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The task is to find the average velocities over two time intervals and then estimate the instantaneous velocity of the car at \( t = 0.2 \) seconds.

### Part (a)
To find the average velocity over the interval \( 0 \leq t \leq 0.2 \) seconds:
The formula for average velocity is given by:
\[
\text{Average Velocity} = \frac{\Delta s}{\Delta t}
\]
where \( \Delta s \) is the change in position and \( \Delta t \) is the change in time.

From the table:
- \( s(0) = 0 \, \text{ft} \)
- \( s(0.2) = 0.5 \, \text{ft} \)
- \( \Delta s = 0.5 - 0 \, \text{ft} = 0.5 \, \text{ft} \)
- \( \Delta t = 0.2 - 0 \, \text{sec} = 0.2 \, \text{sec} \)

Plugging in the values:
\[
\text{Average Velocity} = \frac{0.5 \, \text{ft}}{0.2 \, \text{sec}} = 2.5 \, \text{ft/sec}
\]

### Part (b)
To find the average velocity over the interval \( 0.2 \leq t \leq 0.4 \) seconds:
- \( s(0.2) = 0.5 \, \text{ft} \)
- \( s(0.4) = 1.7 \, \text{ft} \)
- \( \Delta s = 1.7 - 0.5 \, \text{ft} = 1.2 \, \text{ft} \)
- \( \Delta t = 0.4 - 0.2 \, \text{sec} = 0.2 \, \text{sec} \)

Plugging in the values:
\[
\text{Average Velocity} = \frac{1.2 \, \text{ft}}{0.2 \, \text{sec}} = 6 \, \text{ft/sec}
\]

### Part (c)
To estimate the instantaneous velocity at \( t = 0.2 \) seconds, you can use the average velocities obtained from parts (a) and (b). Since the instantaneous velocity at a point is the limit of the average velocity as the interval shrinks around the point, we can average the two average velocities:
\[
\text{Instantaneous Velocity at } t = 0.2 = \frac{2.5 + 6}{2} = 4.25 \, \text{ft/sec}
\]

Would you like to explore more about the calculations or have any other questions on this topic? Here are some options you might consider:

1. How does the average velocity compare to the instantaneous velocity, and why might they differ?
2. What would be the estimated instantaneous velocity at \( t = 0.4 \) seconds?
3. How do you calculate the average acceleration of the car over the interval \( 0 \leq t \leq 0.4 \)?
4. How might the velocity function of the car look like based on these average velocities?
5. Could the actual path of the car be linear or non-linear based on the velocities calculated?

**Tip:** To find the instantaneous velocity more accurately, consider using smaller time intervals if the data is available. This approximates the derivative at a point more closely.

Find the average velocity over the intervals 0 ≤ t ≤ 0.2 seconds and 0.2 ≤ t ≤ 0.4 seconds, and estimate the instantaneous velocity of the car at t = 0.2 seconds.

Explore how to calculate average velocities over specific time intervals and estimate the instantaneous velocity at t = 0.2 seconds for a car's motion. This example illustrates basic concepts of differential calculus applicable in high school level studies.

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