A cyclist travels in a straight line so that his displacement, \( S \) metres, from a fixed point \( O \), is given by \( S=4+15 t-t^{3} \), where \( t \) is the time in seconds after passing a point \( X \) on the line. (i) Find the distance \( O X \). (ii) Find the velocity of the cyclist at \( t=2 \). (iii) Calculate the acceleration of the cyclist when he is instantaneously at rest. (iv) Calculate the distance travelled by the cyclist during the third second of his motion.

Sea \[ f(x)=\left\{\begin{array}{ll}\left(\frac{\sin (2 x)}{x}\right) e^{-x^{2}} & \text { si } x \neq 0 \\ 2 & \text { si } x=0\end{array}\right. \] (a) Demostrar que \( f(x) \) es continua en \( x=0 \). (b) Demostrar que \( f(x) \) es derivable en \( x=0 \) y su derivada es continua en \( x=0 \).

\( L ^ { - 1 } ( \operatorname { Ln } ( \frac { s ^ { 2 } + 2 s + 3 } { s ^ { 2 } + 4 } ) ) \)

\( \lim _ { x \rightarrow 0 } ( \frac { t 9 ( x ) } { x } ) ^ { \frac { 1 } { x ^ { 2 } } } \)

A car's velocity is given by the function \( v(t) = 5t^2 + 3 \). Calculate the distance traveled by the car from time \( t = 0 \) to \( t = 4 \).

\( L ^ { - 1 } ( \operatorname { Ln } ( \frac { s ^ { 2 } + 2 s + 3 } { s ^ { 2 } + 1 } ) ) \)