7.1 Exponential Functions
As this book was being adapted in July 2020, COVID-19, also known as Coronavirus was invading the world. On several dates in June and July, the state of Alaska[1] had an number of 1.2 and this is with mitigating precautions in place. The is the average number of people that one infected people will, in turn, give it to. For example, if 100 people have it right now, in about a week, 120 people will, or 100 times 1.2. This does not sound like a problem that would cause so much disruption to our daily lives. But consider what happens in a month or two, each of the 120 then, in turn, infect 1.2 others, so the next week, the number is 144, then 172, 207 and these are new case, and often, we have people recovering within a week or two. So for our purposes, we can consider these the current active cases at a time, although it is a low estimate. In three months, the 100 cases turn into over 1000 and in 6 months over 10,000 unless more more mitigating precautions are put into place.
In linear growth, we had a constant rate of change – a constant number that the output increased for each increase in input. For example, in the equation , the slope tells us the output increases by three each time the input increases by one. This population scenario is different – we have a percent rate of change rather than a constant number of people as our rate of change. To see the significance of this difference consider these two companies:
Company A has 100 stores, and expands by opening 50 new stores a year.
Company B has 100 stores, and expands by increasing the number of stores by 50% of their total each year.
Looking at a few years of growth for these companies:
| Year | Stores, company A | Stores, company B | |
| 0 | 100 | Starting with 100 each
|
100 |
| 1 | 100 + 50 = 150 | They both grow by 50 stores in the first year.
|
100 + 50% of 100
100 + 0.50(100) = 150 |
| 2 | 150 + 50 = 200 | Store A grows by 50, Store B grows by 75
|
150 + 50% of 150
150 + 0.50(150) = 225 |
| 3 | 200 + 50 = 250 | Store A grows by 50, Store B grows by 112.5
|
225 + 50% of 225
225 + 0.50(225) = 337.5 |
Notice that with the percent growth, each year the company is grows by 50% of the current year’s total, so as the company grows larger, the number of stores added in a year grows as well.
To try to simplify the calculations, notice that after 1 year the number of stores for company B was:
Or equivalently by factoring
We can think of this as “the new number of stores is the original 100% plus another 50%”.
After 2 years, the number of stores was:
Or equivalently by factoring
Now recall that the 150 came from
Substitute that:
After 3 years, the number of stores was:
Or equivalently by factoring
Now recall that the 225 came from
Substitute that:
From this, we can generalize, noticing that to show a 50% increase, each year we multiply by a factor of or , so after years, our equation would be:
or
In this equation, the 100 represented the initial quantity, and the 0.50