Derivatives of exponential functions
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𝑒 and compound interest
Let's say that you are desperate for a dollar. So you come to me the local loan shark, and you say hey I need to borrow a dollar for a year. I tell you I'm in a good mood, I willing to lend you that dollar that you need for a year. I will lend it to you for the low interest of 100% per year. 100% per year. How much would you have to pay me in a year? You're going to have to pay the original principal what I lent you plus 100% of that. Plus one other dollar. Which is clearly going to be equal to $2. You say oh gee, that's a lot to have to pay to pay back twice what I borrowed. There's a possibility that I might have the money in 6 months. What kind of a deal could you get me for that Mr. Loanshark? I say oh gee, if your willing to pay back in 6 months, then I'll just charge you half the interest for half the time. You borrow one dollar, so in 6 months, I will charge you 50% interest. 50% interest over 6 months. This, of course, was 1 year. How much would you have to pay? Well, you would have to pay the original principal what you borrowed. The one dollar plus 50% of that one dollar. Plus 0.50, and that of course is equal to 1.5. That is equal to $1.50. I'll just write it like this. $1.50. Now you say well gee that's I guess better. What happens if I don't have the money then? If I still actually need a year. We actually have a system for that. What I'll do is just say that okay, you don't have the money for me yet. I'll essentially ... we could think about it. I will just lend that amount that you need for you for another 6 months. We'll lend that out. We'll lend that out for another 6 months at the same interest rate at 50% for the next 6 months. Then you'll owe me the principal a $1.50 plus 50% of the principal, plus 75 cents. Plus 75 cents, and that gets us to $2.25. That equals $2.25. Another way of thinking about it is to go from $1 over the first period, you just multiply that times 1.5. If your going to grow something by 50%, you just multiply it times 1.5. If your going to grow it by another 50%, you can multiply by 1.5 again. One way of thinking about it that 50% interest is the same thing as multiplying by 1.5. Multiplying by 1.5. If you start with 1 and multiply by 1.5 twice, this is going to be the same thing. $2.25 is going to be 1 multiplied by 1.5 twice. 1.5 multiplied twice is the same thing as 1.5 squared. You can see the same thing right over here. This is the same thing. 100% is the same thing as multiplying by 2. As we be multiplying 1 plus 1. This is multiplying by 2, so you could do this right over here. You could do this as 1 times 1. 1 times 1 to the first ... I'm sorry 1 times 2 to the first power , because your only doing it over one period over that year. You say once again where's that 2? Well, if someone is asking for 100%, that means over the period you're going have to pay twice. You're going have to pay the principal plus 100%. You're going have to pay twice what you originally borrowed. If someone is charging you 50% over every period, you're going have to pay whatever you borrowed. That's kind of the one part plus 50% of it. So 1.5 times what you borrowed. You multiply times 1.5 every time. If you wanted to see how this actually related to the interest, you could view this as ... this right over here is equal to 1 times, the interest part is 1 plus 100% divided by 1 period to the first power. I know this seems like a crazy way of rewriting what we just wrote over here. Writing 1 plus 1, but you'll see that we can keep writing this as we compound over different periods. This one right over here, we can rewrite. We can write as 1 times 1 plus 100%. Here we took our 100% for the year, and we divided into 2 periods. Two 6 month periods. Each of them at 50%. 1 plus 100% over 2 is the same thing as 1.5, and we compounded it over 2 periods. Let me do that 2 periods into a different color. The periods, let me do in this orange color right over here. You might start to see a pattern forming. Let's say, well gee, I might have the money back in ... and you don't really like this. This is $2.25. That was more than the original $2, so you say, well what if we do this over every 12 months. I say, "Sure. We got a program for that." After every 12 months ... or after every month I should say, I'm just going to charge you 100% divided by 12 interest. This is equal to 8 1/3%. Having to pay back the principal plus 8 1/3%, that's the same thing as multiplying times 1.083 repeating. After 1 month you would have to pay 1.083 repeating. After 2 months ... and this isn't the scale that actually looks more than 2 months, but it's not completely at scale. After 2 months your going have to multiply by this again. Times 1.083 repeating, and so that would get you 1.083 repeating squared. If you went all the way down 12 months ... let me get myself some space here. If you went all the way down 12 months ... let me just. I should way from the beginning 12 months, so another 10 months. What's the total interest you would have to pay over a year if you weren't able to keep coming up with the money? If you had to keep re-borrowing it. I kept compounding that interest. Well, you're going have to pay 1.083 to the ... this is for 1 month. You could view this as to the first power. This is for 2 months, so you're going have to pay this to the 12th power. We have compounded over 12 periods, 8 1/3% over 12 periods. If you wanted to write it in this form right over here, this would be the same thing as the original principal. Our original principal times 1 plus 100% divided by 12. Now we've divided our 100% into 12 periods, and we're going to compound that 12 times. We're going to take that to the 12th power. What is this going to equal to? This buisness over here. We can get a calculator out for that. I'll get my TI-85 out. What is this going to be equal to? We could do it a couple of ways. This is 1.083 repeating. Let's get our calculator out. We could do it a couple of ways. Let me write it this way. Your going to get the same value. I don't have to rewrite this one. I just did that there to kind of hopefully you'd see the kind of structure in this expression. 1 plus ... 100% is the same thing as 1. 1 divided by 12 to the 12th power. 2.613, I'll just round. So approximately 2.613. You say well this is an interesting game you all most forgot about your financial troubles, and you're just intrigued by what happens if we keep going this. Here we compounded just ... we have 100% over here. Here we do 50% every 6 months. Here we do a 12th of 100%, 8 1/3% every 12 months until we get to this number. What happens if we did every day? Every day. If I borrowed a one dollar, and I'd say well gee I'm just going to ... each day I'm going to charge you charge you one three hundred sixty-fifth of a 100%. So, 100% divided 365, and I'm going to compound that 365 times. You're curious mathematically. You say well, what do we get then? What do we get after a year? You have your original principal. Let me scroll over a little bit more to the right, so we have more space. You're going to have your original principal times 1 plus 100% divided by not 12. Now we've divided the 100% into 365 periods. 365 periods. We're going to compound it. Every time we have to multiply by 1 plus 100% over 365 everyday that the loan is not paid. 365th power. You say oh gee taking somebody to 365th power that's going to give me some huge number. Then you say well maybe not so bad, because 100% divided by 365 is going to be a small number. This thing is going to be reasonable close to 1. Obviously, we can raise 1 to whatever power we want, and we don't get anything crazy. Let's see where this one goes. Let's see where this one goes. This is the same thing as 1 plus. 100% is the same thing as 1 divided by 365 to the 365th power. We get 2.71456. Let me put it over here. Then we get ... This is approximately equal to ... this approximate is a very precise approximation, but 2.7 ... but my calculators precision only goes so far. 2.7145675 and it keeps going on and on. This is really really interesting. It looks as if we take larger and larger numbers here, it just doesn't just balloon into some crazy ginormous number. It seems to be approaching some magical and mystical number. It is, in fact, the case. That if you would just take larger and larger ... if you were to take your 100% and divide by larger and larger numbers, but take it to that power, you're going to approach perhaps the most magical and mystical number of all. The number E. You can see it right over here in your calculator. They have this E to the X. I can do that, so E to the ... I'll raise it to the first power so you can look at the calculators internal representation of it. You see all ready raising some ... doing 1 plus 1 over 365 to the 365th power, we got pretty ... we're starting to get really really close to E. I encourage you try this with larger and larger numbers, and your going to get closer and closer to this magical mystery. You almost wouldn't mind paying the loan shark E dollars, because it's such a beautiful number.