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The problem is asking to calculate average velocities over different time intervals for a given position function \( s(t) = -4.9t^2 + 32t + 21 \), and then make a conjecture about the instantaneous velocity at \( t = 3 \).

### Steps to solve:
1. **Average velocity formula**: The average velocity over a time interval \([a, b]\) is given by:
   \[
   \text{Average Velocity} = \frac{s(b) - s(a)}{b - a}
   \]
   Here, \(a = 3\) and \(b\) takes different values like 4, 3.1, 3.01, etc.

2. **Position function evaluation**: You'll need to plug in \(t = a\) and \(t = b\) into the position function \(s(t)\) for each interval.

I'll compute the values for you. Let's start with the first interval \([3, 4]\) and work through each interval step by step.Here are the calculated average velocities for each time interval:

- For the interval \([3, 4]\), the average velocity is approximately \(-2.3\).
- For the interval \([3, 3.1]\), the average velocity is approximately \(2.11\).
- For the interval \([3, 3.01]\), the average velocity is approximately \(2.551\).
- For the interval \([3, 3.001]\), the average velocity is approximately \(2.5951\).
- For the interval \([3, 3.0001]\), the average velocity is approximately \(2.5995\).

### Conjecture:
As the time intervals become smaller and closer to \(t = 3\), the average velocity approaches approximately \(2.6\). Therefore, the instantaneous velocity at \(t = 3\) is likely around \(2.6\).

Would you like further details or clarifications on these calculations?

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Here are 5 follow-up questions to deepen your understanding:
1. How do we compute instantaneous velocity from a position function?
2. Why does the average velocity approach a specific value as the time interval shrinks?
3. Can we find the exact instantaneous velocity at \(t = 3\) using differentiation?
4. How would the velocity change if the position function had a different quadratic coefficient?
5. What happens to the average velocity if we extend the time intervals instead of shrinking them?

**Tip:** The instantaneous velocity is the derivative of the position function at a specific time.

Consider the position function s(t) = -4.9t^2 + 32t + 21. Complete the following table with the appropriate average velocities and make a conjecture about the value of the instantaneous velocity at t = 3.

Learn how to calculate average velocities over different time intervals for the position function s(t) = -4.9t^2 + 32t + 21. This problem includes step-by-step calculations for finding average velocity and making a conjecture about the instantaneous velocity at t = 3. Suitable for calculus students.

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