Let's analyze the problem you provided.

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!