(degree = 2, name = 'f')}
|
||
|
|
Soit $\Var{f.name}$ la fonction définie par
|
||
|
|
\begin{eqnarray*}
|
||
|
|
\Var{f.name} : x \mapsto \Var{f}
|
||
|
|
\end{eqnarray*}
|
||
|
|
|
||
|
|
\begin{parts}
|
||
|
|
\part Quel est le domaine de définition de $\Var{f.name}$? Quel est son domaine de dérivation?
|
||
|
|
\begin{savedSolution}
|
||
|
|
Domaine de définition: $\R$ \quad Domaine de dérivation: $\R$
|
||
|
|
\end{savedSolution}
|
||
|
|
\part Calculer la dérivée de $\Var{f.name}$.
|
||
|
|
\begin{savedSolution}
|
||
|
|
$\Var{f.derivate().name} (x) = \Var{f.derivate()}$
|
||
|
|
\end{savedSolution}
|
||
|
|
\part Déterminer le tableau de variations de $\Var{f.name}$.
|
||
|
|
\begin{savedSolution}
|
||
|
|
\begin{tikzpicture}
|
||
|
|
\tkzTabInit[espcl=2]{$x$/1,Variations de $\Var{f.name}(x)$/2}{$-\infty$, $\Var{f.alpha}$, $+\infty$}
|
||
|
|
\Var{f.tbl_variation()}
|
||
|
|
\end{tikzpicture}
|
||
|
|
\end{savedSolution}
|
||
|
|
\end{parts}
|
||
|
|
|
||
|
|
\question
|
||
|
|
\begin{center}
|
||
|
|
\textbf{Durée : 20min \hspace{3cm} Thème : Variations}
|
||
|
|
\end{center}
|
||
|
|
\Block{set f = Polynom.random(degree = 3, name = 'f', conditions = ["{a}>0", "{4*b**2-12*a*c}>0"])}
|
||
|
|
Soit $\Var{f.name}$ la fonction définie par
|
||
|
|
\begin{eqnarray*}
|
||
|
|
\Var{f.name} : x \mapsto \Var{f}
|
||
|
|
\end{eqnarray*}
|
||
|
|
|
||
|
|
\begin{parts}
|
||
|
|
\part Quel est le domaine de définition de $\Var{f.name}$? Quel est son domaine de dérivation?
|
||
|
|
\begin{savedSolution}
|
||
|
|
Domaine de définition: $\R$ \quad Domaine de dérivation: $\R$
|
||
|
|
\end{savedSolution}
|
||
|
|
\part Calculer la dérivée de $\Var{f.name}$.
|
||
|
|
\begin{savedSolution}
|
||
|
|
\Block{set fp = f.derivate()}%
|
||
|
|
$\Var{fp.name} (x) = \Var{fp}$
|
||
|
|
\end{savedSolution}
|
||
|
|
\part Déterminer le tableau de variations de $\Var{f.name}$.
|
||
|
|
\begin{savedSolution}
|
||
|
|
\begin{tikzpicture}
|
||
|
|
\tkzTabInit[espcl=2.5, lgt=3]{$x$/1,Variations de $\Var{f.name}(x)$/3}\Var{fp.tbl_sgn_header()}
|
||
|
|
\tkzTabVar{-/{}, +/{$f(\Var{fp.roots()[0]})$}, -/{$f(\Var{fp.roots()[1]})$}, +/{} }
|
||
|
|
\end{tikzpicture}
|
||
|
|
\end{savedSolution}
|
||
|
|
\end{parts}
|
||
|
|
|
||
|
|
\question
|
||
|
|
\begin{center}
|
||
|
|
\textbf{Durée : 20min \hspace{3cm} Thème : Variations}
|
||
|
|
\end{center}
|
||
|
|
\Block{set f = Polynom.random(degree = 3, name = 'f', conditions = ["{a}<0", "{4*b**2-12*a*c}>0"])}
|
||
|
|
Soit $\Var{f.name}$ la fonction définie par
|
||
|
|
\begin{eqnarray*}
|
||
|
|
\Var{f.name} : x \mapsto \Var{f}
|
||
|
|
\end{eqnarray*}
|
||
|
|
|
||
|
|
Déterminer les variations de $\Var{f.name}$.
