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Hello and welcome to this last lesson on Growth and Decay.

Exponential growth.

I hope you're ready to go.

We've got our pen and paper and we've got our calculator ready at hand, cause we're going to need that a lot today.

We're going to do a lot of things tending towards numbers and all sorts to do with our calculator and we're going to need to keep it.

Make sure you've got that at hand.

Make sure you've got that quiet space as well, that you're ready to go and that you're distraction free and that you are happy and ready to go.

So without further ado, let's take it away with Mr. Thomas' lesson.

So for our try this what I'd like us to consider is that if we started with one bacterium, how many would there be after one hour, if the bacterium increases by 100% every hour? 50% every half hour, 33.

3 recurring percent every third of an hour, 25% every quarter of an hour, etc.

Keep going with that pattern.

What do you notice? So pause video now, I'm going to give you.

Oh, how long should I give you? I'll give you seven minutes to have a go at that.

Off you go.

Great, so if we have a go at doing this now.

So we start off with one bacterium.

one times 100% every hour means that it's going to increase by two isn't it? So it's going to be two bacteria.

How about 50% every half hour? Well, what's going to happen there is you're going to start off with the one and then multiply by 1.

5 squared because you have 50% every half hour and there's two half hours in an hour.

So if we type that into our calculator, what do we get? We get, what do we get? You're going to beat me to it.

What do we get? We're going to get 2.

25 bacteria.

What about 33.

3 recurring every third of an hour? Well, if you had to type that into your calculator, you'd get, of course, one times.

Well, we don't really like writing out 33333 recurring.

So what we could do is, we could turn that into a fraction if we wanted to, couldn't we? Because we realised that 33.

3% recurring is a third and you can add that onto three thirds, couldn't you? Which would be a whole.

So you get four thirds.

So what we could start to do now, and be quite clever about it, and do four over three.

And then every third of an hour, would be cubed, wouldn't it? What does that equal to if you type that into your calculator? Lets have a look.

Remember those brackets for the whole thing, it's cubed.

And what we end up with is 2.

37037 etc.

25% every quarter of an hour.

What's happening here? 25% every quarter of an hour.

Well, that's going to be one times by, you could do 1.

25, that would work, or you could do four over, sorry five over four.

That could work as well.

So lets go with 1.

25, I quite like that one.

1.

25 and then to the power of four.

And what's that equal to? Type that into your calculator now have a go.

I've got an answer of 2.

441.

I hope you've got that as well.

I'm sure you did.

Etc, etc, etc.

What we're noticing then, what do you notice? Now you've got those results, I'll just really make them clear.

What do you notice about those results? They're tending towards a number, aren't they? They're getting.

If we were to plot them, they'd be looking like this, right? Going gradually they're getting towards a certain number, right? And we're going to explore that idea today.

What's actually going on with those numbers there? 'cause it's really really important you understand that.

So for our connect today, what I want us to explore is this idea that we had just a moment ago but this is a generalisation now I want us to start using.

So you've got this formula, and this is going to be really important to follow throughout.

And all I want you to do is to use that to then take it forward to independent tasks.

Because I think you're very capable now of doing table of values and all sorts of things relating to that.

So I don't think there's actually any real direct teaching I need to do, apart from what I did just a moment ago.

So have a go at the independent task that's coming up, with this in mind.

So we've got our independent task now.

So what I'd like you to do is have a go at those questions for the next 15 minutes please.

Make sure you have a go at them.

They are quite tricky, I'll give you that.

But I reckon you can do it 'cause you've kept up so far.

So pause the video now and have a go please.

Off you go.

Very good, let's go through those answers then.

So we've got our interphase x from one through to nine.

For what values of y? So if we had x is equal to one.

We'd have y is going to be equal to, what would it be? We can see from inspection that would be two of course.

x is equal to two here.

We'd have y is equal to, if we type that into a calculator it's going to be 2.

25 isn't it? x is equal to three.

We're going to have y is equal to something, What would it be? Have you got it? Have you typed it into your calculator? Are you going to beat me? In terms of my calculator speed.

Oh I think you beat me.

What is it? It's going to be 2.

37.

Might just give it a moment to get it right.

What about x is equal to four then? That one there, we're going to get y is equal to, type it into our calculator again.

That is going to be 2.

44.

What about x is equal to five.

y would be equal to, again, you probably beat me already, you're shouting out the answer at your screen.

