00:01
Given the graph of the position function for this problem, we're going to look to see where the slopes are positive, negative, or zero for the position function.
00:10
And that's going to help us determine part a here.
00:14
So where the slopes positive, it looks like from zero up to time is 1 .5, and then it slopes back down negative.
00:25
And then again, from 6 to 7.
00:29
So that's where, therefore, the velocity is positive.
00:34
The particle is moving to the right or up.
00:38
Let's do the next one.
00:40
So velocity is negative.
00:42
That would be between 1 .5 and 2, since the slope is negative there.
00:51
And then again, from 5 to 6.
00:58
Where is the velocity equal to 0? well, that is where the slopes are 0.
01:04
And that occurs at two points at the point 1 .5.
01:10
So at time is 1 .5, at times 6, and then along the whole interval from 2 to 5.
01:20
So the particle was stopped from 2 seconds to 5 seconds.
01:25
And then it temporarily stopped because it turned around that 1 .5 and 6.
01:30
Whenever an object turns around, its velocity is zero just for an instant.
01:36
So using that, we could then go ahead and jump to part c, because c, in order to find that out, we're going to look at the concavity of this graph.
01:46
That will immediately tell us about the acceleration or the second derivative.
01:50
So it's concave down from 0 to 1 .5.
01:55
So that means that its acceleration is negative.
01:59
Since the slopes are decreasing.
02:02
And so we'll go ahead and write that.
02:03
So from 0 to 1 .5, negative acceleration, because the slopes are decreasing.
02:11
Slopes are increasing the whole time from 5 to 7 because they're getting bigger, bigger, bigger, or another way to say that is it's concave up.
02:19
So notice this guy here, concave down.
02:23
The next guy, concave up.
02:24
If it's concave up, acceleration is positive.
02:27
And so that is from 5 to 7.
02:32
And then, of course, in between there, we could say the slope is neither concave up nor concave down from 2 to 5.
02:41
So the acceleration is 0, which makes sense because if it's stopped, like we just said, then it should have no acceleration on it.
02:51
So then we also want to graph the velocity function.
02:55
To do this, again, we want to look at the slopes.
02:58
Here...