https://file.aiquickdraw.com/mathbot/file/17283943960766ql1x1x0.jpg

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### Part a: Approximate \( f'(4.5) \) using the data from the table.

From the image, the table gives the following values:
- \( f(4) = 2.633 \)
- \( f(5) = 3.193 \)

We approximate \( f'(4.5) \) using the average rate of change formula between \( x = 4 \) and \( x = 5 \), which is:
\[
f'(4.5) \approx \frac{f(5) - f(4)}{5 - 4} = \frac{3.193 - 2.633}{1} = 0.56
\]

Thus, \( f'(4.5) \approx 0.56 \).

Would you like more details or have any questions on this? Here are some related questions:

1. How do you determine the horizontal tangent from a graph?
2. What are the steps to find \( f'(x) \) if you're given an equation for \( f(x) \)?
3. How can you improve the accuracy of a derivative approximation from a table?
4. Can the derivative approximation be improved with smaller intervals?
5. Why is the derivative undefined at some points?

**Tip:** When approximating derivatives from a table, always check the spacing between points. The smaller the interval, the better the approximation!

Approximate the value for f'(4.5) using data from the table and show computations.

Learn how to approximate the derivative f'(4.5) using a table of values and the average rate of change formula, f'(x) ≈ (f(b) - f(a)) / (b - a). This AP Calculus BC problem involves differentiation concepts, including horizontal tangents and using data from a table. Suitable for high school students in Grades 11-12.

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