WEBVTT mathematics/pre-calculus/selhorst-jones 00:00:00.000 --> 00:00:02.300 Hi--welcome back to Educator.com. 00:00:02.300 --> 00:00:05.400 Today, we are going to talk about exponential functions. 00:00:05.400 --> 00:00:10.100 Previously, we spent quite a while looking at functions that are based around a variable raised to a number-- 00:00:10.100 --> 00:00:17.500 things like x² or x^47; this is basically the idea of all of those polynomials we have worked with for so long. 00:00:17.500 --> 00:00:19.600 But what if we took that idea and flipped it? 00:00:19.600 --> 00:00:27.200 We could consider functions that are a number raised to a variable, things like 2^x or 47^t, 00:00:27.200 --> 00:00:31.900 where we have some base number that has a variable as its exponent. 00:00:31.900 --> 00:00:36.400 We call functions of this form exponential functions, and we will explore them in this lesson. 00:00:36.400 --> 00:00:40.500 Now, make sure that you have a strong grasp on how exponents work before watching this. 00:00:40.500 --> 00:00:48.500 If you need a refresher on how exponents work, check out the previous lesson, Understanding Exponents, to get a good grounding in how exponents work. 00:00:48.500 --> 00:00:54.100 All right, an exponential function is a function in the form f(x) = a^x, 00:00:54.100 --> 00:01:00.600 where x is any real number, and a is a real number such that a is not equal to 1, and a is greater than 0. 00:01:00.600 --> 00:01:06.500 We call a the base: base is just the name for the thing that is being raised to some exponent. 00:01:06.500 --> 00:01:12.400 So, whatever is being exponentiated--whatever is going through this process of having an exponent--that is called the base, 00:01:12.400 --> 00:01:16.400 because it forms the base, because it is below the exponent. 00:01:16.400 --> 00:01:21.300 We might wonder why there are all of these restrictions on what a can be; well, there are good reasons for each one. 00:01:21.300 --> 00:01:31.500 If a equals 1, we just have this boring constant function, because we would have 1^x, which is just equal to 1 all of the time. 00:01:31.500 --> 00:01:37.400 So, something that is just equal to 1 all of the time is not really interesting, and it is not really going to be an exponential function. 00:01:37.400 --> 00:01:40.100 So, we are not going to consider that case. 00:01:40.100 --> 00:01:44.000 If a equals 0, the function wouldn't be defined for negative values of x. 00:01:44.000 --> 00:01:51.500 If we try to consider what 0^-1 is, well, then we would get 1/0, but we can't do that--we can't divide by 0. 00:01:51.500 --> 00:01:56.200 So, that is not allowed, so that means a = 0--once again, we are not going to allow that one. 00:01:56.200 --> 00:02:01.500 And if we had a < 0, then the function wouldn't be defined for various x-values, like x^1/2. 00:02:01.500 --> 00:02:09.100 For example, if we had -4, and we raised that to the 1/2, well, we know that raising it to the 1/2 is the same thing as taking the square root. 00:02:09.100 --> 00:02:13.700 So, the square root of -4...we can't take the square root of a negative, because that produces imaginary numbers; 00:02:13.700 --> 00:02:17.300 and we are only dealing with real numbers--we are not dealing with the complex numbers right now. 00:02:17.300 --> 00:02:20.300 So, we are going to have to ban anything that is less than 0. 00:02:20.300 --> 00:02:25.200 And that is why we have this restrictions: our base has to be greater than 0, and is not allowed to be 1, 00:02:25.200 --> 00:02:29.100 because otherwise things break down for the exponential function. 00:02:29.100 --> 00:02:34.200 All right, notice that, from the previous lesson, we can compute the value of a given base raised to any exponent. 00:02:34.200 --> 00:02:39.900 We know how exponents work when they are a little more complex (not complex numbers, but just more interesting). 00:02:39.900 --> 00:02:45.300 And so, we can raise things like...4^3/2 = (√4)³, 00:02:45.300 --> 00:02:51.100 which would be equal to...√4 is 2; 2³, 2 times 2 times 2, gives us 8; great. 00:02:51.100 --> 00:02:59.500 If we had 7^-2, well, then that would become 1/7, because we have the negative, so the negative flips it to (1/7)². 00:02:59.500 --> 00:03:04.400 So, 1 squared is 1; 7 squared is 49; we get 1/49. 00:03:04.400 --> 00:03:10.100 So, we can do these things that are a little more difficult than just straight positive integers. 00:03:10.100 --> 00:03:16.100 But we might still find some calculations difficult, like if we had 1.7^6.2--that would probably be pretty hard to do. 00:03:16.100 --> 00:03:20.100 Or (√2)^π--these would be really difficult for us to do. 00:03:20.100 --> 00:03:26.600 So, how do we do them? In practice, we just find these expressions, or a very good approximation, by using a calculator. 00:03:26.600 --> 00:03:34.200 We can end up getting as many digits in our decimal expansion as we want. 00:03:34.200 --> 00:03:40.200 We can just find as many as we need for whatever our application is--whatever the problem asks for--by just using a calculator. 00:03:40.200 --> 00:03:43.500 Any scientific or graphing calculator can do these sorts of calculations. 00:03:43.500 --> 00:03:54.800 There will be some little button that will say x^y, or some sort of _ to the _--some way to raise to some other thing--something random. 00:03:54.800 --> 00:04:10.200 They might have a carat, which says...if I have 3⁶ (not with an a--I accidentally drew that in...oh, I drew it in again), 00:04:10.200 --> 00:04:13.100 then that would be equivalent to us saying 3⁶. 00:04:13.100 --> 00:04:18.900 The carat is saying "go up," so the calculator would interpret 3⁶ as 3⁶. 00:04:18.900 --> 00:04:21.400 There are various ways, depending on if you are using a scientific calculator, 00:04:21.400 --> 00:04:26.200 or if you are using a graphing calculator, to put these things into a calculator and get a number out. 00:04:26.200 --> 00:04:30.100 So, we are able to figure these things out, just by being able to say "use a calculator." 00:04:30.100 --> 00:04:33.000 Now, from a mathematical point of view, that is a terrible statement. 00:04:33.000 --> 00:04:39.600 We don't want to say, "We can deal with this because we have calculators!" because how did you figure it out before you had calculators? 00:04:39.600 --> 00:04:42.700 Calculators didn't just spawn into existence and give us the answers. 00:04:42.700 --> 00:04:47.300 We can't rely on our calculators to do our thinking for us; we have to be able to understand what is going on. 00:04:47.300 --> 00:04:50.500 Otherwise, we don't really have a clue how it works. 00:04:50.500 --> 00:04:56.400 But as you will see as you get into more advanced math classes, there are methods to figure out these values. 00:04:56.400 --> 00:05:03.000 There are ways to do this by hand, because there are various algorithms that give us step-by-step ways to get a few decimals at a time. 00:05:03.000 --> 00:05:10.500 Now, doing it by hand is long, slow, and tedious; it would be hard to get this sort of thing, just because it would be so much calculation to do. 00:05:10.500 --> 00:05:16.400 We could do it, but that is what calculators are for; they are to do lots of calculations very quickly. 