Hi, everyone, welcome to our lecture on exponential functions. We're going to be modeling with these functions. So I thought we could just take a quick minute to review what they actually are. So exponential functions, what do they look like in general? They are of the form y =. Normally we'll put a number in front, then we'll put some base, use any letter we want here, raised to the x. So x is our variable, a lot of letters going on here. So C and a, these are just some real numbers. x is our variable. We call a the base of this function, the base of the function. C could be one, sigma, whatever you want. We tend not to have a = 1. The reason for that is that if a is 1, then 9(x) just is 1 for any number x, and so you just tend to get a constant function, that's a little boring. So we like exponential functions to actually look like an exponential curve. And you've probably heard expressions like things grow or decay exponentially. So let's draw a nice example here. So how about y =, we'll just do a generic one, y = a to the x. We'll just make C be 1 for now. This is the case is exponential growth, we've seen this before. This is when a is bigger than 1, when a is bigger than 1. A couple things about it, it goes right through the y-axis at the point (0, 1). Stare at that for a minute, make sure you convince yourself that that is true. When you plug in 0 for x, you get a to the 0, that's 1. And you never quite touched the x-axis. We say that as x gets small, so as x goes to negative infinity, then the function will tend towards 0. This is asymptotic behavior and we gotta use arrows here, in particular not using equal signs. This is exponential growth, start off low and then you get nice and high. On the flip side of the world, you have exponential decay. This is when you start off with more things and then you have less. Think of this as radioactive decay or something like that, or some population that is shrinking. So in this particular case, same function, the same format, it's exponential. So let's just do y = a to the x, but in this particular case, my value of a is less than 1. We also want a to be positive, I should probably say that, so a is between 0 and 1. Once again, y intercept goes right through the origin (0, 1). That's something that both decay and growth have in common. Again, I'm assuming here just basic function a(x). If you start moving functions around and plus seven or whatever, sure, you're going to move these intercepts. But just get to know the parent function, y = a to the x. What else do we want to say about these functions? What else do they have in common? Their domain for both of these functions, can you see what it is? It's all reals. Plug in any number you want, the domain is all reals. The range is interesting. The range of these functions, stare at for a minute, is 0. So I get these parentheses here because the asymptotic behavior, (0, infinity). I use parentheses on infinity always, always, always. And for growth, I guess I should put it down since I did it for the last one, as x goes to infinity, then the function itself y will head over to 0. So that's the asymptotic behavior as x gets large. All right, so this is our exponential functions. Let's do a couple of examples, things that we're going to see, just some examples of exponential growth. We've seen y = 2 to the x perhaps, a nice exponential growth route [SOUND], up it goes. Exponential decay, how about y = let's do a nice base here that's less than 1. How about one-half, one-half to the x. Just as a heads up, you often see they often bring the exponent in. So if I bring the x to the numerator and the denominator I get 1 to the x, just 1. So you often see this as 1 over 2x. You have to realize that no matter which form you get it in, you're staring at a function that's exponential decay. And then it goes from high to low in that regard. One just word of warning here, and this is the speed up version of exponentials, if you need a review you might want to go back and watch that lecture. These are not functions of the form, say, y = x squared. These functions are called power functions, or polynomials or something else, but they are not exponential functions. The biggest difference with exponential functions is that the base is constant and the variable appears in the exponent. With a power function, or a polynomial function, the base is the variable and then the constant appears in the exponent. So depending on where the location is will dictate the type of function that you're working with. Everything we do in this video and the next one are about exponential functions. This is where I have a constant base and then my exponent is the variable. Now let's talk about the number e. That's a little weird to say, right, e's normally letter. e is a number, we've seen this before, e is about 2.718 and change. It goes on and on and on forever and ever and ever. It's an infinite decimal. We call these guys irrational, good vocabulary word there for