average

Examples of average in the following topics:

• Average Value of a Function

  • The average of a function $f(x)$ over an interval $[a,b]$ is $\bar f = \frac{1}{b-a} \int_a^b f(x) \ dx$.
  • An average is a measure of the "middle" or "typical" value of a data set.
  • However, the concept of average can be extended to functions as well.
  • Note that the average is equal to the area under the curve, $S$, divided by the range:
  • Evaluate the average value of a function over a closed interval using integration

• Tangent and Velocity Problems

  • The average velocity $\bar{\vec{v}}$ of an object moving through a displacement ($\Delta \vec{x}$) during a time interval ($\Delta t$) is described by the formula: $\displaystyle \bar{\vec{v}} = \frac{\Delta \vec{x}}{\Delta t}$.
  • The average velocity becomes instantaneous velocity at time t.
  • Slope of tangent of position or displacement time graph is instantaneous velocity and its slope of chord is average velocity.

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

  • This theorem can be understood intuitively by applying it to motion: If a car travels one hundred miles in one hour, then its average speed during that time was 100 miles per hour.
  • To get at that average speed, the car either has to go at a constant 100 miles per hour during that whole time, or, if it goes slower at one moment, it has to go faster at another moment as well (and vice versa), in order to still end up with an average of 100 miles per hour.
  • 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.

• Applications of Multiple Integrals

  • As is the case with one variable, one can use the multiple integral to find the average of a function over a given set.
  • Given a set $D \subseteq R^n$ and an integrable function $f$ over $D$, the average value of $f$ over its domain is given by:

• Physics and Engeineering: Center of Mass

  • In this case, the distribution of mass is balanced around the center of mass and the average of the weighted position coordinates of the distributed mass defines its coordinates.

• Models Using Differential Equations

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

• Numerical Integration

  • Instead of taking $f(x)$ to be a constant around a chosen $x$, $f(x)$ is approximated as having a constant slope around $x$, where the slope is the average between the chosen points.