1. (10 pts) Give "True" or "False" for each of the following statements. No proof is needed. • $T(n) = 3n^2 + 5n \cdot \log_2 n = O(n)$. • $T(n) = 4^{\log_2 n} + \sqrt{n} = \Omega(n^2)$. • $T(n) = 3n^2 + 9n = O(n^3)$. • $T(n) = 4 \cdot (\log_2 n)^5 + 5\sqrt{n} + 10 = \Theta(\sqrt{n})$. • $T(n) = (\log_2 n)^{\log_2 n} + n^4 = \Theta(n^4)$.
Added by Diamond H.
Close
Step 1
Since $n^2$ grows faster than $n$, and $n \log_2 n$ grows faster than $n$, $3n^2 + 5n \log_2 n$ grows faster than $n$. Therefore, $T(n)$ is not $O(n)$. Answer: False Show more…
Show all steps
Your feedback will help us improve your experience
Vincenzo Zaccaro and 83 other AP CS educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
3. (U & G-required) [40 points] Using the formal definition of the asymptotic notations, prove the following statements: a) n^3 + 10n^2 ∈ O(n^3) b) 5n^3 + 2000n ∈ ̐(n^2) c) n! ∈ O(n^n) d) 10n^2 + 2 ∉ O(n)
Vincenzo Z.
Determine whether or not the following are true and provide a full derivation explaining your answer for each. The domain of the functions of n below is the positive real numbers. For convenience, you may assume that the logs are in the base of your choice, but you should specify what base you are using in your derivation. a. -3 + 17n^2 + 4n is O(n^2) b. 3/n + 11 is O(n) c. 3n log n is O(n^2) d. 1/n + 3 log(log(n)) + 5log(n+1)/2 is O(n) e. log(n^3 + n^-3) is O(n^2) f. (4n - 5)^2 is Θ(n^2) g. 4n^2 + 3n log n + 7n^3 is Δ(n^3) h. 9n^3 is Δ(n^3) Let a be the last digit in your student number, b be the second-to-last digit in your student number, c = 5a, and d = 2b. Compute the following, for the functions f and g defined as: f: Z → Z f(x) = cx - d and g: Z → Z g(x) = dx^2 + cx a. g ∘ f b. f ∘ g c. (f ∘ g) ∘ g d. f ∘ (g ∘ f)
Keondre P.
Please answer parts a - e.
Oswaldo J.
Recommended Textbooks
Computer Science and Information Technology
Introduction to Programming Using Python
Computer Science - An Overview
Transcript
18,000,000+
Students on Numerade
Trusted by students at 8,000+ universities
Watch the video solution with this free unlock.
EMAIL
PASSWORD