imagine this much bacteria in a petri dish now this amount of the same bacteria assuming that each...

Imagine this much bacteria in a Petri dish

Now this amount of the same bacteria

Assuming that each bacterium would reproduce at the same rate, which dish will have a larger rate of growth?

Answer: The second one simply because there are more of them.

So if we presume that the rate of growth is given by:dt

dy

y = amount of bacteria, t = time

And that the population produces at a rate proportional to itself with the proportion represented by the constant k

So if we presume that the rate of growth is given by:dt

dy

y = amount of bacteria, t = time

Then ?dt

dyyk

Use your newfound skills for solving differential equations to solve for y here:

kydt

dy

And that the population produces at a rate proportional to itself with the proportion represented by the constant k

1 dy k dt

y

1 dy k dt

y

ln y kt C

ln y kt Ce e

y eC ekt

kty Ae

Since the initial amount is at t = 0

0Aey

0yA

In this case, y0 is the initial amount

So our equation for this type of growth would be…

0kty y eExponential Change:

If the constant k is positive then the equation represents

growth. If k is negative then the equation represents decay.

There is a similar growth equation used in finance that you may remember from pre-calc…and we’ll talk about that soon

Remember too that we’ve just shown that

kty Ae kydt

dyis the solution to the differential equation

ln 2half-life

kOne straight-forward application of this is

0 0

1

2kty y e

1ln ln

2kte

ln1 ln 2 kt

ln 2 kt

ln 2t

k

Half-life is the period of time it takes for a substance undergoing decay to decrease by half.

In this case, think of y0 as the initial

amount of a substance undergoing decay. To find its half life…

kte2

1

Compounded Interest

If money is invested in a fixed-interest account where the total interest r (which is a % written as a decimal) is broken into k equal portions and added to the account k

times per year, the amount present after t years is:

0 1kt

rA t A

k

If the interest is broken down more and added back more frequently (k is larger), you will make a little more money.

We can add as many times as we want which means we can make k as large as…

Initial investment 100% (initial investment)

% added each time

# times per year over t years

The larger k gets, the more times per year we compound the interest. If we can theoretically compound an infinite number of times, we say that the interest is compounded continuously

We could calculate: 0lim 1kt

k

rA

k

Using an old limit from pre-calc

k

k k

11lim e

rk

ke

k

r

1lim

The larger k gets, the more times per year we compound the interest. If we can theoretically compound an infinite number of times, we say that the interest is compounded continuously

We could calculate: 0lim 1kt

k

rA

k

Just like the exponential growth model we just saw, the interest is directly proportional to the amount present.

Continuously Compounded Interest:

You may also use:

rtA Pe

which turns out to be:

rteA0

Remember PERT?

Same equation, different letters

where is the temperature of the surrounding medium, which is a constant.

sT

Newton’s Law of Cooling

(The colder the air, the faster the coffee cools)

This would give us the differential equation:

Coffee left in a cup will cool to the temperature of the surrounding air. The rate of cooling is proportional to the difference in temperature between the liquid and the air.

s

dTk T T

dt


0kt

s sT T T T e

Don’t be afraid of the size of this equation. It really is not that different from the first exponential growth/decay equation. Don’t forget also that TS and

T0 are constants. Just look at this

comparison…




If we solve the differential equation, we get:


0kt

s sT T T T e




0kt

s sT T T T e

kty AeIt’s just a matter of sorting through the constants

If we solve the differential equation, we get:

The End

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