What is it? 2.

488.

x is equal to six, would be y is equal to, what would that one be? We haven't actually explored that one yet.

That'll be 2.

52.

The next one will be to the power of seven.

That one there will be 2.

55, 2.

46.

There we go, yeah.

x is equal to seven.

y is equal to 2.

546.

We can then have all this.

But the point being, is that how can we organise that into a table? Well, let's have a go at organising it into a table, 'cause this is a little bit messy at the moment, isn't it? Right, so if we have x there and y there.

Apologies for the poorly drawn line there.

But I've got x is equal to one, y is two.

So that's how I organise it into that table, don't I? Two is going to go here, 2.

25.

I'm then going to have x is three, four, five, six, seven and eight.

So you can see it just directly below me, I've got that table of values now.

So then for x is three, I can say 2.

37.

I can then say for x is four, 2.

544.

2.

44, my apologies.

And then 2.

488.

And then number six.

So x is six, it'll be 2.

52.

And then for seven, 2.

546 and then you're going to have it less squashed of course.

And then for eight, it would be 2.

565.

And then for nine, would be equal to, for nine it would be 2.

58.

So we can say that with our tables of values there, it should have been a bit more stretched out, as you can see but it's a lot easier to organise and see as it goes up.

Now, what happens if we try to do it when we got x is zero, well, we get a maths error.

So that is a maths error.

So we can't do that because we cannot divide by zero.

Do you see that? It should result in a maths error or syntax error.

So that's a maths error because we cannot divide by zero.

So what can we say about the results of the graph.

Well, we can say it looks exponential isn't it? It keeps increasing.

But what do we notice? Well, if we draw our graph like that, we can see that when x is one, y is two.

So let's say it's going to be there.

x is two, y is 2.

25.

And it just gradually tails off.

And like I said it tends to a number, doesn't it? So that's what our graph looks like to an certain extent and that's going to be two of course etc.

So you're going to have a more accurate version of that, right? And then with our calculator, find the value of the following, 1.

01 to the power of 100.

That would be 2.

704813829 etc.

I trust you can do that into your calculator.

Eventually we get to that last one.

In the interest of time, I'm not going to do all of them but getting to that last one.

One, two, three, four, five there.

We get 2.

718268237.

So we can see there's quite a lot there.

And then you can continue that and you'll eventually get to this sort of magical number here very close to that one.

And that, if you type into your calculator, it may vary from calculator to calculator.

But if you do shift and then e, if you have this one here, and then raise it to the power of one, we get this exponential function.

And this comes up a lot in A-level maths whereby we're exploring these curves that go increase exponentially.

So it's really important you understand that if you want to go on to do a further study of this.

So for our explore today, I want us to think about that number we just discovered, which is e and how it relates to here.

So its quite hard to spot patterns going on here but I'm sure you can have a go at it, so have a go.

Pause the video now.

And have a go.

Amazing, lets go further.

So for this one, what I'd like you to consider is the fact that we started with one bacterium.

And then we increase every one hour.

So what we notice is this one here, we're increasing it by, if you notice compared to the other one we had, right? Right at the back with the try this.

That tended to our number e.

So the try this tended towards e.

That's what we had right.

What about now? Where we've essentially got to increase it even further, more exponential.

Well, this one is going to be, if you compared the numbers, it's going to be e to the power of two, right? 200% every hour? Well, we're going to have one in a bracket of course, one plus two which gives us and then squared, sorry raised to the power of one, gives us three.

We then going to get 100% every half hour.

Well, that's going to give us one plus one raised to the power of two of course, which gives us four.

We then going to have 1 plus two thirds raised to the power of three.

Which is going to give us 4.

6296 etc.

And then if we do e to the power of two, what we get is 7.

389.

And we can see it increases if we keep following that pattern towards this number here.

Equally, you're now probably realising, Oh brilliant, I can see a pattern here.

This one would be e cubed, and this one would be e to the power of four.

I'll let you explore that 'cause that's a really, really interesting concept to be able to see how exponential can get.

So with that, that's the end of our series on exponential growth and decay, everything.

I just want to say a really big congratulations 'cause some of that is really tricky to do.

Make sure you do that exit quiz and that you smash your learning and prove to us just how much you can do and how much you learnt.

So from me, it's goodbye.

And I shall hopefully see you soon in another series.

Take care for now, bye bye.