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b/TST_sti2d/DS/DS_21_07_08/01_210408_DS8.tex new file mode 100644 index 0000000..d3733db --- /dev/null +++ b/TST_sti2d/DS/DS_21_07_08/01_210408_DS8.tex @@ -0,0 +1,136 @@ +\documentclass[a4paper,10pt]{article} +\usepackage{myXsim} + +% Title Page +\title{DS8 \hfill BAHBAH Zakaria} +\tribe{TST sti2d} +\date{\hfillÀ render pour le vendredi 9 avril à 10h au plus tard} + +\xsimsetup{ + solution/print = false +} + +\begin{document} +\maketitle + +\begin{exercise}[subtitle={Étude de fonction}] + On considère la fonction $f$ définie sur $\intOF{0}{+\infty}$ par $ f(x) = 4x^2 + 72x + 160\ln(x)$ + \begin{enumerate} + \item Démontrer que la dérivée de $f$ est $f'(x) = \frac{8x^2 + 72x + 160}{x}$. + \item Étude du numérateur de $f'(x)$: $N(x) = 8x^2 + 72x + 160$ + \begin{enumerate} + \item Démontrer que $x=- 4$ et $x=- 5$ sont deux racines de $N(x)$.. + \item Proposer une forme factorisée de $N(x)$. + \item Proposer une forme factorisée de $f'(x)$. + \end{enumerate} + \item Étudier le signe de $f'$ et en déduire les variations de $f$. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item pas de correction disponible + \item + \begin{enumerate} + \item \[N(- 4) = 0\] + \[N(- 5) = 0\] + \item \[ + N(x) = 8(x - - 4)(x - - 5) + \] + \item + \[ + f'(x) = \frac{8(x - - 4)(x - - 5)}{x} + \] + \end{enumerate} + \item Pas de correction disponible + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Complexes}] + \begin{enumerate} + \item Mettre le nombre complexe suivant sous forme algébrique $z_1 = \dfrac{2 + 8 i}{-9 + 3 i} $ + \item Mettre le complexe suivante sous forme exponentielle $z_2 = 10 \sqrt{3} - 10 i$ + \item Mettre le complexe suivante sous forme exponentielle $z_3 = 7 + 7 \sqrt{3} i$ + \item Calculer le produit $z_4=z_2\times z_3$ donner le résultat sous forme exponentielle puis algébrique. + \item Calculer le quotient $z_5=\frac{z_2}{z_3}$ donner le résultat sous forme exponentielle puis algébrique. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item $z_1 = \frac{1}{15} - \frac{13 i}{15}$ + \item $z_2 = 20 e^{- \frac{i \pi}{6}}$ + \item $z_3 = 14 e^{\frac{i \pi}{3}}$ + \item $z_4 = 280 e^{\frac{i \pi}{6}} = 140 \sqrt{3} + 140 i = 243.0 + 140.0 i$ + \item $z_5 = \frac{10}{7} e^{- \frac{i \pi}{2}} = - \frac{10 i}{7} = - 1.43 i$ + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Sortie du congélateur}] + Marie a invité quelques amis pour le thé. Elle souhaite leur proposer ses macarons maison. + + Elle les sort de son congélateur à $-17$~\degres C et les place dans une pièce à $16$~\degres C. + + Au bout de 15 minutes, la température des macarons est de $-3$~\degres C. + + \bigskip + + \textbf{Premier modèle} + + \medskip + + On suppose que la vitesse de décongélation est constante : chaque minute la hausse de + température des macarons est la même. + + Estimer dans ce cadre la température au bout de $30$~minutes, puis au bout de $45$~minutes. + + Cette modélisation est-elle pertinente? + + \bigskip + + \textbf{Deuxième modèle} + + \medskip + + On suppose maintenant que la vitesse de décongélation est proportionnelle à la différence + de température entre les macarons et l'air ambiant (il s'agit de la loi de Newton). + + On désigne par $\theta$ la température des macarons à l'instant $t$, et par $\theta'$ la vitesse de décongélation. + + L'unité de temps est la minute et l'unité de température le degré Celsius. + + \smallskip + + On négligera la diminution de température de la pièce et on admettra donc qu'il existe un + nombre réel $a$ tel que, pour $t$ positif : + + \[\theta'(t) = a [\theta(t) - 16]\quad (E)\] + + \medskip + + \begin{enumerate} + \item Vérifier que l'équation $(E)$ a pour solutions $\theta(t) = K e^{at} + 16$ où $K$ est un nombre réel. + + Donner alors, en fonction de $a$, l'ensemble des solutions de $(E)$. + \end{enumerate} + On rappelle que la température des macarons à l'instant $t = 0$ est égale à $-17$~\degres C et que, au bout de $15$~min, elle est de $-3$~\degres C. + \begin{enumerate} + \setcounter{enumi}{1} + \item En utilisant la condition à $t=0$ démontrer que $K = -33$. + \item En utilisant la condition à $t=15$ démontrer que $a \approx -0.04$. + \item En déduire l'expression de la solution de l'équation différentielle puis étudier ses variations. + \item La température idéale de dégustation des macarons étant de $13$~\degres C, Marie estime que + celle-ci sera atteinte au bout de $30$~min. A-t-elle raison ? Justifier la réponse. + + Sinon, combien de temps faudra-t-il attendre ? +\end{enumerate} +\end{exercise} + + + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/TST_sti2d/DS/DS_21_07_08/02_210408_DS8.tex b/TST_sti2d/DS/DS_21_07_08/02_210408_DS8.tex new file mode 100644 index 0000000..a5751e2 --- /dev/null +++ b/TST_sti2d/DS/DS_21_07_08/02_210408_DS8.tex @@ -0,0 +1,136 @@ +\documentclass[a4paper,10pt]{article} +\usepackage{myXsim} + +% Title Page +\title{DS8 \hfill BENALI Ilyas} +\tribe{TST sti2d} +\date{\hfillÀ render pour le vendredi 9 avril à 10h au plus tard} + +\xsimsetup{ + solution/print = false +} + +\begin{document} +\maketitle + +\begin{exercise}[subtitle={Étude de fonction}] + On considère la fonction $f$ définie sur $\intOF{0}{+\infty}$ par $ f(x) = 2.5x^2 + - 95x + 450\ln(x)$ + \begin{enumerate} + \item Démontrer que la dérivée de $f$ est $f'(x) = \frac{5x^2 + - 95x + 450}{x}$. + \item Étude du numérateur de $f'(x)$: $N(x) = 5x^2 - 95x + 450$ + \begin{enumerate} + \item Démontrer que $x=9$ et $x=10$ sont deux racines de $N(x)$.. + \item Proposer une forme factorisée de $N(x)$. + \item Proposer une forme factorisée de $f'(x)$. + \end{enumerate} + \item Étudier le signe de $f'$ et en déduire les variations de $f$. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item pas de correction disponible + \item + \begin{enumerate} + \item \[N(9) = 0\] + \[N(10) = 0\] + \item \[ + N(x) = 5(x - 9)(x - 10) + \] + \item + \[ + f'(x) = \frac{5(x - 9)(x - 10)}{x} + \] + \end{enumerate} + \item Pas de correction disponible + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Complexes}] + \begin{enumerate} + \item Mettre le nombre complexe suivant sous forme algébrique $z_1 = \dfrac{10 + 5 i}{-3 + 10 i} $ + \item Mettre le complexe suivante sous forme exponentielle $z_2 = - 5 \sqrt{3} + 5 i$ + \item Mettre le complexe suivante sous forme exponentielle $z_3 = - 8 \sqrt{2} - 8 \sqrt{2} i$ + \item Calculer le produit $z_4=z_2\times z_3$ donner le résultat sous forme exponentielle puis algébrique. + \item Calculer le quotient $z_5=\frac{z_2}{z_3}$ donner le résultat sous forme exponentielle puis algébrique. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item $z_1 = \frac{20}{109} - \frac{115 i}{109}$ + \item $z_2 = 10 e^{\frac{5 i \pi}{6}}$ + \item $z_3 = 16 e^{- \frac{3 i \pi}{4}}$ + \item $z_4 = 160 e^{\frac{i \pi}{12}} = 40 \sqrt{2} + 40 \sqrt{6} + i \left(- 40 \sqrt{2} + 40 \sqrt{6}\right) = 155.0 + 41.4 i$ + \item $z_5 = \frac{5}{8} e^{\frac{19 i \pi}{12}} = - \frac{5 \sqrt{2}}{32} + \frac{5 \sqrt{6}}{32} + i \left(- \frac{5 \sqrt{6}}{32} - \frac{5 \sqrt{2}}{32}\right) = 0.162 - 0.604 i$ + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Sortie du congélateur}] + Marie a invité quelques amis pour le thé. Elle souhaite leur proposer ses macarons maison. + + Elle les sort de son congélateur à $-17$~\degres C et les place dans une pièce à $17$~\degres C. + + Au bout de 15 minutes, la température des macarons est de $-1$~\degres C. + + \bigskip + + \textbf{Premier modèle} + + \medskip + + On suppose que la vitesse de décongélation est constante : chaque minute la hausse de + température des macarons est la même. + + Estimer dans ce cadre la température au bout de $30$~minutes, puis au bout de $45$~minutes. + + Cette modélisation est-elle pertinente? + + \bigskip + + \textbf{Deuxième modèle} + + \medskip + + On suppose maintenant que la vitesse de décongélation est proportionnelle à la différence + de température entre les macarons et l'air ambiant (il s'agit de la loi de Newton). + + On désigne par $\theta$ la température des macarons à l'instant $t$, et par $\theta'$ la vitesse de décongélation. + + L'unité de temps est la minute et l'unité de température le degré Celsius. + + \smallskip + + On négligera la diminution de température de la pièce et on admettra donc qu'il existe un + nombre réel $a$ tel que, pour $t$ positif : + + \[\theta'(t) = a [\theta(t) - 17]\quad (E)\] + + \medskip + + \begin{enumerate} + \item Vérifier que l'équation $(E)$ a pour solutions $\theta(t) = K e^{at} + 17$ où $K$ est un nombre réel. + + Donner alors, en fonction de $a$, l'ensemble des solutions de $(E)$. + \end{enumerate} + On rappelle que la température des macarons à l'instant $t = 0$ est égale à $-17$~\degres C et que, au bout de $15$~min, elle est de $-1$~\degres C. + \begin{enumerate} + \setcounter{enumi}{1} + \item En utilisant la condition à $t=0$ démontrer que $K = -34$. + \item En utilisant la condition à $t=15$ démontrer que $a \approx -0.04$. + \item En déduire l'expression de la solution de l'équation différentielle puis étudier ses variations. + \item La température idéale de dégustation des macarons étant de $14$~\degres C, Marie estime que + celle-ci sera atteinte au bout de $30$~min. A-t-elle raison ? Justifier la réponse. + + Sinon, combien de temps faudra-t-il attendre ? +\end{enumerate} +\end{exercise} + + + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/TST_sti2d/DS/DS_21_07_08/03_210408_DS8.tex b/TST_sti2d/DS/DS_21_07_08/03_210408_DS8.tex new file mode 100644 index 0000000..cabaf2d --- /dev/null +++ b/TST_sti2d/DS/DS_21_07_08/03_210408_DS8.tex @@ -0,0 +1,136 @@ +\documentclass[a4paper,10pt]{article} +\usepackage{myXsim} + +% Title Page +\title{DS8 \hfill BERNADAT Noah} +\tribe{TST sti2d} +\date{\hfillÀ render pour le vendredi 9 avril à 10h au plus tard} + +\xsimsetup{ + solution/print = false +} + +\begin{document} +\maketitle + +\begin{exercise}[subtitle={Étude de fonction}] + On considère la fonction $f$ définie sur $\intOF{0}{+\infty}$ par $ f(x) = 3.5x^2 + - 21x + - 196\ln(x)$ + \begin{enumerate} + \item Démontrer que la dérivée de $f$ est $f'(x) = \frac{7x^2 + - 21x + - 196}{x}$. + \item Étude du numérateur de $f'(x)$: $N(x) = 7x^2 - 21x - 196$ + \begin{enumerate} + \item Démontrer que $x=- 4$ et $x=7$ sont deux racines de $N(x)$.. + \item Proposer une forme factorisée de $N(x)$. + \item Proposer une forme factorisée de $f'(x)$. + \end{enumerate} + \item Étudier le signe de $f'$ et en déduire les variations de $f$. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item pas de correction disponible + \item + \begin{enumerate} + \item \[N(- 4) = 0\] + \[N(7) = 0\] + \item \[ + N(x) = 7(x - - 4)(x - 7) + \] + \item + \[ + f'(x) = \frac{7(x - - 4)(x - 7)}{x} + \] + \end{enumerate} + \item Pas de correction disponible + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Complexes}] + \begin{enumerate} + \item Mettre le nombre complexe suivant sous forme algébrique $z_1 = \dfrac{8 + 9 i}{-9 + 5 i} $ + \item Mettre le complexe suivante sous forme exponentielle $z_2 = - 8 \sqrt{3} + 8 i$ + \item Mettre le complexe suivante sous forme exponentielle $z_3 = -5 - 5 \sqrt{3} i$ + \item Calculer le produit $z_4=z_2\times z_3$ donner le résultat sous forme exponentielle puis algébrique. + \item Calculer le quotient $z_5=\frac{z_2}{z_3}$ donner le résultat sous forme exponentielle puis algébrique. