WEBVTT mathematics/pre-calculus/selhorst-jones
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Hi--welcome back to Educator.com.
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Today, we are going to talk about exponential functions.
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Previously, we spent quite a while looking at functions that are based around a variable raised to a number--
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things like x² or x^47; this is basically the idea of all of those polynomials we have worked with for so long.
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But what if we took that idea and flipped it?
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We could consider functions that are a number raised to a variable, things like 2^x or 47^t,
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where we have some base number that has a variable as its exponent.
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We call functions of this form **exponential functions**, and we will explore them in this lesson.
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Now, make sure that you have a strong grasp on how exponents work before watching this.
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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.
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All right, an exponential function is a function in the form f(x) = a^x,
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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.
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We call a the **base**: base is just the name for the thing that is being raised to some exponent.
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So, whatever is being exponentiated--whatever is going through this process of having an exponent--that is called the base,
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because it forms the base, because it is below the exponent.
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We might wonder why there are all of these restrictions on what a can be; well, there are good reasons for each one.
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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.
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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.
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So, we are not going to consider that case.
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If a equals 0, the function wouldn't be defined for negative values of x.
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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.
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So, that is not allowed, so that means a = 0--once again, we are not going to allow that one.
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And if we had a < 0, then the function wouldn't be defined for various x-values, like x^1/2.
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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.
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So, the square root of -4...we can't take the square root of a negative, because that produces imaginary numbers;
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and we are only dealing with real numbers--we are not dealing with the complex numbers right now.
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So, we are going to have to ban anything that is less than 0.
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And that is why we have this restrictions: our base has to be greater than 0, and is not allowed to be 1,
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because otherwise things break down for the exponential function.
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All right, notice that, from the previous lesson, we can compute the value of a given base raised to any exponent.
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We know how exponents work when they are a little more complex (not complex numbers, but just more interesting).
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And so, we can raise things like...4^3/2 = (√4)³,
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which would be equal to...√4 is 2; 2³, 2 times 2 times 2, gives us 8; great.
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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.
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So, 1 squared is 1; 7 squared is 49; we get 1/49.
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So, we can do these things that are a little more difficult than just straight positive integers.
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But we might still find some calculations difficult, like if we had 1.7^6.2--that would probably be pretty hard to do.
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Or (√2)^π--these would be really difficult for us to do.
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So, how do we do them? In practice, we just find these expressions, or a very good approximation, by using a calculator.
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We can end up getting as many digits in our decimal expansion as we want.
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We can just find as many as we need for whatever our application is--whatever the problem asks for--by just using a calculator.
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Any scientific or graphing calculator can do these sorts of calculations.
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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.
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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),
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then that would be equivalent to us saying 3⁶.
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The carat is saying "go up," so the calculator would interpret 3⁶ as 3⁶.
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There are various ways, depending on if you are using a scientific calculator,
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or if you are using a graphing calculator, to put these things into a calculator and get a number out.
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So, we are able to figure these things out, just by being able to say "use a calculator."
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Now, from a mathematical point of view, that is a terrible statement.
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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?
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Calculators didn't just spawn into existence and give us the answers.
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We can't rely on our calculators to do our thinking for us; we have to be able to understand what is going on.
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Otherwise, we don't really have a clue how it works.
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But as you will see as you get into more advanced math classes, there are methods to figure out these values.
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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.
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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.
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We could do it, but that is what calculators are for; they are to do lots of calculations very quickly.
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They are to help us get through tedious arithmetic.
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So, since these sorts of calculations take all of this arithmetic, we designed calculators that can do this method for us.
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And that is why we can appeal to a calculator--not because the calculator knows more than us,
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but because, at some point, humans figured out a method to get as many decimals as we wanted to;
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and then, we just built a machine that is able to go through it quickly and rapidly,
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so we can get to the thing that we want to look at, which is more interesting, using this.
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The calculator is a tool; but it is important to realize that we are not just relying on it because it has the knowledge.
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We are relying on it because, at some point, we built it and put these methods into it.
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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.
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All right, now, if we can evaluate at any place--if we can compute what these values of exponential functions are--
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then we can make a graph, because we can plot as many points as we want; we can draw a smooth curve.
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So, let's look at some graphs where the base is greater than 1.
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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.
