The position in feet of a bird flying in a straight line after  
t
  seconds is modeled by the function  
s
(
t
)
=
4
t
3
−
3
 . Find the average velocity of the bird over the following time intervals:  
[
2
,
2.1
]
 ,  
[
2
,
2.01
]
 ,  
[
2
,
2.001
]
 , and  
[
2
,
2.0001
]
 . Use the calculated averages to estimate the instantaneous velocity  
v
inst
  of the bird at  
t
=
2
  seconds.

Round each answer in the table to six decimal places, but enter an integer value for  
v
inst
 .

To find the average velocity of the bird over a given time interval, we use the formula for average velocity over an interval \([t_1, t_2]\):

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

Given the position function \( s(t) = 4t^3 - 3 \), we'll calculate the average velocity for each of the given intervals and then use these results to estimate the instantaneous velocity at \( t = 2 \).

### Step 1: Calculate the position function at relevant points.

1. **Interval \([2, 2.1]\)**:
   - \( t_1 = 2 \), \( t_2 = 2.1 \)
   - Find \( s(2) \) and \( s(2.1) \)

2. **Interval \([2, 2.01]\)**:
   - \( t_1 = 2 \), \( t_2 = 2.01 \)
   - Find \( s(2) \) and \( s(2.01) \)

3. **Interval \([2, 2.001]\)**:
   - \( t_1 = 2 \), \( t_2 = 2.001 \)
   - Find \( s(2) \) and \( s(2.001) \)

4. **Interval \([2, 2.0001]\)**:
   - \( t_1 = 2 \), \( t_2 = 2.0001 \)
   - Find \( s(2) \) and \( s(2.0001) \)

### Step 2: Calculate the average velocities for each interval.

After calculating the positions at the required points, we can determine the average velocities. Then, based on the trend of these averages as the interval gets smaller, we will estimate the instantaneous velocity at \( t = 2 \).

Let me calculate that for you.Here are the calculated positions and average velocities:

1. **Interval \([2, 2.1]\)**:
   - \( s(2) = 29 \)
   - \( s(2.1) = 34.044 \)
   - **Average velocity** = \( 50.440000 \) feet per second

2. **Interval \([2, 2.01]\)**:
   - \( s(2) = 29 \)
   - \( s(2.01) = 29.482404 \)
   - **Average velocity** = \( 48.240400 \) feet per second

3. **Interval \([2, 2.001]\)**:
   - \( s(2) = 29 \)
   - \( s(2.001) = 29.048024004 \)
   - **Average velocity** = \( 48.024004 \) feet per second

4. **Interval \([2, 2.0001]\)**:
   - \( s(2) = 29 \)
   - \( s(2.0001) = 29.004800240004 \)
   - **Average velocity** = \( 48.002400 \) feet per second

### Estimating the Instantaneous Velocity:
As the intervals become smaller, the average velocities approach a value close to 48 feet per second. Thus, we can estimate the instantaneous velocity \( v_{\text{inst}} \) of the bird at \( t = 2 \) seconds to be:

\[
v_{\text{inst}} \approx 48 \text{ feet per second}
\]

Would you like more details on how these values were derived or further clarification on any step?

### Related Questions:
1. What is the significance of instantaneous velocity in physics?
2. How do limits play a role in calculating instantaneous velocity?
3. Can the instantaneous velocity ever be zero for a non-stationary object?
4. How does the derivative of a position function relate to velocity?
5. What is the geometric interpretation of average and instantaneous velocity?

### Tip:
When estimating instantaneous velocity using average velocities over smaller intervals, you're essentially using the concept of limits, a fundamental idea in calculus.

The position in feet of a bird flying in a straight line after t seconds is modeled by the function s(t) = 4t^3 − 3. Find the average velocity of the bird over the following time intervals: [2, 2.1], [2, 2.01], [2, 2.001], and [2, 2.0001]. Use the calculated averages to estimate the instantaneous velocity v_inst of the bird at t = 2 seconds. Round each answer in the table to six decimal places, but enter an integer value for v_inst.

Explore how to calculate the average velocity of a bird flying in a straight line using the position function s(t) = 4t^3 - 3 over different time intervals, and estimate the bird's instantaneous velocity at t = 2 seconds. The solution involves key calculus concepts like derivatives and limits, making it ideal for Grades 11-12.

Math Problem Statement

The position in feet of a bird flying in a straight line after
t seconds is modeled by the function
s ( t )

4 t 3 − 3 . Find the average velocity of the bird over the following time intervals:
[ 2 , 2.1 ] ,
[ 2 , 2.01 ] ,
[ 2 , 2.001 ] , and
[ 2 , 2.0001 ] . Use the calculated averages to estimate the instantaneous velocity
v inst of the bird at
t

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