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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:

| \( t \)  | 12.8  | 12.9  | 13  | 13.1  | 13.2  |
|---------|-------|-------|-----|-------|-------|
| \( f(t) \) | 12.46 | 12.48 | 12.52 | 12.58 | 12.66 |

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.

Use the given table to approximate f'(13) as accurately as you can.

Learn how to approximate f'(13) using the central difference formula with the values from a table. This method provides an accurate derivative estimate based on numerical data points. Suitable for high school or early college students studying calculus.

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