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Now, notice: 2^x, 5^x, and 10^x--all of these end up going through 1, right here,
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because what is happening there is that 2⁰, 5⁰, 10⁰...anything raised to the 0--they all end up being 1.
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Remember, that is one of the basic properties of exponents.
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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.
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And notice that they get very large very quickly.
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By the time 2 is to the fourth, it is already off; and 10 is off by the time it gets to the 1.
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10^x grows very quickly, because it is multiplying by 10, each step it goes forward.
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Notice also: as we go far to the left, it shrinks very quickly.
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Let's consider 10^-3; 10^-3 would be the same thing as 1/10³, which would be 1/1000.
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That is why we end up seeing that this green line is so low.
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It looks like it is almost touching the x-axis; it isn't quite--there is this thin sliver between it.
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But it is being crushed down very, very quickly, because of this negative exponent effect,
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where it gets flipped over, and then it has a really, really large denominator very quickly.
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So, we see, as we go to the left side with these things, that it will crush down to 0.
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And as we go to the right, it becomes very, very big.
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We can change the viewing window, so that we can get a sense for just how big these things get.
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And look at how big: we have gotten up to the size of 1000 by the time we are only out to 10.
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And that is on 2^x; if we look at 10^x, 10^x has already hit 1000 at 10³.
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At x = 3, it has managed to hit 1000 as its height.
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This stuff grows really quickly; this idea of massive growth is so central to the idea of exponential functions.
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We are going to have a story: there is this story that often gets told with exponential functions,
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because it is a great way to get people to understand just how big this stuff gets.
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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.
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The king of the land loved the game of chess so much that he offered the mathematician any reward that the mathematician desired.
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The mathematician was clever, and told the king humbly, "Your Highness, I thank you;
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all I ask for is a meager gift of rice, given day by day on a chessboard."
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"Tomorrow, I would like a single grain of rice give on the first square;
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on the next day, two grains of rice given on the second square; then on the following day, the third day,
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four grains of rice; and so on and so forth, doubling the amount every day until all 64 squares are filled."
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So, the mathematician is asking for the first square, doubled, doubled, doubled, doubled...
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The mathematician drew the king a diagram to help make his request clear.
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On the first day of his gift, he would end up having one grain of rice on the first square.
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On the second day, there would be a total of two grains of rice (1 times 2 becomes 2).
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On the third day, there would be a total of 4 grains of rice (2 times 2 becomes 4).
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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,
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going all the way out to the 64th day, doubling each time we go forward a square on the board.
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The king was delighted by the humble request and agreed to it immediately.
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Grains of rice? You can't get a lot of grains of rice on a single chessboard; "It will be very easy," he thought.
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He ordered that the mathematician would have his daily reward of rice delivered from the royal treasury every day.
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A week later, the king marveled at how the mathematician had squandered his reward.
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After all, he only had to send him 2⁶ = 64 grains of rice that day.
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Notice: on the seventh day, we are at 2⁶--let's see why that is.
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On the first day, we have 1 grain; on the second day, we have 2 grains; on the third day, we have 4 grains.
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On the fourth day, we have 8 grains; on the fifth day, we have 16 grains; on the sixth day, we have 32 grains.
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And thus, on the seventh day, we have 64 grains.
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So, notice: we can express this as 2⁰, 2¹, 2², 2³, 2⁴, 2⁵, and then finally 2⁶ on the seventh day.
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Why is this? Because on the very first day, he just got one grain.
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Every following day, it multiplies by 2--it doubles.
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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.
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So, in general, it is going to be 2 to the (number of day minus 1).
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We will subtract one to figure out the grains on some number of day.
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So now, we have an idea of how we can calculate this pretty quickly and be able to get these things figured out.
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Another week later, on the fourteenth day, the king sent him 2^13 (remember, it is the fourteenth day,
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so we go back one, because it has been multiplied 13 times) grains of rice, which is 8,192 grains.
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And 8,192 grains is just about a very large bowl of rice.
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The king was still amazed at the fantastic deal he was getting.
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But he was glad that the mathematician was at least seeing some small reward.
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He loved the game of chess, after all; and if he ended up feeding the mathematician for a year, that was great.
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It seemed like a wonderful deal; he was willing to give him palaces, jewels, and massive amounts of money.
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He can give him a little bit of rice for the great game of chess.
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At the end of the third week, on the twenty-first day, the king had to send the mathematician a full bag of rice,
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because in the kingdom, a full bag of rice contained precisely 1 million grains.
