3. Use the value of the first integral \( I \) to evaluate the two given integrals. \( I=\int_{0}^{3}\left(x^{3}-4 x\right) d x=\frac{9}{4} \) a. \( \int_{0}^{3}\left(24 x-6 x^{3}\right) d x \) a. \( \int_{0}^{3}\left(24 x-6 x^{3}\right) d x=\frac{-27}{2} \) (Simplify your answer.) \( \int_{3}^{0}\left(4 x-x^{3}\right) d x \) b. \( \int_{3}^{0}\left(4 x-x^{3}\right) d x=\square \frac{9}{4} \) (Simplify your answer.)

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 \( t \).

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

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

a) \( \iint_{\boldsymbol{R}}(\mathbf{5}-\boldsymbol{x}-\mathbf{2 y}) \boldsymbol{d} x \boldsymbol{d y} \) b) \( \iint_{\boldsymbol{R}}(\mathbf{5}-\boldsymbol{x}-\mathbf{2 y}) \boldsymbol{d y} \boldsymbol{d x} \) \[ R=\{(x, y) \mid 1 \leq x \leq 2,0 \leq y \leq 3\} \]

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