Average velocity The position of an object moving along a...

Average velocity The position of an object moving along a line is given by the function $s(t)=-16 t^{2}+128 t .$ Find the average velocity of the object over the following intervals.
c. [1,2]
a. [1,4]
b. [1,3]
d. $[1,1+h],$ where $h>0$ is a real number

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Key Concepts

Position Function

The position function defines the location of an object as a function of time. It is central to problems involving motion, as it allows us to calculate velocities and accelerations by using derivatives or difference quotients.

Interval Analysis

Interval analysis involves evaluating the behavior of a function over a specific range of its domain. By focusing on particular intervals, such as [1,4] or [1,1+h], one can determine how the function's average rate of change, which in the context of motion corresponds to average velocity, varies over different segments of time.

Average Velocity

The average velocity is computed by taking the difference in the object's displacements at the endpoints of a time interval and dividing by the length of that interval. It represents the slope of the secant line connecting two points on the position-time graph, giving a measure of the overall change in position per unit time over that interval.

Difference Quotient

The difference quotient is the algebraic expression (s(b) - s(a))/(b - a) used to calculate the average rate of change of a function between two points. In the context of velocity, this quotient measures how the position function changes with respect to time over a specified interval.

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