Applications of Exponential Functions: Learn It 3

Evaluating Exponential Functions with Base [latex]e[/latex]

As we saw earlier, the amount earned on an account increases as the compounding frequency increases. The table below 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 [latex]$1[/latex] invested at [latex]100 \%[/latex] interest for [latex]1[/latex] year compounded at various frequencies.

Frequency [latex]A\left(t\right)={\left(1+\frac{1}{n}\right)}^{n}[/latex] Value
Annually [latex]{\left(1+\frac{1}{1}\right)}^{1}[/latex] [latex]$2[/latex]
Semiannually [latex]{\left(1+\frac{1}{2}\right)}^{2}[/latex] [latex]$2.25[/latex]
Quarterly [latex]{\left(1+\frac{1}{4}\right)}^{4}[/latex] [latex]$2.441406[/latex]
Monthly [latex]{\left(1+\frac{1}{12}\right)}^{12}[/latex] [latex]$2.613035[/latex]
Daily [latex]{\left(1+\frac{1}{365}\right)}^{365}[/latex] [latex]$2.714567[/latex]
Hourly [latex]{\left(1+\frac{1}{\text{8766}}\right)}^{\text{8766}}[/latex] [latex]$2.718127[/latex]
Once per minute [latex]{\left(1+\frac{1}{\text{525960}}\right)}^{\text{525960}}[/latex] [latex] $2.718279[/latex]
Once per second [latex]{\left(1+\frac{1}{31557600}\right)}^{31557600}[/latex] [latex]$2.718282[/latex]

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

the number [latex]e[/latex]

The letter [latex]e[/latex] represents the irrational number

[latex]{\left(1+\frac{1}{n}\right)}^{n},\text{as }n\text{ increases without bound}[/latex]

 

The letter [latex]e[/latex] is used as a base for many real-world exponential models. To work with base [latex]e[/latex], we use the approximation, [latex]e\approx 2.718282[/latex].

 

The constant was named by the Swiss mathematician Leonhard Euler (1707–1783) who first investigated and discovered many of its properties.

Calculate [latex]{e}^{3.14}[/latex]. Round to five decimal places.

Graph and state the characteristics of [latex]f(x) = e^x[/latex].


To graph the function [latex]f(x) = e^x[/latex], let’s identify and plot key points on the graph.

[latex]x[/latex] [latex]f(x) = e^x[/latex]
[latex]-1[/latex] [latex]f(-1) = e^{-1} \approx 0.37[/latex]
[latex]0[/latex] [latex]f(-1) = e^{0} \approx 1[/latex]
[latex]1[/latex] [latex]f(-1) = e^{1} \approx 2.72[/latex]
[latex]2[/latex] [latex]f(-1) = e^{2} \approx 7.39[/latex]

image

Characteristics:

[latex]\begin{align*} \text{Function:} & \quad f(x) = e^x \\ \text{Domain:} & \quad (-\infty, \infty) \\ \text{Range:} & \quad (0, \infty) \\ \text{Y-intercept:} & \quad (0, 1) \\ \text{Horizontal Asymptote:} & \quad y = 0 \\ \text{Behavior as } x \rightarrow \infty: & \quad f(x) \rightarrow \infty \\ \text{Behavior as } x \rightarrow -\infty: & \quad f(x) \rightarrow 0 \end{align*}[/latex]

Write the equation for the function described below. Give the horizontal asymptote, domain, and range.[latex]f\left(x\right)={e}^{x}[/latex] is vertically stretched by a factor of [latex]2[/latex], reflected across the [latex]y[/latex]-axis, and then shifted up [latex]4[/latex] units.