00:05:16.400 --> 00:05:18.700 They are to help us get through tedious arithmetic. 00:05:18.700 --> 00:05:26.500 So, since these sorts of calculations take all of this arithmetic, we designed calculators that can do this method for us. 00:05:26.500 --> 00:05:31.400 And that is why we can appeal to a calculator--not because the calculator knows more than us, 00:05:31.400 --> 00:05:36.600 but because, at some point, humans figured out a method to get as many decimals as we wanted to; 00:05:36.600 --> 00:05:40.400 and then, we just built a machine that is able to go through it quickly and rapidly, 00:05:40.400 --> 00:05:44.700 so we can get to the thing that we want to look at, which is more interesting, using this. 00:05:44.700 --> 00:05:50.400 The calculator is a tool; but it is important to realize that we are not just relying on it because it has the knowledge. 00:05:50.400 --> 00:05:54.400 We are relying on it because, at some point, we built it and put these methods into it. 00:05:54.400 --> 00:06:01.200 And if you keep going in mathematics, you will eventually see that these are where the methods come from--there is some pretty interesting stuff in calculus. 00:06:01.200 --> 00:06:07.900 All right, now, if we can evaluate at any place--if we can compute what these values of exponential functions are-- 00:06:07.900 --> 00:06:12.500 then we can make a graph, because we can plot as many points as we want; we can draw a smooth curve. 00:06:12.500 --> 00:06:14.800 So, let's look at some graphs where the base is greater than 1. 00:06:14.800 --> 00:06:23.700 If we have 2^x, that would be the one in red; 5^x is the one in blue, and 10^x is the one in green. 00:06:23.700 --> 00:06:32.100 Now, notice: 2^x, 5^x, and 10^x--all of these end up going through 1, right here, 00:06:32.100 --> 00:06:41.400 because what is happening there is that 2⁰, 5⁰, 10⁰...anything raised to the 0--they all end up being 1. 00:06:41.400 --> 00:06:43.600 Remember, that is one of the basic properties of exponents. 00:06:43.600 --> 00:06:48.900 If you raise something to the 0, it just becomes 1; so that is why we see all of them going through the same point. 00:06:48.900 --> 00:06:51.500 And notice that they get very large very quickly. 00:06:51.500 --> 00:06:57.000 By the time 2 is to the fourth, it is already off; and 10 is off by the time it gets to the 1. 00:06:57.000 --> 00:07:01.600 10^x grows very quickly, because it is multiplying by 10, each step it goes forward. 00:07:01.600 --> 00:07:05.600 Notice also: as we go far to the left, it shrinks very quickly. 00:07:05.600 --> 00:07:16.000 Let's consider 10^-3; 10^-3 would be the same thing as 1/10³, which would be 1/1000. 00:07:16.000 --> 00:07:19.700 That is why we end up seeing that this green line is so low. 00:07:19.700 --> 00:07:24.100 It looks like it is almost touching the x-axis; it isn't quite--there is this thin sliver between it. 00:07:24.100 --> 00:07:28.500 But it is being crushed down very, very quickly, because of this negative exponent effect, 00:07:28.500 --> 00:07:34.200 where it gets flipped over, and then it has a really, really large denominator very quickly. 00:07:34.200 --> 00:07:40.600 So, we see, as we go to the left side with these things, that it will crush down to 0. 00:07:40.600 --> 00:07:44.400 And as we go to the right, it becomes very, very big. 00:07:44.400 --> 00:07:48.400 We can change the viewing window, so that we can get a sense for just how big these things get. 00:07:48.400 --> 00:07:54.900 And look at how big: we have gotten up to the size of 1000 by the time we are only out to 10. 00:07:54.900 --> 00:08:03.500 And that is on 2^x; if we look at 10^x, 10^x has already hit 1000 at 10³. 00:08:03.500 --> 00:08:07.700 At x = 3, it has managed to hit 1000 as its height. 00:08:07.700 --> 00:08:14.400 This stuff grows really quickly; this idea of massive growth is so central to the idea of exponential functions. 00:08:14.400 --> 00:08:19.000 We are going to have a story: there is this story that often gets told with exponential functions, 00:08:19.000 --> 00:08:25.300 because it is a great way to get people to understand just how big this stuff gets. 00:08:25.300 --> 00:08:32.700 So, let's check it out: All right, long ago, in a far-off land, there was a mathematician who invented the game of chess. 00:08:32.700 --> 00:08:38.900 The king of the land loved the game of chess so much that he offered the mathematician any reward that the mathematician desired. 00:08:38.900 --> 00:08:43.100 The mathematician was clever, and told the king humbly, "Your Highness, I thank you; 00:08:43.100 --> 00:08:48.100 all I ask for is a meager gift of rice, given day by day on a chessboard." 00:08:48.100 --> 00:08:53.400 "Tomorrow, I would like a single grain of rice give on the first square; 00:08:53.400 --> 00:09:00.800 on the next day, two grains of rice given on the second square; then on the following day, the third day, 00:09:00.800 --> 00:09:09.800 four grains of rice; and so on and so forth, doubling the amount every day until all 64 squares are filled." 00:09:09.800 --> 00:09:16.000 So, the mathematician is asking for the first square, doubled, doubled, doubled, doubled... 00:09:16.000 --> 00:09:19.100 The mathematician drew the king a diagram to help make his request clear. 00:09:19.100 --> 00:09:26.800 On the first day of his gift, he would end up having one grain of rice on the first square. 00:09:26.800 --> 00:09:31.800 On the second day, there would be a total of two grains of rice (1 times 2 becomes 2). 00:09:31.800 --> 00:09:37.800 On the third day, there would be a total of 4 grains of rice (2 times 2 becomes 4). 00:09:37.800 --> 00:09:46.300 On the next day, there will be 8 (4 times 2 becomes 8), and then 16, and then 32, and so on and so on and so on, 00:09:46.300 --> 00:09:53.400 going all the way out to the 64th day, doubling each time we go forward a square on the board. 00:09:53.400 --> 00:09:57.100 The king was delighted by the humble request and agreed to it immediately. 00:09:57.100 --> 00:10:03.300 Grains of rice? You can't get a lot of grains of rice on a single chessboard; "It will be very easy," he thought. 00:10:03.300 --> 00:10:08.800 He ordered that the mathematician would have his daily reward of rice delivered from the royal treasury every day. 00:10:08.800 --> 00:10:12.500 A week later, the king marveled at how the mathematician had squandered his reward. 00:10:12.500 --> 00:10:16.900 After all, he only had to send him 2⁶ = 64 grains of rice that day. 00:10:16.900 --> 00:10:24.000 Notice: on the seventh day, we are at 2⁶--let's see why that is. 00:10:24.000 --> 00:10:30.900 On the first day, we have 1 grain; on the second day, we have 2 grains; on the third day, we have 4 grains. 00:10:30.900 --> 00:10:38.300 On the fourth day, we have 8 grains; on the fifth day, we have 16 grains; on the sixth day, we have 32 grains. 00:10:38.300 --> 00:10:41.700 And thus, on the seventh day, we have 64 grains. 