Calculating and interpreting rates of change for exponential functions

0:00- [Instructor] Okay, so we have a multi-part question here. Let's try to do at least the first part. A concert venue tracks the number of tickets remaining for a limited release sale. The function r gives the number of tickets remaining after t hours since sales opened.

0:15The table shows some values for r, all right? When it opens, there's 10,000 tickets. And then as time goes by, we have fewer and fewer tickets that are remaining to be sold. Calculate the average rate of change over the intervals from 0 to 2 and between 6 and 10.

0:31And then we'll do this last part. So why don't you pause the video and try to figure out these average rates of change. Alright, so between 0 and 2, that's between these two data points right over here. And if we think about it per hour, so we can just think about the change in r over our change in time.

0:54What is our change in r, our change in our tickets? Or you can even say change in r of t if we want. It's -3000.

1:02- 3000 over what time period? Over two hours. We can see here our change in time is two hours, so this is going to be -1500. Or another way to think about it is 1500 tickets per hour, but we're going down, 'cause it's the average rate of change over the, we're just saying the average rate of change of r, the remaining tickets.

1:23So they're going down by 1500 tickets per hour. Now let's do the same thing between 6 and 10. So that's these, that's over this period, right over here. So it's the same notion, change in r of t, over change in time. What's change in r of t? Well, r of t goes from 3440 to 1680.

1:45So let's see, let me see if I can, that went down. If we, let's see, let me just write it down. 3440. Actually I should do the ending point first. So it's really 1680 minus 3440. And the time, our ending point is time, eight hours minus 10 hours, sorry, I should say 10 hours, minus six hours.

2:10So let's see, what is this going to be? It's gonna be the negative of what 3440 minus 1680 is. So let's see, this is going to be 1440, 1440 plus another 320. Let me just do this, 'cause I don't wanna make too many errors in my head. So if I just did 3440 minus 1680, so 0 minus 0 is 0, and I'm gonna have to regroup up here.

2:37So that is a 2, this becomes a 14, and I have to regroup again. That becomes a 13. This becomes a 14. 14 minus 8 is 6. 13 minus 6 is 7. 2 minus 1, so 1,760, but it's the negative of that, since we're going the other way around. So it's negative 1,760 over 10 minus 6 is 4.

3:05So if I do 4 goes into, let me just look into 1760. 4 goes into 17 4 times. 4 times 4 is 16. Subtract, I get a 1, 160, that's going to be 40 more times. And so this is going to be, and this is a negative, so -440 tickets per hour. So now I'll see if we can use that to answer the rest of this question.

3:31So let me scroll down. And so then they are asking us, does the average rate of change over hour 6 to 10 accurately describe how the number of tickets remaining changes for all 10 hours? Explain your reasoning. Well, before I even look at the choices, it's clear that this isn't indicative for all 10 hours.

3:55Just in that last 6 to 10 hours, our rate of sales has slowed down. We're reducing the remaining tickets by 440 tickets per hour, while at the beginning, it was selling much faster. So it's not indicative. So let's see, the tickets sell quickly at first, and then more slowly as fewer are left to buy.

4:12That looks right. This last one is: "No. The tickets sell more slowly at first and then much faster later in sale." No, no, that's not true. We were reducing the remaining tickets very quickly in the beginning, and then it slowed down later on. So C, we like.