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item $z_1 = - \frac{27}{106} - \frac{121 i}{106}$ + \item $z_2 = 16 e^{\frac{5 i \pi}{6}}$ + \item $z_3 = 10 e^{- \frac{2 i \pi}{3}}$ + \item $z_4 = 160 e^{\frac{i \pi}{6}} = 80 \sqrt{3} + 80 i = 139.0 + 80.0 i$ + \item $z_5 = \frac{8}{5} e^{\frac{3 i \pi}{2}} = - \frac{8 i}{5} = - 1.6 i$ + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Sortie du congélateur}] + Marie a invité quelques amis pour le thé. Elle souhaite leur proposer ses macarons maison. + + Elle les sort de son congélateur à $-15$~\degres C et les place dans une pièce à $22$~\degres C. + + Au bout de 15 minutes, la température des macarons est de $-1$~\degres C. + + \bigskip + + \textbf{Premier modèle} + + \medskip + + On suppose que la vitesse de décongélation est constante : chaque minute la hausse de + température des macarons est la même. + + Estimer dans ce cadre la température au bout de $30$~minutes, puis au bout de $45$~minutes. + + Cette modélisation est-elle pertinente? + + \bigskip + + \textbf{Deuxième modèle} + + \medskip + + On suppose maintenant que la vitesse de décongélation est proportionnelle à la différence + de température entre les macarons et l'air ambiant (il s'agit de la loi de Newton). + + On désigne par $\theta$ la température des macarons à l'instant $t$, et par $\theta'$ la vitesse de décongélation. + + L'unité de temps est la minute et l'unité de température le degré Celsius. + + \smallskip + + On négligera la diminution de température de la pièce et on admettra donc qu'il existe un + nombre réel $a$ tel que, pour $t$ positif : + + \[\theta'(t) = a [\theta(t) - 22]\quad (E)\] + + \medskip + + \begin{enumerate} + \item Vérifier que l'équation $(E)$ a pour solutions $\theta(t) = K e^{at} + 22$ où $K$ est un nombre réel. + + Donner alors, en fonction de $a$, l'ensemble des solutions de $(E)$. + \end{enumerate} + On rappelle que la température des macarons à l'instant $t = 0$ est égale à $-15$~\degres C et que, au bout de $15$~min, elle est de $-1$~\degres C. + \begin{enumerate} + \setcounter{enumi}{1} + \item En utilisant la condition à $t=0$ démontrer que $K = -37$. + \item En utilisant la condition à $t=15$ démontrer que $a \approx -0.03$. + \item En déduire l'expression de la solution de l'équation différentielle puis étudier ses variations. + \item La température idéale de dégustation des macarons étant de $19$~\degres C, Marie estime que + celle-ci sera atteinte au bout de $30$~min. A-t-elle raison ? Justifier la réponse. + + Sinon, combien de temps faudra-t-il attendre ? +\end{enumerate} +\end{exercise} + + + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/TST_sti2d/DS/DS_21_07_08/04_210408_DS8.tex b/TST_sti2d/DS/DS_21_07_08/04_210408_DS8.tex new file mode 100644 index 0000000..747506d --- /dev/null +++ b/TST_sti2d/DS/DS_21_07_08/04_210408_DS8.tex @@ -0,0 +1,136 @@ +\documentclass[a4paper,10pt]{article} +\usepackage{myXsim} + +% Title Page +\title{DS8 \hfill BUDIN Nathan} +\tribe{TST sti2d} +\date{\hfillÀ render pour le vendredi 9 avril à 10h au plus tard} + +\xsimsetup{ + solution/print = false +} + +\begin{document} +\maketitle + +\begin{exercise}[subtitle={Étude de fonction}] + On considère la fonction $f$ définie sur $\intOF{0}{+\infty}$ par $ f(x) = 3.5x^2 + - 84x + 140\ln(x)$ + \begin{enumerate} + \item Démontrer que la dérivée de $f$ est $f'(x) = \frac{7x^2 + - 84x + 140}{x}$. + \item Étude du numérateur de $f'(x)$: $N(x) = 7x^2 - 84x + 140$ + \begin{enumerate} + \item Démontrer que $x=10$ et $x=2$ sont deux racines de $N(x)$.. + \item Proposer une forme factorisée de $N(x)$. + \item Proposer une forme factorisée de $f'(x)$. + \end{enumerate} + \item Étudier le signe de $f'$ et en déduire les variations de $f$. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item pas de correction disponible + \item + \begin{enumerate} + \item \[N(10) = 0\] + \[N(2) = 0\] + \item \[ + N(x) = 7(x - 10)(x - 2) + \] + \item + \[ + f'(x) = \frac{7(x - 10)(x - 2)}{x} + \] + \end{enumerate} + \item Pas de correction disponible + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Complexes}] + \begin{enumerate} + \item Mettre le nombre complexe suivant sous forme algébrique $z_1 = \dfrac{8 + 8 i}{-7 + 4 i} $ + \item Mettre le complexe suivante sous forme exponentielle $z_2 = 7 \sqrt{3} - 7 i$ + \item Mettre le complexe suivante sous forme exponentielle $z_3 = - 2 \sqrt{3} + 2 i$ + \item Calculer le produit $z_4=z_2\times z_3$ donner le résultat sous forme exponentielle puis algébrique. + \item Calculer le quotient $z_5=\frac{z_2}{z_3}$ donner le résultat sous forme exponentielle puis algébrique. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item $z_1 = - \frac{24}{65} - \frac{88 i}{65}$ + \item $z_2 = 14 e^{- \frac{i \pi}{6}}$ + \item $z_3 = 4 e^{\frac{5 i \pi}{6}}$ + \item $z_4 = 56 e^{\frac{2 i \pi}{3}} = -28 + 28 \sqrt{3} i = -28.0 + 48.5 i$ + \item $z_5 = \frac{7}{2} e^{- i \pi} = - \frac{7}{2} = -3.5$ + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Sortie du congélateur}] + Marie a invité quelques amis pour le thé. Elle souhaite leur proposer ses macarons maison. + + Elle les sort de son congélateur à $-19$~\degres C et les place dans une pièce à $23$~\degres C. + + Au bout de 15 minutes, la température des macarons est de $3$~\degres C. + + \bigskip + + \textbf{Premier modèle} + + \medskip + + On suppose que la vitesse de décongélation est constante : chaque minute la hausse de + température des macarons est la même. + + Estimer dans ce cadre la température au bout de $30$~minutes, puis au bout de $45$~minutes. + + Cette modélisation est-elle pertinente? + + \bigskip + + \textbf{Deuxième modèle} + + \medskip + + On suppose maintenant que la vitesse de décongélation est proportionnelle à la différence + de température entre les macarons et l'air ambiant (il s'agit de la loi de Newton). + + On désigne par $\theta$ la température des macarons à l'instant $t$, et par $\theta'$ la vitesse de décongélation. + + L'unité de temps est la minute et l'unité de température le degré Celsius. + + \smallskip + + On négligera la diminution de température de la pièce et on admettra donc qu'il existe un + nombre réel $a$ tel que, pour $t$ positif : + + \[\theta'(t) = a [\theta(t) - 23]\quad (E)\] + + \medskip + + \begin{enumerate} + \item Vérifier que l'équation $(E)$ a pour solutions $\theta(t) = K e^{at} + 23$ où $K$ est un nombre réel. + + Donner alors, en fonction de $a$, l'ensemble des solutions de $(E)$. + \end{enumerate} + On rappelle que la température des macarons à l'instant $t = 0$ est égale à $-19$~\degres C et que, au bout de $15$~min, elle est de $3$~\degres C. + \begin{enumerate} + \setcounter{enumi}{1} + \item En utilisant la condition à $t=0$ démontrer que $K = -42$. + \item En utilisant la condition à $t=15$ démontrer que $a \approx -0.05$. + \item En déduire l'expression de la solution de l'équation différentielle puis étudier ses variations. + \item La température idéale de dégustation des macarons étant de $20$~\degres C, Marie estime que + celle-ci sera atteinte au bout de $30$~min. A-t-elle raison ? Justifier la réponse. + + Sinon, combien de temps faudra-t-il attendre ? +\end{enumerate} +\end{exercise} + + + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/TST_sti2d/DS/DS_21_07_08/05_210408_DS8.tex b/TST_sti2d/DS/DS_21_07_08/05_210408_DS8.tex new file mode 100644 index 0000000..28263c9 --- /dev/null +++ b/TST_sti2d/DS/DS_21_07_08/05_210408_DS8.tex @@ -0,0 +1,136 @@ +\documentclass[a4paper,10pt]{article} +\usepackage{myXsim} + +% Title Page +\title{DS8 \hfill CHION Léa} +\tribe{TST sti2d} +\date{\hfillÀ render pour le vendredi 9 avril à 10h au plus tard} + +\xsimsetup{ + solution/print = false +} + +\begin{document} +\maketitle + +\begin{exercise}[subtitle={Étude de fonction}] + On considère la fonction $f$ définie sur $\intOF{0}{+\infty}$ par $ f(x) = 5x^2 + 130x + 300\ln(x)$ + \begin{enumerate} + \item Démontrer que la dérivée de $f$ est $f'(x) = \frac{10x^2 + 130x + 300}{x}$. + \item Étude du numérateur de $f'(x)$: $N(x) = 10x^2 + 130x + 300$ + \begin{enumerate} + \item Démontrer que $x=- 10$ et $x=- 3$ sont deux racines de $N(x)$.. + \item Proposer une forme factorisée de $N(x)$. + \item Proposer une forme factorisée de $f'(x)$. + \end{enumerate} + \item Étudier le signe de $f'$ et en déduire les variations de $f$. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item pas de correction disponible + \item + \begin{enumerate} + \item \[N(- 10) = 0\] + \[N(- 3) = 0\] + \item \[ + N(x) = 10(x - - 10)(x - - 3) + \] + \item + \[ + f'(x) = \frac{10(x - - 10)(x - - 3)}{x} + \] + \end{enumerate} + \item Pas de correction disponible + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Complexes}] + \begin{enumerate} + \item Mettre le nombre complexe suivant sous forme algébrique $z_1 = \dfrac{8 + 10 i}{-3 + 2 i} $ + \item Mettre le complexe suivante sous forme exponentielle $z_2 = - \sqrt{2} - \sqrt{2} i$ + \item Mettre le complexe suivante sous forme exponentielle $z_3 = - 10 \sqrt{2} + 10 \sqrt{2} i$ + \item Calculer le produit $z_4=z_2\times z_3$ donner le résultat sous forme exponentielle puis algébrique. + \item Calculer le quotient $z_5=\frac{z_2}{z_3}$ donner le résultat sous forme exponentielle puis algébrique. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item $z_1 = - \frac{4}{13} - \frac{46 i}{13}$ + \item $z_2 = 2 e^{- \frac{3 i \pi}{4}}$ + \item $z_3 = 20 e^{\frac{3 i \pi}{4}}$ + \item $z_4 = 40 e^{0} = 40 = 40.0$ + \item $z_5 = \frac{1}{10} e^{- \frac{3 i \pi}{2}} = \frac{i}{10} = 0.1 i$ + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Sortie du congélateur}] + Marie a invité quelques amis pour le thé. Elle souhaite leur proposer ses macarons maison. + + Elle les sort de son congélateur à $-20$~\degres C et les place dans une pièce à $24$~\degres C. + + Au bout de 15 minutes, la température des macarons est de $-3$~\degres C. + + \bigskip + + \textbf{Premier modèle} + + \medskip + + On suppose que la vitesse de décongélation est constante : chaque minute la hausse de + température des macarons est la même. + + Estimer dans ce cadre la température au bout de $30$~minutes, puis au bout de $45$~minutes. + + Cette modélisation est-elle pertinente? + + \bigskip + + \textbf{Deuxième modèle} + + \medskip + + On suppose maintenant que la vitesse de décongélation est proportionnelle à la différence + de température entre les macarons et l'air ambiant (il s'agit de la loi de Newton). + + On désigne par $\theta$ la température des macarons à l'instant $t$, et par $\theta'$ la vitesse de décongélation. + + L'unité de temps est la minute et l'unité de température le degré Celsius. + + \smallskip + + On négligera la diminution de température de la pièce et on admettra donc qu'il existe un + nombre réel $a$ tel que, pour $t$ positif : + + \[\theta'(t) = a [\theta(t) - 24]\quad (E)\] + + \medskip + + \begin{enumerate} + \item Vérifier que l'équation $(E)$ a pour solutions $\theta(t) = K e^{at} + 24$ où $K$ est un nombre réel. + + Donner alors, en fonction de $a$, l'ensemble des solutions de $(E)$. + \end{enumerate} + On rappelle que la température des macarons à l'instant $t = 0$ est égale à $-20$~\degres C et que, au bout de $15$~min, elle est de $-3$~\degres C. + \begin{enumerate} + \setcounter{enumi}{1} + \item En utilisant la condition à $t=0$ démontrer que $K = -44$. + \item En utilisant la condition à $t=15$ démontrer que $a \approx -0.03$. + \item En déduire l'expression de la solution de l'équation différentielle puis étudier ses variations. + \item La température idéale de dégustation des macarons étant de $21$~\degres C, Marie estime que + celle-ci sera atteinte au bout de $30$~min. A-t-elle raison ? Justifier la réponse. + + Sinon, combien de temps faudra-t-il attendre ? +\end{enumerate} +\end{exercise} + + + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/TST_sti2d/DS/DS_21_07_08/06_210408_DS8.tex b/TST_sti2d/DS/DS_21_07_08/06_210408_DS8.tex new file mode 100644 index 0000000..d58ccd3 --- /dev/null +++ b/TST_sti2d/DS/DS_21_07_08/06_210408_DS8.tex @@ -0,0 +1,136 @@ +\documentclass[a4paper,10pt]{article} +\usepackage{myXsim} + +% Title Page +\title{DS8 \hfill CLAIN Avinash} +\tribe{TST sti2d} +\date{\hfillÀ render pour le vendredi 9 avril à 10h au plus tard} + +\xsimsetup{ + solution/print = false +} + +\begin{document} +\maketitle + +\begin{exercise}[subtitle={Étude de fonction}] + On considère la fonction $f$ définie sur $\intOF{0}{+\infty}$ par $ f(x) = 3.5x^2 + - 42x + - 280\ln(x)$ + \begin{enumerate} + \item Démontrer que la dérivée de $f$ est $f'(x) = \frac{7x^2 + - 42x + - 280}{x}$. + \item Étude du numérateur de $f'(x)$: $N(x) = 7x^2 - 42x - 280$ + \begin{enumerate} + \item Démontrer que $x=10$ et $x=- 4$ sont deux racines de $N(x)$.. + \item Proposer une forme factorisée de $N(x)$. + \item Proposer une forme factorisée de $f'(x)$. + \end{enumerate} + \item Étudier le signe de $f'$ et en déduire les variations de $f$. