[T] The distance in feet that a ball rolls down an incline...

[T] The distance in feet that a ball rolls down an incline is modeled by the function $s(t)=14 t^{2}, \quad$ where $t$ is seconds after the ball begins rolling. a. Find the average velocity of the ball over the following time intervals: $$ \begin{array}{ll}{\text { i. }} & {[5,5.1]} \\ {\text { ii. }} & {[5,5.01]} \\ {\text { iii. }} & {[5,5.001]} \\ {\text { iv. }} & {[5,5.0001]}\end{array} $$ b. Use the answers from a. to draw a conclusion about the instantaneous velocity of the ball at $t=5$ seconds.

[T] The distance in feet that a ball rolls down an incline is modeled by the function $s(t)=14 t^{2}, \quad$ where $t$ is seconds after the ball begins rolling.
a. Find the average velocity of the ball over the following time intervals:
$$
\begin{array}{ll}{\text { i. }} & {[5,5.1]} \\ {\text { ii. }} & {[5,5.01]} \\ {\text { iii. }} & {[5,5.001]} \\ {\text { iv. }} & {[5,5.0001]}\end{array}
$$
b. Use the answers from a. to draw a conclusion about the instantaneous velocity of the ball at $t=5$
seconds.

Key Concepts

Average Velocity

Average velocity is defined as the total displacement divided by the total time taken over a specific time interval. It is calculated by finding the difference in the position function values at the endpoints of the interval and then dividing by the difference in times. This concept is fundamental in understanding how quickly an object covers a distance over a finite time period.

Instantaneous Velocity

Instantaneous velocity is the speed of an object at a specific moment in time. It is obtained as the limit of the average velocity as the time interval approaches zero. This concept represents the slope of the tangent line to the position-time curve at a given point and is a key idea in the study of motion and calculus.

Derivative

The derivative of a function at a particular point gives the rate at which the function is changing at that point. It is computed as the limit of the difference quotient (average rate of change) as the interval of measurement shrinks to zero. This concept links the idea of instantaneous rates with the calculus methodology, particularly in motion problems.

Limit Process

The limit process involves evaluating the behavior of a function as the input approaches a particular value. In the context of motion and velocity, it is used to transition from average to instantaneous velocity by considering what happens when the time interval becomes infinitesimally small. This concept underpins the fundamental definition of the derivative in calculus.

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