Average value of the function

Definition

The average value of a function over an interval $[a, b]$ is the sum of the function's values at each point in the interval divided by the length of the interval. Mathematically, it is given by $\frac{1}{b-a} \int_{a}^{b} f(x) \, dx$.

5 Must Know Facts For Your Next Test

1. The formula for the average value of a function $f(x)$ over $[a, b]$ is $\frac{1}{b-a} \int_{a}^{b} f(x) \, dx$.
2. This concept involves using definite integrals to find the total area under the curve and then dividing by the interval length.
3. It can be interpreted as finding a constant value that represents the 'average height' of the function over an interval.
4. To compute it, first evaluate the definite integral and then multiply by $\frac{1}{b-a}$.
5. Understanding this concept helps in solving problems related to physical quantities like average speed or average temperature.

Review Questions

• What is the formula for finding the average value of a function on an interval?
• How do you interpret the average value geometrically?
• If $f(x)$ is continuous on $[a, b]$, what does $\frac{1}{b-a} \int_{a}^{b} f(x) \, dx$ represent?

Related terms

Definite Integral:

A definite integral calculates the accumulation of quantities and gives the area under a curve between two points.

$\int_{a}^{b} f(x) \, dx$: Represents the definite integral of $f(x)$ from $a$ to $b$, calculating total accumulation or area under $f(x)$.

$\frac{1}{b-a}$: Represents scaling factor used to find averages, where $(b - a)$ is length of interval from $a$ to $b$.