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item pas de correction disponible + \item + \begin{enumerate} + \item \[N(10) = 0\] + \[N(- 4) = 0\] + \item \[ + N(x) = 7(x - 10)(x - - 4) + \] + \item + \[ + f'(x) = \frac{7(x - 10)(x - - 4)}{x} + \] + \end{enumerate} + \item Pas de correction disponible + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Complexes}] + \begin{enumerate} + \item Mettre le nombre complexe suivant sous forme algébrique $z_1 = \dfrac{3 + 8 i}{-10 + 5 i} $ + \item Mettre le complexe suivante sous forme exponentielle $z_2 = 1 + \sqrt{3} i$ + \item Mettre le complexe suivante sous forme exponentielle $z_3 = -3 + 3 \sqrt{3} i$ + \item Calculer le produit $z_4=z_2\times z_3$ donner le résultat sous forme exponentielle puis algébrique. + \item Calculer le quotient $z_5=\frac{z_2}{z_3}$ donner le résultat sous forme exponentielle puis algébrique. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item $z_1 = \frac{2}{25} - \frac{19 i}{25}$ + \item $z_2 = 2 e^{\frac{i \pi}{3}}$ + \item $z_3 = 6 e^{\frac{2 i \pi}{3}}$ + \item $z_4 = 12 e^{i \pi} = -12 = -12.0$ + \item $z_5 = \frac{1}{3} e^{- \frac{i \pi}{3}} = \frac{1}{6} - \frac{\sqrt{3} i}{6} = 0.167 - 0.289 i$ + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Sortie du congélateur}] + Marie a invité quelques amis pour le thé. Elle souhaite leur proposer ses macarons maison. + + Elle les sort de son congélateur à $-20$~\degres C et les place dans une pièce à $17$~\degres C. + + Au bout de 15 minutes, la température des macarons est de $4$~\degres C. + + \bigskip + + \textbf{Premier modèle} + + \medskip + + On suppose que la vitesse de décongélation est constante : chaque minute la hausse de + température des macarons est la même. + + Estimer dans ce cadre la température au bout de $30$~minutes, puis au bout de $45$~minutes. + + Cette modélisation est-elle pertinente? + + \bigskip + + \textbf{Deuxième modèle} + + \medskip + + On suppose maintenant que la vitesse de décongélation est proportionnelle à la différence + de température entre les macarons et l'air ambiant (il s'agit de la loi de Newton). + + On désigne par $\theta$ la température des macarons à l'instant $t$, et par $\theta'$ la vitesse de décongélation. + + L'unité de temps est la minute et l'unité de température le degré Celsius. + + \smallskip + + On négligera la diminution de température de la pièce et on admettra donc qu'il existe un + nombre réel $a$ tel que, pour $t$ positif : + + \[\theta'(t) = a [\theta(t) - 17]\quad (E)\] + + \medskip + + \begin{enumerate} + \item Vérifier que l'équation $(E)$ a pour solutions $\theta(t) = K e^{at} + 17$ où $K$ est un nombre réel. + + Donner alors, en fonction de $a$, l'ensemble des solutions de $(E)$. + \end{enumerate} + On rappelle que la température des macarons à l'instant $t = 0$ est égale à $-20$~\degres C et que, au bout de $15$~min, elle est de $4$~\degres C. + \begin{enumerate} + \setcounter{enumi}{1} + \item En utilisant la condition à $t=0$ démontrer que $K = -37$. + \item En utilisant la condition à $t=15$ démontrer que $a \approx -0.07$. + \item En déduire l'expression de la solution de l'équation différentielle puis étudier ses variations. + \item La température idéale de dégustation des macarons étant de $14$~\degres C, Marie estime que + celle-ci sera atteinte au bout de $30$~min. A-t-elle raison ? Justifier la réponse. + + Sinon, combien de temps faudra-t-il attendre ? +\end{enumerate} +\end{exercise} + + + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/TST_sti2d/DS/DS_21_07_08/07_210408_DS8.tex b/TST_sti2d/DS/DS_21_07_08/07_210408_DS8.tex new file mode 100644 index 0000000..04a5e00 --- /dev/null +++ b/TST_sti2d/DS/DS_21_07_08/07_210408_DS8.tex @@ -0,0 +1,136 @@ +\documentclass[a4paper,10pt]{article} +\usepackage{myXsim} + +% Title Page +\title{DS8 \hfill COUBAT Alexis} +\tribe{TST sti2d} +\date{\hfillÀ render pour le vendredi 9 avril à 10h au plus tard} + +\xsimsetup{ + solution/print = false +} + +\begin{document} +\maketitle + +\begin{exercise}[subtitle={Étude de fonction}] + On considère la fonction $f$ définie sur $\intOF{0}{+\infty}$ par $ f(x) = 5x^2 + 70x + 60\ln(x)$ + \begin{enumerate} + \item Démontrer que la dérivée de $f$ est $f'(x) = \frac{10x^2 + 70x + 60}{x}$. + \item Étude du numérateur de $f'(x)$: $N(x) = 10x^2 + 70x + 60$ + \begin{enumerate} + \item Démontrer que $x=- 6$ et $x=- 1$ sont deux racines de $N(x)$.. + \item Proposer une forme factorisée de $N(x)$. + \item Proposer une forme factorisée de $f'(x)$. + \end{enumerate} + \item Étudier le signe de $f'$ et en déduire les variations de $f$. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item pas de correction disponible + \item + \begin{enumerate} + \item \[N(- 6) = 0\] + \[N(- 1) = 0\] + \item \[ + N(x) = 10(x - - 6)(x - - 1) + \] + \item + \[ + f'(x) = \frac{10(x - - 6)(x - - 1)}{x} + \] + \end{enumerate} + \item Pas de correction disponible + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Complexes}] + \begin{enumerate} + \item Mettre le nombre complexe suivant sous forme algébrique $z_1 = \dfrac{3 + 5 i}{-4 + 5 i} $ + \item Mettre le complexe suivante sous forme exponentielle $z_2 = 5 + 5 \sqrt{3} i$ + \item Mettre le complexe suivante sous forme exponentielle $z_3 = - 4 \sqrt{3} - 4 i$ + \item Calculer le produit $z_4=z_2\times z_3$ donner le résultat sous forme exponentielle puis algébrique. + \item Calculer le quotient $z_5=\frac{z_2}{z_3}$ donner le résultat sous forme exponentielle puis algébrique. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item $z_1 = \frac{13}{41} - \frac{35 i}{41}$ + \item $z_2 = 10 e^{\frac{i \pi}{3}}$ + \item $z_3 = 8 e^{- \frac{5 i \pi}{6}}$ + \item $z_4 = 80 e^{- \frac{i \pi}{2}} = - 80 i = - 80.0 i$ + \item $z_5 = \frac{5}{4} e^{\frac{7 i \pi}{6}} = - \frac{5 \sqrt{3}}{8} - \frac{5 i}{8} = -1.08 - 0.625 i$ + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Sortie du congélateur}] + Marie a invité quelques amis pour le thé. Elle souhaite leur proposer ses macarons maison. + + Elle les sort de son congélateur à $-20$~\degres C et les place dans une pièce à $21$~\degres C. + + Au bout de 15 minutes, la température des macarons est de $4$~\degres C. + + \bigskip + + \textbf{Premier modèle} + + \medskip + + On suppose que la vitesse de décongélation est constante : chaque minute la hausse de + température des macarons est la même. + + Estimer dans ce cadre la température au bout de $30$~minutes, puis au bout de $45$~minutes. + + Cette modélisation est-elle pertinente? + + \bigskip + + \textbf{Deuxième modèle} + + \medskip + + On suppose maintenant que la vitesse de décongélation est proportionnelle à la différence + de température entre les macarons et l'air ambiant (il s'agit de la loi de Newton). + + On désigne par $\theta$ la température des macarons à l'instant $t$, et par $\theta'$ la vitesse de décongélation. + + L'unité de temps est la minute et l'unité de température le degré Celsius. + + \smallskip + + On négligera la diminution de température de la pièce et on admettra donc qu'il existe un + nombre réel $a$ tel que, pour $t$ positif : + + \[\theta'(t) = a [\theta(t) - 21]\quad (E)\] + + \medskip + + \begin{enumerate} + \item Vérifier que l'équation $(E)$ a pour solutions $\theta(t) = K e^{at} + 21$ où $K$ est un nombre réel. + + Donner alors, en fonction de $a$, l'ensemble des solutions de $(E)$. + \end{enumerate} + On rappelle que la température des macarons à l'instant $t = 0$ est égale à $-20$~\degres C et que, au bout de $15$~min, elle est de $4$~\degres C. + \begin{enumerate} + \setcounter{enumi}{1} + \item En utilisant la condition à $t=0$ démontrer que $K = -41$. + \item En utilisant la condition à $t=15$ démontrer que $a \approx -0.06$. + \item En déduire l'expression de la solution de l'équation différentielle puis étudier ses variations. + \item La température idéale de dégustation des macarons étant de $18$~\degres C, Marie estime que + celle-ci sera atteinte au bout de $30$~min. A-t-elle raison ? Justifier la réponse. + + Sinon, combien de temps faudra-t-il attendre ? +\end{enumerate} +\end{exercise} + + + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/TST_sti2d/DS/DS_21_07_08/08_210408_DS8.tex b/TST_sti2d/DS/DS_21_07_08/08_210408_DS8.tex new file mode 100644 index 0000000..d939261 --- /dev/null +++ b/TST_sti2d/DS/DS_21_07_08/08_210408_DS8.tex @@ -0,0 +1,136 @@ +\documentclass[a4paper,10pt]{article} +\usepackage{myXsim} + +% Title Page +\title{DS8 \hfill EVRARD Jules} +\tribe{TST sti2d} +\date{\hfillÀ render pour le vendredi 9 avril à 10h au plus tard} + +\xsimsetup{ + solution/print = false +} + +\begin{document} +\maketitle + +\begin{exercise}[subtitle={Étude de fonction}] + On considère la fonction $f$ définie sur $\intOF{0}{+\infty}$ par $ f(x) = 5x^2 + 0x + - 360\ln(x)$ + \begin{enumerate} + \item Démontrer que la dérivée de $f$ est $f'(x) = \frac{10x^2 + 0x + - 360}{x}$. + \item Étude du numérateur de $f'(x)$: $N(x) = 10x^2 - 360$ + \begin{enumerate} + \item Démontrer que $x=6$ et $x=- 6$ sont deux racines de $N(x)$.. + \item Proposer une forme factorisée de $N(x)$. + \item Proposer une forme factorisée de $f'(x)$. + \end{enumerate} + \item Étudier le signe de $f'$ et en déduire les variations de $f$. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item pas de correction disponible + \item + \begin{enumerate} + \item \[N(6) = 0\] + \[N(- 6) = 0\] + \item \[ + N(x) = 10(x - 6)(x - - 6) + \] + \item + \[ + f'(x) = \frac{10(x - 6)(x - - 6)}{x} + \] + \end{enumerate} + \item Pas de correction disponible + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Complexes}] + \begin{enumerate} + \item Mettre le nombre complexe suivant sous forme algébrique $z_1 = \dfrac{10 + 8 i}{-5 + 7 i} $ + \item Mettre le complexe suivante sous forme exponentielle $z_2 = 9 \sqrt{2} - 9 \sqrt{2} i$ + \item Mettre le complexe suivante sous forme exponentielle $z_3 = 7 \sqrt{3} + 7 i$ + \item Calculer le produit $z_4=z_2\times z_3$ donner le résultat sous forme exponentielle puis algébrique. + \item Calculer le quotient $z_5=\frac{z_2}{z_3}$ donner le résultat sous forme exponentielle puis algébrique. