$2\mathrm{.}4$ Prove that for any constant $k,lo{g}^{k}N=o(N).$

$2\mathrm{.}5$ Find two functions $f(N)$ and $g(N)$ such that neither $f(N)=O(g(N))$ nor $g(N)=$

$O(f(N)).$

$2\mathrm{.}6$ In a recent court case, a judge cited a city for contempt and ordered a fine of $$2$ for

the first day. Each subsequent day, until the city followed the judge's order, the fine

was squared $($i$.$e$.,$ the fine progressed as follows: $$2,$$$4,$$$16,$$$256,$$$65,536,$dots $).$

a$.$ What would be the fine on day $N?$

b$.$ How many days would it take for the fine to reach $D$ dollars $($a Big$-$Oh answer

will do$)?$

$2\mathrm{.}7$ For each of the following six program fragments:

a$.$ Give an analysis of the running time $($Big$-$Oh will do$).$

b$.$ Implement the code in the language of your choice, and give the running time

for several values of $N.$

c$.$ Compare your analysis with the actual running times.