Answer
$\text {(a) a$_{ave}$ = 7.5 m/s$^2$}$
$\text {(a) a$_{ave}$ $\approx$ -0.373 m/s$^2$}$
Work Step by Step
$\text{(a) The given acceleration is}$
\begin{align}
a(t) = t+ 1; \ 0\leq t \leq 5
\end{align}
\begin{align}
a_{ave} = \frac{1}{5}\int_0^5 (t+1) dt = \frac{1}{5} \times \left[\frac{t^2}{2}+t \right]_0^5 = \frac{1}{5} \times \frac{25}{2}+5 = 7.5 m/s^2
\end{align}
$\text{(a) The given velocity is}$
\begin{align}
v(t) = \cos {t}; \ 0\leq t \leq\frac{\pi}{4}
\end{align}
\begin{align}
& a(t) = \frac{dv(t)}{dt} =-\sin t \\
a_{ave} = -\frac{1}{\frac{\pi}{4}} \int_0^{\frac{\pi}{4}} \sin {t} \ dt =& -\frac{4}{\pi} \times \left[-\cos {t} \right]_0^{\frac{\pi}{4}} = \frac{4}{\pi} \times (\frac{\sqrt 2}{2} - 1) \approx -0.373m/s^2
\end{align}
