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. Exponential functions, what do they look like in general? They are of the form y equals where we normally put a number in front, then we'll put some base, and we'll use any letter we want here, raised to the x. 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 equals 1. The reason for that is that if a is 1, then 1^x, just as 1 for any number x, and so you just tend to get a constant function that's a little boring. We like exponential functions to actually looked like an exponential curve, and you've probably heard expressions like things grow or decay exponentially. Let's draw a nice example here. Let's do it like a generic one, y equals a^x. We'll just make [inaudible] we want for now. This is the case this exponential growth. We've seen this before, This is 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, so you get a^0, that's 1, and you never quite touch the x axis. We say that as x gets small, so as x goes to negative infinity, then the function will tend towards zero. This is asymptotic behavior and we've got to use arrows here. I'm in particular not using equal signs. This is exponential growth. Start off low and 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 like radioactive decay or something like that, or some population that is shrinking. In this particular case, same function, same formats, exponentials, so let's just do y equals a^x, but in this particular case, my value of a is less than one. We also want a to be positive, I should probably say that, so a is between zero and one. 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 plus seven or whatever, sure, you're going to move these intercepts, but just get to know the parent function, y equals a^x. What else do we want to say about these functions? What else do they have in common? They're 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 it for a minute, is zero. I get these parentheses here because the asymptotic behavior zero to infinity. I use parentheses on infinity always. For growth, I guess I should put it down since I did for last one. As x goes to infinity then the function itself y will head over to zero. That's the asymptotic behavior as x gets large. This is our special functions. Let's do a couple of examples, things that we're going to see. Just some examples of exponential growth. We've seen like y equals 2^x. Perhaps a nice exponential growth, up it goes. Exponential decay, how about y equals, let's do a nice base here, is less than one. How about 1/2^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^x, which is 1, so you often see this as 1 over 2^x. 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, again, this is the speedup 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 equals x squared. These functions are called power functions or polynomials or something else, but they're 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 polynomial function, the base is the variable, and then the constant appears in the exponent. 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 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. It's a little weird to say like e is normal letter. E is a number 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. It's two and change, whatever it is. When you work with e, I wouldn't replace it with a decimal form. I would just use it on the calculator as e that's stored in the computer's memory as a lot of decimals. If you introduce. If you call 2.78, you're going to 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 2.78 and to really appreciate e as a function, you're going to need some calculus. It's been argued that e is the most important base for exponentials in calculus. You will see this thing over and over again. The thing that I can talk about now is that a lot of times when you write out any exponential function. If I write out y equals a^x, you can always write to this or rewrite this with a different base. It's like you could change the base for logarithms, you can also change the base. You can always write this with a base e^k and then put an x in here. For your appropriately chosen value of k, you can just say e^k is equal to a. Because that's a basic algebraic expression. Since there's no harm or foul of writing any base, just the cost of just putting this coefficient K in your exponent, we tend to write functions with e^x. So it's very common. The roof function is going to be y e^x, but perhaps a more complicated one is going to be y C e^kx. When I think of an exponential function like this is the default that I go to, some coefficient in front, e^kx. One thing about this coefficient in front, if you plug in x equals 0, so we look at y equals 0 and you get C e^k times 0, k times 0 of course is 0, and e^0 is just 1. This turns out to be like C times 1, which is just C. So C is your initial value. Specifically, in modeling, it's usually like the variable will be time, there's like which are first data point. That's usually what we care about. That's just some interpretation of this coefficient in front and then this k upstairs, this'll be your growth rate. 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 y, but just trust me that you can, with the expression 1 plus 1 over n^n, 1 plus 1 over n, you can almost define e to be this. If you take n to get really large, this will approximately e. 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. Then if you want, you can actually type an e and see how close this get. The numerical way to do if you have any e like five decimal, six 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. Let's talk about interests now. What is interests? Interests helps value to dollar today is worth more than a dollar yesterday and the past and money changes over time. The idea is you can invest that amount and then you will accrue interest on that investment. The times that the investment receives interest is called is when it's compounded. 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 plus my rate. Let's say my rates I get after one year. The nice thing about interest is that the interest accrues interest, so this is after year 1. After year 2, maybe we will write like a function, after year 2, I have the initial amount that I deposited, and then I have my interests from year 1 and then I get my interest again, on that interest. You clean this up and you get, not surprisingly perhaps square to get a nice exponential function and this continues. Maybe after year 3, I have my initial amount, again, I'm assuming I'm not withdrawing anything from this. I have my rate and it's cute, and then it goes on and on and on. The longer you leave the investment in there, then the more interest it accrues. Some people, if you have a savings account or something or checking account, maybe you get interests monthly, maybe you receive dividends on a piece of stock, some stock quarterly. You can receive money at different times, it doesn't have to be just yearly. In particular, you can do things like if interest is compounded semi-annually, twice a year, then half the interest is paid over that period, but the number of period doubles. Let's do that things for example. For example, 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, only getting half the interest. The amount after a full year is going to be the amount that you start with, and then the rate, half the rate you get, and then you get it again, so that starts to take a square power. This has a wonderful year. After t years, we're going to use t as our variable for time. You always have your initial amount and the interest that it accrues per year and then raised to the 2t. If you had monthly compounding in a similar format, this is what I have in the checking account, you get a little interest once a month. Not a ton, but it is what it is, we'll take it. That amount now you're getting whatever rate, but you're getting 12 payments. That means you have to take the interests and divide it by 12. It takes 12 compounding periods, and then times t. 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. Our general formula would be A^t. The amount after a certain period of time is whatever your initial amount you deposited, that's called the principle, one over the rate^nt, and n is how many times the interest is compounded, 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. If you are compounded continuously, this is usually what you pay if you are the receiver of a loan. The price changes, you get the compounding 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. It's the amount that we deposit, here comes the e^r, which is our interest rate times time. This is our formula and perhaps not surprising, the base here e is our friend, this is always number 2.718. We have two formulas here. This is going to be used to compute interest. Once you have the formula's going through problems is just plugging in for the variables and being mindful perhaps with the units. But let's just do an example. Let's say that you deposit $1,000 why not and you want it to sit for six years, so we'll a t be six years. Let's say that the interest rate is five percent. Let's let the compounding be annually. That's going to be m equals 1. 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? I want to know the amount A of t, but t is six years. Then we just plug in the formula here. I told that m equals 1, this is going to be my $1,000 times 1 plus 5 percent is typical to write it as a decimal, so we'll write 0.05. My compounding period is one, I'll write it in anyway. Then we have 1 times 6 of course, which is just six. Clean that up, work that out. I'm expecting this a calculator exercise, of course, but you get 1,340.10 units are in dollars. That's nice, I get more money at the end than what I started with. Perhaps not surprising. You can change what m is. For example, if you want this thing to say compounded daily. Daily would be like m is 365. Well then your amount at the end of six years, again, I leave this as exercise to plug in will be 1,349.83. Last but not least, of course, that it's the interest is compounded continuously, you would plug in all your numbers at the end of six years, and you would get 1,000e^0.05 times 6. Running out of room here. I'll just put the number down, but you can check this is 1,349.86. 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. Well, nice things about these formulas is that oftentimes they give you, you have like an end goal, like I want to have $1,000 today and I want to save up for something. 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 race do I have? How long do I have to wait until this investment accrues enough money for what I want? These formulas are really nice for financial planning. Compound interest or continuously compounded interest is for exponential growth. Let's do one example here, our last example of exponential decay. 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. It was actually found by some tourists, imagine walking by and finding that. What you do, you go get your local scientists and then they start to analyze it. If you want to read about it, definitely 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. We'll give you that the half-life of Carbon-14 is 5,730 years. A half-life remember this is a time that it takes for a mass to decay to get to exactly half of what's left. 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 of t. This will be the amount of carbon-14 present two years after death. This is the amount present. 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. We've got to find a couple of things. I don't know the initial amount and I don't know k, although I can get k from the half-life. Remember they give you half-life, they're always giving you k, why? Let's use the half-life for a minute. The quantity left after the half-life is 5,737. What does that mean? Whatever my initial amount is, I have half of it left. I don't know what the initial amount is, so I will 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 is in years. You 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 that you have 1.5 equals e to the k 5.730. We got to solve for k. This allows us to solve for k. How do you solve for k, where k is trapped inside exponential when we take log of both sides. In that case we get log of a half is log e and it cancel, so k times 5,730, and you can solve for k. K gives you that log of a half divided by 5,730. This is all the calculator says. Obviously you are not supposed to know what log of 1.5 divided by 5.730 is, but you should know how to plug it in a calculator. I would not introduce it as a decimal here, even to 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 like 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 negative 0.000121. That's about k. That number will be used right in our formula for radioactive decay. Now, since we know that 47.590 percent of the carbon had decayed, remember, I want to know what's left, so I have to subtract that from 100 percent or subtract that from 1. In calculator we can do 1 or 100 minus 47.59, and you get 52.41 percent. Here we go. Let's use that. Whatever the initial amount is, maybe I'll switch off colors. Whatever the initial amount is we have, let's do as a decimal, 0.5241. Remember this is equal to 52.41 percent 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. You just subtract the number from 100 percent. That is remaining of my initial amount, whatever that was. 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. Once again, the initial amount 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. You don't need it. Doing the same things to solve for, remember the goal was how old were the remains, we want to find t. We have log of 0.5241 equals log and you cancel. You get the k and t, and now it's just a nice calculator exercise where you have log of 0.5241 divided by k. In the calculator, here's what I'm going to do. I'm going do, let us get k. Let's do log of a half, log of 1 divided by 2, divided by 5,730. I hit "Enter" and I get this small number, just going to leave it. Now to find time, I'm going to do log of 0.5241 divided by second answer. When you do that, I'm going around here just because it's years. I'm going to get 5, 3, 4, I'll say one 5, 3, 4, 1. So 5,341 years give or take. By this method, the skeletal remains of this pre-historic man, they're over 5,000 years old. It's amazing. That means this person was from about 3,000 BCE. Pretty cool. Just measure taking a hike and then finding this thing. Go read up on this person. It's pretty amazing. But more importantly, I guess I care about the math and the use of the exponential function as an application to do radioactive carbon dating This time we find a dinosaur bone, we can use this model to see how old it is. Keep that calculator handy. Great job on this video, I'll see you all next time.