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item $z_1 = \frac{3}{37} - \frac{55 i}{37}$ + \item $z_2 = 18 e^{- \frac{i \pi}{4}}$ + \item $z_3 = 14 e^{\frac{i \pi}{6}}$ + \item $z_4 = 252 e^{- \frac{i \pi}{12}} = 63 \sqrt{2} + 63 \sqrt{6} + i \left(- 63 \sqrt{6} + 63 \sqrt{2}\right) = 243.0 - 65.2 i$ + \item $z_5 = \frac{9}{7} e^{- \frac{5 i \pi}{12}} = - \frac{9 \sqrt{2}}{28} + \frac{9 \sqrt{6}}{28} + i \left(- \frac{9 \sqrt{6}}{28} - \frac{9 \sqrt{2}}{28}\right) = 0.333 - 1.24 i$ + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Sortie du congélateur}] + Marie a invité quelques amis pour le thé. Elle souhaite leur proposer ses macarons maison. + + Elle les sort de son congélateur à $-15$~\degres C et les place dans une pièce à $17$~\degres C. + + Au bout de 15 minutes, la température des macarons est de $-1$~\degres C. + + \bigskip + + \textbf{Premier modèle} + + \medskip + + On suppose que la vitesse de décongélation est constante : chaque minute la hausse de + température des macarons est la même. + + Estimer dans ce cadre la température au bout de $30$~minutes, puis au bout de $45$~minutes. + + Cette modélisation est-elle pertinente? + + \bigskip + + \textbf{Deuxième modèle} + + \medskip + + On suppose maintenant que la vitesse de décongélation est proportionnelle à la différence + de température entre les macarons et l'air ambiant (il s'agit de la loi de Newton). + + On désigne par $\theta$ la température des macarons à l'instant $t$, et par $\theta'$ la vitesse de décongélation. + + L'unité de temps est la minute et l'unité de température le degré Celsius. + + \smallskip + + On négligera la diminution de température de la pièce et on admettra donc qu'il existe un + nombre réel $a$ tel que, pour $t$ positif : + + \[\theta'(t) = a [\theta(t) - 17]\quad (E)\] + + \medskip + + \begin{enumerate} + \item Vérifier que l'équation $(E)$ a pour solutions $\theta(t) = K e^{at} + 17$ où $K$ est un nombre réel. + + Donner alors, en fonction de $a$, l'ensemble des solutions de $(E)$. + \end{enumerate} + On rappelle que la température des macarons à l'instant $t = 0$ est égale à $-15$~\degres C et que, au bout de $15$~min, elle est de $-1$~\degres C. + \begin{enumerate} + \setcounter{enumi}{1} + \item En utilisant la condition à $t=0$ démontrer que $K = -32$. + \item En utilisant la condition à $t=15$ démontrer que $a \approx -0.04$. + \item En déduire l'expression de la solution de l'équation différentielle puis étudier ses variations. + \item La température idéale de dégustation des macarons étant de $14$~\degres C, Marie estime que + celle-ci sera atteinte au bout de $30$~min. A-t-elle raison ? Justifier la réponse. + + Sinon, combien de temps faudra-t-il attendre ? +\end{enumerate} +\end{exercise} + + + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/TST_sti2d/DS/DS_21_07_08/09_210408_DS8.tex b/TST_sti2d/DS/DS_21_07_08/09_210408_DS8.tex new file mode 100644 index 0000000..31ce9a1 --- /dev/null +++ b/TST_sti2d/DS/DS_21_07_08/09_210408_DS8.tex @@ -0,0 +1,136 @@ +\documentclass[a4paper,10pt]{article} +\usepackage{myXsim} + +% Title Page +\title{DS8 \hfill HADJRAS Mohcine} +\tribe{TST sti2d} +\date{\hfillÀ render pour le vendredi 9 avril à 10h au plus tard} + +\xsimsetup{ + solution/print = false +} + +\begin{document} +\maketitle + +\begin{exercise}[subtitle={Étude de fonction}] + On considère la fonction $f$ définie sur $\intOF{0}{+\infty}$ par $ f(x) = 3x^2 + 78x + 240\ln(x)$ + \begin{enumerate} + \item Démontrer que la dérivée de $f$ est $f'(x) = \frac{6x^2 + 78x + 240}{x}$. + \item Étude du numérateur de $f'(x)$: $N(x) = 6x^2 + 78x + 240$ + \begin{enumerate} + \item Démontrer que $x=- 5$ et $x=- 8$ sont deux racines de $N(x)$.. + \item Proposer une forme factorisée de $N(x)$. + \item Proposer une forme factorisée de $f'(x)$. + \end{enumerate} + \item Étudier le signe de $f'$ et en déduire les variations de $f$. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item pas de correction disponible + \item + \begin{enumerate} + \item \[N(- 5) = 0\] + \[N(- 8) = 0\] + \item \[ + N(x) = 6(x - - 5)(x - - 8) + \] + \item + \[ + f'(x) = \frac{6(x - - 5)(x - - 8)}{x} + \] + \end{enumerate} + \item Pas de correction disponible + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Complexes}] + \begin{enumerate} + \item Mettre le nombre complexe suivant sous forme algébrique $z_1 = \dfrac{10 + 5 i}{-6 + 4 i} $ + \item Mettre le complexe suivante sous forme exponentielle $z_2 = 1 + \sqrt{3} i$ + \item Mettre le complexe suivante sous forme exponentielle $z_3 = -7 - 7 \sqrt{3} i$ + \item Calculer le produit $z_4=z_2\times z_3$ donner le résultat sous forme exponentielle puis algébrique. + \item Calculer le quotient $z_5=\frac{z_2}{z_3}$ donner le résultat sous forme exponentielle puis algébrique. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item $z_1 = - \frac{10}{13} - \frac{35 i}{26}$ + \item $z_2 = 2 e^{\frac{i \pi}{3}}$ + \item $z_3 = 14 e^{- \frac{2 i \pi}{3}}$ + \item $z_4 = 28 e^{- \frac{i \pi}{3}} = 14 - 14 \sqrt{3} i = 14.0 - 24.3 i$ + \item $z_5 = \frac{1}{7} e^{i \pi} = - \frac{1}{7} = -0.143$ + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Sortie du congélateur}] + Marie a invité quelques amis pour le thé. Elle souhaite leur proposer ses macarons maison. + + Elle les sort de son congélateur à $-18$~\degres C et les place dans une pièce à $25$~\degres C. + + Au bout de 15 minutes, la température des macarons est de $4$~\degres C. + + \bigskip + + \textbf{Premier modèle} + + \medskip + + On suppose que la vitesse de décongélation est constante : chaque minute la hausse de + température des macarons est la même. + + Estimer dans ce cadre la température au bout de $30$~minutes, puis au bout de $45$~minutes. + + Cette modélisation est-elle pertinente? + + \bigskip + + \textbf{Deuxième modèle} + + \medskip + + On suppose maintenant que la vitesse de décongélation est proportionnelle à la différence + de température entre les macarons et l'air ambiant (il s'agit de la loi de Newton). + + On désigne par $\theta$ la température des macarons à l'instant $t$, et par $\theta'$ la vitesse de décongélation. + + L'unité de temps est la minute et l'unité de température le degré Celsius. + + \smallskip + + On négligera la diminution de température de la pièce et on admettra donc qu'il existe un + nombre réel $a$ tel que, pour $t$ positif : + + \[\theta'(t) = a [\theta(t) - 25]\quad (E)\] + + \medskip + + \begin{enumerate} + \item Vérifier que l'équation $(E)$ a pour solutions $\theta(t) = K e^{at} + 25$ où $K$ est un nombre réel. + + Donner alors, en fonction de $a$, l'ensemble des solutions de $(E)$. + \end{enumerate} + On rappelle que la température des macarons à l'instant $t = 0$ est égale à $-18$~\degres C et que, au bout de $15$~min, elle est de $4$~\degres C. + \begin{enumerate} + \setcounter{enumi}{1} + \item En utilisant la condition à $t=0$ démontrer que $K = -43$. + \item En utilisant la condition à $t=15$ démontrer que $a \approx -0.05$. + \item En déduire l'expression de la solution de l'équation différentielle puis étudier ses variations. + \item La température idéale de dégustation des macarons étant de $22$~\degres C, Marie estime que + celle-ci sera atteinte au bout de $30$~min. A-t-elle raison ? Justifier la réponse. + + Sinon, combien de temps faudra-t-il attendre ? +\end{enumerate} +\end{exercise} + + + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/TST_sti2d/DS/DS_21_07_08/10_210408_DS8.tex b/TST_sti2d/DS/DS_21_07_08/10_210408_DS8.tex new file mode 100644 index 0000000..31686df --- /dev/null +++ b/TST_sti2d/DS/DS_21_07_08/10_210408_DS8.tex @@ -0,0 +1,136 @@ +\documentclass[a4paper,10pt]{article} +\usepackage{myXsim} + +% Title Page +\title{DS8 \hfill HENRIST Maxime} +\tribe{TST sti2d} +\date{\hfillÀ render pour le vendredi 9 avril à 10h au plus tard} + +\xsimsetup{ + solution/print = false +} + +\begin{document} +\maketitle + +\begin{exercise}[subtitle={Étude de fonction}] + On considère la fonction $f$ définie sur $\intOF{0}{+\infty}$ par $ f(x) = 4.5x^2 + - 27x + - 36\ln(x)$ + \begin{enumerate} + \item Démontrer que la dérivée de $f$ est $f'(x) = \frac{9x^2 + - 27x + - 36}{x}$. + \item Étude du numérateur de $f'(x)$: $N(x) = 9x^2 - 27x - 36$ + \begin{enumerate} + \item Démontrer que $x=4$ et $x=- 1$ sont deux racines de $N(x)$.. + \item Proposer une forme factorisée de $N(x)$. + \item Proposer une forme factorisée de $f'(x)$. + \end{enumerate} + \item Étudier le signe de $f'$ et en déduire les variations de $f$. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item pas de correction disponible + \item + \begin{enumerate} + \item \[N(4) = 0\] + \[N(- 1) = 0\] + \item \[ + N(x) = 9(x - 4)(x - - 1) + \] + \item + \[ + f'(x) = \frac{9(x - 4)(x - - 1)}{x} + \] + \end{enumerate} + \item Pas de correction disponible + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Complexes}] + \begin{enumerate} + \item Mettre le nombre complexe suivant sous forme algébrique $z_1 = \dfrac{6 + 5 i}{-6 + 5 i} $ + \item Mettre le complexe suivante sous forme exponentielle $z_2 = 8 \sqrt{3} - 8 i$ + \item Mettre le complexe suivante sous forme exponentielle $z_3 = 5 - 5 \sqrt{3} i$ + \item Calculer le produit $z_4=z_2\times z_3$ donner le résultat sous forme exponentielle puis algébrique. + \item Calculer le quotient $z_5=\frac{z_2}{z_3}$ donner le résultat sous forme exponentielle puis algébrique. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item $z_1 = - \frac{11}{61} - \frac{60 i}{61}$ + \item $z_2 = 16 e^{- \frac{i \pi}{6}}$ + \item $z_3 = 10 e^{- \frac{i \pi}{3}}$ + \item $z_4 = 160 e^{- \frac{i \pi}{2}} = - 160 i = - 160.0 i$ + \item $z_5 = \frac{8}{5} e^{\frac{i \pi}{6}} = \frac{4 \sqrt{3}}{5} + \frac{4 i}{5} = 1.39 + 0.8 i$ + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Sortie du congélateur}] + Marie a invité quelques amis pour le thé. Elle souhaite leur proposer ses macarons maison. + + Elle les sort de son congélateur à $-16$~\degres C et les place dans une pièce à $23$~\degres C. + + Au bout de 15 minutes, la température des macarons est de $0$~\degres C. + + \bigskip + + \textbf{Premier modèle} + + \medskip + + On suppose que la vitesse de décongélation est constante : chaque minute la hausse de + température des macarons est la même. + + Estimer dans ce cadre la température au bout de $30$~minutes, puis au bout de $45$~minutes. + + Cette modélisation est-elle pertinente? + + \bigskip + + \textbf{Deuxième modèle} + + \medskip + + On suppose maintenant que la vitesse de décongélation est proportionnelle à la différence + de température entre les macarons et l'air ambiant (il s'agit de la loi de Newton). + + On désigne par $\theta$ la température des macarons à l'instant $t$, et par $\theta'$ la vitesse de décongélation. + + L'unité de temps est la minute et l'unité de température le degré Celsius. + + \smallskip + + On négligera la diminution de température de la pièce et on admettra donc qu'il existe un + nombre réel $a$ tel que, pour $t$ positif : + + \[\theta'(t) = a [\theta(t) - 23]\quad (E)\] + + \medskip + + \begin{enumerate} + \item Vérifier que l'équation $(E)$ a pour solutions $\theta(t) = K e^{at} + 23$ où $K$ est un nombre réel. + + Donner alors, en fonction de $a$, l'ensemble des solutions de $(E)$. + \end{enumerate} + On rappelle que la température des macarons à l'instant $t = 0$ est égale à $-16$~\degres C et que, au bout de $15$~min, elle est de $0$~\degres C. + \begin{enumerate} + \setcounter{enumi}{1} + \item En utilisant la condition à $t=0$ démontrer que $K = -39$. + \item En utilisant la condition à $t=15$ démontrer que $a \approx -0.04$. + \item En déduire l'expression de la solution de l'équation différentielle puis étudier ses variations. + \item La température idéale de dégustation des macarons étant de $20$~\degres C, Marie estime que + celle-ci sera atteinte au bout de $30$~min. A-t-elle raison ? Justifier la réponse. + + Sinon, combien de temps faudra-t-il attendre ? +\end{enumerate} +\end{exercise} + + + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/TST_sti2d/DS/DS_21_07_08/11_210408_DS8.tex b/TST_sti2d/DS/DS_21_07_08/11_210408_DS8.tex new file mode 100644 index 0000000..61a8951 --- /dev/null +++ b/TST_sti2d/DS/DS_21_07_08/11_210408_DS8.tex @@ -0,0 +1,136 @@ +\documentclass[a4paper,10pt]{article} +\usepackage{myXsim} + +% Title Page +\title{DS8 \hfill HUMBERT Rayan} +\tribe{TST sti2d} +\date{\hfillÀ render pour le vendredi 9 avril à 10h au plus tard} + +\xsimsetup{ + solution/print = false +} + +\begin{document} +\maketitle + +\begin{exercise}[subtitle={Étude de fonction}] + On considère la fonction $f$ définie sur $\intOF{0}{+\infty}$ par $ f(x) = 2.5x^2 + - 25x + - 180\ln(x)$ + \begin{enumerate} + \item Démontrer que la dérivée de $f$ est $f'(x) = \frac{5x^2 + - 25x + - 180}{x}$. + \item Étude du numérateur de $f'(x)$: $N(x) = 5x^2 - 25x - 180$ + \begin{enumerate} + \item Démontrer que $x=- 4$ et $x=9$ sont deux racines de $N(x)$.. + \item Proposer une forme factorisée de $N(x)$. + \item Proposer une forme factorisée de $f'(x)$. + \end{enumerate} + \item Étudier le signe de $f'$ et en déduire les variations de $f$. