Exponential growth and decay
Exponential Growth Exponential growth involving bacteria (some calculus in part c).
⇐ Use this menu to view and help create subtitles for this video in many different languages.
You'll probably want to hide YouTube's captions if using these subtitles.
- Let's do a problem on exponential growth.
- And if you think about it, what we're going to do here with a
- bacteria, it's no different than what we did with compound
- interest when we learned about interest and the number e.
- Because if you think about it-- well, I'll draw the analogy
- as we do this problem.
- Anyway, this is a compound growth problem, or an
- exponential growth problem.
- And it says: a bacteria culture initially contains 100
- cells and grows at a rate proportional to its size.
- After an hour, the population has increased to 420.
- So this first says find an expression for the number
- of bacteria after t hours.
- So in general, anything that-- When you're dealing with
- exponential growth or exponential decay.
- Or continuously compounding growth or continuously
- compounding decay, though you'll very seldom
- hear that term.
- The amount of whatever you have as a function of time will
- be this type of equation.
- I'll say bacteria as a function of time.
- So I'll say b of t.
- And I'll do it here.
- So b of-- Well, let me do it here. b of t.
- Bacteria as a function of time is going to be equal to the
- initial bacteria, or the initial of whatever we have for
- doing compound interest is the amount we start
- with, e to the kt.
- Where k is kind of our compounding rate.
- And so in general, if you have this I naught, the
- initial amount you have.
- And that makes sense, right?
- Because if t equals 0, this whole expression is 1.
- So b of 0 should just be I naught.
- So if you know what you start with, and then you have another
- point, you can solve for k.
- And then you have an expression.
- You have the first thing they want.
- Find an expression for the number bacteria after t hours.
- So my question to you is, first of all, what is I sub zero?
- What is your initial number of bacterial cells?
- Well they tell us.
- They tell us right here.
- The bacteria culture initially contains 100 cells.
- So we know that b of 0 when time equals 0 or 0 hours, that
- is equal to 100 where the unit is cells, right?
- And so, if we substitute here t equals 0, this is 1.
- So we know that I naught is equal to 100.
- Let me do that actually.
- So b of 0 is also equal-- if we look at it there --I naught
- times e to the what? k times 0 times e to the zero is
- equal to I naught.
- So this tells us that the bacteria at time 0 is 100.
- And then when we substitute here, is that the bacteria
- times 0 is equal to I naught.
- So I naught must equal 100.
- OK.
- So we're making some progress.
- So we know that the bacteria as a function of time is
- equal to I naught, 100, times e to the kt.
- Now, if we just knew k, then we would have done this part A,
- find an expression for the number bacteria after t hours.
- Well how can we figure out k?
- Well they give us another data point.
- After an hour, the population has increased to 420.
- So that tells us-- Let me--
- [SIDE COMMENTS]
- Anyway, so what was I doing?
- Oh.
- So this tells us, after an hour the population
- has increased to 420.
- So that tells us that b of 1, right?
- After 1 hour, the population has increased to 420.
- So b of 1 is equal to 420.
- Which is also equal to-- If we put 1 here, that is
- equal to 100 e to the kt.
- Well what's t at b of 1?
- t is 1.
- So e to the k, right?
- So 420 is equal to 100 e to the k.
- And now we can solve for k.
- And let's see what we get.
- So we divide both sides by 100.
- You get 4.2.
- Let me put e to the k on this side.
- e to the k is equal to 4.2.
- To solve for k, we take the natural log of both sides.
- And we get k is equal to the natural log of 4.2, which is
- going to be some bizarre number.
- And we can-- We'll figure that out later with a calculator.
- So there we figured out-- Using the initial condition here,
- we figured out I naught.
- And then using this extra data point, we solved for k.
- And we got k is equal to the ln of the natural log of 4.2.
- So what does that leave us?
- So now we know the equation.
- We know k and we know I naught, so we know that the equation
- is-- So part A, the equation is b of t, bacteria as a function
- of time is equal to the initial number of bacteria, 100
- cells, e to the kt.
- And k is ln of 4.2.
- That says 4.2.
- And all of that times t.
- So that's our function.
