9/16/24, 1:33 PM 8. Use a change of variables to evaluate the definite integral. \( \int_{0}^{2} x \sqrt{4-x^{2}} d x \) \( \int_{0}^{2} x \sqrt{4-x^{2}} d x= \)

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 : \( \forall x \in \mathbb{R} ; E(x)+E\left(x+\frac{1}{2}\right)=E(2 x) \).

2. find Maclurin expansion of (2 Points) * \[ \begin{array}{l}f(x)=\sinh x \\ f(x)=1+\frac{x^{2}}{2!}+\frac{x^{4}}{4!}-\ldots \ldots \\ f(x)=x+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+\ldots \ldots \\ f(x)=x+\frac{x^{3}}{3}+\frac{x^{5}}{5}+\ldots \ldots \\ f(x)=x+\frac{x^{3}}{3!}+\frac{x^{5}}{5!}+\ldots\end{array} \]

\( f ( x ) = \int | x | d x \)

3. find taylor series of (2 Points) * \( \begin{array}{l}f(x)=e^{x} \cos x \quad \text { at } x=0 \\ f(x)=1+x-\frac{2 x^{3}}{3!}+\ldots \\ f(x)=1+x-\frac{x^{3}}{3!}+\ldots \\ f(x)=1+x-\frac{x^{2}}{2!}+\ldots \\ f(x)=1+x-\frac{2 x^{2}}{2!}+\ldots\end{array} \)

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.