QCM Fonctions numériques

$\begin{array}{c}\mathit{QCM}\mathit{Fonctions}\mathit{n}\mathit{u}\mathit{m}\mathit{e}\mathit{r}\mathit{i}\mathit{q}\mathit{u}\mathit{e}\mathit{s}\mathrm{(}\mathit{c}\mathit{o}\mathit{n}\mathit{t}\mathit{i}\mathit{n}\mathit{u}\mathit{i}\mathit{t}\mathrm{é}\mathrm{,}\mathit{d}\mathit{e}\mathit{r}\mathit{i}\mathit{v}\mathit{i}\mathit{b}\mathit{a}\mathit{b}\mathit{i}\mathit{l}\mathit{i}\mathit{t}\mathrm{é}\mathrm{,}\mathrm{.}\mathrm{.}\mathrm{)}\hfill \\ \mathit{Q}\mathit{u}\mathit{e}\mathit{s}\mathit{t}\mathit{i}\mathit{o}\mathit{n}1\mathrm{:}\hfill \\ \left[\mathit{x}\right]\mathit{d}\mathrm{é}\mathit{s}\mathit{i}\mathit{g}\mathit{n}\mathit{e}\mathit{la}\mathit{partie}\mathit{e}\mathit{n}\mathit{t}\mathit{i}\mathrm{è}\mathit{r}\mathit{e}\mathit{de}\mathit{x}\mathrm{.}\hfill \\ \mathit{Soit}\mathit{\phi }\mathit{une}\mathit{fonction}\mathit{d}\mathrm{é}\mathit{f}\mathit{i}\mathit{n}\mathit{i}\mathit{e}\mathit{sur}ℝ-\left\{0\right\}\mathrm{,}\mathit{\phi }\left(\mathit{x}\right)=\frac{1}{\sqrt{\mathit{x}-\left[\mathit{x}\right]}}\mathit{alors}\mathit{on}\mathit{a}\mathrm{:}\hfill \\ ☑\mathit{\phi }\mathit{est}\mathit{b}\mathit{o}\mathit{r}\mathit{n}\mathrm{é}\mathit{e}\hfill \\ \square \mathit{\phi }\mathit{est}\mathit{constante}\mathit{sur}ℝ-\left\{0\right\}\hfill \\ \square \mathit{\phi }\mathit{est}\mathit{croissante}\mathit{sur}ℝ-\left\{0\right\}\hfill \\ \square \mathit{\phi }\mathit{est}\mathit{continue}\mathit{sur}ℝ-\left\{0\right\}\hfill \\ \mathit{Q}\mathit{u}\mathit{e}\mathit{s}\mathit{t}\mathit{i}\mathit{o}\mathit{n}2\mathrm{:}\hfill \\ \mathit{On}\mathit{pose}\mathit{f}\mathrm{(}\mathit{x}\mathrm{)}={\left(\mathit{cos}\left(\mathit{x}\right)\right)}^{\frac{1}{{\mathit{x}}^2}}\mathit{pour}\mathit{tout}\mathit{x}\mathit{non}\mathit{n}\mathit{u}\mathit{l}\mathrm{.}\hfill \\ \mathit{Alors}\mathrm{:}\hfill \\ \square \underset{\mathit{x}\to 0}{\mathit{lim}}\mathit{f}\mathrm{(}\mathit{x}\mathrm{)}=\mathit{e}\hfill \\ \square \underset{\mathit{x}\to 0}{\mathit{lim}}\mathit{f}\mathrm{(}\mathit{x}\mathrm{)}={\mathit{e}}^{-1}\hfill \\ \square \underset{\mathit{x}\to 0}{\mathit{lim}}\mathit{f}\mathrm{(}\mathit{x}\mathrm{)}={\mathit{e}}^{\frac{1}{2}}\hfill \\ ☑\underset{\mathit{x}\to 0}{\mathit{lim}}\mathit{f}\mathrm{(}\mathit{x}\mathrm{)}={\mathit{e}}^{\frac{-1}{2}}\hfill \\ \mathit{Q}\mathit{u}\mathit{e}\mathit{s}\mathit{t}\mathit{i}\mathit{o}\mathit{n}3\mathrm{:}\hfill \\ \mathit{Soit}\mathit{f}\mathit{une}\mathit{fonction}\mathit{d}\mathit{é}\mathit{rivable}\mathit{et}\mathit{croissante}\mathit{sur}\left[0\mathrm{,}1\right]\mathrm{.