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To estimate \( f'(9) \) based on the given table, we can use the **average rate of change** (slope) between the values of \( f(x) \) that are close to \( x = 9 \). The two closest values of \( x \) to 9 are 7 and 10.

We will use the slope formula for these two points to estimate the derivative:

\[
f'(9) \approx \frac{f(10) - f(7)}{10 - 7}
\]

From the table:
- \( f(10) = -99 \)
- \( f(7) = -89 \)

Now, calculate the slope:

\[
f'(9) \approx \frac{-99 - (-89)}{10 - 7} = \frac{-99 + 89}{3} = \frac{-10}{3} \approx -3.33
\]

So, the best estimate for \( f'(9) \) is **A. -3.33**.

Would you like more details or have any questions?

Here are 5 related questions to expand on this topic:
1. How do we interpret the derivative \( f'(x) \) in the context of this function?
2. What other methods could we use to estimate a derivative from a table of values?
3. How would the estimate change if we used points further from \( x = 9 \)?
4. Why is the derivative important in understanding the behavior of a function?
5. How would the process differ if the function were not differentiable?

**Tip:** When estimating derivatives from a table, always choose points as close as possible to the desired \( x \)-value for greater accuracy.

What is the best estimate for f'(9) we can make based on this table?

Learn how to estimate the derivative f'(9) from a table of function values using the average rate of change formula. This problem involves applying key calculus concepts, including derivatives and the slope of a tangent line. Suitable for students in Grades 10-12.

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