$2.$ Let $\backslash ($ f $\backslash )$ be differentiable function with domain $\backslash ($ D$=(-10,10)\backslash ).$ Imagine that a square$-$based cup rides along the graph of $\backslash ($ f $\backslash ),$ by "sitting" on the tangent line to the graph of $\backslash ($ f $\backslash ).$

You're going to imagine what happens when the cup starts off completely full of water $($all the way to the top$),$ and then moves along the graph of $\backslash ($ f $\backslash ).$

For ease of communication let's call the process of filling the cup completely full of water to start, then slowly sliding it along the graph of $\backslash ($ f $\backslash )$ from one end of $\backslash ($ D $\backslash )$ to the other as described above, an $\backslash (\backslash$underline$\{$f $\backslash$text $\{-$transit $\}\}\backslash )$ of the cup. We'll also always use the variables $\backslash ($ w $\backslash )$ and $\backslash ($ h $\backslash )$ to denote the side length of the square base of the cup and the height of the cup, respectively.

$($a$)(5$ marks$)$ Let $\backslash ($ g$($x$)=\backslash$cos $(2$ x$)\backslash ),$ with its domain restricted to $\backslash ($ D $\backslash ).$ Show that if $\backslash ($ h$=2$ w $\backslash ),$ exactly half of the water will spill from the cup after a $\backslash ($ g $\backslash )-$transit.

$($b$)(5$ marks$)$ Find an example of a function $\backslash ($ f $\backslash )$ which is differentiable on $\backslash ($ D $\backslash ),$ and such all the water will spill from any cup during $\backslash ($ f $\backslash )-$transit. Be sure to explain why your example works.

$($c$)(5$ marks$)$ For this problem, let $\backslash ($ h $\backslash )$ and $\backslash ($ w $\backslash )$ be arbitrary but fixed positive numbers $($i$.$e$.,$ the cup is fixed, but you don't know its exact dimensions$).$

Let $\backslash ($ f $\backslash )$ be differentiable function with domain $\backslash ($ D $\backslash ),$ and assume that $\backslash (\backslash$left$|$f$^\{\backslash$prime$\}($x$)\backslash$right$|\backslash )$ achieves some maximum value $\backslash ($ M $\backslash )$ on $\backslash ($ D $\backslash ).$ Determine exactly how much water spill from the cup after an $\backslash ($ f $\backslash )-$transit.

In other words, find a formula for a function $\backslash ($ S $\backslash )$ with domain $\backslash ([0,\backslash$infty$)\backslash )$ such that for any $\backslash ($ M $\backslash$in$[0,\backslash$infty$)\backslash ),\backslash ($ S$($M$)\backslash )$ is the amount of water that spills from the cup during an $\backslash ($ f $\backslash )-$transit, and explain how you got your answer.

Hint: Your formula will probably have to be defined piecewise.