183 lines
7.5 KiB
TeX
183 lines
7.5 KiB
TeX
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\begin{exercise}[subtitle={Calculs de dérivée}, step={1}, origin={Ma tête}, topics={ Fonction dérivée }, tags={ Dérivation }, mode={\trainMode}]
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Calculer les fonctions dérivées des fonctions suivantes
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\Block{
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set fonctions = {
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"f": random_expression("{a}x + {b}", [],),
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"g": random_expression("{a}x + {b}", [],),
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"h": random_expression("{a}x + {b}", [],),
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"i": random_expression("{a}x + {b}", [],),
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"j": random_expression("{a}x + {b}", [],),
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"k": random_expression("{a}x^2 + {b}x + {c}", [],),
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"l": random_expression("{a}x + {b}", [],),
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"m": random_expression("{a}x^2 + {b}x + {c}", [],),
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"n": random_expression("{a}x^2 + {b}x + {c}", [],),
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"o": random_expression("{a}x^2", [],),
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"p": random_expression("{a}x^2 + {b}x", [],),
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"q": random_expression("{a}x^2 + {c}", [],),
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}
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}
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\begin{multicols}{3}
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\begin{enumerate}
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%- for name, f in fonctions.items()
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\item $\Var{name}(x) = \Var{f}$
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%- endfor
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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{multicols}{2}
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\begin{enumerate}
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%- for name, f in fonctions.items()
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\item $\Var{name}(x) = \Var{f}$
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\[
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\Var{name}'(x) = \Var{f.differentiate()}
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\]
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%- endfor
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\end{enumerate}
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\end{multicols}
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\end{solution}
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\begin{exercise}[subtitle={Fonction affines - technique}, step={2}, origin={Ma tête}, topics={ Fonction dérivée }, tags={ Dérivation }, mode={\trainMode}]
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Reprendre l'exercice précédent pour les fonctions suivantes:
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\Block{
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set functions = {
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"f": random_expression("{a}x + {b}", [],),
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"g": random_expression("{a}x + {b}", [],),
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"h": random_expression("{a}x + {b}", [],),
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"i": random_expression("{a}x + {b}", [],),
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}
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}
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\begin{multicols}{2}
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\begin{enumerate}
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%- for name, f in functions.items()
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\item $\Var{name}(x) = \Var{f}$
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%- endfor
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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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%- for name, f in functions.items()
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\item Étude de la fonction $\Var{name}(x) = \Var{f}$
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%- set f1 = f.differentiate()
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\begin{itemize}
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\item Fonction dérivée : $\Var{name}'(x) = \Var{f1}$
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%- if f1 > 0
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\item Comme $\Var{f1} > 0$ la fonction est croissante
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\item
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\begin{center}
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\begin{tikzpicture}
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\tkzTabInit[lgt=3,espcl=10]{$x$/1,Signe de $f'(x)$/2, Variations de $f(x)$/2}{\hspace{2cm}, \hspace{2cm}}%
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\tkzTabLine{,+,}%
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\tkzTabVar{-/ ,+/ }%
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\end{tikzpicture}
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\end{center}
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%- elif f1 < 0
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\item Comme $\Var{f1} < 0$ la fonction est décroissante
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\item
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\begin{center}
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\begin{tikzpicture}
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\tkzTabInit[lgt=3,espcl=10]{$x$/1,Signe de $f'(x)$/2, Variations de $f(x)$/2}{\hspace{2cm}, \hspace{2cm}}%
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\tkzTabLine{,-,}%
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\tkzTabVar{+/ ,-/ }%
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\end{tikzpicture}
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\end{center}
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%- endif
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\end{itemize}
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%- endfor
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\end{enumerate}
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\end{solution}
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\begin{exercise}[subtitle={Fonction affines - technique}, step={2}, origin={Ma tête}, topics={ Fonction dérivée }, tags={ Dérivation }, mode={\trainMode}]
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Reprendre l'exercice précédent pour les fonctions suivantes :
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\Block{
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set functions = {
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"f": random_expression("{a}x^2 + {b}x + {c}", ["a>0"],),
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"g": random_expression("{a}x^2 + {b}x + {c}", ["a>0"],),
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"h": random_expression("{a}x^2 + {b}x + {c}", [],),
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"i": random_expression("{a}x^2 + {b}x + {c}", [],),
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"j": random_expression("{a}x^2 + {b}x + {c}", [],),
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"k": random_expression("{a}x^2 + {b}x + {c}", [],),
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}
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}
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\begin{multicols}{2}
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\begin{enumerate}
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%- for name, f in functions.items()
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\item $\Var{name}(x) = \Var{f}$
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%- endfor
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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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%- for name, f in functions.items()
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\item Étude de la fonction $\Var{name}(x) = \Var{f}$
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%- set f1 = f.differentiate()
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\begin{itemize}
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\item Fonction dérivée : $\Var{name}'(x) = \Var{f1}$
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%- if f1[1] > 0
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\item On résout l'inéquation $\Var{name}'(x) \geq 0$ pour déterminer quand la fonction $\Var{name}'$ est positive.
