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.