Instantaneous velocity The following table gives the...

Instantaneous velocity The following table gives the position $s(t)$ of an object moving along a line at time $t .$ Determine the average velocities over the time intervals $[2,2.01] .[2,2.001]$ and $[2,2.0001] .$ Then make a conjecture about the value of the instantaneous velocity at $t=2.$ $$\begin{array}{|l|c|c|c|c|} \hline t & 2 & 2.0001 & 2.001 & 2.01 \\ \hline s(t) & 56 & 55.99959984 & 55.995984 & 55.9584 \\ \hline \end{array}$$

Instantaneous velocity The following table gives the position $s(t)$ of an object moving along a line at time $t .$ Determine the average velocities over the time intervals $[2,2.01] .[2,2.001]$ and $[2,2.0001] .$ Then make a conjecture about the value of the instantaneous velocity at $t=2.$
$$\begin{array}{|l|c|c|c|c|}
\hline t & 2 & 2.0001 & 2.001 & 2.01 \\
\hline s(t) & 56 & 55.99959984 & 55.995984 & 55.9584 \\
\hline
\end{array}$$

Key Concepts

Difference Quotient

The difference quotient is the expression used to compute the average rate of change of a function over an interval and is given by (s(t + ?t) - s(t)) / ?t. As the time interval ?t becomes smaller, the difference quotient approaches the derivative of the function at time t. This is a fundamental concept in understanding how derivatives represent instantaneous rates of change.

Limit

A limit is a fundamental concept in calculus that describes the value that a function (or sequence) approaches as the input (or index) approaches some value. When calculating instantaneous velocity, one takes the limit of the average velocity as the time interval shrinks to zero, effectively 'zooming in' to a single point in time.

Average Velocity

Average velocity is defined as the total displacement divided by the total time taken over a given time interval. It is calculated by taking the difference in the position function values at the end points of the interval and dividing by the length of the interval. This concept provides a measure of the overall change in position per unit time over that interval, regardless of how the position might vary within it.

Instantaneous Velocity

Instantaneous velocity refers to the velocity of an object at a specific moment in time. It is conceptually the limit of the average velocities calculated over increasingly smaller time intervals that approach the specific moment. In calculus, it is defined as the derivative of the position function with respect to time.

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