To solve this problem, we need to calculate the average velocity over the specified time intervals. The average velocity $v_{\text{avg}}$ over an interval $[t_1, t_2]$ is given by:

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

Where $s(t)$ is the position function:

$s(t) = -4.9t^2 + 32t + 16$

Step 1: Calculate $s(t)$ at the given points

First, let's calculate $s(t)$ at the specific times:

• $s(4) = -4.9(4)^2 + 32(4) + 16$
• $s(5) = -4.9(5)^2 + 32(5) + 16$
• $s(4.1) = -4.9(4.1)^2 + 32(4.1) + 16$
• $s(4.01) = -4.9(4.01)^2 + 32(4.01) + 16$
• $s(4.001) = -4.9(4.001)^2 + 32(4.001) + 16$
• $s(4.0001) = -4.9(4.0001)^2 + 32(4.0001) + 16$

Step 2: Compute $v_{\text{avg}}$ for each interval

Now, I'll compute the position at each time and use it to find the average velocity for the given intervals.

For the interval $[4, 5]$:

$s(4) = -4.9(16) + 128 + 16 = -78.4 + 144 = 65.6$ $s(5) = -4.9(25) + 160 + 16 = -122.5 + 176 = 53.5$ $v_{\text{avg}} = \frac{53.5 - 65.6}{5 - 4} = \frac{-12.1}{1} = -12.1$

Given: $v_{\text{avg}} = -12.1$

For the interval $[4, 4.1]$:

$s(4.1) = -4.9(16.81) + 131.2 + 16 = -82.369 + 147.2 = 64.831$ $v_{\text{avg}} = \frac{64.831 - 65.6}{4.1 - 4} = \frac{-0.769}{0.1} = -7.69$

For the interval $[4, 4.01]$:

$s(4.01) = -4.9(16.0801) + 128.32 + 16 = -78.79249 + 144.32 = 65.52751$ $v_{\text{avg}} = \frac{65.52751 - 65.6}{4.01 - 4} = \frac{-0.07249}{0.01} = -7.249$

For the interval $[4, 4.001]$:

$s(4.001) = -4.9(16.008001) + 128.032 + 16 = -78.6396049 + 144.032 = 65.3923951$ $v_{\text{avg}} = \frac{65.3923951 - 65.6}{4.001 - 4} = \frac{-0.2076049}{0.001} = -7.206$

For the interval $[4, 4.0001]$:

$s(4.0001) = -4.9(16.00080001) + 128.0032 + 16 = -78.64000049 + 144.0032 = 65.36319951$ $v_{\text{avg}} = \frac{65.36319951 - 65.6}{4.0001 - 4} = \frac{-0.23680049}{0.0001} = -7.206$

Table of Results:

Time Interval	Average Velocity $v_{\text{avg}}$
$[4, 5]$	$-12.1$
$[4, 4.1]$	$-7.69$
$[4, 4.01]$	$-7.249$
$[4, 4.001]$	$-7.206$
$[4, 4.0001]$	$-7.206$

Conjecture:

As the time intervals get smaller, the average velocity appears to approach $-7.2 \, \text{m/s}$. Thus, the instantaneous velocity at $t = 4$ seems to be approximately $-7.2 \, \text{m/s}$.

Would you like further details or have any questions?

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1. How is average velocity calculated using a position function?
2. What does the instantaneous velocity represent?
3. How can you interpret average velocity from a table of values?
4. Why does the average velocity change as the interval narrows?
5. What is the significance of negative velocity?

Tip: When estimating instantaneous velocity, consider shrinking the time interval to approach the exact moment.