Find the average rate of change for f(x)=1-6x^2

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 \( -6(a + b) \).

Let me know if you'd like any details or further clarifications!

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.

Discover how to find the average rate of change for the quadratic function f(x) = 1 - 6x^2. This guide covers essential calculus concepts like the rate of change, secant line, and difference quotient, providing a detailed step-by-step solution suitable for students in Grades 10-12.

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