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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.

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