you. It's an irrational real number, and it's two and change, whatever it is. When you work with e, I wouldn't replace it with the decimal form. I would just use it in the calculator as e, that's stored in the computer's memory. It has a lot of decimals. And if you call it 2.78 you can introduce rounding errors, but it's best to think about it as two and change. It helps just give some intuition about what this is. To really understand where and why e is 2.718, and to really appreciate e as a function, you're going to need some calculus. I mean, it's been argued that e is the most important base for exponentials in calculus. So you will see this thing over and over again. The thing that I can sort of talk about now is that a lot of times when you write out any exponential function, right, so if I write out like y = a to the x, you can always, always, always write to this or rewrite this with a different base. Sort of like you could change the base for logarithms, you can also change the base. You can always write this with a base e to the k and then put an x in here. And for your appropriately chosen value of k, you can just say e to the k = a. That's a basic algebraic expression. So since there's no harm or foul of writing any base just at the cost of just putting this coefficient k in your exponent, we tend to write functions with e to the x. So it's very common, the root function is going to be y = e to the x, but perhaps a more complicated one is going to be y = C e to the kx. When I think of an exponential function, this is the default that I go to. Some coefficient in front, e to the kx. And one thing about this coefficient in front, if you plug in x = 0, so if we look at y = 0, then you get Ce to the k times 0. [COUGH] k times 0 of course is 0, and e to the o is just 1. So this turns out to be C times 1, which is just C. So C is your initial value. So C is your initial value, but specifically in modeling it's usually the variable would be time, and it's like what's your first data point? That's usually what we care about. So that's just some interpretation of this coefficient in front. And then this k upstairs, this will be your growth rate. This will help determine how fast or how slow, and so this growth rate will help determine how fast or slow this function is growing. I leave this as a calculator exercise, or maybe even a spreadsheet, but you can numerically approximate e. It's not obvious why, but just trust me that you can, with the expression 1 + 1/n raised to the n, 1 + 1/n, you get almost a find e to be this. This, if you take, and they get really large, this will approximate it. So try it on a calculator, plug in 100. Plug in a million, plug in a billion, whatever you want. This will approximate e, and you start getting closer and closer to 2.718. And then if you want, you can actually type an e, and see how close this gets. So the numerical way to do if you ever need 5 decimal, 6 decimals, whatever, you just plug in larger and larger values of n. The reason why I'm showing you that is for the piece that's going to follow this expression, comes up when we do interest. So let's talk about interest now. So what is interest? Interest helps value. Dollar today is worth more than $1 yesterday in the past, and then money changes over time. The idea is you can invest that amount, and then you will accrue interest on that investment. And so the times that the investment receives interest, this call is when it's compounded. And so if we have some particular value, so let's look at an interest rate. We'll call it R will be our interest rate. If I have some amount after a certain period of time, I get 1 + my rate. And the nice thing about interest, let's say my rates after 1 year. The nice thing about interest is that the interest accrues interest. So this is after year 1, after year 2. Maybe we'll write a function after year 2. I have the initial amount that I deposited, and then I have my interest from the year 1, and then I get my interest again on that interest. And you clean this up, and you get, not surprisingly perhaps, squared. You get a nice exponential function. And this continues, right? This continues. So maybe after year 3, I have my initial amount to get. I'm assuming I'm not withdrawing anything from this. I have my rate, and it's. And then it goes on and on and on. The longer you leave the investment in there, then the more interest that accrues. Some people, if you have a savings account or something, or checking account, maybe you get interest monthly. Maybe you receive dividends on a piece of stock, stock quarterly that you can receive money at different times. It doesn't have to be just yearly. So in particular, you can do things, like if interest is compounded semi-annually, right, twice a year. Then half the interest is paid over that period, but the number of period doubles. So let's do that, for example. So the amount now after half a year will be whatever your initial amount is, and then it's going