Consider the position function s(t)=−16t^2+100t. Complete the following table with the appropriate average velocities. Then make a conjecture about the value of the instantaneous velocity at t=3. <IMAGE>

Verified step by step guidance

1. Step 1: Understand the problem.
   We are given a position function \( s(t) = -16t^2 + 100t \) and need to find average velocities over certain intervals and make a conjecture about the instantaneous velocity at \( t = 3 \).
2. Step 2: Calculate the average velocity over an interval \([a, b]\).
   The average velocity \( v_{avg} \) over an interval \([a, b]\) is given by the formula:
   \[ v_{avg} = \frac{s(b) - s(a)}{b - a} \]
   Apply this formula to the intervals given in the table.
3. Step 3: Evaluate the position function at the endpoints of each interval.
   For example, if the interval is \([3, 3.1]\), calculate \( s(3) \) and \( s(3.1) \).
   \[ s(3) = -16(3)^2 + 100(3) \]
   \[ s(3.1) = -16(3.1)^2 + 100(3.1) \]
4. Step 4: Substitute the values into the average velocity formula.
   Using the example interval \([3, 3.1]\):
   \[ v_{avg} = \frac{s(3.1) - s(3)}{3.1 - 3} \]
   Repeat this process for each interval in the table.
5. Step 5: Make a conjecture about the instantaneous velocity at \( t = 3 \).
   As the intervals get smaller and approach \( t = 3 \), observe the trend in the average velocities. This trend will help you conjecture the instantaneous velocity at \( t = 3 \), which is the derivative \( s'(t) \) evaluated at \( t = 3 \).

Key Concepts

Position Function

A position function describes the location of an object at a given time. In this case, s(t) = -16t² + 100t represents the height of an object in free fall, where 't' is time in seconds. Understanding how to interpret this function is crucial for analyzing motion and calculating velocities.

Average Velocity

Average velocity is defined as the change in position over the change in time, calculated as (s(t2) - s(t1)) / (t2 - t1). It provides a measure of how fast an object is moving over a specific interval. Completing the table with average velocities helps in understanding the object's motion between different time points.

Instantaneous Velocity

Instantaneous velocity is the velocity of an object at a specific moment in time, found by taking the derivative of the position function. It represents the object's speed and direction at that exact time. Making a conjecture about the instantaneous velocity at t=3 involves evaluating the derivative of the position function at that point.