- I know you can't read that, but we'll use it again.
- So it should make a little sense.
- So then they say find-- this is part B --find the number
- of bacteria after 3 hours.
- So that's pretty straight forward.
- We have our function now.
- That at any time t, we can tell you how many
- bacteria there are.
- So let's figure out the number of bacteria after 3 hours.
- So b of 3 is equal to 100 times e to the ln of 4.2 times 3.
- And that number we could calculate if you
- have a calculator.
- e to the ln of-- Let's see.
- That's essentially-- Well what's this going to be?
- This is the same thing.
- We can actually figure this out analytically.
- That equals 100 e to the natural log oh 4.2.
- If you have two things in an exponent multiplied, that's the
- same thing as taking that and then raising it to the
- third power, right?
- If we were simplifying, we'd say oh that's the same thing
- as just multiplying that times this.
- What's e to the ln of 4.2?
- Well it's 4.2, right?
- Because natural log says well, what exponent do I have
- to raise e to to get 4.2?
- So if I raise e to that exponent, I get 4.2.
- So that equals-- I didn't even have to use a calculator, look
- at that --4.2 to the third power.
- And I don't know what that is.
- That's going to be probably 70 something and so
- we'll do that later.
- But that's part B.
- That's just plugging into your calculator.
- That's part B.
- Now what are they asking?
- Find the rate of growth after 3 hours.
- Let me do that in a special color.
- So what are they asking us for?
- Well they're essentially asking us what is the slope of this
- function after 3 hours?
- Or another way you could view it is what is the derivative
- at time is equal to 3?
- Let's do that.
- Let me erase all this other stuff I have.
- I think we can erase all of this because we've written
- down our equation here.
- We can erase that.
- Actually we could even erase all of this because we've
- just answered that.
- That's just plugging into a calculator at this point.
- Just want to keep our original equation.
- So let's do part C.
- Part C, find the rate of growth after 3 hours.
- So we just have to take the derivative of the bacteria
- function with respect to time.
- So let's do that.
- So b prime of t.
- It equals what?
- So what's the derivative of-- Let's just do the chain rule.
- What's the derivative of-- so we can have 100 constant out
- there times the derivative of this inside function.
- Well, that's just a constant times t, so
- it's just a constant.
- So times ln of 4.2 times the derivative of e
- to the whole thing.
- Well, the derivative of e to the x is e to the x.
- So the chain rule tells us it's just the derivative
- of this whole expression.
- So times e to the ln 4.2t.
- So that's the derivative at any time t.
- And then they want to know what the rate of growth
- is after 3 hours.
- So b prime of 3 is equal to 100 ln of 4.2.
- All of that times-- Well, e to the-- As we just
- did this in part B.
- e to the ln 4.2 times 3, that's the same thing as-- Because
- this expression right here is just 4.2.
- This is times 4.2 to the third.
- That's just a little bit of logarithm math we did here.
- And it's the same exact logic.
- All I did is I substituted 3 here and simplified.
- Hopefully that makes sense to you.
- If it doesn't, you might want to use your calculator.
- But this all is actually a pretty important thing to know.
- That e to the ln of x is just equal to x, right?
- Because ln of x is essentially saying what exponent do I
- have to raise e to to get x?
- So if I raise e to that exponent, I get x.
- So that's all I'm saying here.
- That this right here is the same thing as e to the
- ln 4.2 to the t, right?
- Because if you were to simplify, you just take
- this exponent and multiply this and you'd get this.
- And so this is just 4.2 to the t, right?
- Actually that's-- I should have written our original
- equation as that.
- I could rewrite our original bacterial function equation as
- 100 times-- using that same thing --times 4.2
- to the t power.
- Actually, that is a simpler bacterial equation
- than this one.
- And so that would be even a better answer for part A.
- And then that makes part B even easier to solve.
- And then part C.
- Well part C is actually-- it was easier to keep it in this
- form because taking a derivative of e is easier than
- taking a derivative of another base.
- And actually we would probably want to convert backwards.
- So it's good we left it in that form because it made the
- derivative all that easier.
- But we could rewrite the derivative as well.
