The average rate of change of a function over a given interval is a measure of how the function's values change, on average, between two points. It is similar to finding the slope of a secant line connecting two points on the function's graph.

The formula for the average rate of change of a function $f(x)$ over the interval $[a, b]$ is:

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

Steps to Find the Average Rate of Change

1. Identify the function $f(x)$.
2. Determine the endpoints of the interval $[a, b]$.
3. Calculate the values of the function at $a$ and $b$, which are $f(a)$ and $f(b)$.
4. Substitute into the formula:
   $\frac{f(b) - f(a)}{b - a}$
5. Simplify to find the average rate of change.

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Example

Find the average rate of change of $f(x) = x^2$ over the interval $[1, 3]$.

Solution:

1. The function is $f(x) = x^2$.

2. The interval is $[1, 3]$. Here, $a = 1$ and $b = 3$.

3. Compute $f(1)$ and $f(3)$:

   • $f(1) = 1^2 = 1$
   • $f(3) = 3^2 = 9$

4. Use the formula:
   $\text{Average Rate of Change} = \frac{f(3) - f(1)}{3 - 1} = \frac{9 - 1}{3 - 1} = \frac{8}{2} = 4$

Final Answer: The average rate of change is $4$.

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Key Notes:

• The average rate of change measures the "overall" change in the function over an interval.
• It does not describe what happens between the points—only the net effect.
• For a linear function, the average rate of change is constant.

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Do you want me to show more examples or provide explanations for specific functions?

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Related Questions:

1. What is the difference between the average rate of change and instantaneous rate of change?
2. How do you calculate the slope of a secant line for a given function?
3. How is the average rate of change applied in real-world scenarios?
4. Can you explain the average rate of change for nonlinear functions like $f(x) = x^3$?
5. How does the average rate of change relate to derivatives in calculus?

Tip:

For linear functions, the average rate of change is the same as the slope of the line, since the rate of change is constant across all intervals.