Average rate of change

Definition

The average rate of change of a function over an interval is the change in the function's value divided by the change in the input values. It represents the slope of the secant line connecting two points on the graph of the function.

5 Must Know Facts For Your Next Test

1. The formula for average rate of change is $\frac{f(b) - f(a)}{b - a}$ where $a$ and $b$ are points in the domain.
2. It provides a measure of how much a function's output changes, on average, per unit change in input over a specific interval.
3. The concept is similar to finding the slope of a straight line but applied to functions that may not be linear.
4. Average rate of change can be used as an approximation for instantaneous rate of change when intervals are small.
5. Understanding this concept is crucial for interpreting real-world problems involving rates, such as velocity or growth rates.

Review Questions

• What does the average rate of change represent geometrically on a graph?
• How do you calculate the average rate of change between two points?
• Explain how average rate of change differs from instantaneous rate of change.

Related terms

Derivative: A measure of how a function's output value changes as its input value changes. It represents an instantaneous rate of change and is found using limits.

Secant Line: A line that intersects two points on a curve, used to compute the average rate of change between those points.

Instantaneous Rate Of Change: $\frac{d}{dx} f(x)$ at a point gives the derivative, representing how fast $f(x)$ is changing at that particular point.