exponential growth in 109 workshop 5

Exponential growth is coming up again in 109 workshop 5..... perhaps you want me to go over what exponential growth is, and what the formula is that describes it....

First of all, what is exponential growth? Look at this set of numbers...

2, 4, 8, 16, 32, 64, 128

These numbers are increasing exponentially i.e. they are increasing each time by a set factor which is 2. The factor by which these numbers increase is always the same (it is a constant). When numbers keep increasing like this, it is called exponential growth.

We actually came across exponential growth in Application 2 in 109 workshop 3 (Keeping Track of Weeds). Remember, you drew a graph that looked something like this:

So this graph shows how the number of weeds increased over a number of weeks. You can see that it has a curve that increases in steepness as time progresses. This is because the number of weeds are increasing exponentially - i.e. by a set factor each week. In this example the factor by which the number of weeds are increasing is roughly 1.5 - after each week has passed, the number of weeds has increased by approximately 1.5 times, hence the graph keeps getting steeper.

So how could we express this graph using a formula?

Well, lets think first about how we could write the first point on the graph - the number of weeds at week 1 (Number at time 1 or Nt1)....we could say that the number of weeds in week 1 (Nt1) can be calculated by taking the number at time 0 (Nt0) times the factor by which the weeds increase in number each week (lets call that k):

Nt1 = Nt0 x k   (eq. 1)

So, how would we be able to calculate how many weeds there are after two weeks (Nt2)? Well, this time, if we know the number of weeds in week 1 (Nt1) and we know the rate by which the weeds multiply (k) we can calculate the number of weeds in week 2 (Nt2):

Nt2 = Nt1 x k (eq. 2)

And using the same logic we can calculate the number of weeds in week 3 (Nt3) if we know the number of weeds in week 2 (Nt2) and the rate by which the weeds multiply (k):

Nt3 = Nt2 x k (eq. 3)

Lets have a look at this on the graph to help you visualise what I have just written out:

We could re-wright equation 2 above for the number of weeds at week 2 (Nt2 = Nt1 x k) by substituting Nt1 with equation 1 (Nt1 = Nt0 x k):

Nt2 = Nt0 x k x k

And we could condense those two k's to:

                                                       Nt2 = Nt0 x k² (eq. 4)

And we could then re-write equation 3 above for the number of weeds at week 3 (Nt3 = Nt2 x k) by substituting Nt2 with equation 4 (Nt2 = Nt0 x k²):

                                                         Nt3 = Nt0 x k² x k

And again we can combine those k's:

                                                       Nt3 = Nt0 x k³ (eq. 5)

^{Do you see what is happening? If you want to know the number of weeds in week 2, you take the number of weeds in week 0 and times by the rate of increase squared (eq. 4), and if you want to know the number of weeds in week 3 you take the number of weeds in week 0 and times by the rate of increase cubed (eq. 5}). So, it follows that the number of weeds in week 6 could be calculated with: 

                                                          Nt6 = Nt0 x k⁶

This equation can be written in a more general way:

                                                            Nt = Nt0 x k^t 

Where Nt is the number of weeds at time t

Nt0 is the number of weeds at time 0

                                     k^t  is the rate of increase to the power of time t

This is the equation that describes exponential growth, and now you know where it came from!!