mean

(noun)

The average value.

Related Terms

Examples of mean in the following topics:

  • The Mean Value Theorem, Rolle's Theorem, and Monotonicity

    • The MVT states that for a function continuous on an interval, the mean value of the function on the interval is a value of the function.
    • In calculus, the mean value theorem states, roughly: given a planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints .
    • Therefore, the Mean Value Theorem tells us that at some point during the journey, the car must have been traveling at exactly 100 miles per hour; that is, it was traveling at its average speed.
    • The mean value theorem follows from the more specific statement of Rolle's theorem, and can be used to prove the more general statement of Taylor's theorem (with Lagrange form of the remainder term).
    • Use the Mean Value Theorem and Rolle's Theorem to reach conclusions about points on continuous and differentiable functions

  • Average Value of a Function

    • The average of a list of numbers is a single number intended to typify the numbers in the list, which is called the arithmetic mean.
    • If $n$ numbers are given, each number denoted by $a_i$, where $i = 1, \cdots , n$, the arithmetic mean is the sum of all $a_i$ values divided by $n$:
    • The first mean value theorem for integration states that if $G : [a, b] \to R$ is a continuous function and $\varphi$ is an integrable function that does not change sign on the interval $(a, b)$, then there exists a number $x$ in $(a, b)$ such that:
    • The value $G(x)$ is the mean value of $G(t)$ on $[a, b] $ as we saw previously.

  • Derivatives and Rates of Change

    • The word tangent comes from the Latin word tangens, which means touching.
    • But what exactly do we mean by "slope" for a curve?
    • The simplest case is when $y$ is a linear function of x, meaning that the graph of $y$ divided by $x$ is a straight line.
    • where the symbol $\Delta$ (the uppercase form of the Greek letter Delta) means, and is pronounced, "change in."
    • In more precise language, the dependence of $y$ upon x means that $y$ is a function of $x$.

  • Surfaces in Space

    • To say that a surface is "two-dimensional" means that, about each point, there is a coordinate patch on which a two-dimensional coordinate system is defined.

  • Comparison Tests

    • Comparison test may mean either limit comparison test or direct comparison test, both of which can be used to test convergence of a series.
    • Comparison tests may mean either limit comparison tests or direct comparison tests.

  • Area and Arc Length in Polar Coordinates

    • Area and arc length are calculated in polar coordinates by means of integration.

  • Infinite Limits

    • For $f(x)$ a real function, the limit of $f$ as $x$ approaches infinity is $L$, denoted $\lim_{x \to \infty}f(x) = L$, means that for all $\varepsilon > 0$, there exists $c$ such that $\left | f(x) - L \right | < \varepsilon$ whenever $x>c$.
    • Similarly, the limit of $f$ as $x$ approaches negative infinity is $L$, denoted $\lim_{x \to -\infty}f(x) = L$, means that for all $\varepsilon > 0$ there exists $c$ such that $|f(x) - L| < \varepsilon$ whenever $x

  • Models Using Differential Equations

    • The mean lifetime, $\tau$ ("tau"), is the average lifetime of a radioactive particle before decay.
    • The decay constant, $\lambda$ ("lambda"), is the inverse of the mean lifetime.

  • Probability

    • This probability distribution has the mean and variance, denoted by $\mu$ and $\sigma ^2$, respectively.
    • Probability distribution function of a normal (or Gaussian) distribution, where mean $\mu=0 $  and variance $\sigma^2=1$.

  • Inverse Functions

    • More directly, $g(f(x))=x$, meaning $g(x)$ composed with $f(x)$, leaves $x$ unchanged.