}\hfill \\ \mathit{On}\mathit{note}{\mathit{f}}^{-1}\mathit{sa}\mathit{fonction}\mathit{r}\mathrm{é}\mathit{c}\mathit{i}\mathit{p}\mathit{r}\mathit{o}\mathit{q}\mathit{u}\mathit{e}\mathit{et}\mathit{f}\left(0\right)=0\mathrm{.}\hfill \\ \mathit{on}\mathit{pose}\mathit{F}\left(\mathit{x}\right)={\int }_0^{\mathit{x}}\mathit{f}\left(\mathit{t}\right)\mathit{dt}\mathit{et}\mathit{G}\left(\mathit{x}\right)={\int }_0^{\mathit{f}\left(\mathit{x}\right)}{\mathit{f}}^{-1}\left(\mathit{t}\right)\mathit{dt}\mathit{alors}\mathit{on}\mathit{a}\mathrm{:}\hfill \\ \square \mathit{F}\left(\mathit{x}\right)=\mathit{G}\left(\mathit{x}\right)\hfill \\ \square \mathit{F}\left(\mathit{x}\right)=-\mathit{G}\left(\mathit{x}\right)\hfill \\ \square \mathit{F}\left(\mathit{x}\right)=\mathit{xf}\left(\mathit{x}\right)+\mathit{G}\left(\mathit{x}\right)\hfill \\ ☑\mathit{F}\left(\mathit{x}\right)=\mathit{xf}\left(\mathit{x}\right)-\mathit{G}\left(\mathit{x}\right)\hfill \\ \mathit{Q}\mathit{u}\mathit{e}\mathit{s}\mathit{t}\mathit{i}\mathit{o}\mathit{n}4\mathrm{:}\hfill \\ \mathit{On}\mathit{pose}\mathit{g}\left(\mathit{x}\right)=\frac{1-\mathit{cosx}}{\mathit{x}}\mathit{pour}\mathit{tout}\mathit{x}\mathit{different}\mathit{de}0\mathit{et}\mathit{g}\left(0\right)=0\hfill \\ \mathit{et}\mathit{G}\mathit{sa}\mathit{primitive}\mathit{qui}\mathit{s}\mathrm{\text{'}}\mathit{a}\mathit{n}\mathit{n}\mathit{u}\mathit{l}\mathit{e}\mathit{en}0\mathrm{.}\hfill \\ \square \mathit{G}\mathit{est}\mathit{croissante}\mathit{sur}{ℝ}^{*}\hfill \\ ☑\mathit{G}\left(\mathit{x}\right)\le \frac{{\mathit{x}}^2}{4}\hfill \\ \square \mathit{G}\left(\mathit{x}\right)\ge \frac{{\mathit{x}}^2}{4}\hfill \\ \square \mathit{G}\mathit{est}\mathit{impaire}\hfill \\ \mathit{Q}\mathit{u}\mathit{e}\mathit{s}\mathit{t}\mathit{i}\mathit{o}\mathit{n}5\mathrm{:}\hfill \\ \mathit{On}\mathit{pose}\mathit{\phi }\left(\mathit{x}\right)={\int }_{\mathit{x}}^{2\mathit{x}}\frac{\mathit{exp}{\left(\mathit{t}\right)}^{}}{\mathit{t}}\mathit{dt}\mathit{pour}\mathit{tout}\mathit{x}\mathit{de}{ℝ}^{*}\mathrm{.