Question

Asymptotic Growth, sort all the functions below in increasing order of asymptotic (Big-O) growth. If some have the same asymptotic growth, then be sure to indicate that. As usual, log means base 2.

1. 5n
2. 4 log n
3. N⁴
4. 5ⁿ
5. (n / 4)ⁿ/⁴

Answer 1

In increasing order of asymptotic growth, the functions are: 4 log n, 5n, N⁴, (n / 4)ⁿ/⁴, and 5ⁿ, start from the slowest to the fastest growth rates.

Step-by-step explanation:

To sort the given functions in increasing order of asymptotic growth (Big-O), we need to recognize the order each of these types of functions typically has. The order from lowest growth to highest growth is as follows:

1. 4 log n: This function grows logarithmically, which is a very slow rate of growth relative to the others listed.
2. 5n: Linear growth is faster than logarithmic, but slower compared to polynomial and exponential growths.
3. N´: Polynomial growths are generally faster than both linear and logarithmic growths. Among polynomials, higher degrees grow faster.
4. (n / 4)¹⁄₄: This is an example of an exponential growth, but with a smaller base compared to a pure exponential function. Although it grows fast, it's slightly slower than a full exponential function because the base is less than one.
5. 5ⁿ: Exponential growths are the fastest among the functions listed here and within exponential functions, those with a larger base grow faster.

Therefore, in increasing order of asymptotic growth, the functions are: 4 log n, 5n, N´, (n / 4)¹⁄₄, and 5ⁿ.