|
||
|
|
\begin{savedSolution}
|
||
|
|
\Block{set fp = f.derivate()}%
|
||
|
|
\begin{tikzpicture}
|
||
|
|
\tkzTabInit[espcl=2.5, lgt=3]{$x$/1,Variations de $\Var{f.name}(x)$/3}\Var{fp.tbl_sgn_header()}
|
||
|
|
\tkzTabVar{+/{}, -/{$f(\Var{fp.roots()[0]})$}, +/{$f(\Var{fp.roots()[1]})$}, -/{} }
|
||
|
|
\end{tikzpicture}
|
||
|
|
\end{savedSolution}
|
||
|
|
|
||
|
|
\question
|
||
|
|
\begin{center}
|
||
|
|
\textbf{Durée : 20min \hspace{3cm} Thème : Variations}
|
||
|
|
\end{center}
|
||
|
|
\Block{set f = Polynom.random(degree = 3, name = 'f', conditions = ["{a}<0", "{4*b**2-12*a*c}<0"])}
|
||
|
|
Soit $\Var{f.name}$ la fonction définie par
|
||
|
|
\begin{eqnarray*}
|
||
|
|
\Var{f.name} : x \mapsto \Var{f}
|
||
|
|
\end{eqnarray*}
|
||
|
|
|
||
|
|
Déterminer les variations de $\Var{f.name}$.
|
||
|
|
\begin{savedSolution}
|
||
|
|
\begin{tikzpicture}
|
||
|
|
\tkzTabInit[espcl=2.5, lgt=3]{$x$/1,Variations de $\Var{f.name}(x)$/2}{$+\infty$, $-\infty$}
|
||
|
|
\tkzTabVar{+/{}, -/{} }
|
||
|
|
\end{tikzpicture}
|
||
|
|
\end{savedSolution}
|
||
|
|
|
||
|
|
\question
|
||
|
|
\begin{center}
|
||
|
|
\textbf{Durée : 20min \hspace{3cm} Thème : Variations}
|
||
|
|
\end{center}
|
||
|
|
\Block{set f = Polynom.random(degree = 3, name = 'f', conditions = ["{a}<0", "{4*b**2-12*a*c}==0"])}
|
||
|
|
Soit $\Var{f.name}$ la fonction définie par
|
||
|
|
\begin{eqnarray*}
|
||
|
|
\Var{f.name} : x \mapsto \Var{f}
|
||
|
|
\end{eqnarray*}
|
||
|
|
|
||
|
|
Déterminer les variations de $\Var{f.name}$.
|
||
|
|
\begin{savedSolution}
|
||
|
|
\Block{set fp = f.derivate()}%
|
||
|
|
\begin{tikzpicture}
|
||
|
|
\tkzTabInit[espcl=5, lgt=3]{$x$/1,Variations de $\Var{f.name}(x)$/2}{$+\infty$, $-\infty$}
|
||
|
|
\tkzTabVar{+/{}, -/{} }
|
||
|
|
\tkzTabVal{1}{2}{0.5}{$\Var{fp.roots()[0]}$}{$f(\Var{fp.roots()[0]})$}
|
||
|
|
\end{tikzpicture}
|
||
|
|
\end{savedSolution}
|
||
|
|
|
||
|
|
% Suites
|
||
|
|
% ------
|
||
|
|
|
||
|
|
\question
|
||
|
|
\begin{center}
|
||
|
|
\textbf{Durée : 20min \hspace{3cm} Thème : Suites}
|
||
|
|
\end{center}
|
||
|
|
Calculer les 3 premiers termes et le 10ième terme des 4 suites suivantes
|
||
|
|
\begin{multicols}{2}
|
||
|
|
\begin{parts}
|
||
|
|
\Block{set u = Polynom.random(degree = 2, letter = "n", name = "u")}
|
||
|
|
\part $\Var{u.name}_n = \Var{u}$
|
||
|
|
\begin{savedSolution}
|
||
|
|
$\Var{u.name}_0 = \Var{u(0)}$ \quad
|
||
|
|
$\Var{u.name}_1 = \Var{u(1)}$ \quad
|
||
|
|
$\Var{u.name}_2 = \Var{u(2)}$ \quad
|
||
|
|
$\Var{u.name}_{10} = \Var{u(10)}$ \quad
|
||
|
|
\end{savedSolution}
|
||
|
|
\Block{set u = Polynom.random(degree = 2, letter = "n", name = "v")}