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item pas de correction disponible + \item + \begin{enumerate} + \item \[N(- 4) = 0\] + \[N(9) = 0\] + \item \[ + N(x) = 5(x - - 4)(x - 9) + \] + \item + \[ + f'(x) = \frac{5(x - - 4)(x - 9)}{x} + \] + \end{enumerate} + \item Pas de correction disponible + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Complexes}] + \begin{enumerate} + \item Mettre le nombre complexe suivant sous forme algébrique $z_1 = \dfrac{8 + 3 i}{-6 + 6 i} $ + \item Mettre le complexe suivante sous forme exponentielle $z_2 = -4 - 4 \sqrt{3} i$ + \item Mettre le complexe suivante sous forme exponentielle $z_3 = 8 \sqrt{3} + 8 i$ + \item Calculer le produit $z_4=z_2\times z_3$ donner le résultat sous forme exponentielle puis algébrique. + \item Calculer le quotient $z_5=\frac{z_2}{z_3}$ donner le résultat sous forme exponentielle puis algébrique. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item $z_1 = - \frac{5}{12} - \frac{11 i}{12}$ + \item $z_2 = 8 e^{- \frac{2 i \pi}{3}}$ + \item $z_3 = 16 e^{\frac{i \pi}{6}}$ + \item $z_4 = 128 e^{- \frac{i \pi}{2}} = - 128 i = - 128.0 i$ + \item $z_5 = \frac{1}{2} e^{- \frac{5 i \pi}{6}} = - \frac{\sqrt{3}}{4} - \frac{i}{4} = -0.433 - 0.25 i$ + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Sortie du congélateur}] + Marie a invité quelques amis pour le thé. Elle souhaite leur proposer ses macarons maison. + + Elle les sort de son congélateur à $-20$~\degres C et les place dans une pièce à $25$~\degres C. + + Au bout de 15 minutes, la température des macarons est de $-3$~\degres C. + + \bigskip + + \textbf{Premier modèle} + + \medskip + + On suppose que la vitesse de décongélation est constante : chaque minute la hausse de + température des macarons est la même. + + Estimer dans ce cadre la température au bout de $30$~minutes, puis au bout de $45$~minutes. + + Cette modélisation est-elle pertinente? + + \bigskip + + \textbf{Deuxième modèle} + + \medskip + + On suppose maintenant que la vitesse de décongélation est proportionnelle à la différence + de température entre les macarons et l'air ambiant (il s'agit de la loi de Newton). + + On désigne par $\theta$ la température des macarons à l'instant $t$, et par $\theta'$ la vitesse de décongélation. + + L'unité de temps est la minute et l'unité de température le degré Celsius. + + \smallskip + + On négligera la diminution de température de la pièce et on admettra donc qu'il existe un + nombre réel $a$ tel que, pour $t$ positif : + + \[\theta'(t) = a [\theta(t) - 25]\quad (E)\] + + \medskip + + \begin{enumerate} + \item Vérifier que l'équation $(E)$ a pour solutions $\theta(t) = K e^{at} + 25$ où $K$ est un nombre réel. + + Donner alors, en fonction de $a$, l'ensemble des solutions de $(E)$. + \end{enumerate} + On rappelle que la température des macarons à l'instant $t = 0$ est égale à $-20$~\degres C et que, au bout de $15$~min, elle est de $-3$~\degres C. + \begin{enumerate} + \setcounter{enumi}{1} + \item En utilisant la condition à $t=0$ démontrer que $K = -45$. + \item En utilisant la condition à $t=15$ démontrer que $a \approx -0.03$. + \item En déduire l'expression de la solution de l'équation différentielle puis étudier ses variations. + \item La température idéale de dégustation des macarons étant de $22$~\degres C, Marie estime que + celle-ci sera atteinte au bout de $30$~min. A-t-elle raison ? Justifier la réponse. + + Sinon, combien de temps faudra-t-il attendre ? +\end{enumerate} +\end{exercise} + + + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/TST_sti2d/DS/DS_21_07_08/12_210408_DS8.tex b/TST_sti2d/DS/DS_21_07_08/12_210408_DS8.tex new file mode 100644 index 0000000..be5619f --- /dev/null +++ b/TST_sti2d/DS/DS_21_07_08/12_210408_DS8.tex @@ -0,0 +1,136 @@ +\documentclass[a4paper,10pt]{article} +\usepackage{myXsim} + +% Title Page +\title{DS8 \hfill KILINC Suleyman} +\tribe{TST sti2d} +\date{\hfillÀ render pour le vendredi 9 avril à 10h au plus tard} + +\xsimsetup{ + solution/print = false +} + +\begin{document} +\maketitle + +\begin{exercise}[subtitle={Étude de fonction}] + On considère la fonction $f$ définie sur $\intOF{0}{+\infty}$ par $ f(x) = 3.5x^2 + 21x + - 490\ln(x)$ + \begin{enumerate} + \item Démontrer que la dérivée de $f$ est $f'(x) = \frac{7x^2 + 21x + - 490}{x}$. + \item Étude du numérateur de $f'(x)$: $N(x) = 7x^2 + 21x - 490$ + \begin{enumerate} + \item Démontrer que $x=- 10$ et $x=7$ sont deux racines de $N(x)$.. + \item Proposer une forme factorisée de $N(x)$. + \item Proposer une forme factorisée de $f'(x)$. + \end{enumerate} + \item Étudier le signe de $f'$ et en déduire les variations de $f$. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item pas de correction disponible + \item + \begin{enumerate} + \item \[N(- 10) = 0\] + \[N(7) = 0\] + \item \[ + N(x) = 7(x - - 10)(x - 7) + \] + \item + \[ + f'(x) = \frac{7(x - - 10)(x - 7)}{x} + \] + \end{enumerate} + \item Pas de correction disponible + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Complexes}] + \begin{enumerate} + \item Mettre le nombre complexe suivant sous forme algébrique $z_1 = \dfrac{5 + 7 i}{-6 + 4 i} $ + \item Mettre le complexe suivante sous forme exponentielle $z_2 = 1 - \sqrt{3} i$ + \item Mettre le complexe suivante sous forme exponentielle $z_3 = - 8 \sqrt{2} + 8 \sqrt{2} i$ + \item Calculer le produit $z_4=z_2\times z_3$ donner le résultat sous forme exponentielle puis algébrique. + \item Calculer le quotient $z_5=\frac{z_2}{z_3}$ donner le résultat sous forme exponentielle puis algébrique. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item $z_1 = - \frac{1}{26} - \frac{31 i}{26}$ + \item $z_2 = 2 e^{- \frac{i \pi}{3}}$ + \item $z_3 = 16 e^{\frac{3 i \pi}{4}}$ + \item $z_4 = 32 e^{\frac{5 i \pi}{12}} = - 8 \sqrt{2} + 8 \sqrt{6} + i \left(8 \sqrt{2} + 8 \sqrt{6}\right) = 8.28 + 30.9 i$ + \item $z_5 = \frac{1}{8} e^{- \frac{13 i \pi}{12}} = - \frac{\sqrt{6}}{32} - \frac{\sqrt{2}}{32} + i \left(- \frac{\sqrt{2}}{32} + \frac{\sqrt{6}}{32}\right) = -0.121 + 0.0323 i$ + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Sortie du congélateur}] + Marie a invité quelques amis pour le thé. Elle souhaite leur proposer ses macarons maison. + + Elle les sort de son congélateur à $-19$~\degres C et les place dans une pièce à $15$~\degres C. + + Au bout de 15 minutes, la température des macarons est de $-2$~\degres C. + + \bigskip + + \textbf{Premier modèle} + + \medskip + + On suppose que la vitesse de décongélation est constante : chaque minute la hausse de + température des macarons est la même. + + Estimer dans ce cadre la température au bout de $30$~minutes, puis au bout de $45$~minutes. + + Cette modélisation est-elle pertinente? + + \bigskip + + \textbf{Deuxième modèle} + + \medskip + + On suppose maintenant que la vitesse de décongélation est proportionnelle à la différence + de température entre les macarons et l'air ambiant (il s'agit de la loi de Newton). + + On désigne par $\theta$ la température des macarons à l'instant $t$, et par $\theta'$ la vitesse de décongélation. + + L'unité de temps est la minute et l'unité de température le degré Celsius. + + \smallskip + + On négligera la diminution de température de la pièce et on admettra donc qu'il existe un + nombre réel $a$ tel que, pour $t$ positif : + + \[\theta'(t) = a [\theta(t) - 15]\quad (E)\] + + \medskip + + \begin{enumerate} + \item Vérifier que l'équation $(E)$ a pour solutions $\theta(t) = K e^{at} + 15$ où $K$ est un nombre réel. + + Donner alors, en fonction de $a$, l'ensemble des solutions de $(E)$. + \end{enumerate} + On rappelle que la température des macarons à l'instant $t = 0$ est égale à $-19$~\degres C et que, au bout de $15$~min, elle est de $-2$~\degres C. + \begin{enumerate} + \setcounter{enumi}{1} + \item En utilisant la condition à $t=0$ démontrer que $K = -34$. + \item En utilisant la condition à $t=15$ démontrer que $a \approx -0.05$. + \item En déduire l'expression de la solution de l'équation différentielle puis étudier ses variations. + \item La température idéale de dégustation des macarons étant de $12$~\degres C, Marie estime que + celle-ci sera atteinte au bout de $30$~min. A-t-elle raison ? Justifier la réponse. + + Sinon, combien de temps faudra-t-il attendre ? +\end{enumerate} +\end{exercise} + + + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/TST_sti2d/DS/DS_21_07_08/13_210408_DS8.tex b/TST_sti2d/DS/DS_21_07_08/13_210408_DS8.tex new file mode 100644 index 0000000..c9e0d3f --- /dev/null +++ b/TST_sti2d/DS/DS_21_07_08/13_210408_DS8.tex @@ -0,0 +1,136 @@ +\documentclass[a4paper,10pt]{article} +\usepackage{myXsim} + +% Title Page +\title{DS8 \hfill M'BAREK HASNAOUI Bilal} +\tribe{TST sti2d} +\date{\hfillÀ render pour le vendredi 9 avril à 10h au plus tard} + +\xsimsetup{ + solution/print = false +} + +\begin{document} +\maketitle + +\begin{exercise}[subtitle={Étude de fonction}] + On considère la fonction $f$ définie sur $\intOF{0}{+\infty}$ par $ f(x) = 5x^2 + - 40x + - 450\ln(x)$ + \begin{enumerate} + \item Démontrer que la dérivée de $f$ est $f'(x) = \frac{10x^2 + - 40x + - 450}{x}$. + \item Étude du numérateur de $f'(x)$: $N(x) = 10x^2 - 40x - 450$ + \begin{enumerate} + \item Démontrer que $x=9$ et $x=- 5$ sont deux racines de $N(x)$.. + \item Proposer une forme factorisée de $N(x)$. + \item Proposer une forme factorisée de $f'(x)$. + \end{enumerate} + \item Étudier le signe de $f'$ et en déduire les variations de $f$. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item pas de correction disponible + \item + \begin{enumerate} + \item \[N(9) = 0\] + \[N(- 5) = 0\] + \item \[ + N(x) = 10(x - 9)(x - - 5) + \] + \item + \[ + f'(x) = \frac{10(x - 9)(x - - 5)}{x} + \] + \end{enumerate} + \item Pas de correction disponible + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Complexes}] + \begin{enumerate} + \item Mettre le nombre complexe suivant sous forme algébrique $z_1 = \dfrac{2 + 4 i}{-5 + 7 i} $ + \item Mettre le complexe suivante sous forme exponentielle $z_2 = 7 + 7 \sqrt{3} i$ + \item Mettre le complexe suivante sous forme exponentielle $z_3 = 7 \sqrt{2} + 7 \sqrt{2} i$ + \item Calculer le produit $z_4=z_2\times z_3$ donner le résultat sous forme exponentielle puis algébrique. + \item Calculer le quotient $z_5=\frac{z_2}{z_3}$ donner le résultat sous forme exponentielle puis algébrique. