The average rate of change of a continuous function $f(x)$ between two points $x = a$ and $x = b$ is calculated as:

$\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}$

Steps to Calculate

1. Choose Two Points: Identify the interval $[a, b]$ on which the average rate of change is to be calculated.
2. Evaluate the Function: Compute the function values $f(a)$ and $f(b)$ at the endpoints of the interval.
3. Apply the Formula: Plug the values into the formula above.

Interpretation

The average rate of change represents the slope of the secant line that passes through the points $(a, f(a))$ and $(b, f(b))$ on the graph of $f(x)$.

• If the average rate of change is positive, the function is increasing on average over $[a, b]$.
• If it is negative, the function is decreasing on average over $[a, b]$.
• If it is zero, the function has no net change over the interval, indicating a flat or constant behavior.

Example

Suppose $f(x) = x^2$ and you want to find the average rate of change on the interval $[1, 3]$.

1. Evaluate $f(1) = 1^2 = 1$ and $f(3) = 3^2 = 9$.
2. Compute: $\text{Average Rate of Change} = \frac{f(3) - f(1)}{3 - 1} = \frac{9 - 1}{3 - 1} = \frac{8}{2} = 4.$
3. Interpretation: The function $f(x) = x^2$ increases by an average of 4 units for each unit increase in $x$ over the interval $[1, 3]$.

---

Would you like a detailed breakdown of an example or have additional questions?
Here are 5 related questions to deepen your understanding:

1. How is the average rate of change related to the instantaneous rate of change?
2. How do you interpret the average rate of change for real-world problems, such as velocity or cost?
3. What happens to the average rate of change if the interval is very small?
4. Can a function's average rate of change be zero over an interval but still vary within it?
5. How does the secant line compare to the tangent line of a function graphically?

Tip: To visualize the average rate of change, sketch the secant line on the function's graph—it provides an intuitive understanding of the concept!