|
||
|
|
\part $\Var{u.name}_n = \Var{u}$
|
||
|
|
\begin{savedSolution}
|
||
|
|
$\Var{u.name}_0 = \Var{u(0)}$ \quad
|
||
|
|
$\Var{u.name}_1 = \Var{u(1)}$ \quad
|
||
|
|
$\Var{u.name}_2 = \Var{u(2)}$ \quad
|
||
|
|
$\Var{u.name}_{10} = \Var{u(10)}$ \quad
|
||
|
|
\end{savedSolution}
|
||
|
|
\Block{set u = Polynom.random(degree = 3, letter = "n", name = "w")}
|
||
|
|
\part $\Var{u.name}_n = \Var{u}$
|
||
|
|
\begin{savedSolution}
|
||
|
|
$\Var{u.name}_0 = \Var{u(0)}$ \quad
|
||
|
|
$\Var{u.name}_1 = \Var{u(1)}$ \quad
|
||
|
|
$\Var{u.name}_2 = \Var{u(2)}$ \quad
|
||
|
|
$\Var{u.name}_{10} = \Var{u(10)}$ \quad
|
||
|
|
\end{savedSolution}
|
||
|
|
\Block{set u = Polynom.random(degree = 2, letter = "n", name = "l")}
|
||
|
|
\Block{set v = Polynom.random(degree = 2, letter = "n", name = "l")}
|
||
|
|
\part $\Var{u.name}_n = \dfrac{\Var{u}}{\Var{v}}$
|
||
|
|
\begin{savedSolution}
|
||
|
|
$\Var{u.name}_0 = \Var{Expression([u(0),v(0),'/']).simplify()}$ \quad
|
||
|
|
$\Var{u.name}_1 = \Var{Expression([u(1),v(1),'/']).simplify()}$ \quad
|
||
|
|
$\Var{u.name}_2 = \Var{Expression([u(2),v(2),'/']).simplify()}$ \quad
|
||
|
|
$\Var{u.name}_{10} = \Var{Expression([u(10),v(10),'/']).simplify()}$ \quad
|
||
|
|
\end{savedSolution}
|
||
|
|
\end{parts}
|
||
|
|
\end{multicols}
|
||
|
|
|
||
|
|
\question
|
||
|
|
\begin{center}
|
||
|
|
\textbf{Durée : 20min \hspace{3cm} Thème : Suites}
|
||
|
|
\end{center}
|
||
|
|
Calculer les 3 premiers termes et le 10ième terme des 4 suites suivantes
|
||
|
|
\begin{multicols}{2}
|
||
|
|
\begin{parts}
|
||
|
|
\Block{set u = Polynom.random(degree = 2, letter = "n", name = "u")}
|
||
|
|
\part $\Var{u.name}_n = \Var{u}$
|
||
|
|
\begin{savedSolution}
|
||
|
|
$\Var{u.name}_0 = \Var{u(0)}$ \quad
|
||
|
|
$\Var{u.name}_1 = \Var{u(1)}$ \quad
|
||
|
|
$\Var{u.name}_2 = \Var{u(2)}$ \quad
|
||
|
|
$\Var{u.name}_{10} = \Var{u(10)}$ \quad
|
||
|
|
\end{savedSolution}
|
||
|
|
\Block{set u = Polynom.random(degree = 2, letter = "n", name = "v")}
|
||
|
|
\part $\Var{u.name}_n = \Var{u}$
|
||
|
|
\begin{savedSolution}
|
||
|
|
$\Var{u.name}_0 = \Var{u(0)}$ \quad
|
||
|
|
$\Var{u.name}_1 = \Var{u(1)}$ \quad
|
||
|
|
$\Var{u.name}_2 = \Var{u(2)}$ \quad
|
||
|
|
$\Var{u.name}_{10} = \Var{u(10)}$ \quad
|
||
|
|
\end{savedSolution}
|
||
|
|
\Block{set u = Polynom.random(degree = 3, letter = "n", name = "w")}
|
||
|
|
\part $\Var{u.name}_n = \Var{u}$
|
||
|
|
\begin{savedSolution}
|
||
|
|
$\Var{u.name}_0 = \Var{u(0)}$ \quad
|
||
|
|
$\Var{u.name}_1 = \Var{u(1)}$ \quad
|
||
|
|
$\Var{u.name}_2 = \Var{u(2)}$ \quad
|
||
|
|
$\Var{u.name}_{10} = \Var{u(10)}$ \quad
