Average value of the function

Definition

The average value of a function over an interval $[a, b]$ is given by $\frac{1}{b-a} \int_a^b f(x) \, dx$. It represents the mean value of all function outputs in that interval.

5 Must Know Facts For Your Next Test

1. The formula for finding the average value of a function $f(x)$ over an interval $[a, b]$ is $\frac{1}{b-a} \int_a^b f(x) \, dx$.
2. The average value can be interpreted as the height of a rectangle whose area is the same as the area under the curve from $a$ to $b$.
3. If $f(x)$ is continuous on $[a, b]$, then its average value exists and can be calculated using definite integrals.
4. The units of the average value are the same as those of the function $f(x)$.
5. The concept of the average value of a function helps in understanding real-world problems where averages over intervals are considered.

Review Questions

• What is the formula for finding the average value of a function over an interval?
• How can you interpret the geometrical meaning of the average value?
• In what scenario does calculating the average value involve definite integrals?

Related terms

Definite Integral: A definite integral calculates the net area under a curve between two specified points on a graph.

Continuous Function: A function without breaks or jumps; it can be graphed without lifting your pencil from paper.

Mean Value Theorem for Integrals:

This theorem states that if $f(x)$ is continuous on $[a,b]$, there exists at least one point $c \in (a, b)$ such that $f(c) = \frac{1}{b-a} \int_a^b f(x) \, dx$.