The table gives the position s(t)of an object moving along a line at time t, over a two-second interval. Find the average velocity of the object over the following intervals. <IMAGE>

c. $[0, 1]$

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Identify the formula for average velocity over an interval [a, b], which is given by the change in position divided by the change in time: $ v_{avg} = \frac{s(b) - s(a)}{b - a} $.

Determine the values of $ s(0) $ and $ s(1) $ from the table provided. These represent the position of the object at time $ t = 0 $ and $ t = 1 $ respectively.

Substitute the values of $ s(0) $ and $ s(1) $ into the average velocity formula: $ v_{avg} = \frac{s(1) - s(0)}{1 - 0} $.

Simplify the expression by calculating the difference $ s(1) - s(0) $ to find the change in position.

Divide the change in position by the change in time (which is 1 second in this case) to find the average velocity over the interval [0, 1].

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주요 개념

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Average Velocity

Average velocity is defined as the change in position divided by the time interval over which that change occurs. Mathematically, it is expressed as (s(t2) - s(t1)) / (t2 - t1), where s(t) represents the position function. This concept is crucial for understanding how an object's position changes over time and is particularly useful in analyzing motion over specific intervals.

Position Function

The position function, denoted as s(t), describes the location of an object at any given time t. It provides a mathematical representation of the object's movement along a line. Understanding the position function is essential for calculating average velocity, as it allows us to determine the positions at the start and end of the time interval in question.

Time Interval

A time interval is the duration over which an event occurs, typically represented as [t1, t2]. In the context of average velocity, it is the period during which the object's position is measured. Recognizing the significance of the time interval is vital for accurately calculating average velocity, as it directly influences the values used in the formula.

The table gives the position s(t)of an object moving along a line at time t, over a two-second interval. Find the average velocity of the object over the following intervals. <IMAGE>c. [0,1][0, 1][0,1]