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 [1,1.01],[1,1.001] and $[1,1.0001] .$ Then make a conjecture about the value of the instantaneous velocity at $t=1.$ $$\begin{array}{|l|c|c|c|c|} \hline t & 1 & 1.0001 & 1.001 & 1.01 \\ \hline s(t) & 64 & 64.00479984 & 64.047984 & 64.4784 \\ \hline \end{array}$$

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 [1,1.01],[1,1.001] and $[1,1.0001] .$ Then make a conjecture about the value of the instantaneous velocity at $t=1.$
$$\begin{array}{|l|c|c|c|c|}
\hline t & 1 & 1.0001 & 1.001 & 1.01 \\
\hline s(t) & 64 & 64.00479984 & 64.047984 & 64.4784 \\
\hline
\end{array}$$

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Key Concepts

Difference Quotient

The difference quotient is the expression used to calculate the average rate of change over a finite interval. It is given by the ratio of the change in the function's values to the change in the corresponding input values (i.e., (s(t+h)-s(t))/h). As the parameter h approaches zero, the difference quotient becomes the derivative, capturing the instantaneous rate of change.

Average Velocity

Average velocity is a measure of the change in position over a specified time interval. It is calculated by taking the difference between the final and initial positions and dividing by the difference between the final and initial times. This concept provides a basic estimation of how fast an object is moving over an interval, without accounting for variations in speed within that interval.

Instantaneous Velocity

Instantaneous velocity refers to the speed of an object at a specific moment in time. It is defined mathematically as the limit of the average velocity as the time interval approaches zero. In calculus, this is characterized by the derivative of the position function with respect to time, thereby representing the object’s rate of change of position at a precise instant.

Transcript

00:02 The position of our object with time is given in the table in this question and we have to calculate the average velocity for number of cases.

00:11 Okay, so let's just say the first case.

00:13 In first case, you have to calculate the average velocity for time interval from 1 to 1 .01.

00:19 Okay, so that is the first case.

00:21 So we average in this case will be essentially, you already know, pass of 1 .01 position at 1 .01 minus position at 1 .1 of the object.

00:32 Divided by 1 .01 minus 1.

00:35 If you put these values, so the values will be 64 .484.

00:44 So that is s of 1 .01 that is given in the table minus 64, that is s of 1 divided by 0 .01.

00:51 And if we calculate this value, so the value we are getting is 47 .84.

00:59 Okay, so that is the average velocity for the center one.

01:02 Case second we have is t from 1 to 1 .001 okay so in this case the average velocity formula will be same that will be s of 1 .001 minus s of 1 divided by 1 .001 minus 1 .1.

01:26 Put these values from the table so s of 1 .001 which is given in the table is 64 point 0479, 047984 minus 64 divided by 0 .001.

01:49 So the average velocity in this case, this is coming out is 47 .984.

02:03 Okay? third case, you have to calculate the average velocity for t from 1 to 1 to 1 .0 .0 .0.

02:17 0 .001.

02:19 Okay...

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