Exponential function

In mathematics, an exponential function is a function of the form

The input variable x occurs as an exponent – hence the name. A function of the form f(x) = bx ± c is also considered an exponential function, and a function of the form f(x) = a·bx can be re-written as f(x) = bx ± c by the use of logarithms and so is an exponential function.

In contexts where the base b is not specified, especially in more theoretical contexts, the term exponential function is almost always understood to mean the natural exponential function

also written as

where e is Euler's number, a transcendental number approximately 2.718281828. The reason this number e is considered the "natural" base of exponential functions is that this function is its own derivative. Every exponential function is directly proportional to its own derivative, but only when the base is e does the constant of proportionality equal 1.

The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change (i.e. percentage increase or decrease) in the dependent variable. The function is often written as exp(x), especially when it is impractical to write the independent variable as a superscript. The exponential function is widely used in physics, chemistry, engineering, mathematical biology, economics and mathematics.

EX

EX or Ex may refer to:

Relationships

  • Ex (relationship), someone with whom a person was once associated, often romantically
  • Computing

  • ex (text editor), a line editor for the UNIX operating system
  • ex (typography), a CSS unit for the typographic unit x-height
  • Film and television

  • TV Asahi or EX, a TV station in Roppongi, Tokyo, Japan
  • Ex (2009 film), a comedy directed by Fausto Brizzi
  • Ex (2010 film), a 2010 Hong Kong film starring Gillian Chung
  • The Ex (1997 film), a Canadian thriller film by Mark L. Lester
  • The Ex (2006 film), a comedy film
  • "The Ex", an episode of Matlock
  • Mathematics

  • Exponential function (ex)
  • Expected value (E(X)), in probability theory
  • ex (function prefix) (exterior), a prefix for some trigonometric functions in mathematics
  • Music

  • The Ex (band), a Dutch punk band
  • Albums

  • EX (Trigger album), an album by Trigger
  • EX (Plastikman album), an album by Plastikman
  • Songs

  • "The Ex" (song), a song by Billy Talent
  • Other uses

  • Air Santo Domingo (IATA airline code), a former flag carrier airline of the Dominican Republic
  • Expected value

    In probability theory, the expected value of a random variable is intuitively the long-run average value of repetitions of the experiment it represents. For example, the expected value of a six-sided die roll is 3.5 because, roughly speaking, the average of an extremely large number of die rolls is practically always nearly equal to 3.5. Less roughly, the law of large numbers guarantees that the arithmetic mean of the values almost surely converges to the expected value as the number of repetitions goes to infinity. The expected value is also known as the expectation, mathematical expectation, EV, average, mean value, mean, or first moment.

    More practically, the expected value of a discrete random variable is the probability-weighted average of all possible values. In other words, each possible value the random variable can assume is multiplied by its probability of occurring, and the resulting products are summed to produce the expected value. The same works for continuous random variables, except the sum is replaced by an integral and the probabilities by probability densities. The formal definition subsumes both of these and also works for distributions which are neither discrete nor continuous: the expected value of a random variable is the integral of the random variable with respect to its probability measure.

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