4.2: The Natural Exponential Function

e
The number $e$ is defined to be \[\left(1 + \dfrac{1}{n}\right)^n \rightarrow e \quad \text{as} \quad n \rightarrow \infty\]

As a decimal, $e \approx 2.7182818...$

Because $e > 0$, we can use it as the base of an exponential:

natural exponential function
Exponential function with base $e$: \[f(x) = e^x\]

In practice, the natural exponential is useful in Calculus. It will simplify calculations, as you will see in Calculus!

Let's see this function in an example.

A disease begins to spread in a small town of 10,000 people. The total number of people infected on day $t$ is \[v(t) = \dfrac{10,000}{5 + 1245e^{-0.97t}}\]
  1. How many people were infected initially (at time $t = 0$)?
  2. Find the number of people infected on day five.
  3. Analyze $v(t)$ as time progresses ($t\rightarrow \infty$).

Continuously Compounded Interest

Recall the compound interest formula: \[A(t) = P\left(1 + \dfrac{r}{n}\right)^{nt}\] If you apply compounding at shorter and shorter intervals, $n$ grows.

As you let $n\rightarrow\infty$, interest is allowed to compound at every moment in time!

Doing so replaces the interest calculation with a simple form:

continuously compounded interest
Continuously compounded interest is calculated by \[A(t) = Pe^{rt}\] where
Find the amount after 3 years if $1,000 is invested at an interest rate of 12% per year, compounded continuously.