Average rate of change

Definition

The average rate of change of a function over an interval \([a, b]\) is the change in the function's value divided by the change in the input values, represented as $\frac{f(b) - f(a)}{b - a}$. It measures how much the function's output changes per unit increase in input over that interval.

5 Must Know Facts For Your Next Test

1. The average rate of change can be interpreted as the slope of the secant line between two points on a graph.
2. It is calculated using the formula $\frac{\Delta y}{\Delta x} = \frac{f(b) - f(a)}{b - a}$.
3. The units of the average rate of change depend on the units of both the function and its input variable.
4. If $f(x)$ represents position over time, then its average rate of change represents velocity.
5. Unlike instantaneous rate of change, which requires calculus, average rate of change only requires basic algebra.

Review Questions

• How do you interpret the average rate of change geometrically?
• What formula is used to calculate the average rate of change between two points?
• Why does calculating average rate of change not require calculus?

Related terms

Instantaneous Rate of Change: The derivative at a specific point; it measures how a function changes at an exact point.

Secant Line: A line that intersects two points on a curve; its slope represents the average rate of change between those points.

$\Delta y$: $\Delta y$ represents the difference in function values: $f(b) - f(a)$.