Let function $f$ be continuous and decreasing, with values as shown in the table:

$\backslash$table$[[x,2\mathrm{.}5,3\mathrm{.}2,3\mathrm{.}5,4\mathrm{.}0,4\mathrm{.}6,5\mathrm{.}0],[f(x),7\mathrm{.}6,5\mathrm{.}7,-4\mathrm{.}2,3\mathrm{.}8,2\mathrm{.}2,1\mathrm{.}6]]$

$($a$)$ Use the trapezoid method to estimate the area between $f$ and the $x-$axis on the interval $2\mathrm{.}5x5\mathrm{.}0.$

$($b$)$ Find the average rate of change of $f$ on the interval $2\mathrm{.}5x5\mathrm{.}0.$

$($c$)$ Estimate the instantaneous rate of change of $f$ at $x=2\mathrm{.}5.$

$($d$),$ If $g(x)=f^{-1}(x),$ estimate the slope of $g$ at $x=4.$