5. Exprimer \( r \) en fonction de \( f \) et \( f^{\prime} \). Les fonctions \( r \) et \( \theta \) sont fixées ainsi pour la suite de la partie. (a) Montrer que \( \theta^{\prime}=-1+q \sin (\theta) \cos (\theta) \) (b) Montrer que \( r^{\prime}=q r \sin ^{2} \theta \). En déduire la monotonie de \( r \). (c) Etudier les variations de \( t \mapsto r(t) \exp (-\arctan (t)) \) et en déduire que \( 0 \leq r(t) \leq r(0) \exp (\operatorname{arct} \) que \( r \) a une limite strictement positive en \( +\infty \).

\( y ^ { \prime \prime } + 5 y ^ { \prime } + 4 y = 3 + 8 x ^ { 2 } + 2 \cos ( 2 x ) \)

What is the arc length of the parametrically defined curve given by \( x(t) = t, y(t) = t^3 \) for \( t \) in the interval \([0, 2]\)?

(2) Find the limit of a function : \( \begin{array}{llll}\text { a) } \lim _{x \rightarrow \infty} \frac{x^{2}-1}{2 x^{2}+1} & \text { j) } \lim _{x \rightarrow-\infty} \frac{x^{3}+x^{4}-1}{2 x^{5}+x-x^{2}} & \text { s) } \lim _{x \rightarrow \infty} \frac{(x+3)(x+4)(x+5)}{x^{4}+x-11} \\ \text { b) } \lim _{x \rightarrow-\infty} \frac{x^{3}+x^{2}-4}{2 x^{3}+x+11} & \text { k) } \lim _{x \rightarrow \infty} \frac{\left(\sqrt{x^{2}+1}+x\right)^{2}}{\sqrt[3]{x^{6}+1}} & \text { t) } \lim _{x \rightarrow-\infty} \frac{8 x-2 x^{5}+x^{6}}{11 x+5 x^{3}+3 x^{5}} \\ \text { c) } \lim _{x \rightarrow \infty} \frac{3 x^{2}+2 x-1}{x^{3}-x+2} & \text { 1) } \lim _{x \rightarrow-\infty} \frac{x^{6}+7 x^{4}-40}{1-x-5 x^{7}} & \text { c) } \lim _{x \rightarrow \infty}\left(\frac{x^{3}}{2 x^{2}-1}-\frac{x^{2}}{2 x+1}\right) \\ \text { d) } \lim _{x \rightarrow \infty}\left(\frac{x^{3}}{x^{2}+2}-x\right) & \text { m) } \lim _{x \rightarrow \infty} \frac{(x+1)(x-2)}{3 x^{2}+6 x-5} & \text { v) } \lim _{x \rightarrow \infty}\left(x^{2}-\frac{x^{4}-1}{x^{2}-2}\right) \\ \text { e) } \lim _{x \rightarrow \infty} \frac{x^{2}+3 x-4}{3 x^{2}-2 x+5} & \text { n) } \lim _{x \rightarrow \infty} \frac{\sqrt{x^{2}+1}}{x} & \text { w) } \lim _{x \rightarrow \infty} \frac{(x-1)^{100}(6 x+1)^{200}}{(3 x+5)^{300}} \\ \text { f) } \lim _{x \rightarrow \infty} \frac{x(x-1)(x-2)}{x^{2}+6 x-0} & \text { o) } & \lim _{x \rightarrow \infty}\left(\frac{3 x^{2}+2 x+1}{x^{2}}\right)^{4} & \text { x) } \lim _{x \rightarrow \infty} \frac{\sqrt[4]{x^{5}}+\sqrt[5]{x^{3}}+\sqrt[6]{x^{8}}}{3 \sqrt{4}}\end{array} \)

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.

\( \int _ { 5 + 5 - 10 } ^ { \infty } \frac { 10 \sin ( x ) } { 2 x } d x = \)