$($a$)$ Find the average velocity $($in $\frac{m}{s})$ over the given time intervals.

Recall that the average speed between times $t_1$ and $t_2$ is calculated using the following.

$v_{\text{a}\text{v}\text{g}}=\frac{y(t_2)-y(t_1)}{t_2-t_1}$

We are given that $y=5t-1\mathrm{.}86t^2.$ Evaluating at $t=1,$ we have

$y(1)=5(1)-1\mathrm{.}86(1{)}^2=3\mathrm{.}14$

Thus, the average speed between times $1$ and $1+h$ is as follows.

$($:$=\frac{1\mathrm{.}28h-({)}^2}{h}$