Order the following functions by asymptotic growth rate. 4 n log n +2 n 2 10 2 log n 3 n +100log n 4 n 2 n n 2 +10 n n 3 n log n   Show that if d ( n ) is O ( f ( n )), then ad ( n ) is O ( f ( n )),...

Order the following functions by asymptotic growth rate.

4nlog
n+2n
2¹⁰
2<sup>log


n</sup>

3n+100log
n
4n
2
ⁿ

n
²
+10n n
³
nlog
n

Show that if
d(n) is
O(
f
(n)), then
ad(n) is
O(
f
(n)), for any constant

a
> 0.

Show that if
d(n) is
O(
f
(n)) and
e(n) is
O(g(n)), then the product
d(n)e(n)

is
O(
f
(n)g(n)).

Show that if
d(n) is
O(
f
(n)) and
e(n) is
O(g(n)), then
d(n) +e(n) is

O(
f
(n)+g(n)).

Show that if
d(n) is
O(
f
(n)) and
f
(n) is
O(g(n)), then
d(n) is
O(g(n)).