|
||
|
|
\end{savedSolution}
|
||
|
|
\Block{set u = Polynom.random(degree = 2, letter = "n", name = "l")}
|
||
|
|
\Block{set v = Polynom.random(degree = 2, letter = "n", name = "l")}
|
||
|
|
\part $\Var{u.name}_n = \dfrac{\Var{u}}{\Var{v}}$
|
||
|
|
\begin{savedSolution}
|
||
|
|
$\Var{u.name}_0 = \Var{Expression([u(0),v(0),'/']).simplify()}$ \quad
|
||
|
|
$\Var{u.name}_1 = \Var{Expression([u(1),v(1),'/']).simplify()}$ \quad
|
||
|
|
$\Var{u.name}_2 = \Var{Expression([u(2),v(2),'/']).simplify()}$ \quad
|
||
|
|
$\Var{u.name}_{10} = \Var{Expression([u(10),v(10),'/']).simplify()}$ \quad
|
||
|
|
\end{savedSolution}
|
||
|
|
\end{parts}
|
||
|
|
\end{multicols}
|
||
|
|
|
||
|
|
\question
|
||
|
|
\begin{center}
|
||
|
|
\textbf{Durée : 20min \hspace{3cm} Thème : Suites}
|
||
|
|
\end{center}
|
||
|
|
Calculer les 3 premiers termes des 4 suites suivantes
|
||
|
|
\begin{multicols}{2}
|
||
|
|
\begin{parts}
|
||
|
|
\Block{set u = Polynom.random(degree = 1, letter = "u_n", name = "u")}
|
||
|
|
\Block{set a = Expression.random("{a}")}
|
||
|
|
\part $\Var{u.name}_{n+1} = \Var{u}$ et $\Var{u.name}_0 = \Var{a}$.
|
||
|
|
\begin{savedSolution}
|
||
|
|
\Block{set a = u(a)}%
|
||
|
|
$\Var{u.name}_1 = \Var{a}$ \quad
|
||
|
|
\Block{set a = u(a)}%
|
||
|
|
$\Var{u.name}_2 = \Var{a}$ \quad
|
||
|
|
\Block{set a = u(a)}%
|
||
|
|
$\Var{u.name}_3 = \Var{a}$ \quad
|
||
|
|
\end{savedSolution}
|
||
|
|
|
||
|
|
\Block{set u = Polynom.random(degree = 1, letter = "u_n", name = "u")}
|
||
|
|
\Block{set a = Expression.random("{a}")}
|
||
|
|
\part $\Var{u.name}_{n+1} = \Var{u}$ et $\Var{u.name}_0 = \Var{a}$.
|
||
|
|
\begin{savedSolution}
|
||
|
|
\Block{set a = u(a)}%
|
||
|
|
$\Var{u.name}_1 = \Var{a}$ \quad
|
||
|
|
\Block{set a = u(a)}%
|
||
|
|
$\Var{u.name}_2 = \Var{a}$ \quad
|
||
|
|
\Block{set a = u(a)}%
|
||
|
|
$\Var{u.name}_3 = \Var{a}$ \quad
|
||
|
|
\end{savedSolution}
|
||
|
|
|
||
|
|
\Block{set u = Polynom.random(degree = 2, letter = "u_n", name = "u")}
|
||
|
|
\Block{set a = Expression.random("{a}")}
|
||
|
|
\part $\Var{u.name}_{n+1} = \Var{u}$ et $\Var{u.name}_0 = \Var{a}$.
|
||
|
|
\begin{savedSolution}
|
||
|
|
\Block{set a = u(a)}%
|
||
|
|
$\Var{u.name}_1 = \Var{a}$ \quad
|
||
|
|
\Block{set a = u(a)}%
|
||
|
|
$\Var{u.name}_2 = \Var{a}$ \quad
|
||
|
|
\Block{set a = u(a)}%
|
||
|
|
$\Var{u.name}_3 = \Var{a}$ \quad
|
||
|
|
\end{savedSolution}
|
||
|
|
|
||
|
|
\Block{set u = Polynom.random(degree = 1, letter = "u_n", name = "u")}
|
||
|
|
\Block{set a = Expression.random("{a}")}
|
||
|
|
\part $\Var{u.name}_{n+1} = \Var{u}$ et $\Var{u.name}_0 = \Var{a}$.
