The image shows a problem asking to approximate $f'(13)$ as accurately as possible using the provided table of values. Here's how we can solve this:

We are given the following values:

To approximate $f'(13)$, we will use the central difference formula: $f'(13) \approx \frac{f(13.1) - f(12.9)}{13.1 - 12.9}$

Substituting the values from the table: $f'(13) \approx \frac{12.58 - 12.48}{13.1 - 12.9} = \frac{0.10}{0.2} = 0.5$

Thus, the best approximation for $f'(13)$ is $0.5$.

Would you like more details or have any questions?

Here are 5 related questions to consider:

1. How does the central difference formula work in general for derivatives?
2. Can this method be used to approximate derivatives of non-linear functions?
3. What other methods could be used to estimate derivatives if the table spacing is uneven?
4. How accurate is this approximation compared to the actual derivative at $t = 13$?
5. How can second-order differences improve the accuracy of the approximation?

Tip: The central difference method is often more accurate than using forward or backward differences because it takes information from both sides of the point of interest.