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
For each of the following functions, indicate the class Θ(g(n)) the function belongs to. ( Use...
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: ???? ? = ???? ?)???? ?...
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,...
1. (10 pts) For each of the following pairs of functions, indicate whether f = 0(g), f = Ω(g), or both (in which case f-6(1). You do not need to explain your answer. f(n) (n) a) n (b) n-1n+1 (c) 1000n 0.01n2 (d) 10n2 n (lg n)2 21 е) n (f) 3" (g) 4" rl. 72 i-0 2. (12 pts) Sort the following functions by increasing order of growth. For every pair of consecutive functions f(n) and g(n) in the...
Compare the asymptotic orders of growth of the following pairs of functions. log2 n and . n (n+1)/2 and n2. 2n and 3n
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( ?)
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. Justify your answer (1) n(n2+1) and 2000 n2 + 34 n (2) ln n and lg n (3) 2n-1 and 2n (4) 2 n2 and 0.001 n3 – 2 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...
Question 6 !! Thanks Order the following functions according to their order of growth (from the lowest to n!, n lg n, 8 lg (n + 10)^10, 2^3n, 3^2n, n^5 + 10 lg n Prove that a + lg(n^k + c) = Theta (lg n), for every fixed k > 0, a > 0 and c > 0. Determine the complexities of the following recursive functions, where c > 0 is the operations in the functions. (You may assume that...
Prove or disprove the following statements, using the relationship among typical growth-rate functions seen in class. a)n^15log n + n^9 is O(n^9 log n) b) 15^7n^5 + 5n^4 + 8000000n^2 + n is Θ(n^3) c) n^n is Ω (n!) d) 0.01n^9 + 800000n^7 is O(n^9) e) n^14 + 0.0000001n^5 is Ω(n^13) f) n! is O(3n)
For each pair of functions f(n) and g(n), indicate whether f(n) = O(g(n)), f(n) = Ω(g(n)), and/or f(n) = Θ(g(n)), and provide a brief explanation of your reasoning. (Your explanation can be the same for all three; for example, “the two functions differ by only a multiplicative constant” could justify why f(n) = n, g(n) = 2n are related by big-O, big-Omega, and big-Theta.) i. f(n) = n^2 log n, g(n) = 100n^2 ii. f(n) = 100, g(n) = log(log(log...