Please order the function belows by growth rate
n ^ 1.5
n ^ 0.5 + log n
n log ^ 2 n
n log ( n ^ 2 )
n log log n
n ^ 2 + log n
n log n
n
ps: Ordering by growth rate means, as n gets larger and larger, which function will eventually be higher in value than the others.
ps2. I have ordered most of the functions: n , n log log n, n log n, n log^2 n, n log ( n ^ 2 ), n ^ 1.5
I just do not know how to order: n ^ 2 + log n, n ^ 0.5 + log n, these 2 values
Can anyone help me? Thank you
In all of those cases, you're dealing with pairs of functions that themselves have different growth rates.
With that in mind, only the larger one really matters, since it will be most dominant even with a sum. So in each of those function sums, which is the bigger one and how does it compare to the other ones on your larger list?
n0.5 (or n1/2) is the square root of n. So, it grows more slowly than n2.
The key to the answer you seek is that when you sum two functions, their combined "growth rate" is going to be exactly that of the one with the higher growth rate of the two. So, you now know the growth rates of these two functions, since you appear (from knowing the correct ordering of all the others) to know the proper ordering of the growth rates that are in play here.
You can figure this out fairly easily by graphing the functions and seeing which ones get larger (find a graphing calculator, check out Maxima, or try graphing the functions on Wolfram Alpha). Or, or course, you just pick some large value of n and compare the various functions, but graphs can give a bit of a better picture.
Since this is homework, I would suggest that you just try writing these as equations, and either plot them, perhaps in Excel, graphically, or print out numbers, and just see how all of these appear.
You may start with small numbers, perhaps from .001 to 5, going up by .01 perhaps, and then change the range or increment and remove those that you can easily determine.
If you turn in the graph it may help with your explanation.