00:01
So this question gives us a function, f of t, which represents a number of yeast cells in a laboratory culture.
00:09
T is measured in hours.
00:11
And we're told some information that at time zero, the population is 20 cells and it increases at a rate of 12 cells per hour in that moment.
00:20
Our goal is to find a and b and to figure out what happens to f of t in the long run.
00:24
So the first thing that i'd like to do is rewrite this given information in terms of the function f.
00:31
So at t equals zero, population is 20.
00:35
We can instead write this as f of zero is equal to 20.
00:48
And furthermore, it's increasing at a rate of 12 cells per hour.
00:52
This unit here of cells per hour is your indicator that the information being provided is for the derivative.
01:01
So in this instance, they're saying that the rate of change, so f prime, at time zero, is going to equal the 12 cells per hour.
01:13
Notice that it says the rate is increasing.
01:16
That lets you know that the derivative is positive.
01:19
If it said decreasing, you would have to insert a negative into your answer.
01:24
Okay.
01:25
So there we go.
01:26
So we have the given information written in terms of f and f prime.
01:30
And furthermore, we can actually answer the second question first, more or less.
01:39
Again, they're asking what happens in the long run.
01:43
I would like to formulate that mathematically.
01:46
In the long run means as time approaches infinity.
01:50
What they're really asking for here is the limit as t approaches infinity for the population f of t.
02:04
And we can figure this out because the only place that time appears in the function is in the exponent of our e.
02:12
To a power and you'll notice that we have a negative attached to this the negative means that when t goes to infinity that entire exponential is going to approach zero and that means that this limit is only going to be a over one plus zero and that will simplify nicely into a so the idea is that whatever the answer is for a this is going to be what the population approaches in the long run.
02:50
In fact, in biology, this a has a special name for populations.
02:56
It's actually called a carrying capacity, which is, it makes sense.
03:01
It's the highest theoretical population that you can get, which will never actually be achieved.
03:10
If you had a quick little sketch of this curve for the population, it would start at some value.
03:15
It would increase exponentially, but then competition for resources happens, and you'll start to not increase as quickly this horizontal asymptote that you're approaching is actually going to be your carrying capacity.
03:32
Okay, so we've answered the second question, and now our first question then should be, all right, well, now what are the variables a and b in this problem? so to do that, we're going to use the given information, f of, 0 and f prime of 0.
03:50
So let's try f of 0.
03:53
So if we plug in 0 for time, that will cancel our exponential term.
03:59
So f of 0 is actually going to equal a over 1 plus b.
04:14
And according to our given info, this is going to equal 20.
04:19
So we have one equation with two unknowns.
04:23
If you get stuck at a point like this doing a problem in an exam, this should let you know that there's at least one more equation, right? you need the same number of unknowns as equations if you want to figure it out.
04:33
And i'm sure you know at this point that the second equation is going to come from that second piece of given info, f prime.
04:40
So we do need to take a derivative of this guy.
04:44
What i would recommend is i would rewrite f of t as a times one plus the denominator and instead raise it to the negative first power...