00:01
So in this problem, we're given this table of data for the weight gained by brook trout as a as a function of the temperature and degree c.
00:15
And these are on 24 hour days.
00:18
We're asked to find w prime for each of these.
00:23
Make a table of it and draw the graph of it.
00:26
And we're also asked what the units on w prime our hair.
00:30
Well, since w prime is going to be the difference in the weight gain divided by the difference in the temperature, this is going to be in grams for degrees c.
00:40
Because weight gain is in grams, the temperature is in degrees c.
00:47
Okay, so we begin trying to figure out w prime of 15.5.
00:56
Okay, so that's going to be w of 17.7 -w of 15.5 over.
01:10
Oops, this is upside down.
01:12
Let's start over here.
01:14
Yeah, that's right -w of 15.5.
01:19
That's right over 17.7 -15 five.
01:27
Okay, so this is 31.0 -37.2 over 2.2, which is approximately negative 2.8.
01:47
So since 15.5 is the beginning of our table, we only have one interval to use to figure out the derivative of the average rate of change at 15.5.
01:57
So we can only use that one interval there to do that with.
02:00
So we have our first entry here now to do it.
02:06
W prime for 17.7, we're going to do the average rate of change in both directions coming into 17.7 and then averaged them.
02:21
So w prime of 17.7.
02:27
We'll say going going down the table, it's gonna be w of 17.7 -w.
02:38
15.5 over 17.7 -15.5.
02:45
Which is a negative 2.8.
02:49
Actually, let's leave this one is a fraction.
02:53
So this is a negative 6.2 over 2.2.
03:00
Yeah, or a negative 31/11.
03:11
We were to simplify it.
03:13
Then w prime at 17.7.
03:16
Going up the table, it's going to be w at 20 -w at 17.7, divided by 20 point oh -17.7.
03:31
Again, we're just using the average rates of change off the table, they're going both directions.
03:40
So this is 19.8 minus 31 over 2.3.
03:53
And so this is a negative uh what is that? 11.2 over 2.3.
04:04
So that would be a negative 1, 12/23.
04:10
So then mm the average of these then is one half of -31/11 minus 1.
04:26
12/23 which is approximately a negative 3.8.
04:35
Now we have our second entry in the table, so negative 3.8.
04:41
Okay, now we'll do the same thing for 20...