Let f(n) = n 3 − 3n 2 + 50 n + 11 True or False: f(n) = O(n 3 ) True False Suppose f1(n) is O(n 3 ) and Ω(n 2 ). Also suppose that f2(n) is O(n 4 ) and Ω(n). Consider the function f(n) = f1(n) + f2(n). True or false: it must be the case that f(n) = O(n 4 ). True False Problem 18. (2 points) Let f(n) = n · (cos n + 1). Remember that cos n value oscillates between −1 and 1. True or False: f(n) = Θ(n). Show your reasoning that led you to the conclusion True False

Let's analyze each statement carefully:

### Statement 1: 
**True or False: \( f(n) = n^3 - 3n^2 + 50n + 11 \) is \( O(n^3) \)**

To determine whether \( f(n) \) is \( O(n^3) \), we need to check if the function \( f(n) \) is asymptotically bounded above by \( n^3 \) as \( n \) grows large.

- The function \( f(n) = n^3 - 3n^2 + 50n + 11 \) has a leading term \( n^3 \).
- The \( O \) notation captures the dominant term as \( n \) approaches infinity, ignoring lower-order terms and constants.

Since \( n^3 \) is the dominant term, the other terms \( -3n^2 \), \( 50n \), and \( 11 \) become negligible as \( n \) increases. Therefore:

\[
f(n) = O(n^3)
\]

**Answer: True**

### Statement 2:
**True or False: If \( f_1(n) \) is \( O(n^3) \) and \( \Omega(n^2) \), and \( f_2(n) \) is \( O(n^4) \) and \( \Omega(n) \), then \( f(n) = f_1(n) + f_2(n) \) is \( O(n^4) \)**

Let's analyze the given conditions:

- \( f_1(n) \) is \( O(n^3) \), meaning \( f_1(n) \) grows at most like \( n^3 \) as \( n \) increases.
- \( f_1(n) \) is \( \Omega(n^2) \), meaning \( f_1(n) \) grows at least like \( n^2 \) as \( n \) increases.
- \( f_2(n) \) is \( O(n^4) \), meaning \( f_2(n) \) grows at most like \( n^4 \) as \( n \) increases.
- \( f_2(n) \) is \( \Omega(n) \), meaning \( f_2(n) \) grows at least like \( n \) as \( n \) increases.

Now, consider the sum \( f(n) = f_1(n) + f_2(n) \).

- The function \( f_2(n) \) is \( O(n^4) \), meaning it can grow as large as \( n^4 \) in the worst case.
- Since the sum of two functions is dominated by the one with the larger asymptotic growth, \( f(n) \) will be dominated by \( f_2(n) \).

Therefore, \( f(n) \) must be \( O(n^4) \).

**Answer: True**

### Statement 3:
**True or False: \( f(n) = n \cdot (\cos n + 1) \) is \( \Theta(n) \)**

Let's analyze the function \( f(n) = n \cdot (\cos n + 1) \):

- \( \cos(n) \) oscillates between \( -1 \) and \( 1 \), so \( \cos(n) + 1 \) oscillates between \( 0 \) and \( 2 \).
- Therefore, \( f(n) = n \cdot (\cos(n) + 1) \) oscillates between \( 0 \cdot n = 0 \) and \( 2 \cdot n = 2n \).

Since \( f(n) \) can be expressed as \( f(n) = n \cdot (\cos(n) + 1) \) and the multiplicative factor \( (\cos(n) + 1) \) is bounded between 0 and 2, the function \( f(n) \) grows linearly with \( n \), up to a constant factor.

This implies:

\[
f(n) = \Theta(n)
\]

**Answer: True**

---

### Summary of Answers:
1. **True**: \( f(n) = n^3 - 3n^2 + 50n + 11 \) is \( O(n^3) \).
2. **True**: \( f(n) = f_1(n) + f_2(n) \) must be \( O(n^4) \).
3. **True**: \( f(n) = n \cdot (\cos n + 1) \) is \( \Theta(n) \).

---

Would you like further details or have any questions?

---

Here are some related questions you might find interesting:

1. What does it mean for a function to be \( \Theta(n) \) versus \( O(n) \)?
2. How do lower-order terms affect the time complexity analysis?
3. Can a function be both \( O(n^2) \) and \( \Omega(n^3) \)?
4. What are the implications if \( f(n) = \Omega(n^4) \) but not \( O(n^4) \)?
5. How do oscillating functions like \( \cos(n) \) impact time complexity analysis?

**Tip:** Understanding the difference between \( O \), \( \Omega \), and \( \Theta \) is crucial for analyzing algorithms' performance. \( O \) provides an upper bound, \( \Omega \) gives a lower bound, and \( \Theta \) captures the exact growth rate.

Explore the analysis of function growth using asymptotic notations. Learn about O(n), Ω(n), and Θ(n) and their implications through examples. Suitable for college-level mathematics students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation