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>

a. $[0, 2]$

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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(a) $ and $ s(b) $ from the table for the interval [0, 2]. These represent the position of the object at times t = 0 and t = 2, respectively.

Substitute the values of $ s(0) $ and $ s(2) $ into the average velocity formula: $ v_{avg} = \frac{s(2) - s(0)}{2 - 0} $.

Simplify the expression by calculating the difference $ s(2) - s(0) $ to find the change in position.

Divide the change in position by the change in time (which is 2 seconds) to find the average velocity over the interval [0, 2].

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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 fundamental in analyzing motion.

Position Function

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

Time Interval

A time interval is the duration over which motion is observed, typically denoted as [t1, t2]. In this context, it is the period during which the average velocity is calculated. Recognizing the significance of the time interval is important for accurately determining changes in position and understanding the object's motion dynamics.

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>a. [0,2][0, 2][0,2]