Some values of function $f(x)$ are given in the table.

$\backslash$table$[[x,-4,-2,0,2,4,6],[f(x),11\mathrm{.}5,13\mathrm{.}7,14\mathrm{.}75,15\mathrm{.}5,16\mathrm{.}8,17\mathrm{.}5]]$

$(1)$ Find the average rate of change for $f(x^{})$ over $-4,6.$

$(2)$ Estimate the values of $f^{\text{'}}(0)$ and $f^{\text{'}}(6).$

$(3)$ Use the value of $f^{\text{'}}(6)$ to approximate the value of $f(7\mathrm{.}6).$

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