Algebra (all content)
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 find the average rate of descent of a skydiver over a specific time interval.
Teresa went skydiving. The graph below discribes Teresa's height, measured in meters, as a function of time, measured in seconds. So let's look at this graph over here. It's actually quite large, so let me zoom out a little bit. And we can see that at time zero her height is seven hundred meters and then as time increases, as we move to the right her height is decreasing. And her height is decreasing at faster and faster rates as we move to the right so her- her- the rate of decline of her height is quite steep as we approach ten seconds after she jumps And then we can see all of a sudden, then we can see all of a sudden her rate of decline slows down She's still declining as -as we move forward in time, but she's declining at a slower rate. And so we can say she's declining at a --- or she is --- her height is changing at a less negative rate It's -It's quite a negative rate right over here --- seems roughly a fairly negative rate but then it becomes a less negative rate right over here her height is changing at a less negative rate and it makes sense that this is when she deploys the parachute, so after 10 seconds she deploys the parachute so she jumps at 0 seconds, 10 seconds she deploys the parachute. Alright so let's see what they're asking --- actually asking us. They say Complete the following sentence Between 3 seconds and 8 seconds after Teresa jumped, her height decreased, on average, by approximately blank meters per second. So between 3 seconds, and 8 seconds, so at 3 seconds --- so time is 3 right over there So, Let's see what H(3) is. what is her height at 3 seconds and I'm just looking at a graph so I'm going to have to Ballpark it so at 3 seconds at 3 seconds her height looks pretty close it's pretty close we just have to approximate it so her height at the height after 3 seconds it looks like it's about halfway between 600 and 700 so it looks like it's about 650 meters and then we care between 3 seconds and 8 seconds so our height at 8 seconds let's look at that at 8 seconds let's see this looks about halfway it looks about halfway between 350 between 350 and 400 so I'll say her height at 8 seconds actually since I'm approximating it let me put a little squiggly equal sign here her height after 8 seconds looks like it's approximately 300 and looks approximately 375 meters 375 meters so what is her average rate of change her height decreased on average by approximately so we what we want to do is we forgot to figure out the average rate of change which you could view as the slope of the line that connects these two points the slope of this line is going to be her average rate of change so let's think about that her average rate of change her change in height over the change in time for that interval well her change in height after at 8 seconds she is at let me write it this way her height at 8 seconds - her height at 3 seconds so this is going to be her change in height and the change in time is 8 seconds she finishes at 8 seconds - where she started or the interval that we care about starting at 3 seconds and so H of 8 we already said this is approximately 375 H of 3 this is 650 and then of course 8 minus 3 is going to be equal to 5 I just want emphasize this is just her average rate of change for approximate average rate of change over this interval as we as we go from 3 to 8 seconds our height goes from H of 3 to H of 8 so this is going to be let's see 375 - 650 let's see if it was 375 - 675 it would be negative 300 and so this is going to be but then we're not subtracting 675 we're subtracting 650 so it's going to be 25 it's going to be 25 more so this is going to be negative 275 over 5 let me make sure I did that math right let me make sure I did that math right so it's 375 - 650 is negative 275 does that make sense let's see 275 plus 375 would be 2 650 yeah that is right all right so let's just figure out what this is so 5 I'll just figure out what 5 goes into 275 and then we can remember the negative 5 goes into 27 5 times 5 times 5 is 25 subtract we get a to bring down to 5 5 goes into 25 five times and then we're not going to have a remainder so this is going to be equal to negative 55 negative 55 and the unit's our height is given in meters so this part up here this is in meters up here meters per second so her change in height or average change in height over or the average rate of change of height over this five seconds over this five seconds is negative 55 meters per second one way to think about it the slope right over here the slope is equal to negative 55 now it might be tempting to just write negative 55 right over here but let's just think about whether that would be right between 3 seconds and 8 seconds after Teresa jumped her height decreased on average by approximately negative 55 meters per second and decrease is important because they're already saying that it's decreased when this negative is telling us that we're decreasing we're decreasing at a rate of 55 meters per second our average rate of change is a decrease our average rate of high a change of height over time is a decrease of 55 meters per second well the negative is already saying the decrease so we're not decreasing at a negative rate of change we would be decreasing at 55 meters per second so let me just write it 55 meters per second if they asked if they asked her height if we asked her height if we if they asked the average change of height average rate of change let me write this the average rate of change of change of H over or let me say of H from 3 seconds 3 seconds to 8 seconds well now this would be negative 55 meters per second but when they're saying that her height decrease that's re taking the negative into consideration they're saying it's definitely decreasing that's what the negatives already telling us it's decreasing by a rate of 55 meters per second hopefully that makes some sense