Exercises

Fun Class Activities:

1. Find the smallest positive integer \( n \) such that \( 2^n > n^3 \).
2. Compare the growth of \( f(n) = n \log n \) and \( g(n) = n^2 \). Which one grows faster as $n$ keeps growing?
3. Determine whether \( f(n) = 5n^3 + 4n \log n \) is \( O(n^3) \), \( \Omega(n^3) \), or \( \Theta(n^3) \).
4. Is it true that \( f(n) = n^4 \) is \( o(n^5) \)? Explain why or why not.
5. Suppose \( f(n) = 3^n \) and \( g(n) = 2^n \). Is \( f(n) = O(g(n)) \)? Why or why not?
6. Which of the following functions is asymptotically smallest?
   • \( f_1(n) = \log n \)
   • \( f_2(n) = n \)
   • \( f_3(n) = n \log n \)
   • \( f_4(n) = n^2 \)