00:10:41.700 --> 00:10:55.100 So, notice: we can express this as 2⁰, 2¹, 2², 2³, 2⁴, 2⁵, and then finally 2⁶ on the seventh day. 00:10:55.100 --> 00:10:59.100 Why is this? Because on the very first day, he just got one grain. 00:10:59.100 --> 00:11:02.100 Every following day, it multiplies by 2--it doubles. 00:11:02.100 --> 00:11:10.800 So, that means we multiply it by 2; so we count all of the days after the first day, which is why, on the seventh day, we see an exponent of 6. 00:11:10.800 --> 00:11:18.500 So, in general, it is going to be 2 to the (number of day minus 1). 00:11:18.500 --> 00:11:22.700 We will subtract one to figure out the grains on some number of day. 00:11:22.700 --> 00:11:28.900 So now, we have an idea of how we can calculate this pretty quickly and be able to get these things figured out. 00:11:28.900 --> 00:11:34.600 Another week later, on the fourteenth day, the king sent him 2^13 (remember, it is the fourteenth day, 00:11:34.600 --> 00:11:41.700 so we go back one, because it has been multiplied 13 times) grains of rice, which is 8,192 grains. 00:11:41.700 --> 00:11:46.400 And 8,192 grains is just about a very large bowl of rice. 00:11:46.400 --> 00:11:49.600 The king was still amazed at the fantastic deal he was getting. 00:11:49.600 --> 00:11:53.200 But he was glad that the mathematician was at least seeing some small reward. 00:11:53.200 --> 00:11:58.800 He loved the game of chess, after all; and if he ended up feeding the mathematician for a year, that was great. 00:11:58.800 --> 00:12:03.900 It seemed like a wonderful deal; he was willing to give him palaces, jewels, and massive amounts of money. 00:12:03.900 --> 00:12:08.200 He can give him a little bit of rice for the great game of chess. 00:12:08.200 --> 00:12:13.800 At the end of the third week, on the twenty-first day, the king had to send the mathematician a full bag of rice, 00:12:13.800 --> 00:12:18.700 because in the kingdom, a full bag of rice contained precisely 1 million grains. 00:12:18.700 --> 00:12:27.000 So, on the 21st day, we have 2^20 grains of rice, which ends up being 1 million, 48 thousand, 576 grains. 00:12:27.000 --> 00:12:33.100 So we see here: after we jump these first six digits, we have one million plus grains of rice. 00:12:33.100 --> 00:12:41.000 So, he has managed to get one million grains of rice (which is one bag of rice), plus an extra 48000 in change, in grains. 00:12:41.000 --> 00:12:46.100 So, perhaps the mathematician was not as foolish as the king had first thought. 00:12:46.100 --> 00:12:49.000 At the end of the fourth week, the king was starting to get worried. 00:12:49.000 --> 00:12:59.800 On the twenty-eighth day, he had to send him more than 134 bags of rice, because 2^27 is more than 134 million grains of rice. 00:12:59.800 --> 00:13:02.100 So, we are starting to get to some pretty large amounts here. 00:13:02.100 --> 00:13:08.200 Now, the royal treasury has a lot of rice; he is not worried--he has hundreds of thousands of bags of rice. 00:13:08.200 --> 00:13:13.000 So, he is not too worried about it; but he sees that this is starting to grow quite a bit. 00:13:13.000 --> 00:13:18.700 At that moment, the royal accountant bursts into the throne room and says, "Your Highness, I have grave news! 00:13:18.700 --> 00:13:26.100 The mathematician will deplete the royal treasury! On the forty-first day alone, we would have to give one million bags of rice!" 00:13:26.100 --> 00:13:35.700 because 2^40 is here, so we have one million million grains of rice; so we have one million bags of rice, 00:13:35.700 --> 00:13:39.400 which is more than the entirety that the treasury has in rice. 00:13:39.400 --> 00:13:42.900 "And if we kept going--if we let it run all the way to the sixty-fourth day, 00:13:42.900 --> 00:13:47.000 we would have to send him more rice than the total that the world has ever produced, 00:13:47.000 --> 00:13:55.100 because we would be at 2^63, which would come out to be 9 trillion bags of rice." 00:13:55.100 --> 00:14:02.000 Look; we have ones here; we have thousands here; we have millions here, billions here, trillions here, quadrillions here; 00:14:02.000 --> 00:14:11.100 it would be 9 quintillion grains of rice; if we knock off these first ones, we see that we are still at 9 trillion bags of rice. 00:14:11.100 --> 00:14:16.000 That is a lot of rice, and the world doesn't have that much by far. 00:14:16.000 --> 00:14:22.400 So, the mathematician's greed has enraged the king, and the king immediately orders all shipments of rice stopped. 00:14:22.400 --> 00:14:26.500 The mathematician is not getting any more rice, and the mathematician is to be executed! 00:14:26.500 --> 00:14:31.600 Now, the mathematician, being a clever fellow, hears the soldiers coming down the road, and he escapes. 00:14:31.600 --> 00:14:35.600 He fled the kingdom with the few bags of rice that he could manage to carry on his back, 00:14:35.600 --> 00:14:40.500 and he had to find a new place to live, far, far away from the kingdom. 00:14:40.500 --> 00:14:46.000 So, the moral of the story is twofold: first, don't be overly greedy--don't try to trick kings. 00:14:46.000 --> 00:14:55.300 But more importantly than that, exponential functions grow really, really large in a short period of time. 00:14:55.300 --> 00:15:04.600 They get big fast; even if they start at a seemingly very, very small, miniscule amount, they will grow massive if given enough time. 00:15:04.600 --> 00:15:07.300 So, that is the real take-away here from this story. 00:15:07.300 --> 00:15:16.000 Exponential functions get big; they can start small, but given some time, they get really, really big. 00:15:16.000 --> 00:15:18.300 All right, let's see an application of this stuff. 00:15:18.300 --> 00:15:22.300 When you put money in a bank, they will usually give you interest on your money. 00:15:22.300 --> 00:15:31.400 For example, if you had an annual interest rate of 10% (annual just means yearly) on a \$100 principal investment 00:15:31.400 --> 00:15:35.400 (the amount that you put in the bank), the following year you would have that \$100 still 00:15:35.400 --> 00:15:39.900 (they don't take it away from you), plus \$100 times 10%. 00:15:39.900 --> 00:15:48.600 Now, 10% as a decimal is .10; so it is \$100 times .10, so you would get that \$100 that you originally started with, and you would have \$10 in interest. 00:15:48.600 --> 00:15:54.000 Great; but you could leave that interest in the account, and then your interest would also gain interest. 00:15:54.000 --> 00:16:01.300 The interest is going to get interest on top of it; so we would say that the interest is compounded, because we are putting on thing on top of the other. 00:16:01.300 --> 00:16:08.400 So, you have \$110 in your bank account now, because you had \$110 total at the end last time. 00:16:08.400 --> 00:16:18.200 \$110 gets hit by that 10% again; so you still have the \$110, plus...now 10% of \$110 is \$11. 00:16:18.200 --> 00:16:24.900 Notice that \$11 is bigger than 10--your interest is growing. 00:16:24.900 --> 00:16:30.300 Over time, you are getting more and more interest as you keep letting it stay in there. 00:16:30.300 --> 00:16:33.400 You continue to gain larger and larger amounts with each interest. 