}\hfill \\ ☑\underset{\mathit{x}\to 0}{\mathit{lim}}\mathit{\phi }\left(\mathit{x}\right)=\mathit{l}\mathit{n}2\hfill \\ \square \underset{\mathit{x}\to 0}{\mathit{lim}}\mathit{\phi }\left(\mathit{x}\right)=2\mathit{l}\mathit{n}2\hfill \\ \square \underset{\mathit{x}\to 0}{\mathit{lim}}\mathit{\phi }\left(\mathit{x}\right)=2\hfill \\ \square \underset{\mathit{x}\to 0}{\mathit{lim}}\mathit{\phi }\left(\mathit{x}\right)=0\hfill \\ \square \underset{\mathit{x}\to 0}{\mathit{lim}}\mathit{\phi }\left(\mathit{x}\right)=-2\mathit{l}\mathit{n}2\hfill \\ \mathit{Q}\mathit{u}\mathit{e}\mathit{s}\mathit{t}\mathit{i}\mathit{o}\mathit{n}6\mathrm{:}\hfill \\ \mathit{Soit}\mathit{la}\mathit{fonction}\mathit{f}\mathit{definie}\mathit{sur}\mathit{son}\mathit{domaine}\mathit{de}\mathit{definition}{\mathit{D}}_{\mathit{f}}\mathit{par}\hfill \\ \mathit{f}\left(\mathit{x}\right)=\frac{1}{\left[\frac{3-\mathit{x}}{\mathit{x}}\right]}\mathit{avec}\left[\mathit{x}\right]\mathit{d}\mathrm{é}\mathit{s}\mathit{i}\mathit{g}\mathit{n}\mathit{e}\mathit{la}\mathit{partie}\mathit{e}\mathit{n}\mathit{t}\mathit{i}\mathrm{è}\mathit{r}\mathit{e}\mathit{de}\mathit{x}\mathrm{.}\hfill \\ \mathit{On}\mathit{a}\mathrm{:}\hfill \\ \square {\mathit{D}}_{\mathit{f}}={ℝ}^{*}\mathit{et}\underset{\mathit{x}\to 0}{\mathit{lim}}\mathit{f}\left(\mathit{x}\right)=0\hfill \\ \square {\mathit{D}}_{\mathit{f}}=\left.\right]-\infty \mathrm{,}0\left[\bigcup \left.\right]3\mathrm{,}+\infty \left[\right.\right.\mathit{et}\underset{\mathit{x}\to 0}{\mathit{lim}}\mathit{f}\left(\mathit{x}\right)=0\hfill \\ ☑{\mathit{D}}_{\mathit{f}}=\left.\right]-\infty \mathrm{,}0\left[\bigcup \left.\right]0\mathrm{,}3/2\left.\right]\bigcup \left.\right]3\mathrm{,}+\infty \left[\right.\right.\mathit{et}\underset{\mathit{x}\to 0}{\mathit{lim}}\mathit{f}\left(\mathit{x}\right)=0\hfill \\ \square {\mathit{D}}_{\mathit{f}}=\left.\right]-\infty \mathrm{,}0\left[\bigcup \left.\right]3\mathrm{,}+\infty \left[\right.\right.\mathit{et}\underset{\mathit{x}\to 0}{\mathit{lim}}\mathit{f}\left(\mathit{x}\right)=1\hfill \\ \mathit{Q}\mathit{u}\mathit{e}\mathit{s}\mathit{t}\mathit{i}\mathit{o}\mathit{n}7\mathrm{:}\hfill \\ \mathit{la}\mathit{limite}\mathit{de}\varphi \left(\mathit{x}\right)=\frac{1}{{\mathit{x}}^2}-\frac{1}{{\mathit{sin}}^2\left(\mathit{x}\right)}\mathit{quand}\mathit{x}\mathit{tend}\mathit{vers}0\mathit{vaut}\mathrm{:}\hfill \\ \square 0\hfill \\ \square -1/6\hfill \\ ☑-1/3\hfill \\ \square 1/6\hfill \\ \square +\infty \hfill \\ \mathit{Question}8\hfill \\ \mathit{On}\mathit{consid}\mathit{é}\mathit{re}\mathit{la}\mathit{fonction}\mathit{\phi }\left(\mathit{t}\right)=\arctan \mathit{t}+\arctan \frac{1}{\mathit{t}}\mathrm{;}\mathit{t}\ne 0\hfill \\ \mathit{on}\mathit{a}\mathrm{:}\hfill \\ \square \mathit{la}\mathit{fonction}\mathit{\phi }\mathit{est}\mathit{cons}\tan \mathit{te}\hfill \\ ☑\mathit{la}\mathit{fonction}\mathit{\phi }\mathit{ne}\mathit{prend}\mathit{que}\mathit{deux}\mathit{valeurs}\mathit{distincts}\mathit{sur}\mathit{son}\mathit{domaine}\mathit{de}\mathit{d}\mathit{é}\mathit{finition}\hfill \\ \square \mathit{la}\mathit{fonction}\mathit{\phi }\mathit{prend}\mathit{plus}\mathit{de}\mathit{deux}\mathit{valeurs}\mathit{distincts}\mathit{sur}\mathit{son}\mathit{domaine}\mathit{de}\mathit{d}\mathit{é}\mathit{finition}\hfill \\ \square \mathit{la}\mathit{fonction}\mathit{\phi }\mathit{est}\mathit{paire}\hfill \\ \hfill \\ \mathit{Question}9\hfill \\ \underset{\mathit{x}\to \frac{\mathit{\pi }}{4}}{\lim }\frac{{\left(1-\mathit{tan}\left(\mathit{x}\right)\right)}^2}{1+\mathit{cos}\left(4\mathit{x}\right)}\mathit{vaut}\mathrm{:}\hfill \\ \square \frac{1}{2\sqrt{2}}\hfill \\ ☑\frac{1}{2}\hfill \\ \square 2\sqrt{2}\hfill \\ \square \frac{1}{\sqrt{2}}\hfill \\ \mathit{Q}\mathit{u}\mathit{e}\mathit{s}\mathit{t}\mathit{i}\mathit{o}\mathit{n}10\hfill \\ \mathit{Soit}\mathit{g}\mathit{une}\mathit{fonction}\mathit{definie}\mathit{d}\mathit{e}\mathrm{]}-1\mathrm{,}+\infty \mathrm{[}àℝ\mathit{admettant}\mathit{l}\mathrm{\text{'}}\mathit{a}\mathit{l}\mathit{l}\mathit{u}\mathit{r}\mathit{e}\mathit{r}\mathit{e}\mathit{p}\mathit{r}\mathit{e}\mathit{s}\mathit{e}\mathit{n}\mathit{t}\mathrm{é}\mathit{comme}\mathit{s}\mathit{u}\mathit{i}\mathit{t}\mathrm{:}\hfill \\ \mathit{A}\mathit{l}\mathit{o}\mathit{r}\mathit{s}\mathrm{:}\hfill \\ \square \mathit{g}\left(\mathit{x}\right)=\mathit{xln}\left(1+\frac{1}{\mathit{x}}\right)\hfill \\ \square \mathit{g}\left(\mathit{x}\right)=\mathit{x}+\mathit{ln}\left(\mathit{x}+1\right)\hfill \\ \square \mathit{g}\left(\mathit{x}\right)=\frac{\mathit{x}}{\mathit{ln}\left(\mathit{x}+1\right)}\hfill \\ ☑\mathit{g}\left(\mathit{x}\right)=\mathit{xln}\left(\mathit{x}+1\right)\hfill \end{array}$