Polynomial Functions

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This lecture segment introduces polynomial functions and their growth rates as n approaches infinity. The instructor presents a core example problem requiring students to arrange six specific functions in increasing order of growth: sqrt(n), n, n^2, n log n, n^3, and n^k (where k >= 1). The teaching method emphasizes rewriting radical expressions as fractional exponents to facilitate direct comparison of polynomial powers. For instance, the square root function is explicitly converted from sqrt(n) to n^0.5 or n^(1/2). The instructor systematically compares terms by analyzing their exponents and logarithmic factors, establishing a hierarchy where lower powers grow slower than higher powers. The visual progression shows the instructor underlining terms, circling intermediate comparisons like 5n versus n, and highlighting the final sorted sequence in a pink box. Key concepts include identifying that polynomial growth dominates logarithmic-linear terms, and that higher exponents indicate faster asymptotic growth.

Chapters

1. 0:00 – 2:00 00:00-02:00

   The instructor introduces the topic of polynomial functions and presents an example problem asking to arrange a set of functions in increasing order of growth as n approaches infinity. The visible text on screen lists the target functions: sqrt(n), n, n^2, n log n, n^3, and n^k. The instructor begins by analyzing the first function, square root of n, rewriting it as a power of n (n^0.5) to facilitate comparison with the other polynomial and logarithmic terms. This conversion is a critical step shown in the notes, where n^0.5 = n^(1/2) is written to standardize the notation for exponent comparison.

2. 2:00 – 3:30 02:00-03:30

   The instructor continues the analysis by comparing linear terms against logarithmic factors to establish the hierarchy between n, n log n, and n^2. The visual notes show comparisons like 5n versus n and the analysis of n log n versus n^2 to determine their relative magnitudes. The progression shows a step-by-step comparison of terms, circling them to indicate order. Finally, the sorted sequence is highlighted in a pink box on screen: sqrt(n), n, n log n, n^2, n^3, n^k. This final arrangement demonstrates the hierarchy of growth rates from the lowest power (n^0.5) up to the highest polynomial term (n^k), confirming that polynomial functions with higher exponents grow faster than those with lower exponents or logarithmic multipliers.

The lecture effectively demonstrates the method for comparing asymptotic growth rates of polynomial and related functions. The central technique involves converting all terms to a standard exponential form, such as rewriting sqrt(n) as n^0.5, which allows for direct comparison of the exponents. The instructor establishes a clear hierarchy where n^0.5 grows slower than n, which in turn grows slower than n log n, followed by higher powers like n^2 and n^3. The final sorted sequence provided in the pink box serves as a definitive reference for students to understand that among polynomial functions, the magnitude of the exponent dictates the growth rate. This example reinforces the fundamental concept that as n approaches infinity, higher powers dominate lower powers and logarithmic factors.