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item $z_1 = \frac{9}{37} - \frac{17 i}{37}$ + \item $z_2 = 14 e^{\frac{i \pi}{3}}$ + \item $z_3 = 14 e^{\frac{i \pi}{4}}$ + \item $z_4 = 196 e^{\frac{7 i \pi}{12}} = - 49 \sqrt{6} + 49 \sqrt{2} + i \left(49 \sqrt{2} + 49 \sqrt{6}\right) = -50.7 + 189.0 i$ + \item $z_5 = 1 e^{\frac{i \pi}{12}} = \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4} + i \left(- \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right) = 0.966 + 0.259 i$ + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Sortie du congélateur}] + Marie a invité quelques amis pour le thé. Elle souhaite leur proposer ses macarons maison. + + Elle les sort de son congélateur à $-15$~\degres C et les place dans une pièce à $25$~\degres C. + + Au bout de 15 minutes, la température des macarons est de $0$~\degres C. + + \bigskip + + \textbf{Premier modèle} + + \medskip + + On suppose que la vitesse de décongélation est constante : chaque minute la hausse de + température des macarons est la même. + + Estimer dans ce cadre la température au bout de $30$~minutes, puis au bout de $45$~minutes. + + Cette modélisation est-elle pertinente? + + \bigskip + + \textbf{Deuxième modèle} + + \medskip + + On suppose maintenant que la vitesse de décongélation est proportionnelle à la différence + de température entre les macarons et l'air ambiant (il s'agit de la loi de Newton). + + On désigne par $\theta$ la température des macarons à l'instant $t$, et par $\theta'$ la vitesse de décongélation. + + L'unité de temps est la minute et l'unité de température le degré Celsius. + + \smallskip + + On négligera la diminution de température de la pièce et on admettra donc qu'il existe un + nombre réel $a$ tel que, pour $t$ positif : + + \[\theta'(t) = a [\theta(t) - 25]\quad (E)\] + + \medskip + + \begin{enumerate} + \item Vérifier que l'équation $(E)$ a pour solutions $\theta(t) = K e^{at} + 25$ où $K$ est un nombre réel. + + Donner alors, en fonction de $a$, l'ensemble des solutions de $(E)$. + \end{enumerate} + On rappelle que la température des macarons à l'instant $t = 0$ est égale à $-15$~\degres C et que, au bout de $15$~min, elle est de $0$~\degres C. + \begin{enumerate} + \setcounter{enumi}{1} + \item En utilisant la condition à $t=0$ démontrer que $K = -40$. + \item En utilisant la condition à $t=15$ démontrer que $a \approx -0.03$. + \item En déduire l'expression de la solution de l'équation différentielle puis étudier ses variations. + \item La température idéale de dégustation des macarons étant de $22$~\degres C, Marie estime que + celle-ci sera atteinte au bout de $30$~min. A-t-elle raison ? Justifier la réponse. + + Sinon, combien de temps faudra-t-il attendre ? +\end{enumerate} +\end{exercise} + + + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/TST_sti2d/DS/DS_21_07_08/14_210408_DS8.tex b/TST_sti2d/DS/DS_21_07_08/14_210408_DS8.tex new file mode 100644 index 0000000..870ac75 --- /dev/null +++ b/TST_sti2d/DS/DS_21_07_08/14_210408_DS8.tex @@ -0,0 +1,136 @@ +\documentclass[a4paper,10pt]{article} +\usepackage{myXsim} + +% Title Page +\title{DS8 \hfill MERCIER Almandin} +\tribe{TST sti2d} +\date{\hfillÀ render pour le vendredi 9 avril à 10h au plus tard} + +\xsimsetup{ + solution/print = false +} + +\begin{document} +\maketitle + +\begin{exercise}[subtitle={Étude de fonction}] + On considère la fonction $f$ définie sur $\intOF{0}{+\infty}$ par $ f(x) = 2.5x^2 + - 50x + 120\ln(x)$ + \begin{enumerate} + \item Démontrer que la dérivée de $f$ est $f'(x) = \frac{5x^2 + - 50x + 120}{x}$. + \item Étude du numérateur de $f'(x)$: $N(x) = 5x^2 - 50x + 120$ + \begin{enumerate} + \item Démontrer que $x=6$ et $x=4$ sont deux racines de $N(x)$.. + \item Proposer une forme factorisée de $N(x)$. + \item Proposer une forme factorisée de $f'(x)$. + \end{enumerate} + \item Étudier le signe de $f'$ et en déduire les variations de $f$. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item pas de correction disponible + \item + \begin{enumerate} + \item \[N(6) = 0\] + \[N(4) = 0\] + \item \[ + N(x) = 5(x - 6)(x - 4) + \] + \item + \[ + f'(x) = \frac{5(x - 6)(x - 4)}{x} + \] + \end{enumerate} + \item Pas de correction disponible + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Complexes}] + \begin{enumerate} + \item Mettre le nombre complexe suivant sous forme algébrique $z_1 = \dfrac{10 + 10 i}{-2 + 3 i} $ + \item Mettre le complexe suivante sous forme exponentielle $z_2 = - 7 \sqrt{2} - 7 \sqrt{2} i$ + \item Mettre le complexe suivante sous forme exponentielle $z_3 = - 9 \sqrt{2} + 9 \sqrt{2} i$ + \item Calculer le produit $z_4=z_2\times z_3$ donner le résultat sous forme exponentielle puis algébrique. + \item Calculer le quotient $z_5=\frac{z_2}{z_3}$ donner le résultat sous forme exponentielle puis algébrique. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item $z_1 = \frac{10}{13} - \frac{50 i}{13}$ + \item $z_2 = 14 e^{- \frac{3 i \pi}{4}}$ + \item $z_3 = 18 e^{\frac{3 i \pi}{4}}$ + \item $z_4 = 252 e^{0} = 252 = 252.0$ + \item $z_5 = \frac{7}{9} e^{- \frac{3 i \pi}{2}} = \frac{7 i}{9} = 0.778 i$ + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Sortie du congélateur}] + Marie a invité quelques amis pour le thé. Elle souhaite leur proposer ses macarons maison. + + Elle les sort de son congélateur à $-16$~\degres C et les place dans une pièce à $22$~\degres C. + + Au bout de 15 minutes, la température des macarons est de $3$~\degres C. + + \bigskip + + \textbf{Premier modèle} + + \medskip + + On suppose que la vitesse de décongélation est constante : chaque minute la hausse de + température des macarons est la même. + + Estimer dans ce cadre la température au bout de $30$~minutes, puis au bout de $45$~minutes. + + Cette modélisation est-elle pertinente? + + \bigskip + + \textbf{Deuxième modèle} + + \medskip + + On suppose maintenant que la vitesse de décongélation est proportionnelle à la différence + de température entre les macarons et l'air ambiant (il s'agit de la loi de Newton). + + On désigne par $\theta$ la température des macarons à l'instant $t$, et par $\theta'$ la vitesse de décongélation. + + L'unité de temps est la minute et l'unité de température le degré Celsius. + + \smallskip + + On négligera la diminution de température de la pièce et on admettra donc qu'il existe un + nombre réel $a$ tel que, pour $t$ positif : + + \[\theta'(t) = a [\theta(t) - 22]\quad (E)\] + + \medskip + + \begin{enumerate} + \item Vérifier que l'équation $(E)$ a pour solutions $\theta(t) = K e^{at} + 22$ où $K$ est un nombre réel. + + Donner alors, en fonction de $a$, l'ensemble des solutions de $(E)$. + \end{enumerate} + On rappelle que la température des macarons à l'instant $t = 0$ est égale à $-16$~\degres C et que, au bout de $15$~min, elle est de $3$~\degres C. + \begin{enumerate} + \setcounter{enumi}{1} + \item En utilisant la condition à $t=0$ démontrer que $K = -38$. + \item En utilisant la condition à $t=15$ démontrer que $a \approx -0.05$. + \item En déduire l'expression de la solution de l'équation différentielle puis étudier ses variations. + \item La température idéale de dégustation des macarons étant de $19$~\degres C, Marie estime que + celle-ci sera atteinte au bout de $30$~min. A-t-elle raison ? Justifier la réponse. + + Sinon, combien de temps faudra-t-il attendre ? +\end{enumerate} +\end{exercise} + + + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/TST_sti2d/DS/DS_21_07_08/15_210408_DS8.tex b/TST_sti2d/DS/DS_21_07_08/15_210408_DS8.tex new file mode 100644 index 0000000..738238e --- /dev/null +++ b/TST_sti2d/DS/DS_21_07_08/15_210408_DS8.tex @@ -0,0 +1,136 @@ +\documentclass[a4paper,10pt]{article} +\usepackage{myXsim} + +% Title Page +\title{DS8 \hfill MOUFAQ Amine} +\tribe{TST sti2d} +\date{\hfillÀ render pour le vendredi 9 avril à 10h au plus tard} + +\xsimsetup{ + solution/print = false +} + +\begin{document} +\maketitle + +\begin{exercise}[subtitle={Étude de fonction}] + On considère la fonction $f$ définie sur $\intOF{0}{+\infty}$ par $ f(x) = 5x^2 + 0x + - 810\ln(x)$ + \begin{enumerate} + \item Démontrer que la dérivée de $f$ est $f'(x) = \frac{10x^2 + 0x + - 810}{x}$. + \item Étude du numérateur de $f'(x)$: $N(x) = 10x^2 - 810$ + \begin{enumerate} + \item Démontrer que $x=- 9$ et $x=9$ sont deux racines de $N(x)$.. + \item Proposer une forme factorisée de $N(x)$. + \item Proposer une forme factorisée de $f'(x)$. + \end{enumerate} + \item Étudier le signe de $f'$ et en déduire les variations de $f$. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item pas de correction disponible + \item + \begin{enumerate} + \item \[N(- 9) = 0\] + \[N(9) = 0\] + \item \[ + N(x) = 10(x - - 9)(x - 9) + \] + \item + \[ + f'(x) = \frac{10(x - - 9)(x - 9)}{x} + \] + \end{enumerate} + \item Pas de correction disponible + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Complexes}] + \begin{enumerate} + \item Mettre le nombre complexe suivant sous forme algébrique $z_1 = \dfrac{8 + 4 i}{-3 + 5 i} $ + \item Mettre le complexe suivante sous forme exponentielle $z_2 = - 2 \sqrt{2} + 2 \sqrt{2} i$ + \item Mettre le complexe suivante sous forme exponentielle $z_3 = 6 + 6 \sqrt{3} i$ + \item Calculer le produit $z_4=z_2\times z_3$ donner le résultat sous forme exponentielle puis algébrique. + \item Calculer le quotient $z_5=\frac{z_2}{z_3}$ donner le résultat sous forme exponentielle puis algébrique. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item $z_1 = - \frac{2}{17} - \frac{26 i}{17}$ + \item $z_2 = 4 e^{\frac{3 i \pi}{4}}$ + \item $z_3 = 12 e^{\frac{i \pi}{3}}$ + \item $z_4 = 48 e^{\frac{13 i \pi}{12}} = - 12 \sqrt{6} - 12 \sqrt{2} + i \left(- 12 \sqrt{6} + 12 \sqrt{2}\right) = -46.4 - 12.4 i$ + \item $z_5 = \frac{1}{3} e^{\frac{5 i \pi}{12}} = - \frac{\sqrt{2}}{12} + \frac{\sqrt{6}}{12} + i \left(\frac{\sqrt{2}}{12} + \frac{\sqrt{6}}{12}\right) = 0.0863 + 0.322 i$ + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Sortie du congélateur}] + Marie a invité quelques amis pour le thé. Elle souhaite leur proposer ses macarons maison. + + Elle les sort de son congélateur à $-15$~\degres C et les place dans une pièce à $20$~\degres C. + + Au bout de 15 minutes, la température des macarons est de $-3$~\degres C. + + \bigskip + + \textbf{Premier modèle} + + \medskip + + On suppose que la vitesse de décongélation est constante : chaque minute la hausse de + température des macarons est la même. + + Estimer dans ce cadre la température au bout de $30$~minutes, puis au bout de $45$~minutes. + + Cette modélisation est-elle pertinente? + + \bigskip + + \textbf{Deuxième modèle} + + \medskip + + On suppose maintenant que la vitesse de décongélation est proportionnelle à la différence + de température entre les macarons et l'air ambiant (il s'agit de la loi de Newton). + + On désigne par $\theta$ la température des macarons à l'instant $t$, et par $\theta'$ la vitesse de décongélation. + + L'unité de temps est la minute et l'unité de température le degré Celsius. + + \smallskip + + On négligera la diminution de température de la pièce et on admettra donc qu'il existe un + nombre réel $a$ tel que, pour $t$ positif : + + \[\theta'(t) = a [\theta(t) - 20]\quad (E)\] + + \medskip + + \begin{enumerate} + \item Vérifier que l'équation $(E)$ a pour solutions $\theta(t) = K e^{at} + 20$ où $K$ est un nombre réel. + + Donner alors, en fonction de $a$, l'ensemble des solutions de $(E)$. + \end{enumerate} + On rappelle que la température des macarons à l'instant $t = 0$ est égale à $-15$~\degres C et que, au bout de $15$~min, elle est de $-3$~\degres C. + \begin{enumerate} + \setcounter{enumi}{1} + \item En utilisant la condition à $t=0$ démontrer que $K = -35$. + \item En utilisant la condition à $t=15$ démontrer que $a \approx -0.03$. + \item En déduire l'expression de la solution de l'équation différentielle puis étudier ses variations. + \item La température idéale de dégustation des macarons étant de $17$~\degres C, Marie estime que + celle-ci sera atteinte au bout de $30$~min. A-t-elle raison ? Justifier la réponse. + + Sinon, combien de temps faudra-t-il attendre ? +\end{enumerate} +\end{exercise} + + + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/TST_sti2d/DS/DS_21_07_08/16_210408_DS8.tex b/TST_sti2d/DS/DS_21_07_08/16_210408_DS8.tex new file mode 100644 index 0000000..9027713 --- /dev/null +++ b/TST_sti2d/DS/DS_21_07_08/16_210408_DS8.tex @@ -0,0 +1,136 @@ +\documentclass[a4paper,10pt]{article} +\usepackage{myXsim} + +% Title Page +\title{DS8 \hfill NARDINI Kakary} +\tribe{TST sti2d} +\date{\hfillÀ render pour le vendredi 9 avril à 10h au plus tard} + +\xsimsetup{ + solution/print = false +} + +\begin{document} +\maketitle + +\begin{exercise}[subtitle={Étude de fonction}] + On considère la fonction $f$ définie sur $\intOF{0}{+\infty}$ par $ f(x) = 3x^2 + - 18x + - 108\ln(x)$ + \begin{enumerate} + \item Démontrer que la dérivée de $f$ est $f'(x) = \frac{6x^2 + - 18x + - 108}{x}$. + \item Étude du numérateur de $f'(x)$: $N(x) = 6x^2 - 18x - 108$ + \begin{enumerate} + \item Démontrer que $x=6$ et $x=- 3$ sont deux racines de $N(x)$.. + \item Proposer une forme factorisée de $N(x)$. + \item Proposer une forme factorisée de $f'(x)$. + \end{enumerate} + \item Étudier le signe de $f'$ et en déduire les variations de $f$. