The e^x Function: To e or not to e - At A Glance

Do you know what continuous growth means? Picture a balloon being blown up at a constant rate, rather than in little increments. Pop! Some exponential functions that model continuous growth use the e function. In this section we are going to take a look at a few different types of e functions.

We're going to talk about population growth, bacteria over-growth (eww), carbon dating, virus spread, and even more financial exponential growth. You know that age old question, "What does this have to do with the real world?" The real world is all over the place in this section.

Looking back at the equation for compounding money into a savings account, what happens if we compounded not just daily, but every hour, every second, and even faster than that? The universe would shatter to pieces, right? No, actually this is a real thing. It is called is continuous growth or decay.

Looking at the equation again, what we really want to know is when n approaches infinity. Remember that n is the number of times per year that the investment is compounded.

Sample Problem

In the following example, look what happens to the , when we increase the n values (let's say we invested $1 at a rate of 5% for 1 year):

What happened there is we took the following limit:

One of math's forefathers, Leonhard Euler, in the 1700's, discovered this and named this the exp function (or the e function).

f(x) = e^x

You have been introduced to the e value before (2.718281828...) which some people call Euler's number. It's an irrational number like π.

What is interesting with the above example is that if the rate was 100% (meaning 100% growth, not 5% growth rate, look what happens!).

Here is 5% rate of continuous compounding for one year if we started with $1:

f(x) = 1e^0.05 = 1.051271096

If we had 100% growth rate:

f(x) = e¹ = 2.718281828

This is Euler's number! The general equation for continuous growth, whether it be population, or zombie virus spread, or money is:

A = Pe^rt

where A is ending amount, P is the initial amount, r is growth rate, t is time.

Sample Problem

How many people will live in New York City in 2024 if there is continuous growth, the growth rate is 2% and the population was 8,336,697 in 2012?

A = 8,336,697e^0.02(10)
A = 10,182,640

10 million people. eeeeeeek! This is just an estimate based on previous year growth rate but you can do this with any type of calculation that calls for continuous growth.

Sample Problem

If a bacteria growing on a water fountain grows continuously and the number of bacteria is defined by the equation f(t) = 200e^0.415t where t is the time in hours, how many bacteria did we start with and how many will there be after 6 hours?

The first thing to remember is that 200 is our initial count of drinking fountain bacteria. Remember back to our equation A = Pe^rt. Don't let the different variables freak you out. P is the same as our 200. We didn't have to do any math to figure out the first part of the problem! Now let's solve for how many bacteria we will have after 6 hours of continuous growth. (Warning: you may not ever drink from a water fountain again.)

f(t) = 200e^0.415t
f(6) = 200e^0.415(6)
f(6) ≈ 2413

Gross! In just 6 hours, our nasty bacteria colony started a city.

Sample Problem

In 2415, an archeologist finds a cell phone from 2012. Carbon continuously decays and can help determine the age of materials. Using the carbon dating technique, how much carbon was present using the following equation: C = 18e^-0.0000132t?

Since carbon is the main element in plastic, there is a lot of carbon in cell phones. Our t value is the number of years the cell phone has been laying in the alley, which is 2415 – 2012 = 403 years. Did you notice that the exponent is negative? This is also another type of exponential decay.

C = 18e^-0.0000132t
C = 18e^{-0.0000132(403)}
C ≈ 17.9

As you can see, we started with 18 and after 403 years, we didn't lose much carbon. That's because it takes 50,000 years to really start breaking down.