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%- set cst = -f1[0]
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%- set coef = f1[1]
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%- set racine = cst / coef
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\begin{align*}
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\Var{name}(x) & \geq 0 \\
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\Var{f1} & \geq 0 \\
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\Var{f1} + \Var{cst} &\geq 0 + \Var{cst} \\
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\Var{f1 + cst} &\geq \Var{0 + cst} \\
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\frac{\Var{f1 + cst}}{\Var{coef}} &\geq \frac{\Var{cst}}{\Var{coef}} \\
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x &\geq \Var{racine.simplify()} \\
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\end{align*}
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Donc $\Var{name}(x)$ est positif quand $x$ est plus \textbf{grand} que $\Var{racine.simplify()}$
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%- set racine = racine.simplify()
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%- set img_racine = f(racine)
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\item
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\begin{center}
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\begin{tikzpicture}
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\tkzTabInit[lgt=3,espcl=4]{$x$/1,Signe de $f'(x)$/2, Variations de $f(x)$/2}{, $\Var{racine}$ ,}%
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\tkzTabLine{, -, z, +, }
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\tkzTabVar{+/ ,-/$f(\Var{racine}) = \Var{img_racine}$ , +/}%
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\end{tikzpicture}
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\end{center}
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%- elif f1[1] < 0
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\item On résout l'inéquation $\Var{name}'(x) \geq 0$ pour déterminer quand la fonction $\Var{name}'$ est positive.
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%- set cst = -f1[0]
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%- set coef = f1[1]
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%- set racine = cst / coef
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\begin{align*}
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\Var{name}(x) & \geq 0 \\
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\Var{f1} & \geq 0 \\
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\Var{f1} + \Var{cst} &\geq 0 + \Var{cst} \\
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\Var{f1 + cst} &\geq \Var{0 + cst} \\
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\frac{\Var{f1 + cst}}{\Var{coef}} &\leq \frac{\Var{cst}}{\Var{coef}} \\
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x &\leq \Var{racine.simplify()} \\
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\end{align*}
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Donc $\Var{name}(x)$ est positif quand $x$ est plus \textbf{petit} que $\Var{racine.simplify()}$
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%- set racine = racine.simplify()
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%- set img_racine = f(racine)
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\item
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\begin{center}
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\begin{tikzpicture}
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\tkzTabInit[lgt=3,espcl=4]{$x$/1,Signe de $f'(x)$/2, Variations de $f(x)$/2}{, $\Var{racine}$ ,}%
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\tkzTabLine{, +, z, -, }
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\tkzTabVar{-/ ,+/$f(\Var{racine}) = \Var{img_racine}$ , -/}%
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\end{tikzpicture}
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\end{center}
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%- endif
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\end{itemize}
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%- endfor
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\end{enumerate}
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\end{solution}
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