periodicity

Examples of periodicity in the following topics:

• 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:$f(x + P) = f(x)$.

• Interest Compounded Continuously

  • The amount of interest earned increases with each compounding period.
  • The equation representing investment value as a function of principal, interest rate, period and time is:
  • The more frequent the compounding periods the more interest is accrued.
  • In this situation the amount of money in the account will be given by $(1+\frac{1}{n})^n$ where $n$ is the number of compounding periods and $\frac{1}{n}$ is the rate per compounding period.
  • Graph of interest accrued under differing number of compounding periods per year

• Population Growth

  • Namely, it is given by the formula $P(r, t, f)=P_i(1+r)^\frac{t}{f}$ where $P{_i}$ represents the initial population, r is the rate of population growth (expressed as a decimal), t is elapsed time, and f is the period over which time population grows by a rate of r.
  • If we multiply the $PGR$ by $100$ we arrive at the percentage growth relative to the population at the beginning of the time period.

• Tangent as a Function

  • 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.

• Defining Trigonometric Functions on the Unit Circle

  • The unit circle demonstrates the periodicity of trigonometric functions.
  • Periodicity refers to the way trigonometric functions result in a repeated set of values at regular intervals.
  • This is an indication of the periodicity of the cosine function.

• Inverse Trigonometric 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.

• Solving General Problems with Logarithms and Exponents

  • For this example n represents a period of 2 years, therefore the n is halved for this purpose.

• Applications of the Parabola

  • Aircraft used to create a weightless state for purposes of experimentation, such as NASA's "Vomit Comet," follow a vertically parabolic trajectory for brief periods in order to trace the course of an object in free fall, which, for most purposes, produces the same effect as zero gravity.

• Exponential Decay

  • Use the exponential decay formula to calculate how much of something is left after a period of time

• Addition and Subtraction; Scalar Multiplication

  • Matrix addition, subtraction and scalar multiplication can be used to find such things as: the sales of last month and the sales of this month, the average sales for each flavor and packaging of soda in the $2$-month period.