|
||
|
|
\begin{savedSolution}
|
||
|
|
\Block{set a = u(a)}%
|
||
|
|
$\Var{u.name}_1 = \Var{a}$ \quad
|
||
|
|
\Block{set a = u(a)}%
|
||
|
|
$\Var{u.name}_2 = \Var{a}$ \quad
|
||
|
|
\Block{set a = u(a)}%
|
||
|
|
$\Var{u.name}_3 = \Var{a}$ \quad
|
||
|
|
\end{savedSolution}
|
||
|
|
\end{parts}
|
||
|
|
|
||
|
|
\end{multicols}
|
||
|
|
|
||
|
|
\question
|
||
|
|
\begin{center}
|
||
|
|
\textbf{Durée : 20min \hspace{3cm} Thème : Suites}
|
||
|
|
\end{center}
|
||
|
|
Calculer les 3 premiers termes des 4 suites suivantes
|
||
|
|
\begin{multicols}{2}
|
||
|
|
\begin{parts}
|
||
|
|
\Block{set u = Polynom.random(degree = 1, letter = "u_n", name = "u")}
|
||
|
|
\Block{set a = Expression.random("{a}")}
|
||
|
|
\part $\Var{u.name}_{n+1} = \Var{u}$ et $\Var{u.name}_0 = \Var{a}$.
|
||
|
|
\begin{savedSolution}
|
||
|
|
\Block{set a = u(a)}%
|
||
|
|
$\Var{u.name}_1 = \Var{a}$ \quad
|
||
|
|
\Block{set a = u(a)}%
|
||
|
|
$\Var{u.name}_2 = \Var{a}$ \quad
|
||
|
|
\Block{set a = u(a)}%
|
||
|
|
$\Var{u.name}_3 = \Var{a}$ \quad
|
||
|
|
\end{savedSolution}
|
||
|
|
|
||
|
|
\Block{set u = Polynom.random(degree = 1, letter = "u_{n-1}", name = "u")}
|
||
|
|
\Block{set a = Expression.random("{a}")}
|
||
|
|
\part $\Var{u.name}_{n} = \Var{u}$ et $\Var{u.name}_0 = \Var{a}$.
|
||
|
|
\begin{savedSolution}
|
||
|
|
\Block{set a = u(a)}%
|
||
|
|
$\Var{u.name}_1 = \Var{a}$ \quad
|
||
|
|
\Block{set a = u(a)}%
|
||
|
|
$\Var{u.name}_2 = \Var{a}$ \quad
|
||
|
|
\Block{set a = u(a)}%
|
||
|
|
$\Var{u.name}_3 = \Var{a}$ \quad
|
||
|
|
\end{savedSolution}
|
||
|
|
|
||
|
|
\Block{set u = Polynom.random(degree = 1, letter = "u_{n-1}", name = "u")}
|
||
|
|
\Block{set a = Expression.random("{a}")}
|
||
|
|
\part $\Var{u.name}_n = \Var{u}$ et $\Var{u.name}_0 = \Var{a}$.
|
||
|
|
\begin{savedSolution}
|
||
|
|
\Block{set a = u(a)}%
|
||
|
|
$\Var{u.name}_1 = \Var{a}$ \quad
|
||
|
|
\Block{set a = u(a)}%
|
||
|
|
$\Var{u.name}_2 = \Var{a}$ \quad
|
||
|
|
\Block{set a = u(a)}%
|
||
|
|
$\Var{u.name}_3 = \Var{a}$ \quad
|
||
|
|
\end{savedSolution}
|
||
|
|
|
||
|
|
\Block{set u = Polynom.random(degree = 1, letter = "u_n", name = "u")}
|
||
|
|
\Block{set P = Polynom.random([0,"{a}"], conditions=["{a}>0"], letter = 'n')}
|
||
|
|
\Block{set a = Expression.random("{a}")}
|
||
|
|
\part $\Var{u.name}_{n+1} = \Var{u} + \Var{P}$ et $\Var{u.name}_0 = \Var{a}$.
