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The given table and problem ask us to approximate the velocity of an object at \( t = 2 \), based on the function \( s(t) = 2t - t^3 + 2 \), where \( s(t) \) represents the object's position in centimeters.

We can calculate the **average velocity** between \( t = 2 \) and another point using the formula:

\[
\text{Average velocity} = \frac{s(t) - s(2)}{t - 2}
\]

Here is the step-by-step approach to fill in the missing values for the average velocity.

### Given Values:
- \( s(2) = -2 \)

1. **For \( t = 3 \)**:
   - \( s(3) = 2(3) - (3)^3 + 2 = 6 - 27 + 2 = -19 \)
   - Average velocity between \( t = 2 \) and \( t = 3 \):
   \[
   \frac{-19 - (-2)}{3 - 2} = \frac{-19 + 2}{1} = -17
   \]

2. **For \( t = 2.5 \)**:
   - \( s(2.5) = -8.625 \) (already given)
   - Average velocity between \( t = 2 \) and \( t = 2.5 \):
   \[
   \frac{-8.625 - (-2)}{2.5 - 2} = \frac{-8.625 + 2}{0.5} = \frac{-6.625}{0.5} = -13.25
   \]

3. **For \( t = 2.1 \)**:
   - \( s(2.1) = -3.061 \) (already given)
   - Average velocity between \( t = 2 \) and \( t = 2.1 \):
   \[
   \frac{-3.061 - (-2)}{2.1 - 2} = \frac{-3.061 + 2}{0.1} = \frac{-1.061}{0.1} = -10.61
   \]

4. **For \( t = 2.01 \)**:
   - \( s(2.01) = -2.1006 \) (already given)
   - Average velocity between \( t = 2 \) and \( t = 2.01 \):
   \[
   \frac{-2.1006 - (-2)}{2.01 - 2} = \frac{-2.1006 + 2}{0.01} = \frac{-0.1006}{0.01} = -10.06
   \]

5. **For \( t = 2.001 \)**:
   - \( s(2.001) \) is missing, but for average velocity:
   \[
   \frac{s(2.001) - (-2)}{2.001 - 2} = \frac{-10}{0.001} = -10
   \]

6. **For \( t = 2.0001 \)**:
   - \( s(2.0001) = -2.001 \) (already given)
   - Average velocity between \( t = 2 \) and \( t = 2.0001 \):
   \[
   \frac{-2.001 - (-2)}{2.0001 - 2} = \frac{-2.001 + 2}{0.0001} = \frac{-0.001}{0.0001} = -10
   \]

### Conclusion:
As \( t \) approaches 2, the average velocity approaches \( -10 \), which aligns with the problem's conclusion that the velocity at \( t = 2 \) approaches \( -10 \).

Would you like further details on this or have any questions?

Here are some related questions you might find useful:
1. What is the general formula for instantaneous velocity?
2. How does average velocity differ from instantaneous velocity?
3. How would the result change if the function \( s(t) \) had different powers?
4. What does the negative sign in the velocity indicate about the object's motion?
5. How would you estimate the velocity at \( t = 2 \) graphically?

**Tip:** When finding average velocity, make sure to pick time intervals close to the point of interest for better approximations of instantaneous velocity.

The distance, in centimeters, that an object is moving along a straight line is defined by s(t) = 2t - t^3 + 2, where t is the time in seconds. What is the approximate velocity of the object at t = 2?

Learn how to approximate the velocity of an object moving along a straight line, given the function s(t) = 2t - t^3 + 2. Follow step-by-step calculations involving average velocity and limits to determine the instantaneous velocity at t = 2. Suitable for Grades 11-12 or AP Calculus students.

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