periodic function

Examples of periodic function in the following topics:

• Examples

  • It's because we've assume the function is periodic on the interval $[-1,1]$.
  • The {\it periodic extension} of $f(x) = x$ must therefore have a sort of sawtooth appearance.
  • In other words any non-periodic function defined on a finite interval can be used to generate a periodic function just by cloning the function over and over again.
  • Figure~\ref{sawtooth} shows the periodic extension of the function $f(x) = x$ relative to the interval $[0,1]$.
  • It's a potentially confusing fact that the same function will give rise to different periodic extensions on different intervals.

• Sine and Cosine as Functions

  • In the graphs for both sine and cosine functions, the shape of the graph repeats after $2\pi$, which means the functions are periodic with a period of $2\pi$.
  • A periodic function is a function with a repeated set of values at regular intervals.
  • When this occurs, we call the smallest such horizontal shift with $P>0$ the period of the function.
  • The diagram below shows several periods of the sine and cosine functions.
  • The sine and cosine functions are periodic, meaning that a specific horizontal shift, $P$, results in a function equal to the original function:$f(x + P) = f(x)$.

• Tangent as a Function

  • Characteristics of the tangent function can be observed in its graph.
  • The tangent function can be graphed by plotting $\left(x,f(x)\right)$ points.
  • As with the sine and cosine functions, tangent is a periodic function.
  • The period of the tangent function is $\pi$ because the graph repeats itself on $x$-axis intervals of $k\pi$, where $k$ is a constant.
  • The graph of the tangent function is symmetric around the origin, and thus is an odd function.

• Inverse Trigonometric Functions

  • Each trigonometric function has an inverse function that can be graphed.
  • To use inverse trigonometric functions, we need to understand that an inverse trigonometric function “undoes” what the original trigonometric function “does,” as is the case with any other function and its inverse.
  • However, the sine, cosine, and tangent functions are not one-to-one functions.
  • In fact, no periodic function can be one-to-one because each output in its range corresponds to at least one input in every period, and there are an infinite number of periods.
  • As with other functions that are not one-to-one, we will need to restrict the domain of each function to yield a new function that is one-to-one.

• Supply Function

  • Like the demand function, supply can be viewed from two perspectives.
  • Supply is a schedule of quantities that will be produced and offered for sale at a schedule of prices in a given time period, ceteris paribus.
  • Figure III.A.5 is a graphical representation of a supply function.
  • The equation for this supply function is Qsupplied= -10 + 2P.
  • Table III.A5 also represents this supply function.

• Five-Part Rondo

  • Hybrid themes generally combine the features of sentences and periods.
  • See the atoms on the sentence, the period, and Classical theme functions for more information on the elements included in hybrid themes.
  • On the large scale, the antecedent phrase functions like a big presentation function zone (like the presentation phrase does).
  • The CBI expresses presentation function, followed by a continuation phrase that expresses continuation and cadential functions.
  • Hybrid 4 is almost identical to a period, with the exception that the first phrase does not end with a cadence.

• Harmonic Wave Functions

  • The vibration of particle is represented by a harmonic sine or cosine function.
  • We know that time period in SHM is equal to time taken by the particle to complete one oscillation.
  • We can determine speed of the wave by noting that wave travels a linear distance " in one period (T).
  • Harmonic waves are described by sinusoidal functions.
  • Express relationship between the wave number and the wavelength, and frequency and period, of the harmonic wave function

• Derivatives of Trigonometric Functions

  • Derivatives of trigonometric functions can be found using the standard derivative formula.
  • The trigonometric functions (also called the circular functions) are functions of an angle.
  • Trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications.
  • The most familiar trigonometric functions are the sine, cosine, and tangent.
  • The same procedure can be applied to find other derivatives of trigonometric functions.

• Multi-Period Investment

  • Multi-period investments are investments with more than one period, so n (or t) is greater than one.
  • The number of periods, however, is not 24--it is 2.
  • When using a financial calculator, we enter our known values followed by their corresponding function key.
  • Enter 5 and then press the "n" key (or whatever function key corresponds with the number of periods).
  • Enter 0.03 and press the "i" key (or whatever function key corresponds with the investment rate).

• Development of the Respiratory System

  • Lung development can be divided into distinct stages: the pseudoglandular period, the canalicular period, and the terminal saccular period.
  • The pseudoglandular period (also known as "glandular period") spans weeks six to 16 and during this time the developing lung resembles an endocrine gland.
  • The terminal saccular period spans from week 26 to birth.
  • At birth, the respiratory system becomes fully functional upon exposure to air, although some lung development and growth continues throughout childhood.
  • The lungs of pre-term infants therefore may not function well because the lack of surfactant leads to increased surface tension within the alveoli leading to alveoli collapse and no gas exchange, a condition known as respiratory distress syndrome.