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- In this video, I want to introduce you to the idea of
- an exponential function and really just show you how fast
- these things can grow.
- So let's just write an example exponential function here.
- So let's say we have y is equal to 3 to the x power.
- Notice, this isn't x to the third power, this
- is 3 to the x power.
- Our independent variable x is the actual exponent.
- So let's make a table here to see how quickly this thing
- grows, and maybe we'll graph it as well.
- So let's take some x values here.
- Let's start with x is equal to negative 4.
- Then we'll go to negative 3, negative 2, 0, 1, 2, 3, and 4.
- And let's figure out what our y-values are going to be for
- each of these x-values.
- Now, here, y is going to be 3 to the negative 4 power, which
- is equal to 1 over 3 to the fourth power.
- 3 to the third is 27 times 3 again is 81.
- So this is equal to 1/81.
- When x is equal to negative 3, y is 3.
- We'll do this in a different color.
- This color is hard to read.
- y is 3 to the negative 3 power.
- Well, that's 1 over 3 to the third power,
- which is equal to 1/27.
- So we're going from a super-small number to a less
- super-small number.
- And then 3 to the negative 2 power is
- going to be 1/9, right?
- 1 over 3 squared, and then we have 3 to the 0 power, which
- is just equal to 1.
- So we're getting a little bit larger, a little bit larger,
- but you'll see that we are about to explode.
- Now, we have 3 to the first power.
- That's equal to 3.
- So we have 3 to the second power, right? y is equal to 3
- to the second power.
- That's 9.
- 3 to the third power, 27.
- 3 to the fourth power, 81.
- If we were to put the fifth power, 243.
- Let's graph this, just to get an idea of how
- quickly we're exploding.
- Let me draw my axes here.
- So that's my x-axis and that is my y-axis.
- And let me just do it in increments of 5, because I
- really want to get the general shape of the graph here.
- So let me just draw as straight a line as I can.
- Let's say this is 5, 10, 15.
- Actually, I won't get to 81 that way.
- I want to get to 81.
- Well, that's good enough.
- Let me draw it a little bit differently
- than I've drawn it.
- So let me draw it down here because all of these values,
- you might notice, are positive values because I have a
- positive base.
- So let me draw it like this.
- Good enough.
- And then let's say I have 10, 20, 30, 40, 50, 60, 70, 80.
- That is 80 right there.
- That's 10.
- That's 30.
- That'll be good for approximation.
- And then let's say that this is negative 5.
- This is positive 5 right here.
- And actually, let me stretch it out a little bit more.
- Let's say this is negative 1, negative 2, negative 3,
- negative 4.
- Then we have 1, 2, 3, and 4.
- So when x is equal to 0, we're equal to 1, right?
- When x is equal to 0, y is equal to 1, which is right
- around there.
- When x is equal to 1, y is equal to 3, which is right
- around there.
- When x is equal to 2, y is equal to 9, which is right
- around there.
- When x equal to 3, y is equal to 27, which is
- right around there.
- When x is equal to 4, y is equal to 81.
- You see very quickly this is just exploding.
- If I did 5, we'd go to 243, which wouldn't
- even fit on my screen.
- When you go to negative 1, we get smaller and smaller.
- So at negative 1, we're at 1/9.
- Eventually, you're not even going to see this.
- It's going to get closer and closer to zero.
- As this approaches larger and larger negative numbers, or I
- guess I should say smaller negative numbers, so 3 to the
- negative thousand, 3 to the negative million, we're
- getting numbers closer and closer to zero without
- actually ever approaching zero.
- So as we go from negative infinity, x is equal to
- negative infinity, we're getting very close to zero,
- we're slowly getting our way ourselves away from
- zero, but then bam!
- Once we start getting into the positive
- numbers, we just explode.
- We just explode, and we keep exploding at an
- ever-increasing rate.
- So the idea here is just to show you that exponential
- functions are really, really dramatic.
- Well, you can always construct a faster expanding function.
- For example, you could say y is equal to x to the x, even
- faster expanding, but out of the ones that we deal with in
- everyday life, this is one of the fastest. So given that,
- let's do some word problems that just give us an
- appreciation for exponential functions.
- So let's say that someone sends out a chain
- letter in week 1.
- In week 1, someone sent a chain letter to 10 people.
- And the chain letter says you have to now send this chain
- letter to 10 more new people, and if you don't, you're going
- to have bad luck, and your hair is going to fall out, and
- you'll marry a frog, or whatever else.
- So all of these people agree and they go and each send it
- to 10 people the next week.
- So in week 2, they go out and each send
- it to 10 more people.
- So each of those original 10 people are each sending out 10
- more of the letters.
- So now 100 people have the letters, right?
- Each of those 10 sent out 10, so that 100
- letters were sent out.
- 10 were sent.
- Here, 100 were sent.
- In week 3, what's going to happen?
- Each of those 100 people who got this one, they're each
- going to send out 10, assuming that everyone is really into
- chain letters.
- So 1,000 people are going to get it.
- And so the general pattern here is, the people who
- receive it, so in week n where n is the week we're talking
- about, how many people received the letter?
- In week n, we have 10 to the nth people receive-- i before
- e except after c-- the letter.
- So if I were to ask you, how many people are getting the
- letter on the sixth week?
- How many people are actually going to receive that letter?
- Well, what's 10 to the sixth power?
- 10 to the sixth is equal to 1 with six zeroes, which is 1
- million people are going to receive that letter in just 6
- weeks, just sending out 10 letters each.
- And obviously, in the real world, most people chuck these
- in the basket, so you don't have this good of a hit rate.
- But if you did, if every 10 people you sent it to also
- sent it to 10 people and so on and so forth, by the sixth
- week, you would have a million people.
- And by the ninth week, you would have a billion people.
- And frankly, the week after that, you
- would run out of people.
- I'll see you in the next video.

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