|
||
|
|
\begin{savedSolution}
|
||
|
|
\Block{set a = u(a) + P(1)}%
|
||
|
|
$\Var{u.name}_1 = \Var{a}$ \quad
|
||
|
|
\Block{set a = u(a) + P(2)}%
|
||
|
|
$\Var{u.name}_2 = \Var{a}$ \quad
|
||
|
|
\Block{set a = u(a) + P(3)}%
|
||
|
|
$\Var{u.name}_3 = \Var{a}$ \quad
|
||
|
|
\end{savedSolution}
|
||
|
|
\end{parts}
|
||
|
|
|
||
|
|
\end{multicols}
|
||
|
|
|
||
|
|
\question
|
||
|
|
\begin{center}
|
||
|
|
\textbf{Durée : 20min \hspace{3cm} Thème : Suites}
|
||
|
|
\end{center}
|
||
|
|
Calculer les 3 premiers termes des 4 suites suivantes
|
||
|
|
\begin{multicols}{2}
|
||
|
|
\begin{parts}
|
||
|
|
\Block{set u = Polynom.random(degree = 1, letter = "u_n", name = "u")}
|
||
|
|
\Block{set a = Expression.random("{a}")}
|
||
|
|
\part $\Var{u.name}_{n+1} = \Var{u}$ et $\Var{u.name}_0 = \Var{a}$.
|
||
|
|
\begin{savedSolution}
|
||
|
|
\Block{set a = u(a)}%
|
||
|
|
$\Var{u.name}_1 = \Var{a}$ \quad
|
||
|
|
\Block{set a = u(a)}%
|
||
|
|
$\Var{u.name}_2 = \Var{a}$ \quad
|
||
|
|
\Block{set a = u(a)}%
|
||
|
|
$\Var{u.name}_3 = \Var{a}$ \quad
|
||
|
|
\end{savedSolution}
|
||
|
|
|
||
|
|
\Block{set u = Polynom.random(degree = 1, letter = "u_{n-1}", name = "u")}
|
||
|
|
\Block{set a = Expression.random("{a}")}
|
||
|
|
\part $\Var{u.name}_{n} = \Var{u}$ et $\Var{u.name}_0 = \Var{a}$.
|
||
|
|
\begin{savedSolution}
|
||
|
|
\Block{set a = u(a)}%
|
||
|
|
$\Var{u.name}_1 = \Var{a}$ \quad
|
||
|
|
\Block{set a = u(a)}%
|
||
|
|
$\Var{u.name}_2 = \Var{a}$ \quad
|
||
|
|
\Block{set a = u(a)}%
|
||
|
|
$\Var{u.name}_3 = \Var{a}$ \quad
|
||
|
|
\end{savedSolution}
|
||
|
|
|
||
|
|
\Block{set u = Polynom.random(degree = 1, letter = "u_{n-1}", name = "u")}
|
||
|
|
\Block{set a = Expression.random("{a}")}
|
||
|
|
\part $\Var{u.name}_n = \Var{u}$ et $\Var{u.name}_0 = \Var{a}$.
|
||
|
|
\begin{savedSolution}
|
||
|
|
\Block{set a = u(a)}%
|
||
|
|
$\Var{u.name}_1 = \Var{a}$ \quad
|
||
|
|
\Block{set a = u(a)}%
|
||
|
|
$\Var{u.name}_2 = \Var{a}$ \quad
|
||
|
|
\Block{set a = u(a)}%
|
||
|
|
$\Var{u.name}_3 = \Var{a}$ \quad
|
||
|
|
\end{savedSolution}
|
||
|
|
|
||
|
|
\Block{set u = Polynom.random(degree = 1, letter = "u_n", name = "u")}
|
||
|
|
\Block{set P = Polynom.random([0,"{a}"], conditions=["{a}>0"], letter = 'n')}
|
||
|
|
\Block{set a = Expression.random("{a}")}
|
||
|
|
\part $\Var{u.name}_{n+1} = \Var{u} + \Var{P}$ et $\Var{u.name}_0 = \Var{a}$.
|
||
|
|
\begin{savedSolution}
|
||
|
|
\Block{set a = u(a) + P(1)}%
|
||
|
|
$\Var{u.name}_1 = \Var{a}$ \quad
|
||
|
|
\Block{set a = u(a) + P(2)}%
|
||
|
|
$\Var{u.name}_2 = \Var{a}$ \quad
|
||
|
|
\Block{set a = u(a) + P(3)}%
|
||
|
|
$\Var{u.name}_3 = \Var{a}$ \quad
|
||
|
|
\end{savedSolution}
|
||
|
|
\end{parts}
|
||
|
|
|
||
|
|
\end{multicols}
|
||
|
|
|
||
|
|
|
||
|
|
|