In Problems 11-14, a position function is provided, wheres...

In Problems $11-14$, a position function is provided, wheres represents miles and t represents hours. Find the average velocity on the four intervals provided, then estimate the instantaneous velocity at the time that begins each interval. In Problems $11-14,$ a position function is provided, wheres represents miles and t represents hours. Find the average velocity on the four intervals provided, then estimate the instantaneous velocity at the time that begins each interval. $s(t)=t^{2}+12 t ; \quad[1,2],[1,1.5],[1,1.1],[1,1.01]$

In Problems $11-14$, a position function is provided, wheres represents miles and t represents hours. Find the average velocity on the four intervals provided, then estimate the instantaneous velocity at the time that begins each interval.
In Problems $11-14,$ a position function is provided, wheres represents miles and t represents hours. Find the average velocity on the four intervals provided, then estimate the instantaneous velocity at the time that begins each interval.
$s(t)=t^{2}+12 t ; \quad[1,2],[1,1.5],[1,1.1],[1,1.01]$

Key Concepts

Difference Quotient

The difference quotient is a crucial mathematical tool used to estimate the instantaneous rate of change of a function. By calculating the average rate of change over a small interval and then taking the limit as the interval shrinks to zero, it formally leads to the derivative, which represents the instantaneous velocity for motion analysis.

Instantaneous Velocity

Instantaneous velocity refers to the velocity of an object at a specific moment in time. It is obtained by taking the limit of the average velocity as the time interval approaches zero, effectively providing the derivative of the position function with respect to time. This concept is crucial in understanding how an object's velocity changes at each moment during its motion.

Derivative

The derivative of a function represents the rate at which the function's value changes with respect to changes in its input, and in the context of motion, it gives the instantaneous velocity. This concept connects the foundational ideas of calculus with practical applications in kinematics, enabling a precise description of how motion occurs.

Position Function

A position function describes the location of an object as it varies over time, typically relating the distance or displacement to the time elapsed. It serves as the foundation for analyzing motion by providing a mathematical model that can be differentiated or integrated to extract further information about the motion of the object.

Average Velocity

Average velocity is defined as the total displacement divided by the total time interval over which the movement occurs. It is calculated by taking the difference in position values at the endpoints of an interval and dividing by the difference in the corresponding times. This concept provides a general idea of the object's speed and direction over that specific time period.

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