To complete the table, we need to calculate the average velocity over each time interval $[1, t]$ using the position function $s(t) = -16t^2 + 103t$. The average velocity $v_{\text{avg}}$ over an interval $[1, t]$ is given by:

$v_{\text{avg}} = \frac{s(t) - s(1)}{t - 1}$

First, let's calculate the position at $t = 1$:

$s(1) = -16(1)^2 + 103(1) = -16 + 103 = 87$

Now, for each $t$ value in the table, we calculate $s(t)$, then use the formula above to find the average velocity.

For $t = 2$:

$s(2) = -16(2)^2 + 103(2) = -16(4) + 206 = -64 + 206 = 142$ $v_{\text{avg}} = \frac{142 - 87}{2 - 1} = \frac{55}{1} = 55 \text{ units/time}$

For $t = 1.5$:

$s(1.5) = -16(1.5)^2 + 103(1.5) = -16(2.25) + 154.5 = -36 + 154.5 = 118.5$ $v_{\text{avg}} = \frac{118.5 - 87}{1.5 - 1} = \frac{31.5}{0.5} = 63 \text{ units/time}$

For $t = 1.1$:

$s(1.1) = -16(1.1)^2 + 103(1.1) = -16(1.21) + 113.3 = -19.36 + 113.3 = 93.94$ $v_{\text{avg}} = \frac{93.94 - 87}{1.1 - 1} = \frac{6.94}{0.1} = 69.4 \text{ units/time}$

For $t = 1.01$:

$s(1.01) = -16(1.01)^2 + 103(1.01) = -16(1.0201) + 103.03 = -16.3216 + 103.03 = 86.7084$ $v_{\text{avg}} = \frac{86.7084 - 87}{1.01 - 1} = \frac{-0.2916}{0.01} = -29.16 \text{ units/time}$

For $t = 1.001$:

$s(1.001) = -16(1.001)^2 + 103(1.001) = -16(1.002001) + 103.103 = -16.032016 + 103.103 = 87.070984$ $v_{\text{avg}} = \frac{87.070984 - 87}{1.001 - 1} = \frac{0.070984}{0.001} = 70.984 \text{ units/time}$

Table of Average Velocities

Time Interval	Average Velocity (units/time)
[1, 2]	55
[1, 1.5]	63
[1, 1.1]	69.4
[1, 1.01]	69.16
[1, 1.001]	70.984

Conjecture about the Instantaneous Velocity at $t = 1$:

The instantaneous velocity at $t = 1$ appears to be approaching a value close to 71 units/time as $t$ gets closer to 1.

Would you like more details or have any questions?

Here are 5 related questions for further exploration:

1. How do you calculate the instantaneous velocity at $t = 1$ using calculus?
2. What does the average velocity over an interval tell us about the motion of an object?
3. How does the concept of a derivative relate to instantaneous velocity?
4. What happens to the average velocity as the interval gets infinitely small?
5. How would the velocity change if the position function had a different form?

Tip: Remember, the average velocity over smaller intervals approximates the instantaneous velocity more closely. The derivative is the exact measure of instantaneous velocity at a point.