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### Course: Integrated math 1>Unit 8

Lesson 12: Average rate of change word problems

# Average rate of change word problem: table

Average rate of change tells us how much the function changed per a single time unit, over a specific interval. It has many real-world applications. In this video, we compare the average rate of change of temperature over different time periods.

## Want to join the conversation?

• Why is the rate of change "change in temperature over change in time?" Why isn't it "change in time over change in temperature?"
• Well, you could do that but it wouldn't sound reasonable as time is the main thing. Everything changes over time. Even the time which the clock shows changes over time ( although that is not a good e.g. ). Also, we want to calculate something in terms of something which is fix and steady. Right. Well, I am not going to stop you from trying it. But I would say this that it is going to be completely crazy and it would be pretty hard to make heads or tails out of it as the x-axis is going to be the temperature.

Hope that helped.
• why is this sooo hard
• Average rate of change is just an application of the slope formula. You would have learned about slope when you did linear equations and the slope of lines. Unlike lines where the slope is always the same, curved lines have slopes that change (this is what causes the curve). So, we use the slope formula to find an average rate of change (or an average slope) across a section of the curve. The table gives you points along the curve. The problem tells you what interval to use. Pick the 2 points from the table that match the requested start and end values for the interval. Then use the slope formula: (y2-y1)/(x2-x1) to calculate the average rate of change.

Hope this helps.
• Why is Delta written/drawn as a triangle? What is its history?
• Delta is a Greek letter which is used to represent change or difference. The letter itself looks like a triangle, but it does not signify a triangle in math. When we write Δy or Δx, we are indicating the change in y and the change in x.
• So this comparison is non-linear?
• Yes, it is because we are dealing with the temperature here. It does not steadily increase. We have no sun in the night but then suddenly the sun pops up and the temperature rapidly increases. Therefore the rate of change is not the same over time.

Hope that helped.
• How come 3/2 is equivalent to 1.5?
• draw two circles (like pizzas) and divide each one in halves. Then shade in 3 of those halves. You will find that you shaded in one full circle and .5 (1/2) or the other circle.
• Why does the average rate of change matter ?
• why can't "m" be "s"? wouldn't that make more sense?
• At , Sal writes ΔT/Δt. Should it be ΔT(t)/Δt since T is a function, not the output?
• Because the problem says that, "Let T(t) denote the temperature T " that means T(t) = T
Just like y = f(x)
(1 vote)
• I didnt get why when he wrote change in T over change in t he didnt just compare 19/6, 9/25 and 13/31
(1 vote)
• Just like when we study from 7 am to 10 am and then we want to know how many hours of study we have done so we subtract the initial time from the final time (10-7). So when he wrote change in T over change in t he means T/t = change in the temperature T(t) over change in time in hours
= {T(9)-T(6)}/(9-6)
= (25-19)/(9-6)
= 6/3
= 2
(1 vote)
• why can't "m" be "s"? wouldn't that make more sense?
(1 vote)

## Video transcript

Let T of T, so capital T of lowercase T denote the temperature capital T in Windhoek, Namibia measured in degrees Celsius when it's T lowercase T hours after midnight on a given day. The table below lists a few selected values of the function. Alright, when it's 6 hours after midnight our temperature is 19 degrees Celsius Nine hours after midnight or 9 a.m. 25 degrees Celsius. 13 hours after midnight, which is the same thing as 1 p.m. Our temperature is 31 degrees Celsius. When did the temperature increase faster? So this is between 6 & 9 a.m.. So 6 & 9 a.m.. 9 a.m.. and 1 p.m. So that's between these two points, or did the temperature increase at the same rate over both intervals? Well if we want to figure out the rate of increase- so let's see rate of temperature change temperature change Well this is going to be equal to, our change in temperature capital T, over our change in time. This triangle, that's the Greek letter Delta, represents change in. So change in temperature over change in time. So let's think about what this rate of temperature change is, between these different intervals. So between 6:00 a.m. and 9:00 a.m.. So let me really see Change in temperature over change in time So what was our change in temperature? Our temperature went up, by 6 degrees Celsius. Let me just write it here 6 degrees Celsius. And what is our change in time? Our time went up by 3 hours, plus 3 hours. So you're going to have a change of 6 degrees Celsius, positive change of 6 degrees Celsius over a positive change, we've Gone 3 hours into the future, Over 3 hours we increased our temperature by 6 degrees or you could say it's an average rate of change of 6 divided by 3 is 2 degrees Celsius, per hour. Notice, I'm just keeping the unit's the same degrees Celsius per hour, so that's the rate of change between 6 a.m. And 9:00 a.m. Now let's ask ourselves the same question between 9:00 a.m.. and 1:00 p.m.. So our change in temperature over change in time What is our change in temperature? Well our temperature goes up from 25 to 31 degrees Celsius. That's another 6 degrees Celsius. But it's no longer 3 hours to go from 9 hours after minute to 13 hours after midnight We're now doing it over 4 hours. So Plus 4 hours. So 6 degrees Celsius over 4 hours and We actually don't even have to calculate you see that you've had you've had the same change But you've had to do it over more hours So this is a lower Rate of change the temperature is increasing slower here. It took it four hours to increase 6 degrees Celsius well over here it took it only 3 hours. So immediately you might recognize that this is going to be faster. If you increase 6 degrees Celsius over 3 hours that's faster than increasing 6 degrees Celsius Over 4 hours. But just to make the comparison a little bit clearer Let's actually just do the math here. 6 divided by 4 well that's going to be 1.5 1.5 degrees Celsius per hour. Here you can make the direct comparison. 2 degrees Celsius per hour is faster than 1.5 degrees Celsius per hour. So that's why we liked this choice up here