The point $P(4,2)$ is on the graph of $f(x)=\sqrt[\phantom{2}]{x}.$ Let $Q$ be the point $(x,f(x)).$

$($a$)$ Find the slopes of the secant lines $PQ$ for the following values of $x$:$4\mathrm{.}5,4\mathrm{.}1,4\mathrm{.}01,$ and $4\mathrm{.}001.$

$($b$)$ Using the results of $($a$),$ what appears to the slope of the tangent to the graph of $f$ at $P?$

$($c$)$ Find an equation of the tangent line to the graph of $f$ at $P.$

$($d$)$ Sketch the graph of $f$ and the tangent line at $P.$