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

$\text{Average rate of change} = \frac{f(b) - f(a)}{b - a}$

Now let's calculate the average rate of change for each specified interval:

(a) Interval $[2, 5]$

From the table:

• $f(2) = 14$
• $f(5) = 28$

$\text{Average rate of change} = \frac{f(5) - f(2)}{5 - 2} = \frac{28 - 14}{5 - 2} = \frac{14}{3} \approx 4.67$

(b) Interval $[3.8, 4]$

From the table:

• $f(3.8) = 13$
• $f(4) = 10$

$\text{Average rate of change} = \frac{f(4) - f(3.8)}{4 - 3.8} = \frac{10 - 13}{4 - 3.8} = \frac{-3}{0.2} = -15$

Summary:

• (a) The average rate of change over $[2, 5]$ is approximately 4.67.
• (b) The average rate of change over $[3.8, 4]$ is -15.

Would you like more details or have any questions?

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Here are five related questions to expand on this topic:

1. How do you interpret the meaning of the average rate of change in real-world applications?
2. How would the average rate of change differ if the function was decreasing instead of increasing?
3. What is the significance of the sign (positive or negative) of the average rate of change?
4. How do you find the instantaneous rate of change at a point for a function?
5. How does the average rate of change relate to the slope of a secant line on a graph?

Tip: The average rate of change gives the slope of the line connecting two points on the graph of the function.