00:01
Hi there, so for this problem, we are given this information about the rate at which the radius of a balloon is changing at any given time.
00:17
Okay, so we need to use this information to find an overestimate and an underestimate for the amount of the radius that has increased over the 15 minutes.
00:28
With that set, first of all, i'm going to try to draw the behavior of this rate of change.
00:42
It should be something like this.
00:46
Okay, we know that at zero it starts a value that is equal to 5 .7.
00:52
Let's put it right here.
00:54
So this is the rate of change versus the time.
00:58
Then it starts decreasing at 4 to 1 .2, 0 .0.
01:10
And so as you can see, the function should have a behavior that is something like this.
01:21
Well, let me put it in blue.
01:26
Okay, something like this.
01:28
And then we know that an over an underestimate, is when we estimate the area of this function to be less than the xzat 1.
01:49
Then we know that in this case if we start from 0, we will obtain an overestimate.
01:57
So what we need to do in this case is to start at the point at the first point in here.
02:09
So we will have to calculate the areas of all of this.
02:15
So that is an underestimate, okay? so when we do that, we start at the first point, but not at zero.
02:31
Then the, let me call this the r under is just the area for each one of this, and then we add this together.
02:42
Now, we know that the intervals of time is the same for everyone.
02:46
So as you can see the difference between the point and the previous one is always three.
02:55
So we will have that then three times.
02:59
We don't include the point at zero, but we will include the other points...