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So, on the 21st day, we have 2^20 grains of rice, which ends up being 1 million, 48 thousand, 576 grains.
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So we see here: after we jump these first six digits, we have one million plus grains of rice.
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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.
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So, perhaps the mathematician was not as foolish as the king had first thought.
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At the end of the fourth week, the king was starting to get worried.
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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.
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So, we are starting to get to some pretty large amounts here.
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Now, the royal treasury has a lot of rice; he is not worried--he has hundreds of thousands of bags of rice.
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So, he is not too worried about it; but he sees that this is starting to grow quite a bit.
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At that moment, the royal accountant bursts into the throne room and says, "Your Highness, I have grave news!
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The mathematician will deplete the royal treasury! On the forty-first day alone, we would have to give one million bags of rice!"
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because 2^40 is here, so we have one million million grains of rice; so we have one million bags of rice,
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which is more than the entirety that the treasury has in rice.
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"And if we kept going--if we let it run all the way to the sixty-fourth day,
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we would have to send him more rice than the total that the world has ever produced,
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because we would be at 2^63, which would come out to be 9 trillion bags of rice."
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Look; we have ones here; we have thousands here; we have millions here, billions here, trillions here, quadrillions here;
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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.
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That is a lot of rice, and the world doesn't have that much by far.
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So, the mathematician's greed has enraged the king, and the king immediately orders all shipments of rice stopped.
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The mathematician is not getting any more rice, and the mathematician is to be executed!
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Now, the mathematician, being a clever fellow, hears the soldiers coming down the road, and he escapes.
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He fled the kingdom with the few bags of rice that he could manage to carry on his back,
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and he had to find a new place to live, far, far away from the kingdom.
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So, the moral of the story is twofold: first, don't be overly greedy--don't try to trick kings.
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But more importantly than that, exponential functions grow really, really large in a short period of time.
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They get big fast; even if they start at a seemingly very, very small, miniscule amount, they will grow massive if given enough time.
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So, that is the real take-away here from this story.
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Exponential functions get big; they can start small, but given some time, they get really, really big.
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All right, let's see an application of this stuff.
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When you put money in a bank, they will usually give you interest on your money.
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For example, if you had an annual interest rate of 10% (annual just means yearly) on a $100 principal investment
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(the amount that you put in the bank), the following year you would have that $100 still
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(they don't take it away from you), plus $100 times 10%.
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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.
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Great; but you could leave that interest in the account, and then your interest would also gain interest.
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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.
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So, you have $110 in your bank account now, because you had $110 total at the end last time.
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$110 gets hit by that 10% again; so you still have the $110, plus...now 10% of $110 is $11.
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Notice that $11 is bigger than 10--your interest is growing.
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Over time, you are getting more and more interest as you keep letting it stay in there.
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You continue to gain larger and larger amounts with each interest.
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Compound interest is a common and excellent way to invest money, because over time,
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your interest gains interest, and gains interest, and gains interest.
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And eventually, it can manage to get large enough to be even larger than the principal investment,
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and be the thing that is really earning you money--the time that you have spent letting it compound.
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We can describe the amount of money, A, in such an account with an exponential function: A(t) = P[(1 + r)/n]nt.
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Let's unpack that: P is the principal in the account--the amount that is originally placed in the account.
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So, in our example, that would be $100 put in; so our principal would be 100 in that last example.
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r is the annual rate of interest, and we give that as a decimal: here is our r, right up here.
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In the last one, that was 10%, so it was expressed as .10.
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n is the number of times a year that the interest compounds; n is the number of times that we see compounding.
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So, n = 1 would be yearly; n = 4 would be quarterly; n = 12 would be monthly; n = 365 would be daily.
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In our last one, it compounded annually, every year; so it compounded just once a year, so n was equal to 1.
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Notice that n also shows up up here; it is n times t.
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And then finally, t is just the number of years that we have gone through; so it is times t.
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So, let's understand why this is the case.
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Well, if we looked at 10%, just on the $100, we would have $100 times 1 + 10%.
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So, $100 times 1.1 equals $110.
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Now, if we wanted to have this multiple times, well, the next time it is $110 times 1.1, again.
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We would get another number out of it; and then, if we wanted to keep hitting it...
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we can just think of it as (100^1.1)^t, and that will just give us the amount of times
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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!