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item pas de correction disponible + \item + \begin{enumerate} + \item \[N(6) = 0\] + \[N(- 3) = 0\] + \item \[ + N(x) = 6(x - 6)(x - - 3) + \] + \item + \[ + f'(x) = \frac{6(x - 6)(x - - 3)}{x} + \] + \end{enumerate} + \item Pas de correction disponible + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Complexes}] + \begin{enumerate} + \item Mettre le nombre complexe suivant sous forme algébrique $z_1 = \dfrac{9 + 6 i}{-3 + 6 i} $ + \item Mettre le complexe suivante sous forme exponentielle $z_2 = -4 - 4 \sqrt{3} i$ + \item Mettre le complexe suivante sous forme exponentielle $z_3 = 9 \sqrt{2} + 9 \sqrt{2} i$ + \item Calculer le produit $z_4=z_2\times z_3$ donner le résultat sous forme exponentielle puis algébrique. + \item Calculer le quotient $z_5=\frac{z_2}{z_3}$ donner le résultat sous forme exponentielle puis algébrique. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item $z_1 = \frac{1}{5} - \frac{8 i}{5}$ + \item $z_2 = 8 e^{- \frac{2 i \pi}{3}}$ + \item $z_3 = 18 e^{\frac{i \pi}{4}}$ + \item $z_4 = 144 e^{- \frac{5 i \pi}{12}} = - 36 \sqrt{2} + 36 \sqrt{6} + i \left(- 36 \sqrt{6} - 36 \sqrt{2}\right) = 37.3 - 139.0 i$ + \item $z_5 = \frac{4}{9} e^{- \frac{11 i \pi}{12}} = - \frac{\sqrt{6}}{9} - \frac{\sqrt{2}}{9} + i \left(- \frac{\sqrt{6}}{9} + \frac{\sqrt{2}}{9}\right) = -0.429 - 0.115 i$ + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Sortie du congélateur}] + Marie a invité quelques amis pour le thé. Elle souhaite leur proposer ses macarons maison. + + Elle les sort de son congélateur à $-18$~\degres C et les place dans une pièce à $17$~\degres C. + + Au bout de 15 minutes, la température des macarons est de $2$~\degres C. + + \bigskip + + \textbf{Premier modèle} + + \medskip + + On suppose que la vitesse de décongélation est constante : chaque minute la hausse de + température des macarons est la même. + + Estimer dans ce cadre la température au bout de $30$~minutes, puis au bout de $45$~minutes. + + Cette modélisation est-elle pertinente? + + \bigskip + + \textbf{Deuxième modèle} + + \medskip + + On suppose maintenant que la vitesse de décongélation est proportionnelle à la différence + de température entre les macarons et l'air ambiant (il s'agit de la loi de Newton). + + On désigne par $\theta$ la température des macarons à l'instant $t$, et par $\theta'$ la vitesse de décongélation. + + L'unité de temps est la minute et l'unité de température le degré Celsius. + + \smallskip + + On négligera la diminution de température de la pièce et on admettra donc qu'il existe un + nombre réel $a$ tel que, pour $t$ positif : + + \[\theta'(t) = a [\theta(t) - 17]\quad (E)\] + + \medskip + + \begin{enumerate} + \item Vérifier que l'équation $(E)$ a pour solutions $\theta(t) = K e^{at} + 17$ où $K$ est un nombre réel. + + Donner alors, en fonction de $a$, l'ensemble des solutions de $(E)$. + \end{enumerate} + On rappelle que la température des macarons à l'instant $t = 0$ est égale à $-18$~\degres C et que, au bout de $15$~min, elle est de $2$~\degres C. + \begin{enumerate} + \setcounter{enumi}{1} + \item En utilisant la condition à $t=0$ démontrer que $K = -35$. + \item En utilisant la condition à $t=15$ démontrer que $a \approx -0.06$. + \item En déduire l'expression de la solution de l'équation différentielle puis étudier ses variations. + \item La température idéale de dégustation des macarons étant de $14$~\degres C, Marie estime que + celle-ci sera atteinte au bout de $30$~min. A-t-elle raison ? Justifier la réponse. + + Sinon, combien de temps faudra-t-il attendre ? +\end{enumerate} +\end{exercise} + + + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/TST_sti2d/DS/DS_21_07_08/17_210408_DS8.tex b/TST_sti2d/DS/DS_21_07_08/17_210408_DS8.tex new file mode 100644 index 0000000..5f07d55 --- /dev/null +++ b/TST_sti2d/DS/DS_21_07_08/17_210408_DS8.tex @@ -0,0 +1,136 @@ +\documentclass[a4paper,10pt]{article} +\usepackage{myXsim} + +% Title Page +\title{DS8 \hfill ONAL Yakub} +\tribe{TST sti2d} +\date{\hfillÀ render pour le vendredi 9 avril à 10h au plus tard} + +\xsimsetup{ + solution/print = false +} + +\begin{document} +\maketitle + +\begin{exercise}[subtitle={Étude de fonction}] + On considère la fonction $f$ définie sur $\intOF{0}{+\infty}$ par $ f(x) = 5x^2 + - 40x + - 450\ln(x)$ + \begin{enumerate} + \item Démontrer que la dérivée de $f$ est $f'(x) = \frac{10x^2 + - 40x + - 450}{x}$. + \item Étude du numérateur de $f'(x)$: $N(x) = 10x^2 - 40x - 450$ + \begin{enumerate} + \item Démontrer que $x=9$ et $x=- 5$ sont deux racines de $N(x)$.. + \item Proposer une forme factorisée de $N(x)$. + \item Proposer une forme factorisée de $f'(x)$. + \end{enumerate} + \item Étudier le signe de $f'$ et en déduire les variations de $f$. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item pas de correction disponible + \item + \begin{enumerate} + \item \[N(9) = 0\] + \[N(- 5) = 0\] + \item \[ + N(x) = 10(x - 9)(x - - 5) + \] + \item + \[ + f'(x) = \frac{10(x - 9)(x - - 5)}{x} + \] + \end{enumerate} + \item Pas de correction disponible + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Complexes}] + \begin{enumerate} + \item Mettre le nombre complexe suivant sous forme algébrique $z_1 = \dfrac{5 + 2 i}{-10 + 2 i} $ + \item Mettre le complexe suivante sous forme exponentielle $z_2 = 6 - 6 \sqrt{3} i$ + \item Mettre le complexe suivante sous forme exponentielle $z_3 = 10 \sqrt{3} + 10 i$ + \item Calculer le produit $z_4=z_2\times z_3$ donner le résultat sous forme exponentielle puis algébrique. + \item Calculer le quotient $z_5=\frac{z_2}{z_3}$ donner le résultat sous forme exponentielle puis algébrique. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item $z_1 = - \frac{23}{52} - \frac{15 i}{52}$ + \item $z_2 = 12 e^{- \frac{i \pi}{3}}$ + \item $z_3 = 20 e^{\frac{i \pi}{6}}$ + \item $z_4 = 240 e^{- \frac{i \pi}{6}} = 120 \sqrt{3} - 120 i = 208.0 - 120.0 i$ + \item $z_5 = \frac{3}{5} e^{- \frac{i \pi}{2}} = - \frac{3 i}{5} = - 0.6 i$ + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Sortie du congélateur}] + Marie a invité quelques amis pour le thé. Elle souhaite leur proposer ses macarons maison. + + Elle les sort de son congélateur à $-18$~\degres C et les place dans une pièce à $19$~\degres C. + + Au bout de 15 minutes, la température des macarons est de $-2$~\degres C. + + \bigskip + + \textbf{Premier modèle} + + \medskip + + On suppose que la vitesse de décongélation est constante : chaque minute la hausse de + température des macarons est la même. + + Estimer dans ce cadre la température au bout de $30$~minutes, puis au bout de $45$~minutes. + + Cette modélisation est-elle pertinente? + + \bigskip + + \textbf{Deuxième modèle} + + \medskip + + On suppose maintenant que la vitesse de décongélation est proportionnelle à la différence + de température entre les macarons et l'air ambiant (il s'agit de la loi de Newton). + + On désigne par $\theta$ la température des macarons à l'instant $t$, et par $\theta'$ la vitesse de décongélation. + + L'unité de temps est la minute et l'unité de température le degré Celsius. + + \smallskip + + On négligera la diminution de température de la pièce et on admettra donc qu'il existe un + nombre réel $a$ tel que, pour $t$ positif : + + \[\theta'(t) = a [\theta(t) - 19]\quad (E)\] + + \medskip + + \begin{enumerate} + \item Vérifier que l'équation $(E)$ a pour solutions $\theta(t) = K e^{at} + 19$ où $K$ est un nombre réel. + + Donner alors, en fonction de $a$, l'ensemble des solutions de $(E)$. + \end{enumerate} + On rappelle que la température des macarons à l'instant $t = 0$ est égale à $-18$~\degres C et que, au bout de $15$~min, elle est de $-2$~\degres C. + \begin{enumerate} + \setcounter{enumi}{1} + \item En utilisant la condition à $t=0$ démontrer que $K = -37$. + \item En utilisant la condition à $t=15$ démontrer que $a \approx -0.04$. + \item En déduire l'expression de la solution de l'équation différentielle puis étudier ses variations. + \item La température idéale de dégustation des macarons étant de $16$~\degres C, Marie estime que + celle-ci sera atteinte au bout de $30$~min. A-t-elle raison ? Justifier la réponse. + + Sinon, combien de temps faudra-t-il attendre ? +\end{enumerate} +\end{exercise} + + + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/TST_sti2d/DS/DS_21_07_08/18_210408_DS8.tex b/TST_sti2d/DS/DS_21_07_08/18_210408_DS8.tex new file mode 100644 index 0000000..aab89a8 --- /dev/null +++ b/TST_sti2d/DS/DS_21_07_08/18_210408_DS8.tex @@ -0,0 +1,136 @@ +\documentclass[a4paper,10pt]{article} +\usepackage{myXsim} + +% Title Page +\title{DS8 \hfill RADOUAA Saleh} +\tribe{TST sti2d} +\date{\hfillÀ render pour le vendredi 9 avril à 10h au plus tard} + +\xsimsetup{ + solution/print = false +} + +\begin{document} +\maketitle + +\begin{exercise}[subtitle={Étude de fonction}] + On considère la fonction $f$ définie sur $\intOF{0}{+\infty}$ par $ f(x) = 3x^2 + - 54x + 108\ln(x)$ + \begin{enumerate} + \item Démontrer que la dérivée de $f$ est $f'(x) = \frac{6x^2 + - 54x + 108}{x}$. + \item Étude du numérateur de $f'(x)$: $N(x) = 6x^2 - 54x + 108$ + \begin{enumerate} + \item Démontrer que $x=3$ et $x=6$ sont deux racines de $N(x)$.. + \item Proposer une forme factorisée de $N(x)$. + \item Proposer une forme factorisée de $f'(x)$. + \end{enumerate} + \item Étudier le signe de $f'$ et en déduire les variations de $f$. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item pas de correction disponible + \item + \begin{enumerate} + \item \[N(3) = 0\] + \[N(6) = 0\] + \item \[ + N(x) = 6(x - 3)(x - 6) + \] + \item + \[ + f'(x) = \frac{6(x - 3)(x - 6)}{x} + \] + \end{enumerate} + \item Pas de correction disponible + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Complexes}] + \begin{enumerate} + \item Mettre le nombre complexe suivant sous forme algébrique $z_1 = \dfrac{7 + 9 i}{-3 + 7 i} $ + \item Mettre le complexe suivante sous forme exponentielle $z_2 = -6 + 6 \sqrt{3} i$ + \item Mettre le complexe suivante sous forme exponentielle $z_3 = 10 + 10 \sqrt{3} i$ + \item Calculer le produit $z_4=z_2\times z_3$ donner le résultat sous forme exponentielle puis algébrique. + \item Calculer le quotient $z_5=\frac{z_2}{z_3}$ donner le résultat sous forme exponentielle puis algébrique. