Consider the position function s(t)=−16t^2+128t (Exercise 13)....

Question

Consider the position function s(t)=−16t^2+128t (Exercise 13). Complete the following table with the appropriate average velocities. Then make a conjecture about the value of the instantaneous velocity at t=1.

Accepted Answer

Hi, everyone. Let's take a look at this park's problem dealing with velocities. This problem says, consider the position function S of T is equal to minus 12 T squared plus 60 T approximate the instantaneous velocity at T equal to one by completing the given table with the average velocities below the question, given a table that contains two rows. The first row has the label of time interval and the remain columns contain time intervals which are 1 to 21 to 1.51 to 1.11 to 1.01 and 1 to 1.001. The second row has a label of average velocities and the rest of the columns are left blank. We're also looking for possible choices as our answers. For choice A we have the average velocity of 2430 47.8 47.88 and 47.988. And the instates velocity of 48. For choice B, we have the average velocities of 2430 71.8 71.88 71.988. And the instantaneous velocity of 72. For choice C, we have average velocities of 2430 34.8 35.88 35.988 and the instantaneous velocity of 24. And for choice D, we have the average velocities of 2430 34.8 35.88 and 35.988. And it entertains velocity of 36. Now, the first thing we need to do is determine the average velocities for each of the given time intervals. So we're gonna start off with the first time interval which is going from 1 to 2. And so here we need to calculate the average velocity on this time interval. And so recall your definition for average velocity that the average velocity which will label as the average is equal to our change in position divided by our change in time. So our change of position in this case is going to be s of two minus s of one. That quantity is gonna be divided by our change in time, which is two minus one where s here is our function or our position at a given time. So we need to evaluate that function when T is equal to two and when T is equal to one. So we're gonna start off with setting T equal to one. So we'll have s of one is equal to and here we'll just um substitute in the value into the expression that we're given the problem with T equal to one. So that's gonna be minus 12 multiplied by one squared plus 60 multiplied by one. And when I evaluate this expression, this is gonna be equal to 48 we're gonna do something similar for, um, s of two will have S of two. So here we'll just be setting T equal to. So S of two is gonna be equal to minus 12, multiplied by two squared plus 60 multiplied by two. And this is going to be equal to 72. So we can now substitute those values back into our formula for the average velocity. So we'll have the average is equal to or S of two, we have 72 minus S of one, which is 48 that quantity is divided by two minus one, which is just one. So evaluating this expression, this is going to be equal to 24. And so that is our average velocity for the first time interval. And so we'll come back up to our table and add that value into the table in that first column. So now we need to look at our next time interval, which is gonna be our time interval going from 1 to 1.5. And again, we're just going to repeat the same calculation or our average velocity. So we have the average equal to s of 1.5 minus s of one. And that quantity is gonna be divided by the quantity of 1.5 minus one here. Since we already have s of one calculated, we just need to evaluate our final for s of 1.5 we have S of 1.5 is equal to minus 12 multiplied by 1.5 squared plus 60 multiplied by 1.5. And evaluating that expression, we will get 63. So substituting that value back in or average velocity formula, we have the average, it's gonna be equal to rest of 1.5. That's gonna be 63 minus s of one which is 48. That quantity is gonna be divided by the um 1.5 minus one, which is gonna be 0.5. So valuating this expression, this is going to be equal to 30. So that is our average velocity for the second time interval. So we'll go back up and add that to our table in the next column. So now we just need to look at our next time interval, which is going from one 21.1. And so again, we're going to calculate the average velocity using our average velocity formula. So we'll have the average, this could be equal to s of 1.1 minus s of one. And that quantity could be divided by the quantity of 1.1 minus one. So again, we need to calculate our position at t equal to 1.1. So we'll have s of 1.1 is equal to minus 12, multiplied by 1.1 squared plus 60 multiplied by 1.1. That's gonna be equal to 51.48. So we'll take the value and substitute back into our average velocity formula. So we'll have the average equal to or s of 1.1. We'll have the 51.48 minus 48. That quantity is gonna be divided by 0.1. So 1.1 minus one is 0.1. So evaluating this expression, we're going to get this equal to 34.8. So that's gonna be our answer for this time interval. And so again, we'll come back up and write that into our table. The 34 0.8 goes in the next column. So now we need to look at the next time interval, which is our interval going from one 21.01. Again, we're going to calculate our average velocity will have the average that's gonna be equal to s of 1.01 minus s of one. That quantity is gonna be divided by the quantity of 1.01 minus one. So we need to calculate our position at t equal to 1.01. So we'll have s of 1.01 is equal to minus 12 multiplied by 1.01 squared plus 60 multiplied by 1.0. 1. And then that is going to be equal to 48 0.3588. So substituting in that value and for our average velocity formula will have the average is equal to 48.3588 minus 48 divided by, you will have 1.01 minus one which is 0.01. So evaluating this expression, this is going to be equal to 35.88. So that is our average velocity for this time interval. So we'll go ahead and add that to our table in the next column. So we'll write in 35 0.88. So now we need to just look at our final time interval. And so that time interval, it's going to be going from one 21.001. Again, we'll calculate the average velocity just as we've done before will have, the average is equal to s of 1.001 minus s of one. That quantity is gonna be divided by the quantity of 1.001 minus one. Again, we need to calculate our position at T equal to 1.00. 1 will have s of 1.001. That's gonna be equal to minus 12, multiplied by 1.001 squared plus 60 multiplied by 1.001. I wanna evaluate that expression. It's gonna be equal to 48 0.0. 35988. So we take that value and substitute it back into our average velocity formula. We'll have, the average is equal to 48.035988 minus 48. That quantity is divided by 1.001 minus one which is 0.001. Evaluating this expression, we will get 35.988. So that is going to be our average velocity for this time interval. And so we'll come back up and add that value to the table. So we'll have 35 wait 988 in the last column. So now at this point, I have the table filled out. And so I need to evaluate what the instantaneous velocity will be at T equal to one. And so recall that the T uh instantaneous velocity occurs when the time interval over which the positions are measured actually approaches zero. So as the time inter interval gets smaller and smaller, we should be approaching our instantaneous velocity. So if I look at the table here as the time interval gets smaller and smaller, the average velocity approaches the value of 36 so we can approximate our instantaneous velocity at one second or at T equal to one with V is equal to 36. That's at E equal to one. So that is our value for instantaneous velocity. And if we look at our answer choices, the correct answer choice corresponds to answer choice D. So just a quick recap of what we've done here for the average velocity, we calculated the average velocity over each time interval. And we had to do this five times once we had all the average velocities, we looked at how they were trending as the time interval got smaller and smaller. And so we could approximate that our instantaneous velocity in this case was going to be equal to 36 at t equal to one. So I hope that this explanation has been useful and I'll see you in the next video.

Consider the position function s(t)=−16t^2+128t (Exercise 13). Complete the following table with the appropriate average velocities. Then make a conjecture about the value of the instantaneous velocity at t=1. <IMAGE>

Key Concepts

Position Function

Average Velocity

Instantaneous Velocity

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