00:16:33.400 --> 00:16:38.600 Compound interest is a common and excellent way to invest money, because over time, 00:16:38.600 --> 00:16:41.800 your interest gains interest, and gains interest, and gains interest. 00:16:41.800 --> 00:16:45.800 And eventually, it can manage to get large enough to be even larger than the principal investment, 00:16:45.800 --> 00:16:51.500 and be the thing that is really earning you money--the time that you have spent letting it compound. 00:16:51.500 --> 00:17:04.500 We can describe the amount of money, A, in such an account with an exponential function: A(t) = P[(1 + r)/n]nt. 00:17:04.500 --> 00:17:12.000 Let's unpack that: P is the principal in the account--the amount that is originally placed in the account. 00:17:12.000 --> 00:17:18.200 So, in our example, that would be \$100 put in; so our principal would be 100 in that last example. 00:17:18.200 --> 00:17:25.500 r is the annual rate of interest, and we give that as a decimal: here is our r, right up here. 00:17:25.500 --> 00:17:29.900 In the last one, that was 10%, so it was expressed as .10. 00:17:29.900 --> 00:17:37.600 n is the number of times a year that the interest compounds; n is the number of times that we see compounding. 00:17:37.600 --> 00:17:46.500 So, n = 1 would be yearly; n = 4 would be quarterly; n = 12 would be monthly; n = 365 would be daily. 00:17:46.500 --> 00:17:52.300 In our last one, it compounded annually, every year; so it compounded just once a year, so n was equal to 1. 00:17:52.300 --> 00:17:56.700 Notice that n also shows up up here; it is n times t. 00:17:56.700 --> 00:18:03.800 And then finally, t is just the number of years that we have gone through; so it is times t. 00:18:03.800 --> 00:18:05.900 So, let's understand why this is the case. 00:18:05.900 --> 00:18:16.000 Well, if we looked at 10%, just on the \$100, we would have \$100 times 1 + 10%. 00:18:16.000 --> 00:18:22.000 So, \$100 times 1.1 equals \$110. 00:18:22.000 --> 00:18:28.800 Now, if we wanted to have this multiple times, well, the next time it is \$110 times 1.1, again. 00:18:28.800 --> 00:18:32.100 We would get another number out of it; and then, if we wanted to keep hitting it... 00:18:32.100 --> 00:18:39.400 we can just think of it as (100^1.1)^t, and that will just give us the amount of times 00:18:39.400 --> 00:18:45.100 that the interest has hit, over and over and over--our principal times the 1 00:18:45.100 --> 00:18:50.700 (because the bank lets you keep what you started with), plus the interest in decimal form, 00:18:50.700 --> 00:18:54.600 all raised to the t--the number of years that have elapsed. 00:18:54.600 --> 00:18:56.400 Now, what about that "divide by n" part? 00:18:56.400 --> 00:19:00.100 Well, let's say that we compounded it twice a year; so they didn't just give you your interest 00:19:00.100 --> 00:19:04.200 in a lump sum at the end of the year--they gave it to you in bits and pieces. 00:19:04.200 --> 00:19:10.800 So, the first time it compounds, if they did it twice a year (let's say they did it semiannually, two times a year), 00:19:10.800 --> 00:19:18.000 then it would be 1 (because they let you keep the amount of money), plus .1/2 (because they are going to do it twice in a year). 00:19:18.000 --> 00:19:28.800 So, the first time in the year, we would get 100 times (1 + 0.05), 100 times 1.05. 00:19:28.800 --> 00:19:33.200 The first time in the year it gets hit, you would get \$105 out of that. 00:19:33.200 --> 00:19:40.700 Now, they could do it again, and we would have \$105 get hit with another one of 1.05, and then we could calculate that again. 00:19:40.700 --> 00:19:42.900 And that would be the total amount that you would have over the year. 00:19:42.900 --> 00:19:52.200 Now, notice: 105 times 1.05 is going to be a little bit extra, because we are getting that 5 times 1.05, in addition to what we would have ended up having. 00:19:52.200 --> 00:20:00.100 We would have 105 times 1.05; 5.25...so we will end up getting 5.25 out of this. 00:20:00.100 --> 00:20:07.800 So, we will have a total of 110 dollars and 25 cents. 00:20:07.800 --> 00:20:14.200 So, by compounding twice in a year, we end up getting 25 cents more than we did by compounding just once in a year. 00:20:14.200 --> 00:20:19.300 So, the more times we compound, we get more chances to earn interest on interest on interest. 00:20:19.300 --> 00:20:25.200 1 + .1, divided by 2...it is going to happen twice in a year; so since it happens twice in a year, 00:20:25.200 --> 00:20:29.800 we have to have the number of times that it is happening in a year, times the number of years. 00:20:29.800 --> 00:20:40.300 So, at the twice-in-a-year scale, we would see 1.05 to the 2 times number of years, because it happens twice every year. 00:20:40.300 --> 00:20:46.400 And this method continues the whole time; so that is why we have the divide by n, because the rate has to be split up that many times. 00:20:46.400 --> 00:20:52.600 But then, it also has to get multiplied that many times extra, because it happens that many times extra in the year. 00:20:52.600 --> 00:20:55.500 So, that is where we see this whole thing coming from. 00:20:55.500 --> 00:21:00.100 Now, we noticed, over the course of doing that, that the more times it compounded, the better. 00:21:00.100 --> 00:21:07.100 We earn more interest if it is calculated more often; the more often our account compounds, 00:21:07.100 --> 00:21:11.100 the more interest we earn, because we have more chances to earn interest on top of interest. 00:21:11.100 --> 00:21:17.700 So, we would prefer if it compounded as often as possible--every minute--every second--every instant-- 00:21:17.700 --> 00:21:20.900 if we had it happening continuously--absolutely constantly. 00:21:20.900 --> 00:21:27.900 This idea of having it happen more and more often leads to the idea of the natural base, which we denote with the letter e. 00:21:27.900 --> 00:21:35.300 The number e comes from evaluating 1 + 1/n to the n as n approaches infinity--as this becomes larger and larger-- 00:21:35.300 --> 00:21:42.100 because remember: the structure last time was 1 plus this rate, divided by n to the n times t. 00:21:42.100 --> 00:21:49.800 So, if we forget about the times of the year that it is occurring, and forget about the rate, we get just down to (1 + 1/n) to the n. 00:21:49.800 --> 00:21:54.200 So, we can see what happens as n goes out to infinity--what number does this become? 00:21:54.200 --> 00:21:57.200 It does stabilize to a number, as you can see from this graph here. 00:21:57.200 --> 00:22:05.400 So, by the time it has gotten to 40, it starts to look pretty stable; it has this asymptote that it is approaching, so it is starting to become pretty stable. 00:22:05.400 --> 00:22:08.500 We can look at some numbers as we plug in various values of n. 00:22:08.500 --> 00:22:22.300 At 1, we get 2; at 10, we have 2.594; at 100, we have 2.705; at 1000, 2.717; at 10000, 2.718; at 100000, it is still at 2.718. 00:22:22.300 --> 00:22:26.700 And there are other decimals there; but we see that it ends up stabilizing. 