Okay, so we're going to look at a little bit of basic modeling with exponential functions. So for example, suppose a population of an animal is known to grow exponentially given certain environmental conditions. And it is found that a population of 1000 doubles in a matter of two years.
Find an equation that models the population size after t years. What are we going to do here? So we're going to assume we have the following form, f of t is y0, and we'll explain why we have this y0 kind of, hmm.
index under the y there and We have two pieces of information we start off with a population of a thousand which means f of zero Is a thousand and it doubles in a matter of two years so that means after two years so t is in years here We end up with twice Whatever we started with So in that case we started with 1000 So this is 2000. And that will allow us to solve for two unknowns that we want. We want to solve for y0, and we want to solve for the base b here. By the way, once we get a little bit more traction on what are called logarithms, we'll be able to see also a similar equation that we won't use now, but it's also very commonly used. f of t equals y0 e to the kt. or this k corresponds to kind of a rate of growth.
So, but we won't do that now. We're going to use that maybe later on. For now, we're going to stick to the y0 and b to the t equation.
So what's the question is, what is the base and what is this y0? So let's start off by using this information. Well, this means that time 0, f of 0 was 1,000.
So we can have 1,000 on the left side. which is f of 0, and that's y0 b to the 0th power. Now b to the 0th power is 1, so this just means that y0 itself is 1,000. And that's why this is y sub 0. It's kind of like what is the quantity at time 0? So that's the little sub 0 there is indicating essentially what's happening at time 0. So this guy actually, this y0 is always the initial quantity.
So this guy, okay, is. is in general the initial quantity of whatever you're looking at. In this case, we're looking at the population of an animal. Okay. So we know that f of t is 1000 b to the t.
That is the form of f of t. And so now we plug in our second piece of information, which is that f of 2 is 2000. So. So 2000 is 1000b squared.
So in particular, if you divide by 1000, b squared is 2. So b is just the square root of 2. And so now we have our final equation that models the growth of this population. 1000 squared of 2 to the t-th power. And then you can, you know, if you wanted to plug in and know what the population is, you know.
know so after say 10 years you would have a population of 1000 times square root of 2 to the 10th power. So let's just see what that is. So we know that square root of 2 squared is 2. So this is just 2 to the 5th power.
That's 32, so we'd have 32,000. Okay, so this is kind of how the process goes for this kind of modeling equation. We'll do a similar one, just in a different context.
So, suppose that... money is invested in a certain bank that pays compound interest and the account grows exponentially it grows from $1,000 to 13 16 and 7 years if the growth function is of the form so by the way even if the that you could also right now we're looking at two growth examples you could also look at things that decline exponentially and the equations would be exactly the same you would still use something like f of T equals y 0 B to the T but in a declines if you're decaying if you're going down then you would have a B end up when you solve for it, you would end up with b being between 0 and 1. Because remember, the decay equations have the property that the base is less than 1. So if the base is bigger than 1, then we're going to have a base of 0. we're in the growth situation. So right now we're in the exponential growth situation.
But if we were in a decline, the procedure would still be the same. It's just that your b value would end up being between 0 and 1. Okay, that's just a little side fact. Let's go ahead and do this. So what do we have here? So we have the same modeling equation wherein we start off with $1,000.
So at time 0... we have $1,000, and in seven years, we have $1,316. And the question is, how much money is there after 12 years?
So again, f of 0 is 1,000. We'd plug that in here, so we get 1,000 is y0, b to the 0. b to the 0 is just 1, so y0, again, is this initial value of 1,000. So we have f of t being... 1000 b to the t and we know that after seven years we have three one thousand three hundred and sixteen dollars so f of seven is this guy so we have 1316 is 1000 b to the seventh power so we have 1316 divided by a thousand is b to the seventh and we take the seventh root or we can essentially put both powers to 1 7th and what you end up with is this is rounded to 2 sig figs so this is rounded, or two decimal places so we have a rounded 1.04 so f of t is 1,001.04 to the t, so if you plug in 12 years you get this This is 1,001.04 to the 12th power.
And if you punch that in, you get about $1,601. All right, so this is some examples of modeling exponential growth. Again, if you wanted to do a decay problem, what would happen is you would have a similar, so for example, just to illustrate.
Suppose you had a bank where you're losing money or an investment where you're losing money. Maybe you started off with $1,000 and you ended up with, you know, $850 in two years or something like that. and you know there's a projection that you're going to have exponential loss of money, well, how much are you going to lose in like say three years? It will be the same situation. So you would have Y of or F of T being 1,000.
That would be the initial quantity. B to the T. After two years, you're at 850. So 850 would be 1,000 B to the T. And here, or B to the, B squared in this case. so B squared because after two years B squared would be 850 over a thousand so B squared is 0.85 if you took the square root of that That would be b, and that would be, notice that b now is between 0 and 1. And that's indicative that you're in an exponential decay.
So your equation would be f of t is 1,000 squared of 0.85. to the t's power. Alright, so just to illustrate a situation where you would have a b declining or an exponential decay, your b indeed would be between zero and one. Okay, so this is a little bit about exponential decay.
We're going to stop here and we'll continue with more material in the next video.