Average Rate of Change

Definition

The average rate of change of $y = f(x)$ with respect to $x$ over the interval $[x_1, x_2]$ is: $\frac{\Delta y}{\Delta x} = \frac{f(x_2) - f(x_1)}{x_2 - x_1} = \frac{f(x_1 + h) - f(x_1)}{h}, \quad h \neq 0$

• How to read: “The change in y divided by the change in x is equal to the difference f of x two minus f of x one, all over the difference x two minus x one. This is also equal to the difference f of the sum x one plus h, minus f of x one, all divided by h.”
• Meaning: How much $y$ changes per unit change in $x$ over a finite interval.

Why It Matters

They are the stepping stones to the derivative, allowing us to estimate trends in data before we have a complete continuous model. This is the foundation of data analysis for understanding the trajectory of a system.

Core Concepts

In the context of calculus, the average rate of change generalizes the concept of average speed to any function $y = f(x)$.

• Geometric Interpretation: Geometrically, the average rate of change is equivalent to the slope of the secant line passing through the points $P(x_1, f(x_1))$ and $Q(x_2, f(x_2))$ on the graph of the function.

Connected Concepts

• Secant Lines
• Derivative as a Rate of Change
• Rates of Change in Scientific Applications
• Slope