00:01
So in this problem, we're given this data on the height of a typical pine tree grown for lumber at a managed site.
00:13
And so we have these different heights from the arkansas forestry commission for the different ages of the trees.
00:22
And we're asked to determine, first of all, to construct a table of estimated values for h prime, the derivative, and then sketch the graph of it.
00:33
H prime, the derivative, is the average rate of change for each of the intervals.
00:43
And so h prime of 14 will be h at 21 minus h at 14, divided by 21 minus 14, which will be 54 minus 41 over 7 and this is 13 over 7.
01:16
So this one is 13 over 7.
01:20
Now to do h prime at 21, first of all we'll do the forward interval and then we'll do the reverse interval and average the two.
01:36
We already have the forward interval here.
01:41
So all we need now is h prime at 21 on the reverse interval.
01:47
I'll do it like that.
01:49
And so that's h of 28 minus h of 21 divided by 28 minus 21.
02:00
And so this is 64 minus 54 over 7.
02:07
So this is 10 over 7.
02:12
And now we do the average of this.
02:17
So h prime of 21 is 1 half times 13 over 7 plus 10 over 7.
02:30
So what is that? that is 23 over 14.
02:38
So this one is 23 over 14.
02:44
14.
02:47
Okay, now to do h prime at 28, we do the forward interval and then the reverse interval.
02:56
But we already have the forward interval right here.
03:00
So we just need the reverse interval.
03:03
So h prime at 28 on the reverse interval is h at 35 minus h at 28 over 35 minus h at 28 over 35, minus 28.
03:27
And so that's 72 minus 64 over 7, which is 8 over 7.
03:40
And so then h prime at 28, then is the average of these two...