Consider the position function 

​s(t)equals=negative 4.9 t squared plus 31 t plus 22−4.9t2+31t+22. 

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.

Question content area bottom

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]

enter your response here

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

Part 2

The value of the instantaneous velocity at 

t equals 4t=4 

is 

enter your response here.

​(Round to the nearest tenth as​ needed.)

The position function is given as \( s(t) = -4.9t^2 + 31t + 22 \).

### Part 1: Calculating the Average Velocities

The average velocity over a time interval \([t_1, t_2]\) is given by:

\[
\text{Average Velocity} = \frac{s(t_2) - s(t_1)}{t_2 - t_1}
\]

We will use this formula to calculate the average velocities for each time interval.

1. **For the interval \([4, 5]\):**

\[
\text{Average Velocity} = \frac{s(5) - s(4)}{5 - 4}
\]
First, calculate \( s(5) \) and \( s(4) \):
\[
s(5) = -4.9(5^2) + 31(5) + 22 = -4.9(25) + 155 + 22 = -122.5 + 177 = 54.5
\]
\[
s(4) = -4.9(4^2) + 31(4) + 22 = -4.9(16) + 124 + 22 = -78.4 + 124 + 22 = 67.6
\]
\[
\text{Average Velocity} = \frac{54.5 - 67.6}{1} = -13.1
\]

2. **For the interval \([4, 4.1]\):**

\[
\text{Average Velocity} = \frac{s(4.1) - s(4)}{4.1 - 4}
\]
First, calculate \( s(4.1) \):
\[
s(4.1) = -4.9(4.1^2) + 31(4.1) + 22 = -4.9(16.81) + 127.1 + 22 = -82.369 + 127.1 + 22 = 66.731
\]
\[
\text{Average Velocity} = \frac{66.731 - 67.6}{0.1} = -8.69
\]

3. **For the interval \([4, 4.01]\):**

\[
\text{Average Velocity} = \frac{s(4.01) - s(4)}{4.01 - 4}
\]
First, calculate \( s(4.01) \):
\[
s(4.01) = -4.9(4.01^2) + 31(4.01) + 22 = -4.9(16.0801) + 124.31 + 22 = -78.79249 + 124.31 + 22 = 67.51751
\]
\[
\text{Average Velocity} = \frac{67.51751 - 67.6}{0.01} = -8.249
\]

4. **For the interval \([4, 4.001]\):**

\[
\text{Average Velocity} = \frac{s(4.001) - s(4)}{4.001 - 4}
\]
First, calculate \( s(4.001) \):
\[
s(4.001) = -4.9(4.001^2) + 31(4.001) + 22 = -4.9(16.008001) + 124.031 + 22 = -78.4392049 + 124.031 + 22 = 67.5917951
\]
\[
\text{Average Velocity} = \frac{67.5917951 - 67.6}{0.001} = -8.2049
\]

5. **For the interval \([4, 4.0001]\):**

\[
\text{Average Velocity} = \frac{s(4.0001) - s(4)}{4.0001 - 4}
\]
First, calculate \( s(4.0001) \):
\[
s(4.0001) = -4.9(4.0001^2) + 31(4.0001) + 22 = -4.9(16.00080001) + 124.0031 + 22 = -78.403920049 + 124.0031 + 22 = 67.599179951
\]
\[
\text{Average Velocity} = \frac{67.599179951 - 67.6}{0.0001} = -8.2005
\]

### Table of Average Velocities:

| Time Interval         | Average Velocity |
|-----------------------|------------------|
| \([4, 5]\)            | \(-13.1\)        |
| \([4, 4.1]\)          | \(-8.69\)        |
| \([4, 4.01]\)         | \(-8.249\)       |
| \([4, 4.001]\)        | \(-8.2049\)      |
| \([4, 4.0001]\)       | \(-8.2005\)      |

### Part 2: Instantaneous Velocity at \( t = 4 \)

To estimate the instantaneous velocity, we observe the trend in the average velocities as the time intervals get smaller. The values approach \(-8.2\), suggesting that the instantaneous velocity at \( t = 4 \) is approximately \(-8.2\) m/s.

The instantaneous velocity at \( t = 4 \) is \(-8.2\) m/s (rounded to the nearest tenth).

---

Would you like more details or have any further questions? Here are 5 related questions to think about:

1. How would the instantaneous velocity change if we altered the coefficients in the position function?
2. What is the physical significance of a negative instantaneous velocity?
3. Can you determine the velocity at other points, such as \( t = 2 \) or \( t = 6 \)?
4. How would you find the acceleration at \( t = 4 \)?
5. How is the concept of instantaneous velocity related to the derivative of the position function?

**Tip:** Instantaneous velocity is the derivative of the position function, and you can find it by taking the first derivative of \( s(t) \) with respect to \( t \).

Consider the position function s(t) = -4.9t^2 + 31t + 22. Complete the following table with the appropriate average velocities and make a conjecture about the value of the instantaneous velocity at t = 4.

Learn how to calculate average and instantaneous velocity from the position function s(t) = -4.9t^2 + 31t + 22. This example uses calculus to compute average velocities over various intervals and estimate the instantaneous velocity at t = 4. Suitable for students in Grades 11-12 studying calculus.

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