Definition:Mean Value of Function

Definition

Let $f$ be an integrable function on some closed interval $\closedint a b$.

The 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 known as

The mean value of a function is also known as its average value.

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 $\closedint a b$.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): mean value (of a function)
• 2005: Roland E. Larson, Robert P. Hostetler and Bruce H. Edwards: Calculus (8th ed.): $\S 4.4$
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): mean value (of a function)
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): mean value