interval

Examples of interval in the following topics:

• Interval Notation

  • The two numbers are called the endpoints of the interval.
  • Bounded intervals are also commonly known as finite intervals.
  • For example, the interval $(1,10)$ is considered bounded; the interval $(- \infty, + \infty)$ is considered unbounded.
  • Representations of open and closed intervals on the real number line.
  • Use interval notation to show how a set of numbers is bounded

• Increasing, Decreasing, and Constant Functions

  • We say that a function is increasing on an interval if the function values increase as the input values increase within that interval.
  • Similarly, a function is decreasing on an interval if the function values decrease as the input values increase over that interval.
  • There are no intervals where this curve is constant.
  • The interval notation is written as: $(−∞, −2)∪(2, ∞)$.  
  • The function is decreasing on on the interval: $ (−2, 2)$.  

• Relative Minima and Maxima

  • A function $f$ has a relative (local) maximum at  $x=b$ if there exists an interval  $(a,c)$ with $ainterval $(a,c)$, $f(x)≤f(b)$.
  • Likewise, $f$ has a relative (local) minimum at $x=b$ if there exists an interval $(a,c)$ with $ainterval $(a,c)$, $f(x)≥f(b)$.
  • The graph attains a local maximum at $(1,2)$ because it is the highest point in an open interval around $x=1$.  
  • The graph attains a local minimum at $(-1,-2)$ because it is the lowest point in an open interval around $x=-1$. 

• Piecewise Functions

  • A piecewise function is a function defined by multiple subfunctions that are each applied to separate intervals of the input.
  • Piecewise functions are defined using the common functional notation, where the body of the function is an array of functions and associated intervals.  
  • An open circle at the end of an interval means that the end point is not included in the interval, i.e. strictly less than or strictly greater than.

• Tangent as a Function

  • As with the sine and cosine functions, tangent is a periodic function; its values repeat at regular intervals.
  • The period of the tangent function is $\pi$ because the graph repeats itself on intervals of $k\pi$ where $k$ is a constant.
  • If we look at any larger interval, we will see that the characteristics of the graph repeat.

• Common Bases of Logarithms

  • They describe musical intervals, appear in formulas counting prime numbers, inform some models in psychophysics, and can aid in forensic accounting.
  • Logarithms are related to musical tones and intervals.
  • In equal temperament, the frequency ratio depends only on the interval between two tones, not on the specific frequency, or pitch, of the individual tones.
  • The interval between A and B-flat is a semitone, as is the one between B-flat and B (frequency 493 Hz).
  • Therefore, logarithms can be used to describe the intervals: an interval is measured in semitones by taking the base-21/12 logarithm of the frequency ratio.

• Inverse Trigonometric Functions

  • For angles in the interval $[-\frac{\pi}{2}, \frac{\pi}{2}]$, if $\sin y = x$, then $\sin^{−1} x=y$.
  • For angles in the interval $[0, \pi]$, if $\cos y = x$, then $\cos^{-1} x = y$.
  • For angles in the interval $(-\frac{\pi}{2}, \frac{\pi}{2})$, if $\tan y = x$, then $\tan^{-1}x = y$.

• Sine and Cosine as Functions

  • By thinking of the sine and cosine values as coordinates of points on a unit circle, it becomes clear that the range of both functions must be the interval [-1, 1].
  • A periodic function is a function with a repeated set of values at regular intervals.

• The Intermediate Value Theorem

  • Stated in the language of algebra supported by : If f is a real-valued continuous function on the interval [a, b], and u is a number between f(a) and f(b), then there is a c ∈ [a, b] such that f(c) = u.

• Rational Inequalities

  • Thus we can conclude that for $x$ values on the open interval from $-\infty$ to $-3$, the rational expression is negative.