Exponential Decay Functions

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Exponential Decay Functions

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In the last video on exponential functions, we saw
that functions of the form, y is equal to some
number to the x.
Say y is equal to 3x, which was what we
used in the last video.
We saw that these grow very, very, very quickly.
So the shape of the graph that we got in the last video looks
something like this.
It gets very close to 0, slowly pulls away from
0, and then, bam!
It starts to explode.
This, of course, is when x is equal to 0.
So the y-intercept there is y is equal to 1.
But the important thing is it just grows super fast. But
this was the case where the base in the exponential
function is greater then 1.
Here, it's 3.
If it's 2, it'll still grow quickly.
If it was 10, it would grow even faster.
What I want to talk a little bit about in this video, and
really just introduce you to the idea, is the notion of an
exponential decay function.
So this is an exponential growth function.
An exponential decay function has essentially, as its base,
a number less than 1.
So an example of an exponential decay function
would be y is to 1/3 to the x power.
This would be exponential decay.
And notice, another way I could write this, 1/3 is the
same thing as 3 to the negative 1.
So it's 3 to the negative 1 to the x power.
I could also write this as 3 to the negative x power.
This and this are equivalent.
That's another interesting way to think about exponential
decay functions.
It could either be kind of a standard positive exponent,
with a base that's less than 1, or you could have a base
larger than 1, but you're going to have a negative as
the exponent.
So let's plot this, just to get the general shape.
And the shape will always be consistent.
I'm going to do it for 1/3, but it's going to have a
similar shape for 1/2, or for 1/10, or for 1/100.
It'll just be scaled up or down.
So let's plot a few.
Let's have some x values and let's have some y values.
So let's start with some small values.
So let's say that x is equal to negative 3.
Well, what is 1/3 to the negative 3 power?
1/3 to the negative 3, that's the same thing as 3 to the
third power, which is equal to 81.
Let's do another point.
What about negative 2?
1/3 to the negative 2 power.
It's the same thing as 1 over-- Sorry, it's equal to 1
over 1/3 squared, which the same thing as 3 squared.
You just take the inverse of the number, get rid of the
negative and that is equal to equal to 9.
Oh, I got this one wrong up here.
3 cubed isn't 81.
That's 3 to the fourth.
3 times 3 times 3 is 27.
My brain is malfunctioning.
What about when x is equal to negative 1, then you have 1/3
to the negative 1 power.
which is just 3.
What about when x is equal to 0?
When x is equal to 0, remember, anything to the 0
power is 1.
And then let's go to a couple of larger numbers.
When x is equal to 1, what is y?
1/3 to the 1 power is 1/3.
Let's do one more.
When x is equal to 2, we have 1/3 squared, which is 1/9.
So let's plot this exponential decay function right here.
I want to draw it reasonably neatly.
And all of these values, once again, are positive because
our base was positive.
So it looks like that.
So let's say that this is 10, 20, 30.
Let's start at negative 3, negative 2,
negative 1, 0, 1, 2.
So when x was negative 3, y was 27.
Right about there.
When x was negative 2, y was 9.
Which is right about there.
When x is negative 1, y is 3.
Right about there. x is 0, y is 1.
x is 1, y is 1/3, until we decline like this.
Let me draw it a little bit neater than that.
The decay functions are harder for me to draw.
We'll draw it in reverse order.
No, that's even worse.
One more shot.
I think you get the general idea.
That's my best shot right there.
So you see, a decay function is kind of
reversed around the y-axis.
We start from a large quantity and we decay
very, very, very quickly.
So in either case, exponential functions, whether we have
exponential growth or exponential decay-- Let me
make sure that you understand this growth applies to this
thing up here.
This is exponential decay.
And either way, in exponential functions,
things happen quickly.
Now, using what we know about exponential decay functions,
let's do an actual word problem.
Let me read it here.
It says, a person is infected by a bacterial infection.
And they say, when he goes to the doctor, the population of
bacteria is 2 million.
So at the start, he's got 2 million little critters, 2
million bacteria.
The doctor prescribes an antibiotic which kills
bacteria, that reduces the bacteria to 1/4 of its
population each day.
So we have the antibiotic and it cuts down the population.
So it reduces population to 1/4 of the previous
population each day.
So they ask us a couple of things.
Draw the graph of the size of the bacteria population
against time in days.
So let's draw this.
So day, and then population.
So let's say on day 0, before you went to the doctor, your
population is going to be 2 million.
Now on day 1, what happens?
On day 1 you've taken their antibiotic, right?
The antibiotic is starting right there.
So it reduces it to 1/4 of your previous population.
So now your population is going to be 1/4 times 2
million, which is equal to 500,000.
Then on day 2, what happens?
On day 2, it's going to be 1/4 of this.
It's going to be 1/4 of your previous day.
1/4 of 1/4 of your previous 2 million.
So it's 1/4 of this 500,000 which is 125,000.
I think you're starting to see the pattern that's emerging.
On day 3, what's it going to be?
It's going to be 1/4 times this thing.
So it's going to be 1/4 times this 1/4 times 1/4 times 1/4
times 2 million.
Which is essentially 125,000 divided by 4, but the
important thing is, I want you to see this pattern.
In general, the population at day n-- Let me write
this in a new color.
The population on day n is going to be equal to what?
It's going to be equal to 1/4 to the n power, right?
This is day n.
On the third day, it was 1/4 to the third
power times 2 million.
On day 2, it was 1/4 to the second power.
On day 1 it was 1/4 to the first power.
On day 0, there is actually 1/4 to the 0 power, right?
That's 1.
So it's 1/4 to the nth power times our starting population,
times 2 million.
And if we were to graph it, using at least just the points
that we have here, we're only going to be dealing with
positive x's.
So on day 0, we start at 2 million.
So let's say this is a million.
So on day 0, we're at 2 million.
On day 1, we're at 1/4 of that.
So let's say that's 1 million, so then that is 500,000.
On day 2, we're 1/4 of this, so this is
250,000, that's 125,000.
And then on day 3, we're 1/4 of that.
So you see the population of our bacteria really goes down
very quickly.
It never gets quite to 0.
We're always going to have some bacteria.
Well, eventually you're going to have 1 bacteria, and you
can get the antibiotic.
And it's going to have a 1/4 chance of hitting it so,
maybe, eventually you'll get to 0, but it slowly gets to 0.
But the absolute number decreases very, very fast.
Now they also ask us, find the size of the bacteria
population 10 days after the drug was first taken.
So that's just n is equal to 10.
So it's going to be equal to 1/4 to the tenth
power times 2 million.
Let's take our calculator out to calculate to this.
So let's take 1 divided by 4 to the tenth power.
That's equal to this very small number times 2 million.
1, 2, 3, 1, 2, 3, is equal to-- we're only going to have
a 1.9 bacteria after the tenth day.
This is going to be equal to 1.9 bacteria, so pretty much
most of the bacteria is gone by the tenth day.