T $22.$ Make a table of values for the function $F(x)=\frac{x+2}{x-2}$ at the points $x=1\mathrm{.}2,x=\frac{11}{10},x=\frac{101}{100},x=\frac{1001}{1000},x=\frac{10001}{10000},$ and $x=1.$

a$.$ Find the average rate of change of $F(x)$ over the intervals $1,x$ for each $x1$ in your table.

b$.$ Extending the table if necessary, try to determine the rate of change of $F(x)$ at $x=1.$

T $23.$ Let $g(x)=\sqrt[\phantom{2}]{x}$ for $x0.$

a$.$ Find the average rate of change of $g(x)$ with respect to $x$ over the intervals $1,2$

$1,1\mathrm{.}5$ and $1,1+h.$

b$.$ Make a table of values of the average rate of change of $g$ with respect to $x$ over the interval $1,1+h$ for some values of $h$ approaching zero, say $h=0\mathrm{.}1,0\mathrm{.}01,0\mathrm{.}001,0\mathrm{.}0001,0\mathrm{.}00001,$ and $0\mathrm{.}000001.$

c$.$ What does your table indicate is the rate of change of $g(x)$ with respect to $x$ at $x=1?$

T $24.$ Let $f(t)=\frac{1}{t}$ for $t0.$

a$.$ Find the average rate of change of $f$ with respect to $t$ over the intervals $($i$)$ from $t=2$ to $t=3,$ and $($ii$)$ from $t=2$ to $t=T.$

b$.$ Make a table of values of the average rate of change of $f$ with respect to $t$ over the interval $2,T,$ for some values of $T$ approaching $2,$ say $T=2\mathrm{.}1,2\mathrm{.}01,2\mathrm{.}001,2\mathrm{.}0001,2\mathrm{.}00001,$ and $2\mathrm{.}000001.$

c$.$ What does your table indicate is the rate of change of $f$ with respect to $t$ at $t=2?$

$25.$ The accompanying graph shows the total distance $s$ traveled by a bicyclist after $t$ hours.

a$.$ Estimate the bicyclist's average speed over the time intervals $0,1$

$1,2\mathrm{.}5,$ and $2\mathrm{.}5,3\mathrm{.}5.$

b$.$ Estimate the bicyclist's instantaneous speed at the times $t=\frac{1}{2},t=2,$ and $t=3.$

c$.$ Estimate the bicyclist's maximum speed and the specific time at which it occurs.