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

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

Steps to Calculate:

1. Identify the interval $[a, b]$.

   • $a$: The starting $x$-value of the interval.
   • $b$: The ending $x$-value of the interval.

2. Find the function values at $x = a$ and $x = b$.

   • $f(a)$: Substitute $a$ into the function $f(x)$.
   • $f(b)$: Substitute $b$ into the function $f(x)$.

3. Apply the formula: Substitute $f(a)$, $f(b)$, $a$, and $b$ into the formula and simplify.

Example:

Suppose the function is $f(x) = 3x^2 + 2x$, and you want the average rate of change over the interval $[1, 3]$.

1. $a = 1$, $b = 3$.
2. Find $f(1)$ and $f(3)$: $f(1) = 3(1)^2 + 2(1) = 3 + 2 = 5$ $f(3) = 3(3)^2 + 2(3) = 27 + 6 = 33$
3. Apply the formula: $\text{Average Rate of Change} = \frac{f(3) - f(1)}{3 - 1} = \frac{33 - 5}{2} = \frac{28}{2} = 14$

Thus, the average rate of change is $14$.

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Do you want to work through a specific function or interval? Let me know!
Here are some related questions to explore further:

1. What is the significance of the average rate of change in real-world contexts?
2. How does the average rate of change relate to the slope of a line?
3. Can you find the instantaneous rate of change using a similar approach?
4. What happens to the average rate of change as the interval shrinks to a single point?
5. How does the formula change for different types of functions, like logarithmic or exponential?

Tip: Always plot the function to visualize the change over the interval!