Definition:Average Value of Function
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Definition
Let $f$ be an integrable function on some closed interval $\closedint a b$.
The average value of $f$ (or mean value of $f$) on $\closedint a b$ is defined as:
- $\ds \frac 1 {b - a} \int_a^b \map f x \rd x$
Also see
- Mean Value Theorem for Integrals which proves that $f$ attains this value on $\closedint a b$, provided additionally that $f$ is continuous on $\closedint a b$.
Note on Terminology
The word average is generally considered to be too vague for use in mathematics, as it could mean one of a number of kinds of average.
For serious mathematics it is considered preferable to use the term mean value rather than average value.
However, this is a significant elementary concept which has applications across a wide range of applied mathematics and soft-science subjects, and the popular terminology in such circumstances takes precedence.
Sources
- 2005: Roland E. Larson, Robert P. Hostetler and Bruce H. Edwards: Calculus (8th ed.): $\S 4.4$
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): mean value