Use the definition of Θ in order to show the following: a. 5n^3 + 2n^2 + 3n = Θ (n^3) b. sqroot (7n^2 + 2n − 8) = Θ( ?)
Use the definition of Θ in order to show the following: a. 5n^3 + 2n^2 +...
Use the definition of 0 to show that 5n^5 +4n^4 + 3n^3 + 2n^2 + n 0(n^5).Use the definition of 0 to show that 2n^2 - n+ 3 0(n^2).Let f,g,h : N 1R*. Use the definition of big-Oh to prove that if/(n) 6 0(g{n)) and g(n) 0(h{n)) then/(n) 0(/i(n)). You should use different letters for the constants (i.e. don't use c to denote the constant for each big-Oh).
Show using definition of Θ that 1/2 n2 − 5n = Θ(n2)
For each of the following functions, indicate the class Θ(g(n)) the function belongs to. ( Use the simplest g(n) possible in your answers). Prove your assertions. a. ( n2 + 1)10 b. 2n+1 + 3n-1 c. [ log2 n ] d. 2n lg(n+2)2 + ( n+2)2 lg n/2 e. ( 10n2 + 7n + 3)1/2
Let f(n) = 5n^2. Prove that f(n) = O(n^3). Let f(n) = 7n^2. Prove that f(n) = Ω(n). Let f(n) = 3n. Prove that f(n) =ꙍ (√n). Let f(n) = 3n+2. Prove that f(n) = Θ (n). Let k > 0 and c > 0 be any positive constants. Prove that (n + k)c = O(nc). Prove that lg(n!) = O(n lg n). Let g(n) = log10(n). Prove that g(n) = Θ(lg n). (hint: ???? ? = ???? ?)???? ?...
2. Asymptotic Notation (8 points) Show the following using the definitions of O, Ω, and Θ. (1) (2 points) 2n 3 + n 2 + 4 ∈ Θ(n 3 ) (2) (2 points) 3n 4 − 9n 2 + 4n ∈ Θ(n 4 ) (Hint: careful with the negative number) (3) (4 points) Suppose f(n) ∈ O(g1(n)) and f(n) ∈ O(g2(n)). Which of the following are true? Justify your answers using the definition of O. Give a counter example if...
Find the limit of the sequence whose nth term is 2n + ln(n) An= 5n Show all of your work. Hint. Use l'Hospital's Rule.
Given the following statements, mark those correct statements as True and mark those incorrect statements as False. n^2 = O (7n^2 + 3 log n +22) True False 2^n = O (n^3 + 3 n^2 + 7 log n + 2) True False 5n + 3 log n + 1 = O (n log n) True False 7n log n + 3n = O (11 n + 5 log n + 7) True False 2n^2 + 3 n log n...
Order the following functions by asymptotic growth rate. 2n log n + 2n, 210, 2 log n, 3n + 100 log n, 4n, 2n, n2 + 10n, n3, n log n2
Please note n's are superscripted. (a) Use mathematical induction to prove that 2n+1 + 3n+1 ≤ 2 · 4n for all integers n ≥ 3. (b) Let f(n) = 2n+1 + 3n+1 and g(n) = 4n. Using the inequality from part (a) prove that f(n) = O(g(n)). You need to give a rigorous proof derived directly from the definition of O-notation, without using any theorems from class. (First, give a complete statement of the definition. Next, show how f(n) =...
Need help with 1,2,3 thank you. 1. Order of growth (20 points) Order the following functions according to their order of growth from the lowest to the highest. If you think that two functions are of the same order (Le f(n) E Θ(g(n))), put then in the same group. log(n!), n., log log n, logn, n log(n), n2 V, (1)!, 2", n!, 3", 21 2. Asymptotic Notation (20 points) For each pair of functions in the table below, deternme whether...