to be your rate but half the rate. You're only getting half the interest. And the amount after a full year is going to be the amount you start with, and then the rate, half the rate you get, and then you get it again. So that starts to take square power. And this adds year. So after t years, we're going to use t as our variable for time. You always have your initial amount, and the interest that accrues per year, and then time raised to the 2 t. If you had monthly compounding In a similar format, you would get this is what I have in the checking account. Do you get a little interest? Once a month, so not not a ton, but you know it is what it is. We'll take it. So that amount, now you're getting whatever rate, but you're getting 12 payments. So that means you have to take the interest, and divide it by 12. And it takes a full year, takes 12 compounding periods, and then times t. And if you had daily, you could replace the 12 with 365. If you had some other weird thing, whatever you want, you can just keep replacing it in general. So our general formula would be a to the t. So the amount after a certain period time is whatever your initial amount you deposited. That's called the principal, 1 over the rate, to the n, to the n t. And n is your how many times the interest is compounded. Okay, and t is in the years. You can start to see that there's an expression in here that's starting to look like that expression for e. And if you are compounded continuously, and this is usually what you pay if you are the receiver of a loan, right? The price changes, you get compounded continuously. That is basically saying I want the number of compounded to continue to go to infinity. In that particular case, we have a new formula. And it's the amount that we deposit. Here comes the e, e to the r, which is our interest rate, times time. So this is our formula, and perhaps not surprising, the base here, e, is our friend. This is Oilers number 2.718. So we have two formulas here. And this is going to be used to compute interest. Once you have the formulas, going through problems is just plugging in for the variables, and being mindful perhaps of the units. But let us just do an example. Let's say that you deposit $1,000. Why not? And you want it to sit for 6 years. So t be 6 years. And let's say that the interest rate is 5%. Let's let the compounding be annually. So it's going to be m = 1. And what I'd like to know is I'm trying to do some financial planning here. What is going to be my amount at the end of this investment? So I want to know the amount, a(t), but t is 6 years. And then we just plug in the formula here. So what I'm told that m = 1. So this is going to be my $1000 times 1 + 5%. It's typical to write it as a decimal, so we'll write 0.05. My compounding period is 1, I'll write it in. I'll write it in anyway. And then we have 1 times 6, of course, which is just 6. Clean that up, work that out. This is a calculator exercise, of course. But you get 1340.10. Units are in dollars. That's nice. I get more money at the end than what I started with, perhaps not surprising. And you can change what m is. For example, if you wanted this thing to say compounded daily. So daily would be like m is 365, well then your amount at the end of 6 years, again, I leave this as exercise to plug in, would be 1349.83. And last but not least, of course, if you plugged in, and you wanted that the interest is compounded continuously, you would plug in all your numbers at the end of 6 years. And you would get 1000 e to the 0.05 times 6. Running out of room here, so I'll just put the number down. But you can check this is 1349.86. So there's very little difference between the amount produced by daily compounding, and continuously compounding. But it's good to know if you see these things, especially if you're going to get a loan, or who knows for what? You should know what they work. The nice things about these formulas is that oftentimes, they give you an end goal. I want to have $1000 today, and I want to save up for something. And then you can use these formulas to solve for t, or you can solve for r, and you say, well, what rate do I need? Or how long at the given rates do I have? How long do I have to wait until this investment accrues enough money for what I want? So these formulas are really nice for financial planning. All right, so compound interest or continuously compounded interest is for exponential growth. Let's do one example here, last example of exponential decay. So, the remains of a prehistoric man, which is now called Otzi, also called the Iceman was found in September 1991 near the border of Italy and Switzerland. That was actually found by some tourists as in, they're just walking by and finding that. So what do you do, you go meet your local scientists and then they start to analyze it. And if you want to read about it, definitely can search the Wikipedia article on this thing, it's pretty fascinating. But one of the things that you can do is