The exercise asks us to calculate the average velocity of an object over various time intervals using the function $s(t) = t^3 - 2t^2$, which represents the position of the object at time $t$. The average velocity is calculated as the change in position over the change in time for each interval.

The average velocity formula is:

$v_{\text{avg}} = \frac{s(t_2) - s(t_1)}{t_2 - t_1}$

where $[t_1, t_2]$ is the given time interval. Let’s calculate the average velocities for each interval.

1. For the interval $[1, 1.5]$:

   • $t_1 = 1$, $t_2 = 1.5$
   • Calculate $s(1)$ and $s(1.5)$: $s(1) = 1^3 - 2 \times 1^2 = 1 - 2 = -1$ $s(1.5) = (1.5)^3 - 2 \times (1.5)^2 = 3.375 - 4.5 = -1.125$
   • Average velocity: $v_{\text{avg}} = \frac{-1.125 - (-1)}{1.5 - 1} = \frac{-0.125}{0.5} = -0.25 \, \text{ft/s}$

2. For the interval $[1, 1.1]$:

   • $t_1 = 1$, $t_2 = 1.1$
   • Calculate $s(1)$ and $s(1.1)$: $s(1.1) = (1.1)^3 - 2 \times (1.1)^2 = 1.331 - 2.42 = -1.089$
   • Average velocity: $v_{\text{avg}} = \frac{-1.089 - (-1)}{1.1 - 1} = \frac{-0.089}{0.1} = -0.89 \, \text{ft/s}$

3. For the interval $[0.5, 1]$:

   • $t_1 = 0.5$, $t_2 = 1$
   • Calculate $s(0.5)$ and $s(1)$: $s(0.5) = (0.5)^3 - 2 \times (0.5)^2 = 0.125 - 0.5 = -0.375$
   • Average velocity: $v_{\text{avg}} = \frac{-1 - (-0.375)}{1 - 0.5} = \frac{-0.625}{0.5} = -1.25 \, \text{ft/s}$

4. For the interval $[0.9, 1]$:

   • $t_1 = 0.9$, $t_2 = 1$
   • Calculate $s(0.9)$ and $s(1)$: $s(0.9) = (0.9)^3 - 2 \times (0.9)^2 = 0.729 - 1.62 = -0.891$
   • Average velocity: $v_{\text{avg}} = \frac{-1 - (-0.891)}{1 - 0.9} = \frac{-0.109}{0.1} = -1.09 \, \text{ft/s}$

Based on these calculations, the reasonable conjecture for the instantaneous velocity at $t = 1$ is around $-1 \, \text{ft/s}$.

Would you like further details or explanations?

Related Questions:

1. How do you find the instantaneous velocity from a function?
2. What is the difference between average and instantaneous velocity?
3. How does the derivative relate to velocity?
4. Can the average velocity be used to estimate instantaneous velocity?
5. What is the physical interpretation of negative velocity?

Tip: To find the instantaneous velocity at a specific point, calculate the derivative of the position function and evaluate it at that point.