Find the average rate of chang of g(x)=8$$ \sqrt{x+14} $$ over the interval [-1,6]

To find the average rate of change of the function $ g(x) = 8\sqrt{x + 14} $ over the interval $[-1, 6]$, we use the formula:

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

where $ a = -1 $ and $ b = 6 $.

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

\[
g(-1) = 8\sqrt{-1 + 14} = 8\sqrt{13}
\]

2. Next, calculate $ g(6) $:

\[
g(6) = 8\sqrt{6 + 14} = 8\sqrt{20}
\]

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

\[
\text{Average rate of change} = \frac{g(6) - g(-1)}{6 - (-1)} = \frac{8\sqrt{20} - 8\sqrt{13}}{7}
\]

4. Simplify the expression:

\[
\text{Average rate of change} = \frac{8(\sqrt{20} - \sqrt{13})}{7}
\]

So, the average rate of change of the function $ g(x) = 8\sqrt{x + 14} $ over the interval $[-1, 6]$ is:

\[
\boxed{\frac{8(\sqrt{20} - \sqrt{13})}{7}}
\]

This expression gives the exact average rate of change over the interval.

Would you like more details or have any questions?

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Here are some related questions:
1. What is the significance of the average rate of change in a real-world context?
2. How would the rate of change differ if the interval was [0, 6] instead of [-1, 6]?
3. What happens to the average rate of change if the interval includes a point where the function is undefined?
4. How can you approximate the average rate of change using a graph?
5. What is the relationship between the average rate of change and the derivative of the function?

**Tip:** The average rate of change is a useful concept for understanding how a function behaves over a certain interval, and it can give you insight into the overall trend of the function.

Learn how to calculate the average rate of change for the function g(x) = 8sqrt(x + 14) over the interval [-1, 6]. This problem involves using the formula for average rate of change, square roots, and basic algebraic operations. Suitable for high school students.

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