# Recitation 10: Quiz 1 Review | Summary and Q&A

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January 14, 2013
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MIT OpenCourseWare
Recitation 10: Quiz 1 Review

## TL;DR

Analyzing the growth rates of different functions.

## Key Insights

• ☠️ Functions can be simplified or compared using mathematical formulas to determine their growth rates.
• ☠️ The growth rates of n^3 and n^log(n) can be compared by taking the logarithm of n^log(n).
• ☠️ The growth rates of n+log^4(n) and 2^n can be compared to determine which function grows faster.

## Transcript

The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: All right. So I brought a few problems. They're... Read More

### Q: How can the growth rates of different functions be compared?

One way to compare the growth rates of functions is by simplifying them or using mathematical formulas to determine their respective growth rates.

### Q: What is the relationship between the growth rates of n^3 and n^log(n)?

By taking the logarithm of n^log(n), it simplifies to log(n)log(n), which has a smaller growth rate than n^3. Therefore, n^3 grows faster than n^log(n).

### Q: What is the relationship between the growth rates of n+log^4(n) and 2^n?

The growth rate of n+log^4(n) is n, which is smaller than the growth rate of 2^n. Therefore, 2^n grows faster than n+log^4(n).

### Q: How can the growth rate of the nC3 function be determined?

By using the binomial coefficient formula, nC3 simplifies to n(n-1)(n-2)/2. By expanding the formula, the growth rate of nC3 can be determined.

## Summary & Key Takeaways

• The first function is n^4.

• The second function is nC3 (n choose 3), which simplifies to n^3.

• The third function is n + log^4(n), which can be simplified to n.

• The fourth function is n^log(n), whose growth rate can be determined by taking the logarithm.

• The fifth function is 2^n, which has the largest growth rate.