00:22:26.700 --> 00:22:36.200 As we put more and more decimal digits, as n becomes larger and larger and larger, we see more and more decimal digits that e is going towards. 00:22:36.200 --> 00:22:44.100 e is stabilizing to a single value, and we see more and more of its digits, every time we keep going with this decimal expansion. 00:22:44.100 --> 00:22:47.500 So, as we continue this pattern, e stabilizes to a single number. 00:22:47.500 --> 00:22:52.200 Now, it doesn't stabilize to a single number where we have finished figuring it; we keep finding new decimals. 00:22:52.200 --> 00:22:55.700 But we see that decimals we have found so far aren't going to change. 00:22:55.700 --> 00:23:02.300 e is 2.718281828...and that decimal expansion will keep going forever. 00:23:02.300 --> 00:23:09.000 Just like π, the number e is an irrational number; its decimal expansion continues forever, never repeating. 00:23:09.000 --> 00:23:18.800 So, that decimal expansion just keeps going forever, just like π isn't 3.14 (it is 3.141...it just keeps going forever and ever and ever). 00:23:18.800 --> 00:23:26.600 So, e is the same thing, where we can find many of the decimals, but we can't find all of the decimals, because it goes on infinitely long. 00:23:26.600 --> 00:23:33.800 Now, also, just like π, the number e is deeply connected to some fundamental things in math and the nature of the universe. 00:23:33.800 --> 00:23:38.300 e is connected to the very fabric of the way that the universe, and just things, work. 00:23:38.300 --> 00:23:43.500 So, π is fundamentally connected to how circles work; circles show up a lot in nature, in the universe. 00:23:43.500 --> 00:23:50.400 π is connected to circles, and e is connected to things that are continuously growing-- 00:23:50.400 --> 00:23:58.000 things that are always growing, that don't take this break between growth spurts, but that are just always, always, always growing. 00:23:58.000 --> 00:24:02.700 e gives us things that are doing this continual growth; e has this deep connection; 00:24:02.700 --> 00:24:08.500 and if you continue on in math, you will see e a lot (and also if you continue on in science). 00:24:08.500 --> 00:24:13.800 One application of e is to see how an account would grow if it was being compounded every single instant. 00:24:13.800 --> 00:24:19.000 That idea, that we are not just doing it every year; not just every day; not just every minute; not just every second; 00:24:19.000 --> 00:24:26.000 but every single instant--that gives us P (our principal amount) times ert. 00:24:26.000 --> 00:24:33.700 The amount in our account is P times ert; we can also just remember this as "Pert"; Pert is the mnemonic for remembering this. 00:24:33.700 --> 00:24:38.500 P is the principal, or we can just think of it as the starting amount--however much we started with. 00:24:38.500 --> 00:24:43.300 r is the annual rate of interest, and it can even be used for things that aren't just annual rates, 00:24:43.300 --> 00:24:47.100 but r is the annual rate of interest; and remember: we give that as a decimal. 00:24:47.100 --> 00:24:49.600 If we give it as a percent, things will not end up working out. 00:24:49.600 --> 00:24:52.900 And t is the number of years elapsed. 00:24:52.900 --> 00:24:58.700 Now, this above equation, this one right here--this "Pert" thing--this can be used for a wide variety of things 00:24:58.700 --> 00:25:03.600 that grow or decay continuously--things that are constantly growing or constantly decaying. 00:25:03.600 --> 00:25:07.800 You will see it show up a lot in math and science as you go further and further into it. 00:25:07.800 --> 00:25:16.200 It is very, very important--this idea of some principal amount, times e to the rate times the amount of time elapsed. 00:25:16.200 --> 00:25:21.000 You can use it for a lot of things; and while we will end up, in these next few examples, using some other things 00:25:21.000 --> 00:25:25.100 than just ert (with the exception of the examples that involve continuously compounded interest), 00:25:25.100 --> 00:25:30.100 you can actually bend a lot of stuff that you have in exponents into using e. 00:25:30.100 --> 00:25:35.200 So, it is easiest to end up just remembering this one, and then changing how you base your r around it. 00:25:35.200 --> 00:25:39.900 Now, don't get too confused about that right now; we will see it more as we get into other things and logarithms, 00:25:39.900 --> 00:25:42.000 and also just as you get further and further into math. 00:25:42.000 --> 00:25:50.400 You will see how Pert is a really fundamental thing that gives us all of the stuff that is doing the growth. 00:25:50.400 --> 00:25:56.200 Finally, exponential decay: so far, we have only seen exponential functions that grow as we go forward-- 00:25:56.200 --> 00:26:02.300 f(x) = a^x, where our base, a, is greater than 1; so it gets bigger and bigger as we march forward. 00:26:02.300 --> 00:26:06.600 But we can also see decay, if we look at 0 < a < 1-- 00:26:06.600 --> 00:26:11.300 if a is between 0 and 1--it is a fraction--it is smaller than 1. 00:26:11.300 --> 00:26:15.700 Here are some examples: if we have 4/5^x, we see that one in red; 00:26:15.700 --> 00:26:22.900 1/2^x--we see that one in blue; 1/10^x--we see that one in green. 00:26:22.900 --> 00:26:30.500 Notice how quickly the functions become very small as they repeatedly lose value because of the fraction compounding on them. 00:26:30.500 --> 00:26:38.600 1/10 becomes very small by the time it has gotten to just 2; we have 1/10², which is equal to 1/100. 00:26:38.600 --> 00:26:44.000 So, it becomes very, very small: by the time we are at (1/10)^10, we are absolutely tiny. 00:26:44.000 --> 00:26:47.900 Once again, it looks like it touches the x-axis, but that is just because it is a picture. 00:26:47.900 --> 00:26:51.500 It never actually quite gets there; there is always a thin sliver of numbers between it. 00:26:51.500 --> 00:27:00.900 But it gets very, very, very close; they will all become very, very small as the fraction on fraction on fraction compounds over and over. 00:27:00.900 --> 00:27:05.700 Bits get eaten away each time the fraction hits, so it gets smaller and smaller and smaller. 00:27:05.700 --> 00:27:14.900 But notice: if we go the other direction, we end up getting very large, just like normal exponential functions that grow, where a was greater than 1. 00:27:14.900 --> 00:27:18.900 They got small when they went negative; they grew when they went positive, 00:27:18.900 --> 00:27:22.500 because when they went negative, they flipped; we have that same idea of flipping. 00:27:22.500 --> 00:27:30.700 If we have 1/10 to the -2, well, that is going to be 10/1 squared, which is equal to 100. 00:27:30.700 --> 00:27:37.300 And that is why we see it blow up so quickly--it becomes very, very large, because we go negative for decay things. 00:27:37.300 --> 00:27:41.300 But we will normally be looking at it as we go forward in time, which is why we talk about decay, 00:27:41.300 --> 00:27:44.300 and things that are greater than 1 being growth, because we are normally looking at it 00:27:44.300 --> 00:27:48.100 as we go forward--as we go to the right on our horizontal axis. 00:27:48.100 --> 00:27:49.800 All right, let's look at some examples. 