To better understand why the definition of Big-O focuses on the behavior of functions for large values of $n$, choose two functions with different growth rates where the faster-growing function is initially lower but eventually becomes larger.

Write a short program that periodically compares the values of these two functions and identifies the point at which the faster-growing function overtakes the slower-growing one.

As an example, consider the following two functions:
- $ f(n) = 500n^2 + 15n + 1000 $
- $ g(n) = 2n^3 $

Shown below is a table of the values of both functions for small values of $n$:

\[
\begin{array}{|c|c|c|}
\hline
n & f(n) & g(n) \\
\hline
10 & 51150 & 2000 \\
20 & 201300 & 16000 \\
30 & 451450 & 54000 \\
40 & 801600 & 128000 \\
50 & 1251750 & 250000 \\
60 & 1801900 & 432000 \\
70 & 2452050 & 686000 \\
80 & 3202200 & 1024000 \\
90 & 4052350 & 1458000 \\
100 & 5002500 & 2000000 \\
\hline
\end{array}
\]

The provided program compares the values of two functions, f(n) = 500n^2 + 15n + 1000 and g(n) = 2n^3, for different values of n. It illustrates the point at which the faster growing function, g(n), overtakes the slower growing function, f(n), despite initially having smaller values for smaller values of n.

The program computes the values of both functions for a range of values of n, starting from 10 and incrementing by 10 up to 100. It compares the values of f(n) and g(n) at each step and prints the results in a table format.

By analyzing the table, we can observe that for small values of n, f(n) has larger values compared to g(n). However, as n increases, the growth rate of g(n) surpasses that of f(n), causing g(n) to eventually become larger. This demonstrates the concept of Big-O notation, where the focus is on the behavior of functions for large values of n. In this case, even though f(n) has lower values initially, g(n) eventually dominates and has a higher growth rate.

Learn more about Big-O notation here: brainly.com/question/13257594

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