\( \left. \begin{array} { l } { M = x y \cos x y + \sin x y } \\ { \frac { \partial M } { \partial y } = ? } \end{array} \right. \)

Exercice 1: Soit \( f \) la fonction numérique définie sur \( \mathbb{R} \) par: \( f(x)=E(x)+E\left(x+\frac{1}{2}\right)-E(2 x) \) 1) Montrer que : \( \forall x \in\left[0, \frac{1}{2}[; f(x)=0\right. \). 2) Montrer que \( f \) est \( \frac{1}{2} \)-périodique. 3) Montrer que : \( \forall x \in \mathbb{R}, \exists n \in \mathbb{Z} ; x-n \frac{1}{2} \in\left[0, \frac{1}{2}[\right. \). 4) En déduire que : \( \forall x \in \mathbb{R} ; E(x)+E\left(x+\frac{1}{2}\right)=E(2 x) \).

container in the shape of an inverted cone of radius 3 metres and vertical height 4.5 metres is initially illed with liquid fertiliser. This fertiliser is released through a hole in the bottom of the container at a rate f \( 0.01 \mathrm{~m}^{3} \) per second. At time \( t \) seconds the fertiliser remaining in the container forms an inverted cone of eight \( h \) metres. The volume of a cone is \( V=\frac{1}{3} \pi r^{2} h \).] (i) Show that \( h^{2} \frac{\mathrm{~d} h}{\mathrm{~d} t}=-\frac{9}{400 \pi} \). (ii) Express \( h \) in terms of \( t \).

II- On considère la fonction \( f \) définie sur \( / \) par : \( f(x)=x+\frac{1}{x}-(\ln x)^{2}-2 \) El Soit \( \left(C_{f}\right) \) la courbe représentative de \( f \) dans un repère orthonormé \( (O, \vec{i}, \bar{j}) \). 1 Calculer : \( \lim _{x \rightarrow+\infty} f(x) \) 2 Vérifier que : \( (\forall x \in I), \quad f\left(\frac{1}{x}\right)=f(x) \) 3 Calculer : \( \lim _{x \rightarrow 0^{+}} f(x) \) puis interpréter graphiquement ce résultat. 4 Étudier la branche infinie de \( f \) au voisinage de \( +\infty \).

\( \frac{d}{dx}(\ln (\arctan (x+1))-\ln (\arctan (x))) \)

A container in the shape of an inverted cone of radius 3 metres and vertical height 4.5 metres is initially filled with liquid fertiliser. This fertiliser is released through a hole in the bottom of the container at a rate of \( 0.01 \mathrm{~m}^{3} \) per second. At time \( t \) seconds the fertiliser remaining in the container forms an inverted cone of height \( h \) metres. [The volume of a cone is \( V=\frac{1}{3} \pi r^{2} h \).] (i) Show that \( h^{2} \frac{\mathrm{~d} h}{\mathrm{~d} t}=-\frac{9}{400 \pi} \). (ii) Express \( h \) in terms of \( t \). (iii) Find the time it takes to empty the container, giving your answer to the nearest minute.