00:27:49.800 --> 00:27:58.300 A bank account is opened with a principal of \$5000; the account has an interest rate of 4.5%, compounded semiannually (which is twice a year). 00:27:58.300 --> 00:28:00.600 How much money is in the account after 20 years? 00:28:00.600 --> 00:28:04.700 So, what do we need? We go back and figure out the function we are using. 00:28:04.700 --> 00:28:15.700 The formula is the one for interest compounded; so it is our principal, times 1 plus the rate, but divided by the number of times it occurs, 00:28:15.700 --> 00:28:21.400 and then also raised to the number of times it occurs in the year, times the number of years that pass. 00:28:21.400 --> 00:28:23.200 So, what are the numbers we are dealing with here? 00:28:23.200 --> 00:28:37.300 We have a principal of \$5000; we have a percentage rate of 4.5%, but we need that in decimal, so we have 0.045. 00:28:37.300 --> 00:28:42.100 And what is the amount of time? The amount of time is 20 years. 00:28:42.100 --> 00:28:49.800 If we do this with it going semiannually, twice a year, when we look at that, it will be n = 2. 00:28:49.800 --> 00:29:03.500 a at 20 =...what is our principal? \$5000, times 1 + the rate, 0.045 divided by the number of times it occurs in the year; 00:29:03.500 --> 00:29:11.500 it occurred twice; n = 2, so divide by 2; raise it to the 2, times how many years? 20 years. 00:29:11.500 --> 00:29:23.400 We go through that with a calculator; it comes out to 12175 dollars and 94 cents. 00:29:23.400 --> 00:29:30.500 Now, what if we wanted to compound more often--what if it had been compounded quarterly or monthly or daily or continuously? 00:29:30.500 --> 00:29:39.000 If it was compounded quarterly, it would occur four times in the year--every quarter of the year, every season--so n = 4. 00:29:39.000 --> 00:29:52.400 So, we have 5000 times 1 + 0.045/4; that will be 4 times 20; we use a calculator to figure this out. 00:29:52.400 --> 00:30:00.400 It comes out to 12236 dollars and 37 cents. 00:30:00.400 --> 00:30:05.500 So, notice that we end up making a reasonable amount more than we did when it was compounded just twice in the year. 00:30:05.500 --> 00:30:10.100 We are making about 50 dollars more--a little bit more than 50 dollars. 00:30:10.100 --> 00:30:13.600 What if we have it do it monthly? How many months are there in a year? 00:30:13.600 --> 00:30:16.700 There are 12 months in a year, so that would be n = 12. 00:30:16.700 --> 00:30:29.000 5000 is our initial principal, times 1 + our rate, over 12 (I am losing room)...12 to the t...12 times t; so what is our t? 00:30:29.000 --> 00:30:36.100 Our t was 20; sorry about that...12 times 20. 00:30:36.100 --> 00:30:45.000 That will come out to be 12277 dollars and 33 cents. 00:30:45.000 --> 00:30:49.100 What if we have it at daily--how many days are there in the year? 00:30:49.100 --> 00:30:55.100 There are 365 days in a year, so that will be an n of 365. 00:30:55.100 --> 00:31:11.400 So, at 365, we have 5000 times 1 + 0.045/365 (the number of times it occurs--365--the number of times it occurs in the year); 00:31:11.400 --> 00:31:21.000 we had 20 years total; we simplify that out; we get 12297 dollars and 33 cents. 00:31:21.000 --> 00:31:26.000 And what if we managed to do it every single instant--we actually had it compounding continuously? 00:31:26.000 --> 00:31:30.600 Well, if n is equal to infinity, we are no longer using this formula here. 00:31:30.600 --> 00:31:37.700 We change away from this formula, and we switch to the Pert formula, because that is what we do for compounded continuously. 00:31:37.700 --> 00:31:45.700 That is going to be 5000 times e; what is our rate? 0.045; how many years? 20 years. 00:31:45.700 --> 00:31:49.600 Once again, we punch that into a calculator: there will be an e key on the calculator-- 00:31:49.600 --> 00:31:55.200 you don't have to worry about memorizing that number that we saw earlier, because there is always an e key. 00:31:55.200 --> 00:32:06.000 5000 times...oops, let's just get a number here; we are not going to end up doing this number, because it would be hard to do. 00:32:06.000 --> 00:32:08.500 We will use a calculator; so let's just hop right to our answer. 00:32:08.500 --> 00:32:15.300 We get 12298 dollars and 2 cents. 00:32:15.300 --> 00:32:20.500 Finally, I would like to point out: notice that we ended up seeing reasonable amounts of growth 00:32:20.500 --> 00:32:25.400 when we jumped from going only semiannually (twice a year) to four times a year. 00:32:25.400 --> 00:32:33.200 And we also saw an appreciable amount of increase when we went from four times a year to twelve times a year-- when we went to monthly. 00:32:33.200 --> 00:32:35.700 We got a jump of a little over 40 dollars. 00:32:35.700 --> 00:32:40.100 When we managed to make it up to daily, we got a jump of about 20 dollars. 00:32:40.100 --> 00:32:44.400 But going from daily to every single instant forever only got us a dollar. 00:32:44.400 --> 00:32:50.400 So, we get better returns the more often it happens; but they end up eventually coming to an asymptote. 00:32:50.400 --> 00:32:56.000 It increases asymptotically to this horizontal...it eventually stabilizes at a single value. 00:32:56.000 --> 00:33:03.800 So, you won't see much difference between an account that compounds every single day and an account that compounds every single instant. 00:33:03.800 --> 00:33:05.900 There won't be a whole lot of difference. 00:33:05.900 --> 00:33:12.200 It is much better to have daily versus yearly, but daily versus continuously is not really that noticeable. 00:33:12.200 --> 00:33:16.000 The second example: The day a child is born, a trust fund is opened. 00:33:16.000 --> 00:33:19.300 The fund has an interest rate of 6% and is compounded continuously. 00:33:19.300 --> 00:33:23.900 It is opened with a principal of \$14000; what is the fund worth on the child's eighteenth birthday? 00:33:23.900 --> 00:33:26.700 What formula will we be using? We will be using Pert. 00:33:26.700 --> 00:33:34.100 The amount that we have in the end is equal to the principal that we started with, times e to the rate that we are at times t. 00:33:34.100 --> 00:33:43.100 What is our principal? Our principal is 14000 dollars. What is our rate? Our rate was 6%. 00:33:43.100 --> 00:33:49.800 We can't just use it as a 6; we have to change it to a decimal form, because 6 percent says to divide by 100; so we get 0.06. 00:33:49.800 --> 00:33:52.100 Finally, what is the amount of time that we have? 00:33:52.100 --> 00:33:58.800 In our first one, we are looking at a time of the eighteenth birthday--so 18 years; t = 18. 00:33:58.800 --> 00:34:09.900 A principal of \$14000 times e to our rate, 0.06, times the amount of years, 18 years-- 00:34:09.900 --> 00:34:21.700 we plug that into a calculator, and we see that, on his eighteenth birthday, the child has managed to get 41225 dollars and 51 cents. 00:34:21.700 --> 00:34:26.900 So, that is pretty good; but what if the child managed to not need the money--didn't really want the money-- 00:34:26.900 --> 00:34:33.100 wanted to save it and maybe use it to buy a house when he was 30 (or put down a good down payment on a house when he was 30)? 