To find the average rate of change of the function $f(x) = 2x^5$ over the interval $[-2, 1]$, we use the following formula for the average rate of change:

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

Where:

• $a = -2$
• $b = 1$

Step 1: Calculate $f(a) = f(-2)$

$f(-2) = 2(-2)^5 = 2(-32) = -64$

Step 2: Calculate $f(b) = f(1)$

$f(1) = 2(1)^5 = 2(1) = 2$

Step 3: Apply the formula for the average rate of change

$\text{Average Rate of Change} = \frac{f(1) - f(-2)}{1 - (-2)} = \frac{2 - (-64)}{1 + 2} = \frac{2 + 64}{3} = \frac{66}{3} = 22$

Thus, the average rate of change is $22$.

Let me know if you'd like further details or have any questions.

Here are 5 related questions to further explore this concept:

1. How is the average rate of change related to the slope of a secant line?
2. What is the geometric interpretation of the average rate of change?
3. How does the average rate of change differ from the instantaneous rate of change?
4. Can you compute the average rate of change for a quadratic function?
5. How would the average rate of change formula change if the interval was larger?

Tip: The average rate of change is similar to the slope of a straight line connecting two points on a curve.