Instantaneous velocity Consider the position function...

Instantaneous velocity Consider the position function $s(t)=-16 t^{2}+100 t .$ Complete the following table with the appropriate average velocities. Then make a conjecture about the value of the instantaneous velocity at $t=3.$ $$\begin{array}{|l|l|} \hline \text { Time interval } & \text { Average velocity } \\ \hline[2,3] & \\ \hline[2.9,3] & \\ \hline[2.99,3] & \\ \hline[2.999,3] & \\ \hline[2.9999,3] & \\ \hline \end{array}$$

Instantaneous velocity Consider the position function $s(t)=-16 t^{2}+100 t .$ Complete the following table with the appropriate average velocities. Then make a conjecture about the value of the instantaneous velocity at $t=3.$
$$\begin{array}{|l|l|}
\hline \text { Time interval } & \text { Average velocity } \\
\hline[2,3] & \\
\hline[2.9,3] & \\
\hline[2.99,3] & \\
\hline[2.999,3] & \\
\hline[2.9999,3] & \\
\hline
\end{array}$$

Key Concepts

Difference Quotient

The difference quotient is the expression used to compute the average rate of change, typically written as [f(a+h) - f(a)]/h. This formulation is essential in calculus because it allows for the computation of derivatives by analyzing the behavior of functions as the interval h becomes infinitesimally small.

Limit Process

The limit process is a fundamental concept in calculus that involves analyzing the behavior of a function as the input approaches a particular value. In the context of finding instantaneous rates of change, it is used to rigorously define the derivative by determining the value that the difference quotient approaches as the interval shrinks to zero.

Average Rate of Change

This concept involves measuring how a quantity changes, on average, over a defined interval. In calculus, it is calculated by taking the difference between the function values at two points and dividing by the difference of the input values. It forms the initial step in understanding how quantities change over time and sets the stage for defining instantaneous change.

Instantaneous Rate of Change

Also referred to as instantaneous velocity or the derivative at a point, this concept represents the rate at which a quantity is changing at an exact moment. It is defined as the limit of the average rate of change (difference quotient) as the interval approaches zero, providing a precise measure of change at a specific point in time.

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