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Let's break down this problem based on the table and the derivative estimation.

The table provides values for the function \( f \) at specific points \( t \):

\[
\begin{array}{c|cccccc}
t & 0 & 2 & 4 & 6 & 9 & 12 \\
\hline
f(t) & 9 & 17 & 22 & 29 & 31 & 36 \\
\end{array}
\]

### a) Does \( f'(9) \) appear to be positive or negative?

To estimate \( f'(9) \), we look at the function values around \( t = 9 \). The derivative \( f'(9) \) can be approximated by the slope between \( t = 6 \) and \( t = 12 \):

\[
f'(9) \approx \frac{f(12) - f(6)}{12 - 6} = \frac{36 - 29}{12 - 6} = \frac{7}{6} \approx 1.17
\]

Since this value is positive, the derivative \( f'(9) \) appears to be **positive**.

### b) Estimate \( f'(6) \)

To approximate \( f'(6) \), we calculate the slope between \( t = 4 \) and \( t = 9 \):

\[
f'(6) \approx \frac{f(9) - f(4)}{9 - 4} = \frac{31 - 22}{9 - 4} = \frac{9}{5} = 1.8
\]

Thus, \( f'(6) \approx 1.8 \).

### c) \( f'(4) \)

The problem already provides \( f'(4) \approx 3 \), so no further calculation is needed.

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Would you like to go into more detail about any of these calculations? Here are five related questions you might find helpful:

1. How do you calculate the derivative from a table of values?
2. What does a positive or negative derivative tell you about the function's behavior?
3. How can you estimate the derivative if you don't have values immediately around the target point?
4. Why do we use a difference quotient to estimate the derivative?
5. How can these methods help us understand the overall trend of a function?

**Tip:** When approximating a derivative from a table, always choose the points closest to the target value for the best estimate.

Does f'(9) appear to be positive or negative? Estimate f'(6) and check if f'(4) is approximately 3.

Learn how to estimate derivatives from a table of function values using the difference quotient. This example examines the function at t=9, t=6, and t=4 to estimate f'(9), f'(6), and confirm that f'(4) is approximately 3. Suitable for students learning introductory calculus.

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