Question

in my c++ class i need help with these question please

Question 1. Indicate whether the first function of each of the following pairs has a smaller, same, or larger order of growth (to within a constant multiple) than the second function. Use the correct notation to indicate the order of growth (f(n) ∈O(g(n)), Ω(g(n)), or Θ(g(n)) as applicable). Prove your statement using limits. (a) (lnn)2 and lnn2 (b) 42n+1 and 42n

Question 2. Use the formal definitions of O, Ω, and Θ to prove the following: (a) lnn ∈ Θ(log2 n) (b) 100n3 ∈ Ω(n2)

Question 3. Order the following functions according to their order of growth from lowest to highest. Ties, if any, must be clearly stated. (a) log2 n10 (b) 6 √n (c) log2 n (d) (n + 1000)! (e) 3n (f) 1.93n (g) n8 1000 (h) n−6000 (i) 0.0001n2 + 105n (j) nlogn Question 4. For each of the following program fragments, give an analysis of the running time (Big-Oh will do). Give the exact value leading to the Big-Oh conclusion, and show your work. sum = 0; //1 for( i = 0; i < n; ++i ) ++sum; sum = 0; //2 for( i = 0; i < n; ++i ) for( j = 0; j < n; ++j ) ++sum; sum = 0; //3 for( i = 0; i < n; ++i ) for( j = 0; j < n * n; ++j ) ++sum;

Question 5. Define a Rectangle class that provides getLength and getWidth. Write a main that creates a vector of Rectangles and finds the largest Rectangle on the basis of area. For this part of the assignment, you need to submit a separate .cpp source code file as well as screenshots in your homework submission indicating successful testing of the code. No credit will be given for code that does not compile or segfaults.

Answer 1

Solution:

1 is solved as per the HOMEWORKLIB RULES, please repost others.

1)

a)

(log₂n)² and log₂ n²

===> log₂² n = log₂ n * log₂ n and

and

log₂ n² = 2 log n.

(According to the logarithmic property)

which means

$(logn_2 n)^2 = \Omega (log n^2)$

b)

4^(2n+1) and 4^2n

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