- We could rewrite this as b prime of t is
- equal to 100 ln 4.2.
- All of that times 4.2 to the t, right?
- I just took a this-- Sorry.
- No.
- I took this and I replaced it with this.
- Anyway, I got a little bit messier than I wanted to.
- But then their final question is when will the
- population reach 10,000?
- So let me see erase what we did with part C.
- When will the population reach 10,000?
- Let me write what we have already done a
- little bit neater.
- We know-- We knew that, part A, b of t is equal to 100 e to
- the natural log of 4.2t.
- And I said that's the same thing as 100 times
- 4.2 to the t power.
- And they say when will the population reach 10,000?
- So they're essentially saying when is-- at what time t
- is b of t equal to 10,000?
- So we can just say 10,000 is equal to this: 100-- And
- actually, I'm going to use this one because I want to take the
- natural log of both sides.
- And it just makes things simpler.
- 100e to the ln 4.2t.
- So you could divide both sides by 100.
- And you get 100 is equal to e to the ln 4.2t.
- And now take the natural logs of both sides of
- this equation, right?
- So then we get what?
- We get-- let me draw a divide-- Well, I'll do
- it in a different color.
- We get-- taking the natural log of both sides of this --the
- natural log of 100 is equal to-- Well if you take the
- natural log of e to something, you're just left
- with the something.
- And you might want to review all the logarithms if you found
- that a little confusing.
- But that-- If you take the natural log of this
- side, you're just left with the exponent.
- So that equals ln of 4.2t.
- And if you want to solve for t, you just divide both
- sides by ln of 4.2.
- And you get t is equal to ln of 100 divided by
- the natural log of 4.2.
- And that'll be an hour.
- So how many hours does it take?
- You just plug this in your calculator and you will get a
- number for the number of hours.
- Hopefully you didn't find that too confusing.
- And actually since-- Just to hit another point home, we
- could have used this equation.
- And it's the same thing, right?
- And what would we would have had?
- We would have had 100 times 4.2 to t power is equal to 10,000.
- Divide both sides by 100 and you get 4.2 to the
- t is equal to 100.
- And so, to solve this-- So you're essentially saying log
- base-- You would essentially want to take log base 4.2
- of both sides of this.
- So you would say t is equal to log base 4.2 of 100.
- And I went over this in the exponent proper-- in the
- logarithm properties video.
- But it's a pretty useful thing to know in general.
- How do you figure out-- Your calculator has two bases.
- It has log base 10 and it has log base e, which
- is the natural log.
- How do you figure out log base of another number of 100?
- Well, the simple answer is you could take natural log of this
- and divide it by the natural log of this.
- You could also take log base ten of this and divide it by
- log ten base of this-- log base ten of this.
- Anyway, hopefully I didn't further confuse you.
- And I will do-- This was actually exponential growth.
- And you could replace these words.
- You could say-- Instead of saying bacteria culture, you
- could say a bank account initially contains $100 and
- grows at a rate proportional to its size, which would
- mean compound interest.
- And then you could say, after an hour the population has
- increased to, whatever, $4.20.
- And then find the expression for the number of-- So
- essentially, you'd be trying to figure out the compound
- interest rate and all of that.
- But this is the exact same phenomenon.
- That the larger you grow, you're going to grow by an
- absolute value more than that.
- Anyway, I'll do more of these problems because they can
- be a little confusing.
- And I'll do one on exponential decay as well.
- See you soon.
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
|
Have something that's not a question about this content? |
This discussion area is not meant for answering homework questions.
Discuss the site
For general discussions about Khan Academy, visit our Reddit discussion page.
Flag inappropriate posts
Here are posts to avoid making. If you do encounter them, flag them for attention from our Guardians.
abuse
- disrespectful or offensive
- an advertisement
not helpful
- low quality
- not about the video topic
- soliciting votes or seeking badges
- a homework question
- a duplicate answer
- repeatedly making the same post
wrong category
- a tip or feedback in Questions
- a question in Tips & Feedback
- an answer that should be its own question
about the site
Share a tip
Suggest a fix
Have something that's not a tip or feedback about this content?
This discussion area is not meant for answering homework questions.