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

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

For the function $f(x) = 1 - 6x^2$, the average rate of change over an interval $[a, b]$ would be:

$\text{Average rate of change} = \frac{(1 - 6b^2) - (1 - 6a^2)}{b - a}$

Simplifying:

$\text{Average rate of change} = \frac{1 - 6b^2 - 1 + 6a^2}{b - a}$ $\text{Average rate of change} = \frac{6a^2 - 6b^2}{b - a}$ $\text{Average rate of change} = \frac{6(a^2 - b^2)}{b - a}$ $\text{Average rate of change} = \frac{6(a - b)(a + b)}{b - a}$ Since $b - a = -(a - b)$, we get:

$\text{Average rate of change} = -6(a + b)$

So, the average rate of change for $f(x) = 1 - 6x^2$ over an interval $[a, b]$ is $-6(a + b)$.

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Here are 5 related questions:

1. How would you find the instantaneous rate of change for this function?
2. What is the average rate of change of $f(x)$ over the interval [1, 3]?
3. Can you explain how to interpret the average rate of change geometrically?
4. How does the average rate of change differ from the derivative at a point?
5. What happens to the average rate of change as $a$ and $b$ get closer?

Tip: The average rate of change is the slope of the secant line between two points on the graph.