Periodic Functions Practice Notes

A set of vocabulary-style flashcards covering the determination of fundamental periods for various trigonometric and composite functions from the lecture notes.

Constant Function Period ($f(x) = 1$)

A function that is periodic with any positive real number as its period, but its fundamental period is defined as nd (not defined).

Period of $f(x) = \sec^{2}(x) - \tan^{2}(x)$

This function simplifies to $1$ wherever it is defined, making it periodic, but its fundamental period is not defined.

Period of $f(x) = \cos(x + \sin(x))$

The period is $2\pi$ because $f(2\pi + x) = \cos(x + \sin(x))$, satisfying the periodicity condition.

Period of $f(x) = \cos^{4}(x) + \sin^{4}(x)$

The period is $\frac{\pi}{2}$ as identified by testing values such as $f(\frac{\pi}{2} + x) = f(x)$.

Period of $f(x) = \frac{\sin(x) + \sin(3x)}{\cos(x) + \cos(3x)}$

The period is $\frac{\pi}{2}$ because the expression simplifies to $\tan(2x)$, and the period of $\tan(ax)$ is $\frac{\pi}{|a|}$.

Non-linear Arguments and Periodicity

Functions with non-linear arguments, such as $f(x) = \cos(\sqrt{x})$, are categorized as aperiodic or non-periodic.

Period of $f(x) = 5\sin(2\sqrt{2}x) + 7\cos(3\sqrt{2}x)$

The period is calculated using the LCM of individual periods $\frac{2\pi}{2\sqrt{2}}$ and $\frac{2\pi}{3\sqrt{2}}$, which results in $\frac{\pi}{\sqrt{2}}$.

Fundamental Period of Cumulative Sum Functions

In the function $f(x) = [x] + [2x] + [3x] + \dots + [nx] - \frac{n(n+1)}{2}x$, the result is expressed in terms of fractional parts $-(\{x\} + \{2x\} + \{3x\} + \dots + \{nx\})$, which relates to periodicity.