00:34:33.100 --> 00:34:38.900 At that point, if he was 30 before he took out the money, the child would have 14000; 00:34:38.900 --> 00:34:43.600 it is the same setup, but we are going to have a different number of years--times 30. 00:34:43.600 --> 00:34:52.300 That would end up coming out to 84000; it has more than doubled since he was 18--pretty good. 00:34:52.300 --> 00:34:57.100 So, it has more than doubled; he has managed to make \$84000 there. 00:34:57.100 --> 00:35:01.700 That is not bad--he could get a good down payment on a house with that, so it is pretty useful. 00:35:01.700 --> 00:35:04.700 But if he really didn't need the money--if he managed to not spend that money, 00:35:04.700 --> 00:35:10.400 and he said, "I will use it as a retirement fund; that way I won't have to invest for my retirement at all--I already have it set up." 00:35:10.400 --> 00:35:14.000 How much would he end up having at the age of 65? 00:35:14.000 --> 00:35:24.100 We have 14000--the same setup as before--times e to our rate, 0.06, times our new number of years we are doing--it is 65 years. 00:35:24.100 --> 00:35:33.800 And you would manage to have a huge 691634 dollars and 29 cents. 00:35:33.800 --> 00:35:37.100 So, this points out just how powerful compound interest was. 00:35:37.100 --> 00:35:41.700 We managed to start at 14000 dollars; but if we can avoid touching that money, 00:35:41.700 --> 00:35:48.600 if we can just leave it for a very long time, we can get to very large values as the interest compounds on itself over and over again. 00:35:48.600 --> 00:36:02.600 In 65 years, which is a very long time, we managed to grow from 14000 dollars to 691634 dollars--a lot of money. 00:36:02.600 --> 00:36:08.500 And this gives us an appreciation for how important it is to make investments for retirement at an early age. 00:36:08.500 --> 00:36:11.900 It is difficult when you are young; but if you manage to invest when you are young-- 00:36:11.900 --> 00:36:17.400 if you can wait on spending that money now--it can grow to very large amounts by the time you want to spend it to retire. 00:36:17.400 --> 00:36:21.000 So, that is the benefit of investing early--being able to do that. 00:36:21.000 --> 00:36:24.900 Also, it shows just how great, how useful, an interest rate is. 00:36:24.900 --> 00:36:29.300 If that 6% was bumped up to 8% or 10%, we would see massive increases. 00:36:29.300 --> 00:36:34.400 You can get a lot of increase if you can just get that percentage rate up another point or two--it is pretty impressive. 00:36:34.400 --> 00:36:39.000 All right, the third example: The population of yeast cells doubles every 14 hours. 00:36:39.000 --> 00:36:44.700 If the population starts with 100 cells, how many cells will there be left in two weeks? 00:36:44.700 --> 00:36:49.800 So, this isn't compound interest, and it isn't continual growth, like we had before. 00:36:49.800 --> 00:36:51.700 We might want to build our own here. 00:36:51.700 --> 00:36:56.700 The population is doubling, so let's say n is the number of cells after some time. 00:36:56.700 --> 00:37:00.900 We will set it up as a function--that makes sense; we are in "Exponential Function Land" right now. 00:37:00.900 --> 00:37:05.700 So, n(t) is equal to...well, how many cells did we start with? 00:37:05.700 --> 00:37:10.800 We started out with 100 cells, and we were told that it doubles. 00:37:10.800 --> 00:37:14.100 So, we are going to have some "times 2," because we multiply it by 2. 00:37:14.100 --> 00:37:17.400 How often does it do that? It does it every 14 hours. 00:37:17.400 --> 00:37:29.500 So, if we have our number of hours, t = number of hours, t divided by 14 will be how many times it has managed to double. 00:37:29.500 --> 00:37:32.100 After 14 hours, we have multiplied by 2 once. 00:37:32.100 --> 00:37:38.300 After 28 hours, we have multiplied by 2 twice; we have 2 times 2 at 28 hours. 00:37:38.300 --> 00:37:42.000 So, let's do a quick check and make sure that this is working out so far. 00:37:42.000 --> 00:37:53.100 So, if we had n at 14 hours, we would have 100 times 2^14/14, which would simplify to 100 times 2¹. 00:37:53.100 --> 00:37:56.100 So, we would get 200, so that part checks out. 00:37:56.100 --> 00:38:05.200 Let's try one more, just to be sure: n(28)...if we had double double, then we know that we should be at 400, so we can see what is coming there. 00:38:05.200 --> 00:38:15.000 So, 2 times 28/14...that simplifies to 100 times 2², which is equal to 100 times 4, or 400. 00:38:15.000 --> 00:38:20.600 So, that checks out, as well; it passes muster--this makes sense as a way of looking at things. 00:38:20.600 --> 00:38:24.600 So, as long as we have the amount of time we spent and the number of hours, 00:38:24.600 --> 00:38:28.200 then we can see how many cells we have after that number of hours. 00:38:28.200 --> 00:38:33.000 Now, we were told to figure out how many there will be in 2 weeks. 00:38:33.000 --> 00:38:36.200 And we can assume that none of the cells die off, so the number just keeps increasing. 00:38:36.200 --> 00:38:40.300 It is a question of how many times the population has gotten to double. 00:38:40.300 --> 00:38:44.100 If that is the case, what number are we plugging in--it is n of how many hours? 00:38:44.100 --> 00:38:50.300 Is it 2? No, no, it is not 2! Well, how many weeks...oh, 14 days? No, it is not 14. 00:38:50.300 --> 00:38:54.500 What were we setting this up in? t was set in number of hours. 00:38:54.500 --> 00:38:57.600 So, the question is how many hours we have on hand. 00:38:57.600 --> 00:39:04.100 Let's first see how many hours 2 weeks is: how many days is that? 00:39:04.100 --> 00:39:09.500 Well, that is going to be 2 times...how many days in a week?...7, so that is 2 times 7 days. 00:39:09.500 --> 00:39:20.900 How many hours is that? 2 times 7 times 24, or 14 times 24 hours, which we could then figure out with a calculator, and get a number of hours. 00:39:20.900 --> 00:39:25.500 But we can actually just leave it like that, which (we will see in just a few moments) is a useful thing to do, 00:39:25.500 --> 00:39:32.800 because we notice that there is a divide by 14 coming up; maybe it would be useful to just leave it as 14 times 24--a little less work for us. 00:39:32.800 --> 00:39:44.600 So, 14 times 24...now notice: 14 times 24 is the number of hours in 2 weeks. 00:39:44.600 --> 00:39:48.500 That is why we are plugging that in, because once again, the function we built, 00:39:48.500 --> 00:39:52.600 our n(t) function that we built, was based on hours going into it. 00:39:52.600 --> 00:39:55.100 We can't use any other time format. 00:39:55.100 --> 00:40:05.900 100 times 2 to the 14 times 24 (is the number for t), divided by 14; look at that--the 14's cancel out. 00:40:05.900 --> 00:40:08.500 We can be a little bit lazier--that is nice. 00:40:08.500 --> 00:40:25.000 100 times 2 to the 24: we plug that into a calculator, and we get a huge 1677721600 cells. 00:40:25.000 --> 00:40:32.600 That is more than one and a half billion cells: ones, thousands, millions, billions. 