Average rate of change word problem: table

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