For the following position functions, make a table of average ...

Question

For the following position functions, make a table of average velocities similar to those in Exercises 19–20 and make a conjecture about the instantaneous velocity at the indicated time.

a. s(t)=−16t^2+80t+60 at t=3

Accepted Answer

Hi, everyone. Let's take a look at this practice problem dealing with velocities. This problem says, consider the position function S of T is equal to nine T squared minus 45 T plus 100. Approximately the instantaneous velocity at T equal to two by finding the average velocities at T equal to 2.52 0.12 0.01 and 2.00. 1 also provide the table for average velocities and time intervals. Now, the first thing we want to do to solve this equation is to create our table. So we're gonna have two rows at this table. The first row is gonna be our time intervals will label that row as intervals. And then the second row is going to be our average velocities. We'll label those as velocity. Now, we were actually given the time intervals in the problem. So since we're looking for the instantaneous velocity at two, our time intervals are all gonna start at T equal to two and then it's gonna go to a different final time. So for the first interval, we're going to be going from 2 to 2.5 the next interval it's gonna be going from 2 to 2.1. The next interval is gonna be going from two to 2.01 and the final time interval is gonna be going from two, 22 point 001. And so we need to calculate the average velocities for all or time intervals. And then we'll use our table here to estimate the instantaneous velocity at T equal to two. So we're gonna start off by calculating the average velocity for a first time interval, which is gonna be going from 2 to 2.5. And so here we're going to use our definition of average velocity, which is our change in position divided by our change in time over the time interval. So we'll label our average velocity as the average and this is gonna be equal to our change in position, which is gonna be our position at T equal to 2.5 which is S of 2.5 minus our position at time T equal to two, which is gonna be S of two. And then that quantity is going to be divided by our change in time, which is 2.5 minus two. So what we need to do is find our position at T equal to two and T equal to 2.5. And we're going to use the equation that we are given in the problem for our position to calculate these two values. So we're gonna start off with T equal to two. So that's gonna be S of two and that could be equal to, based upon the equation that we were given in the problem nine multiplied by two squared minus 45 multiplied by two plus 100. And when we evaluate this expression, we will get 46 we're going to use the same expression except set T equal to 2.5. So we're gonna have S of 2.5 and that is going to be equal to nine, multiplied by 2.5 squared minus 45 multiplied by 2.5 plus 100. And this is gonna be equal to 43.75. So we'll take these two values and substitute them back into our formula for the average velocity. So we'll have the average is equal to or s of 2.5. That's gonna be the 43.75. Then we'll have minus for s of two, we'll have 46 and that quantity is going to be divided by the denominator. We have 2.5 minus two, which is equal to 0.5. So now we need to just evaluate this expression. So we'll have our V average, it is equal to minus 4.5. And so this will be our average velocity for this time intervals. So we'll add that to our table in the first empty column. We'll write minus 4.5 in that space. So we're going to repeat this calculation for the other three time intervals. So the next time interval then we look at is two, going to 2.1. And so again, we'll use our same expression for the average velocity, we'll have the average, it's going to be equal to. And in this case, we'll have s of 2.1 minus s of two. That quantity is gonna be divided by the quantity of 2.1 minus two. Now, we've already calculated the value for S of two. And so that's not going to change. So the only value that we need to calculate is S of 2.1. We have S of 2.1 and that's gonna be equal to nine, multiplied by 2.1 squared minus 45 multiplied by 2.1 plus 100. I wanna evaluate this expression. I will get this equal to 45 0.19. So I'll substitute this value into our formula for the average velocity. So we'll get the average is equal to you d 45.19 minus 46. And that quantity is divided by 2.1 minus two, which is 0.1. So we can evaluate this expression. So we'll get the average is equal to minus 8.1. So that is going to be our average velocity for this time interval. And we'll add that into our table. So the next empty column underneath the interval of 2 to 2.1 we're going to have minus 8.1. So now we'll look at our next time interval which is going from two, two, 2.01. And so now we just need to recalculate our average velocity using this new upper time interval limit. So we'll have, the average is equal to s of 2.01 minus s of two. The quantity is gonna be divided by the quantity of 2.01 minus two. So again, we just need to recalculate our position with T equal to 2.01. We'll have s of 2.01. That's gonna be equal to nine, multiplied by 2.01 squared minus 45 multiplied by 2.01 plus 100. And evaluating that expression, we will get 45 0.9109. So we'll substitute this value into our for, for the average velocity. So we'll have the average equal to 45.9109 minus 46. That quantity is divided by 2.01 minus two, which is 0.01. Evaluating this expression, we will have the average is equal to minus 8.91. So that is gonna be our average velocity for this time interval. And so now we will come back up and add that into our table. We'll have minus 8.91 in the next empty column. And so now we need to just calculate our average velocity over our final time interval. And so that time interval is going to be going from 222.001. Again, we'll calculate our average velocity the same way. So we'll have the average is equal to s of 2.001 minus s of two. That quantity is going to be divided by the quantity of 2.001 minus two. So we'll now need to calculate S of 2.001. That's gonna be equal to nine, multiplied by 2.001 squared minus 45 multiplied by 2.001 plus 100. And when I evaluate this um expression, we will get 45 0.991009. So we'll substitute this value into our formula for the average velocity. And so we'll get the average is equal to 45.991009 minus 46. That quantity is divided by 2.001 minus two is 0.001. And so we can evaluate this expression and we will get the average is equal to minus 8.991 just be a average velocity for this time interval. And so we can come back up to our table and add that value into the last column. So we'll have minus 8.991 as the value in the last column. So now at this point, we have our table filled up with the time intervals and the average velocities. So we can use this table to approximate the instantaneous velocity at T equal to two. Do we call our instantaneous velocity? V instantaneous? And recall that your instantaneous velocity occurs when the time interval between our position measurements approaches zero. So if I look at the table for each of the time intervals, the the interval over which the, the time interval over which the positions were measured as actually getting smaller, smaller. So our time interval is getting smaller and smaller. And if I note that the average velocity is actually approaching a value of minus nine as the interval gets smaller and smaller. So it starts out at minus 4.5 eventually gets to minus 8.991. And so we can approximate our instantaneous velocity by setting it equal to minus nine and that's gonna occur at T equal to two. So that is gonna be the value of our instantaneous velocity. So now we've calculated the um average velocities over all the intervals as well as approximated are instantaneous velocity, which is what the question was asking for. So just a quick little recap of what we've done here. So we started off by creating a table that contained our time intervals as well as um our velocities. And we just left with the velocity spaces blank. And what we did was we calculated the average velocities for each one of those time intervals. And we repeated this four times to fill in the table. We then use the table to determine the instantaneous velocity at T equal to two. And we noticed that as our time interval got smaller, that the average velocity approached a value of minus nine, which we assume will be our instantaneous velocity in this case at T equal to two. So I hope that this um explanation has been useful and I'll see you in the next video.

For the following position functions, make a table of average velocities similar to those in Exercises 19–20 and make a conjecture about the instantaneous velocity at the indicated time.

a. s(t)=−16t^2+80t+60 at t=3

Key Concepts

Average Velocity

Instantaneous Velocity

Position Function

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