Financial Applications of Exponential Functions

In the last section on exponential functions, you will learn how to apply the compound interest formula and explore continuous growth.

Evaluating Functions with Base e

As we saw earlier, the amount earned on an account increases as the compounding frequency increases. Table 5 shows that the increase from annual to semi-annual compounding is larger than the increase from monthly to daily compounding. This might lead us to ask whether this pattern will continue.

Examine the value of $1 invested at 100% interest for 1 year, compounded at various frequencies, listed in Table 5.

Frequency	$A(n)=(1+ \frac{1}{n})^n$	Value
Annually	$(1+ \frac{1}{1})^1$	$2
Semiannually	$(1+ \frac{1}{2})^2$	$2.25
Quarterly	$(1+ \frac{1}{4})^4$	$2.441406
Monthly	$(1+ \frac{1}{12})^{12}$	$2.613035
Daily	$(1+ \frac{1}{365})^{365}$	$2.714567
Hourly	$(1+ \frac{1}{8760})^{8760}$	$2.718127
Once per minute	$(1+ \frac{1}{525600})^{525600}$	$2.718279
Once per second	$(1+ \frac{1}{31536000})^{31536000}$	$2.718282

Table 5

These values appear to be approaching a limit as $n$ increases without bound. In fact, as $n$ gets larger and larger, the expression $(1+ \frac{1}{n})^n$ approaches a number used so frequently in mathematics that it has its own name: the letter $e$ . This value is an irrational number, which means that its decimal expansion goes on forever without repeating. Its approximation to six decimal places is shown below.

THE NUMBER $e$

The letter $e$ represents the irrational number

$(1+ \frac{1}{n})^n$ , as $n$ increases without bound

The letter $e$ is used as a base for many real-world exponential models. To work with base e, we use the approximation, $e \approx 2.718282$ . The constant was named by the Swiss mathematician Leonhard Euler (1707–1783) who first investigated and discovered many of its properties.

EXAMPLE 10

Using a Calculator to Find Powers of $e$

Calculate $e^{3.14}$ . Round to five decimal places.

Solution

On a calculator, press the button labeled $\left[e^{x}\right]$ . The window shows $\left[e^{\wedge}(]\right.$ . Type 3.14 and then close parenthesis, [)]. Press [ENTER]. Rounding to 5 decimal places, $e^{3.14} \approx 23.10387$ . Caution: Many scientific calculators have an "Exp" button, which is used to enter numbers in scientific notation. It is not used to find powers of $e$ .