A particle moves in a straight line so that, \( t \) seconds after leaving a fixed point O , its velocity, \( v \mathrm{~m} / \mathrm{s} \), is given by \( v=\mathrm{e}^{2 t}-4 \mathrm{e}^{t}+12 \). (i) Find the time at which the particle has a velocity of \( 17 \mathrm{~m} / \mathrm{s} \). (ii) Is the acceleration of the particle always positive? How can you tell?

\( F ^ { \prime } ( x ) = \lim _ { h \rightarrow 0 } \frac { \log _ { a } ( x + h ) - \log _ { a } ( x ) } { h } \)

\[ f(x)=\frac{x^{2}+3 x+2}{x^{2}-2 x-3} \] a) Find the domain of \( f \) and deduce the vertical asymptote. b) Find \( x \)-axis intercepts, the \( y \)-axis intercepts and the horizontal asymptote ( check the intersection with the horizontal asymptote). c) Compute the derivative \( f^{\prime} \) of \( f \) and analyze it d) Compute the second derivative \( f^{\prime \prime} \) of \( f \) and analyze it. e) sketch the graph of \( f \)

Determine the area between the curves \( y = x^2 \) and \( y = 4 - x^2 \) for the interval where they intersect.

12. Si una bola de nieve se derrite de tal modo que el área superficial disminuye a razón de \( 1 \mathrm{~cm}^{2} / \mathrm{min} \), calcule la rapidez con la que disminuye el diámetro cuando éste es 10 cm .

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 \( t = 4 \).