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item $z_1 = \frac{21}{29} - \frac{38 i}{29}$ + \item $z_2 = 12 e^{\frac{2 i \pi}{3}}$ + \item $z_3 = 20 e^{\frac{i \pi}{3}}$ + \item $z_4 = 240 e^{i \pi} = -240 = -240.0$ + \item $z_5 = \frac{3}{5} e^{\frac{i \pi}{3}} = \frac{3}{10} + \frac{3 \sqrt{3} i}{10} = 0.3 + 0.52 i$ + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Sortie du congélateur}] + Marie a invité quelques amis pour le thé. Elle souhaite leur proposer ses macarons maison. + + Elle les sort de son congélateur à $-20$~\degres C et les place dans une pièce à $16$~\degres C. + + Au bout de 15 minutes, la température des macarons est de $4$~\degres C. + + \bigskip + + \textbf{Premier modèle} + + \medskip + + On suppose que la vitesse de décongélation est constante : chaque minute la hausse de + température des macarons est la même. + + Estimer dans ce cadre la température au bout de $30$~minutes, puis au bout de $45$~minutes. + + Cette modélisation est-elle pertinente? + + \bigskip + + \textbf{Deuxième modèle} + + \medskip + + On suppose maintenant que la vitesse de décongélation est proportionnelle à la différence + de température entre les macarons et l'air ambiant (il s'agit de la loi de Newton). + + On désigne par $\theta$ la température des macarons à l'instant $t$, et par $\theta'$ la vitesse de décongélation. + + L'unité de temps est la minute et l'unité de température le degré Celsius. + + \smallskip + + On négligera la diminution de température de la pièce et on admettra donc qu'il existe un + nombre réel $a$ tel que, pour $t$ positif : + + \[\theta'(t) = a [\theta(t) - 16]\quad (E)\] + + \medskip + + \begin{enumerate} + \item Vérifier que l'équation $(E)$ a pour solutions $\theta(t) = K e^{at} + 16$ où $K$ est un nombre réel. + + Donner alors, en fonction de $a$, l'ensemble des solutions de $(E)$. + \end{enumerate} + On rappelle que la température des macarons à l'instant $t = 0$ est égale à $-20$~\degres C et que, au bout de $15$~min, elle est de $4$~\degres C. + \begin{enumerate} + \setcounter{enumi}{1} + \item En utilisant la condition à $t=0$ démontrer que $K = -36$. + \item En utilisant la condition à $t=15$ démontrer que $a \approx -0.07$. + \item En déduire l'expression de la solution de l'équation différentielle puis étudier ses variations. + \item La température idéale de dégustation des macarons étant de $13$~\degres C, Marie estime que + celle-ci sera atteinte au bout de $30$~min. A-t-elle raison ? Justifier la réponse. + + Sinon, combien de temps faudra-t-il attendre ? +\end{enumerate} +\end{exercise} + + + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/TST_sti2d/DS/DS_21_07_08/19_210408_DS8.tex b/TST_sti2d/DS/DS_21_07_08/19_210408_DS8.tex new file mode 100644 index 0000000..75d5d9c --- /dev/null +++ b/TST_sti2d/DS/DS_21_07_08/19_210408_DS8.tex @@ -0,0 +1,136 @@ +\documentclass[a4paper,10pt]{article} +\usepackage{myXsim} + +% Title Page +\title{DS8 \hfill TAVERNIER Joanny} +\tribe{TST sti2d} +\date{\hfillÀ render pour le vendredi 9 avril à 10h au plus tard} + +\xsimsetup{ + solution/print = false +} + +\begin{document} +\maketitle + +\begin{exercise}[subtitle={Étude de fonction}] + On considère la fonction $f$ définie sur $\intOF{0}{+\infty}$ par $ f(x) = 4.5x^2 + - 144x + 567\ln(x)$ + \begin{enumerate} + \item Démontrer que la dérivée de $f$ est $f'(x) = \frac{9x^2 + - 144x + 567}{x}$. + \item Étude du numérateur de $f'(x)$: $N(x) = 9x^2 - 144x + 567$ + \begin{enumerate} + \item Démontrer que $x=9$ et $x=7$ sont deux racines de $N(x)$.. + \item Proposer une forme factorisée de $N(x)$. + \item Proposer une forme factorisée de $f'(x)$. + \end{enumerate} + \item Étudier le signe de $f'$ et en déduire les variations de $f$. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item pas de correction disponible + \item + \begin{enumerate} + \item \[N(9) = 0\] + \[N(7) = 0\] + \item \[ + N(x) = 9(x - 9)(x - 7) + \] + \item + \[ + f'(x) = \frac{9(x - 9)(x - 7)}{x} + \] + \end{enumerate} + \item Pas de correction disponible + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Complexes}] + \begin{enumerate} + \item Mettre le nombre complexe suivant sous forme algébrique $z_1 = \dfrac{10 + 6 i}{-4 + 5 i} $ + \item Mettre le complexe suivante sous forme exponentielle $z_2 = - 8 \sqrt{3} + 8 i$ + \item Mettre le complexe suivante sous forme exponentielle $z_3 = 5 - 5 \sqrt{3} i$ + \item Calculer le produit $z_4=z_2\times z_3$ donner le résultat sous forme exponentielle puis algébrique. + \item Calculer le quotient $z_5=\frac{z_2}{z_3}$ donner le résultat sous forme exponentielle puis algébrique. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item $z_1 = - \frac{10}{41} - \frac{74 i}{41}$ + \item $z_2 = 16 e^{\frac{5 i \pi}{6}}$ + \item $z_3 = 10 e^{- \frac{i \pi}{3}}$ + \item $z_4 = 160 e^{\frac{i \pi}{2}} = 160 i = 160.0 i$ + \item $z_5 = \frac{8}{5} e^{\frac{7 i \pi}{6}} = - \frac{4 \sqrt{3}}{5} - \frac{4 i}{5} = -1.39 - 0.8 i$ + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Sortie du congélateur}] + Marie a invité quelques amis pour le thé. Elle souhaite leur proposer ses macarons maison. + + Elle les sort de son congélateur à $-19$~\degres C et les place dans une pièce à $21$~\degres C. + + Au bout de 15 minutes, la température des macarons est de $-4$~\degres C. + + \bigskip + + \textbf{Premier modèle} + + \medskip + + On suppose que la vitesse de décongélation est constante : chaque minute la hausse de + température des macarons est la même. + + Estimer dans ce cadre la température au bout de $30$~minutes, puis au bout de $45$~minutes. + + Cette modélisation est-elle pertinente? + + \bigskip + + \textbf{Deuxième modèle} + + \medskip + + On suppose maintenant que la vitesse de décongélation est proportionnelle à la différence + de température entre les macarons et l'air ambiant (il s'agit de la loi de Newton). + + On désigne par $\theta$ la température des macarons à l'instant $t$, et par $\theta'$ la vitesse de décongélation. + + L'unité de temps est la minute et l'unité de température le degré Celsius. + + \smallskip + + On négligera la diminution de température de la pièce et on admettra donc qu'il existe un + nombre réel $a$ tel que, pour $t$ positif : + + \[\theta'(t) = a [\theta(t) - 21]\quad (E)\] + + \medskip + + \begin{enumerate} + \item Vérifier que l'équation $(E)$ a pour solutions $\theta(t) = K e^{at} + 21$ où $K$ est un nombre réel. + + Donner alors, en fonction de $a$, l'ensemble des solutions de $(E)$. + \end{enumerate} + On rappelle que la température des macarons à l'instant $t = 0$ est égale à $-19$~\degres C et que, au bout de $15$~min, elle est de $-4$~\degres C. + \begin{enumerate} + \setcounter{enumi}{1} + \item En utilisant la condition à $t=0$ démontrer que $K = -40$. + \item En utilisant la condition à $t=15$ démontrer que $a \approx -0.03$. + \item En déduire l'expression de la solution de l'équation différentielle puis étudier ses variations. + \item La température idéale de dégustation des macarons étant de $18$~\degres C, Marie estime que + celle-ci sera atteinte au bout de $30$~min. A-t-elle raison ? Justifier la réponse. + + Sinon, combien de temps faudra-t-il attendre ? +\end{enumerate} +\end{exercise} + + + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/TST_sti2d/DS/DS_21_07_08/20_210408_DS8.tex b/TST_sti2d/DS/DS_21_07_08/20_210408_DS8.tex new file mode 100644 index 0000000..53d713b --- /dev/null +++ b/TST_sti2d/DS/DS_21_07_08/20_210408_DS8.tex @@ -0,0 +1,136 @@ +\documentclass[a4paper,10pt]{article} +\usepackage{myXsim} + +% Title Page +\title{DS8 \hfill ZAHORE Zahiri} +\tribe{TST sti2d} +\date{\hfillÀ render pour le vendredi 9 avril à 10h au plus tard} + +\xsimsetup{ + solution/print = false +} + +\begin{document} +\maketitle + +\begin{exercise}[subtitle={Étude de fonction}] + On considère la fonction $f$ définie sur $\intOF{0}{+\infty}$ par $ f(x) = 4.5x^2 + - 54x + - 63\ln(x)$ + \begin{enumerate} + \item Démontrer que la dérivée de $f$ est $f'(x) = \frac{9x^2 + - 54x + - 63}{x}$. + \item Étude du numérateur de $f'(x)$: $N(x) = 9x^2 - 54x - 63$ + \begin{enumerate} + \item Démontrer que $x=7$ et $x=- 1$ sont deux racines de $N(x)$.. + \item Proposer une forme factorisée de $N(x)$. + \item Proposer une forme factorisée de $f'(x)$. + \end{enumerate} + \item Étudier le signe de $f'$ et en déduire les variations de $f$. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item pas de correction disponible + \item + \begin{enumerate} + \item \[N(7) = 0\] + \[N(- 1) = 0\] + \item \[ + N(x) = 9(x - 7)(x - - 1) + \] + \item + \[ + f'(x) = \frac{9(x - 7)(x - - 1)}{x} + \] + \end{enumerate} + \item Pas de correction disponible + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Complexes}] + \begin{enumerate} + \item Mettre le nombre complexe suivant sous forme algébrique $z_1 = \dfrac{7 + 4 i}{-4 + 8 i} $ + \item Mettre le complexe suivante sous forme exponentielle $z_2 = 4 \sqrt{2} + 4 \sqrt{2} i$ + \item Mettre le complexe suivante sous forme exponentielle $z_3 = - 4 \sqrt{2} - 4 \sqrt{2} i$ + \item Calculer le produit $z_4=z_2\times z_3$ donner le résultat sous forme exponentielle puis algébrique. + \item Calculer le quotient $z_5=\frac{z_2}{z_3}$ donner le résultat sous forme exponentielle puis algébrique. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item $z_1 = \frac{1}{20} - \frac{9 i}{10}$ + \item $z_2 = 8 e^{\frac{i \pi}{4}}$ + \item $z_3 = 8 e^{- \frac{3 i \pi}{4}}$ + \item $z_4 = 64 e^{- \frac{i \pi}{2}} = - 64 i = - 64.0 i$ + \item $z_5 = 1 e^{i \pi} = -1 = -1.0$ + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Sortie du congélateur}] + Marie a invité quelques amis pour le thé. Elle souhaite leur proposer ses macarons maison. + + Elle les sort de son congélateur à $-18$~\degres C et les place dans une pièce à $21$~\degres C. + + Au bout de 15 minutes, la température des macarons est de $4$~\degres C. + + \bigskip + + \textbf{Premier modèle} + + \medskip + + On suppose que la vitesse de décongélation est constante : chaque minute la hausse de + température des macarons est la même. + + Estimer dans ce cadre la température au bout de $30$~minutes, puis au bout de $45$~minutes. + + Cette modélisation est-elle pertinente? + + \bigskip + + \textbf{Deuxième modèle} + + \medskip + + On suppose maintenant que la vitesse de décongélation est proportionnelle à la différence + de température entre les macarons et l'air ambiant (il s'agit de la loi de Newton). + + On désigne par $\theta$ la température des macarons à l'instant $t$, et par $\theta'$ la vitesse de décongélation. + + L'unité de temps est la minute et l'unité de température le degré Celsius. + + \smallskip + + On négligera la diminution de température de la pièce et on admettra donc qu'il existe un + nombre réel $a$ tel que, pour $t$ positif : + + \[\theta'(t) = a [\theta(t) - 21]\quad (E)\] + + \medskip + + \begin{enumerate} + \item Vérifier que l'équation $(E)$ a pour solutions $\theta(t) = K e^{at} + 21$ où $K$ est un nombre réel. + + Donner alors, en fonction de $a$, l'ensemble des solutions de $(E)$. + \end{enumerate} + On rappelle que la température des macarons à l'instant $t = 0$ est égale à $-18$~\degres C et que, au bout de $15$~min, elle est de $4$~\degres C. + \begin{enumerate} + \setcounter{enumi}{1} + \item En utilisant la condition à $t=0$ démontrer que $K = -39$. + \item En utilisant la condition à $t=15$ démontrer que $a \approx -0.06$. + \item En déduire l'expression de la solution de l'équation différentielle puis étudier ses variations. + \item La température idéale de dégustation des macarons étant de $18$~\degres C, Marie estime que + celle-ci sera atteinte au bout de $30$~min. A-t-elle raison ? Justifier la réponse. + + Sinon, combien de temps faudra-t-il attendre ? +\end{enumerate} +\end{exercise} + + + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/TST_sti2d/DS/DS_21_07_08/all_210408_DS8.pdf b/TST_sti2d/DS/DS_21_07_08/all_210408_DS8.pdf new file mode 100644 index 0000000000000000000000000000000000000000..404867965e23894476d366a468f11aa46e73649f GIT binary patch literal 199390 zcmd43Wo%u|mY{8BX7({d95ctv%#Jx`W@cuHnVFd#GegV}Gcz+Yjg$Aj_ug;1HQgh9 zEscMas!H3-YuDMbw4d{=MJ6XC@|l5_1%~X`%+CcFR%QS_z*^4?hKmb^PRh{A*uezA z$^`hw4-B1%sfB}~9e_^6Lf64i$WY(fzz~Lq2gcsP&QRA9#$|~~i5^Hm5L^qC8&psq z$}lH7eMEWCu-F&~tPlxY%p!zTiBO4eooRkluA40D6A~0W+LUkZ8JHjm zKO%k1erLhT@NelTx!M>4=u~C(%nbD%VCWPb^&I}}h*;ZMeq{DB{<(qyhE7gNSe5o8 zRYMtFOG5x9os+H|or9$fosEH!nXct$VKzNEc2_!kYezeM!;hQ)*rxi&FUjc|8_GCZ z>V4ef-*LpPjI4!B^&J4LY#+%ez|aW@Si1l;=sy-L94xeqY^(rwW(Hbz4lNiu1;f9B z+5>){pP;pcwVk4kuD;=)rH!$Xjh(R`K!Z+B*Wu$7t>}adolNx&6+{KJJ`VBk-QU-K ztb|=0L=_!APWc~e5o;@lkLkzJv#p$iU8~`?szxP=H z?2Lc!vjf;!{@!N>u(SU?X8>^gKI`8x8Q1_EzaN7?@mT;Ie{LJ_@hQ{E*;(r=8ae

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