Draw the recursion tree when n = 12, where n represents the length of the array, for the following recursive method:

1. Given the following two functions:
 f(n) = 6n3 + 4n2 + 2
 g(n) = 5n2 + 9
Use L’Hopital’s rule and limits to prove or disprove each of the following:
 f  (g)
 g  (f)
2. Rank the following functions from lowest asymptotic order to highest. List any two or
more that are of the same order on the same line.
 2𝑛2 + 10𝑛 + 5
 3𝑛 log2 𝑛
 4𝑛 + 10
 3√𝑛
 2𝑛
 𝑛2 + 6𝑛
 2 log2 𝑛
 2𝑛3 + 𝑛2 + 6
 4n
 log4 𝑛
3. Draw the recursion tree when n = 12, where n represents the length of the array, for the
following recursive method:
int sumSquares(int[] array, int first, int last) {
if (first == last)
return array[first] * array[first];
int mid = (first + last) / 2;
return sumSquares(array, first, mid) +
sumSquares(array, mid + 1, last);
}
 Determine a formula that counts the numbers of nodes in the recursion tree.
 What is the Big- for execution time?
 Determine a formula that expresses the height of the tree.
 What is the Big- for memory?
 Write an iterative solution for this same problem and compare its efficiency with this
recursive solution.
4. Using the recursive method in problem 3 and assuming n is the length of the array.

 Modify the recursion tree from the previous problem to show the amount of work on
each activation and the row sums.
 Determine the initial conditions and recurrence equation.
 Determine the critical exponent.
 Apply the Little Master Theorem to solve that equation.
 Explain whether this algorithm opti

Draw the recursion tree when n = 12, where n represents the length of the array, for the following recursive method:

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