The distance (in feet) of an object from a point is given by...

The distance (in feet) of an object from a point is given by $s(t)=t^{2},$ where time $t$ is in seconds.
(a) What is the average velocity of the object between $t=2$ and $t=5 ?$
(b) By using smaller and smaller intervals around $2,$ estimate the instantaneous velocity at time $t=2$

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Key Concepts

Average Velocity

Average velocity is a measure of the overall rate of change of an object's position with respect to time over a specific time interval. It is calculated by taking the difference in the object's position at the end and the beginning of the interval, and then dividing by the length of the time interval.

Instantaneous Velocity

Instantaneous velocity represents the speed and direction of an object at a specific moment in time. Conceptually, it is the limit of the average velocity as the time interval considered approaches zero, providing a precise rate of change at that specific point.

Difference Quotient

The difference quotient is an expression used to compute the average rate of change of a function over an interval. It is fundamental to the concept of derivatives and is calculated as the change in the function's value divided by the change in the input, forming the basis for estimating instantaneous rates of change.

Limit

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a specific value. In the context of instantaneous velocity, limits are used to formally define the derivative by considering the behavior of the difference quotient as the interval between time points approaches zero.

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