5. Given that \( y=(2 x+3)^{5}(x+2)^{8} \), find the value of \( \frac{\mathrm{d} y}{\mathrm{~d} x} \) when \( x=-1 \)

Determina expresau a. \( \int 8 x\left(4 x^{2}-3\right) \) b. \( \int 2 t^{-3}\left(4 t^{-2}\right)^{-} \) c. \( \int_{1}^{3}\left(8 x-2 x^{2}\right) \)

\begin{tabular}{|l|}\hline Enunciado \\ \hline Determine si los siguientes enunciados son verdaderos \( (V) \) o faso \( (F) \) : \\ a. Si \( f(x)=3 x^{3}+6 \) cuando \( x \) varia de -3 a 1 su incremento es -24 \\ b. Si \( f(x)=3 x-2 \) cuando \( x \) varia de 2 a 2.5 no es un ejemplo de decremento \\ \hline\end{tabular}

The motion of a particle pertorming damped vibrations s given by \( \gamma=e^{-t} \operatorname{Sin} 2 t, \gamma \) being the displacement rom its mean position at time \( t \). Show that \( \gamma \) is a maximum when \( t=\frac{1}{2} \tan ^{-1}(2) \) and determine this maximum displacement to hree significant tigures.

(v) The value of \( \int_{-\infty}^{\infty} \delta(t) d t \) is: (a) 0 (b) 1 (c) -1 (d) \( \infty \) (vi) Solution of \( \frac{d^{2} x}{d x^{2}}-k x=0 \), when \( k=0 \) is: (a) \( c_{1} x+c_{2} \) (b) \( c_{1} \cos x+c_{2} \sin x \) (c) \( c_{1} e^{\lambda x}+c_{2} e^{-\lambda x} \) (d) None of these (vii) Equation \( u_{t u}-c^{2} u_{x z}=0 \) is: (a) one dimensional wave equation P.T.O.

The function \( f(x)=\frac{x}{x+1} \) is concave up on