00:40:32.600 --> 00:40:39.800 So we are at 1.6 billion cells--actually, closer to 1.7 billion cells. 00:40:39.800 --> 00:40:43.500 This gives us a sense of just how fast small populations are able to grow. 00:40:43.500 --> 00:40:49.200 And that is how populations grow: they grow exponentially, because each cell splits in half. 00:40:49.200 --> 00:40:53.400 So, if we have one cell split in half to 2, and then each of those splits in half to 4, 00:40:53.400 --> 00:40:57.700 and each of those splits in half to 8, this is going to do this process of exponentiation. 00:40:57.700 --> 00:41:01.300 We are doing this through doubling, so we are going to see very, very fast growth. 00:41:01.300 --> 00:41:04.000 And we actually see this in the real world. 00:41:04.000 --> 00:41:13.800 We could also write this, for ease, as...we have 1, 2, 3, 4, 5, 6, 7, 8, 9...so that is the same thing... 00:41:13.800 --> 00:41:21.800 we could write it as approximately 1.67x10⁹ cells, 00:41:21.800 --> 00:41:25.700 so that we can encapsulate that information without having to write all of those digits. 00:41:25.700 --> 00:41:28.600 That is scientific notation for us; all right. 00:41:28.600 --> 00:41:35.100 The fourth and final example: The radioactive isotope uranium-237 has a half-life of 6.75 days. 00:41:35.100 --> 00:41:38.700 Now, what is half-life? We would have to go figure that out, but luckily, they gave it to us right here. 00:41:38.700 --> 00:41:48.300 Half-life is the time that it takes for one-half of the material of our isotope to decay and break down--to go through a process of decay. 00:41:48.300 --> 00:41:55.200 If you start with one kilogram of U-237, how much will have not decayed after a year? 00:41:55.200 --> 00:42:02.000 So, we are saying that, after 6.75 days, we will have half of a kilogram. 00:42:02.000 --> 00:42:08.900 We start with one kilogram, and we know that, after every 6.75 days, we will have lost half of our starting material. 00:42:08.900 --> 00:42:12.900 So, we will go down from one kilogram to half of a kilogram that has not decayed. 00:42:12.900 --> 00:42:17.500 So, let's see if we can figure out a way to turn this into another function. 00:42:17.500 --> 00:42:24.800 The...let's make it amount...the amount of our isotope that has not decayed, based on time, 00:42:24.800 --> 00:42:31.400 is equal to...how much did we start with? We started with 1 kilogram; times...what happens every cycle? 00:42:31.400 --> 00:42:39.100 1/2...we halve it every time we put it through a cycle; so how fast is a cycle? 00:42:39.100 --> 00:42:46.700 The number of days--we will make t into the number of days, because we see that we are dealing with days, based on this here. 00:42:46.700 --> 00:42:52.600 So, t divided by 6.75--let's do a really quick check. 00:42:52.600 --> 00:43:00.000 We check, because we know that, after 6.75 days, we should have 1/2 of a kilogram. 00:43:00.000 --> 00:43:12.000 So, let's check that by plugging it in: a(6.75) is going to be 1 times 1/2 raised to the 6.75 over 6.75, 00:43:12.000 --> 00:43:16.500 which is the same thing as just 1/2 to the 1, which equals 1/2. 00:43:16.500 --> 00:43:21.300 So sure enough, it checks out--it seems like the way we have set this up passes muster, 00:43:21.300 --> 00:43:24.600 because it is going to divide by half every time the 6.75 days pass. 00:43:24.600 --> 00:43:30.000 So, if we plugged in double 6.75, it would divide by half twice, because it would be 1/2 squared. 00:43:30.000 --> 00:43:36.100 It seems to make sense; we have set it up well; and we can see that this also can be just written as 1/2 times... 00:43:36.100 --> 00:43:41.100 let's just leave it as it is; it gives us a better idea of how this works in general, for half-life breakdowns. 00:43:41.100 --> 00:43:45.300 So, now we are going to ask ourselves how long--what is the time that we are dealing with? 00:43:45.300 --> 00:43:51.700 In our case, t is one year; what is one year in days (because we set up our units as days, 00:43:51.700 --> 00:43:54.100 because that is what our half-life was given to us in)? 00:43:54.100 --> 00:44:03.500 One year is 365 days; so at the end of that, we plug in 365 = 1 (the amount that we started with), 00:44:03.500 --> 00:44:12.000 times...the half-life will occur every 6.75 days (and we are still having 365 days go in). 00:44:12.000 --> 00:44:28.900 We plug that all into a calculator, and we get the amazingly tiny number of 5.273x10^-17 kilograms--a really, really, really small number. 00:44:28.900 --> 00:44:32.000 To appreciate how small that is, let's try to expand it a bit more. 00:44:32.000 --> 00:44:40.900 1 kilogram is 1000 grams; so that means that a kilogram is 10³ grams. 00:44:40.900 --> 00:44:55.300 We could also write this as 5.273x10...if it is 1000 grams for a kilogram, then that means we are going to increase by 3 in our scientific exponent. 00:44:55.300 --> 00:45:01.100 So, in the scientific notation, we are now at 5.237x10^-14 grams of our material. 00:45:01.100 --> 00:45:20.600 which, if we wanted to write this whole thing out...we would be able to write it as 0.00000 (five so far) 00000 (10 so far) 000 (13)... 00:45:20.600 --> 00:45:27.800 and let's see why that is the case--we can stop writing there--because if we were to bring that 10^-14 here, 00:45:27.800 --> 00:45:40.400 (and remember, it is in grams, because we had grams here), that would count as 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 00:45:40.400 --> 00:45:45.600 because we can move the decimal places 14 times to the right by having 10^-14. 00:45:45.600 --> 00:45:48.900 And that is how that scientific notation there is working. 00:45:48.900 --> 00:46:07.400 Or alternatively, we could also write this with kilograms as the incredibly tiny 0.00000000000000005273 kilograms. 00:46:07.400 --> 00:46:18.800 And if we counted that one out as well, we would have 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17... 00:46:18.800 --> 00:46:23.500 so we have that 5.273x10^-17 kilograms there, as well. 00:46:23.500 --> 00:46:28.200 So, it is much easier to write it with scientific notation; that is also probably what a calculator would put out, 00:46:28.200 --> 00:46:32.300 because it is hard to write a number like this, this long, on a calculator. 00:46:32.300 --> 00:46:39.500 So, we are much more likely to see it in scientific notation, 5.273x10^-17 kilograms, 00:46:39.500 --> 00:46:45.100 which is an absolutely miniscule amount of radioactive material left, considering that we started at 1 kilogram. 00:46:45.100 --> 00:46:46.800 That shows us how decay works. 00:46:46.800 --> 00:46:49.700 All right, cool: we have a pretty good base in exponential functions. 00:46:49.700 --> 00:46:54.100 Next, we will see logarithms, and see how logarithms allow us to flip this idea of exponentiation. 00:46:54.100 --> 00:46:59.300 And then, in a little while, we will see how logarithms and exponential functions...how we can oppose the two against each other. 00:46:59.300 --> 00:47:01.400 It is pretty cool--we can find out a lot of stuff with this. 00:47:01.400 --> 00:47:04.000 All right, we will see you at Educator.com later--goodbye!