Consider the position function ​s(t)equals=negative 4.9 t squared plus 32 t plus 16−4.9t2+32t+16. Complete the following table with the appropriate average velocities and then make a conjecture about the value of the instantaneous velocity at t equals 4t=4.
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Part 1
Complete the following table.
​(Type exact answers. Type integers or​ decimals.)
Time Interval
Average Velocity
left bracket 4 comma font size decreased by 5 5 right bracket[4, 5]
negative 12.1−12.1
left bracket 4 comma font size decreased by 5 4.1 right bracket[4, 4.1]
enter your response here
left bracket 4 comma font size decreased by 5 4.01 right bracket[4, 4.01]
enter your response here
left bracket 4 comma font size decreased by 5 4.001 right bracket[4, 4.001]
enter your response here
left bracket 4 comma font size decreased by 5 4.0001 right bracket[4, 4.0001]
enter your response here


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?

---

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.

Consider the position function s(t) = -4.9t^2 + 32t + 16. Complete the following table with the appropriate average velocities and then make a conjecture about the value of the instantaneous velocity at t = 4.

Time Interval | Average Velocity
[4, 5]        | -12.1
[4, 4.1]      | ?
[4, 4.01]     | ?
[4, 4.001]    | ?
[4, 4.0001]   | ?

Learn how to calculate average velocity using the position function s(t) = -4.9t^2 + 32t + 16 over various time intervals and make a conjecture about the instantaneous velocity at t = 4. This problem covers key calculus concepts and is suitable for students in Grades 11-12.

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