Find the average rate of chang of g(x)=$$ \frac{1}{-x} $$ over the interval [-2,8]

To find the average rate of change of the function $ g(x) = \frac{1}{-x} $ over the interval $[-2, 8]$, we use the formula:

\[
\text{Average rate of change} = \frac{g(b) - g(a)}{b - a}
\]

where $ a = -2 $ and $ b = 8 $.

1. First, calculate $ g(-2) $:

\[
g(-2) = \frac{1}{-(-2)} = \frac{1}{2}
\]

2. Next, calculate $ g(8) $:

\[
g(8) = \frac{1}{-8} = -\frac{1}{8}
\]

3. Now, substitute these values into the formula for the average rate of change:

\[
\text{Average rate of change} = \frac{g(8) - g(-2)}{8 - (-2)} = \frac{-\frac{1}{8} - \frac{1}{2}}{10}
\]

4. Simplify the expression:

\[
\text{Average rate of change} = \frac{-\frac{1}{8} - \frac{4}{8}}{10} = \frac{-\frac{5}{8}}{10} = -\frac{5}{80} = -\frac{1}{16}
\]

So, the average rate of change of the function $ g(x) = \frac{1}{-x} $ over the interval $[-2, 8]$ is:

\[
\boxed{-\frac{1}{16}}
\]

Would you like more details or have any questions?

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Here are some related questions:
1. What does a negative average rate of change imply about the behavior of the function over the interval?
2. How would the rate of change differ if the interval was [-2, 2] instead of [-2, 8]?
3. What is the relationship between the average rate of change and the slope of the secant line connecting two points on the curve?
4. How does the function $ g(x) = \frac{1}{-x} $ behave as $ x $ approaches zero?
5. How can the concept of average rate of change be applied to analyze the speed of a moving object?

**Tip:** The average rate of change can help you understand whether a function is increasing or decreasing over an interval, and it can provide insights into the overall behavior of the function in different contexts.

Learn how to calculate the average rate of change for the function g(x) = 1/(-x) over the interval [-2, 8]. Understand the step-by-step process with detailed calculations and insights into function behavior. Suitable for advanced high school students and calculus enthusiasts.

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