you can find out how much Carbon-14 is left or remaining because it is radioactive isotope of carbon, and you can calculate how old something is. This is how they do carbon dating as you may have heard. So I'll give you that the half life of carbon-14 is 5,730 years. The half-life remember, this is a time that it takes for a mass to decay, to get exactly half of what's left. So what we're going to do is we're going to set up a model for this equation. We're going to switch variables here a little bit called Q(t). This will be the amount of carbon-14 present two years after death. So this is the amount present. Now, this is an exponential decay function, so we're going to set it up with our initial quantity. We're going to set up our base to be e, we have some decay rate k, that we don't know, and time that's left. Okay, so we gotta find a couple things. I don't know the initial amount and I don't know k, although I think get k from the half life. Remember if they give you half life, they're always giving you k. Why, so let's use the half life for a minute. So the quantity left after the half life is 5,737. Okay, so what does that mean? Whatever my initial amount is, I have half of it left, half of it left. I don't know the initial amount is, so we'll just call it Q naught, and then I set it equal to this other formula Q naught e to the k, and the time here is 5,730. Everything's in years. You'll say, well, wait a minute, I don't know what the initial amount is. That's okay because it cancels, completely don't need it. And then that allows you the nice cleaner equation, they have one half equals e to the k 5,730. So we're going to solve for k. This allows us to solve for k. How do you solve for k, well, k is trapped inside an exponential. When we take log of both sides, in that case, we get log of half is log and e cancel. So k times 5730, and you can solve for k. So k gives you the log of a half divided by 5,730. This is all a calculator says, obviously you're not supposed to know a log of one half divided by 5730 is, but you should know how to plug in a calculator. I would not introduce it as a decimal here, even the three or four decimal places. Hold off on that urge, you're going to introduce a rounding error. Let the number live inside the memory of the calculator, store it somewhere or just use it like a second answer. But I don't want to call it a decimal at this point. If you want to make sure you're plugging it in right, just so we can check, it's about -0.000121. That's about k. So that number will be used right in a formula for radioactive decay. Now, since we know that 47.590% of the carbon had decayed, remember I want to know what's left. So I have to subtract that from 100% or subtract that from 1. So that is what a calculator can do. 1 or 100 minus 47.59, and you get 52.41%. So here we go. Let's use that. So whatever the initial amount is, maybe I'll switch up colors. Whatever the initial amount is, we have, let's do as a decimal, 0.5241. So remember this is equal to 52.41% remaining. As always with word problems, you just got to be very careful. Notice I'm not using the number that's decayed, I want the amount that's left, so you just subtract the number from 100%. So that is remaining of my initial amount, whatever that was. And that's going to equal by the formula Q naught e to this k value. I'm just going to write k for now, but you know it's some decimal that's known, times t. And once again, the initial amounts which we don't know cancel, which means I don't need to know this. A lot of students get stuck here, they say but I don't know the initial amount of Carbon-14 inside the person. It doesn't matter, don't need it. Doing the same sort of things to solve for, remember the goal is how old were the remains, we want to find t. So we have log of 0.5241 equals, log and e cancel, so you get k and t. And now it's just a nice calculator exercise, where you have log of 0.5241 divided by k. So in the calculator, in the calculator, here's what I'm going to do. I'm going to do, first let's get k. So let's do log of a half, log of 1 divided by 2, divided by 5730. I hit Enter, and I get this small number, is going to leave it, okay? Now to find time, I'm going to do log of 0.5241 divided by second answer. When you do that, be around here, just because it's years, I'm going to get 534, and I'll say 1, 5341. So 5,341 years, give or take. So, by this method, the skeletal remains of this prehistoric man, over 5,000 years old. It's amazing, right? That means this person was from about 3,000 BCE. Pretty cool. So, just imagine taking a hike and finding this thing. So go read up on this person, is pretty amazing. But more importantly, I guess they care about the math and the use of the exponential function as an application to do radioactive carbon dating. So next time you find a dinosaur bone, you can use this model to see how old it is. All right, keep that calculator handy. Great job on this video. We'll see you all next time.
