Refresher - Exponential Functions

Refresher

There is a particular value of base b that is useful in economic analysis and problems involving natural growth and decay. This base is “e” named after the Swiss mathematician Leonard Euler (1707-1783) who investigated the number and discovered many of its properties. This exponential function with base e, $f(x)=e^x$, is called the natural exponential function.

“e” is defined as $\underset{n\to \infty }{{\displaystyle \lim ?}}{\left(1+\frac{1}{n}\right)}^n$. Euler was the first to prove that this limit exists. The table shows some approximations for ${\left(1+\frac{1}{n}\right)}^n$as n gets larger.

n	10	100	1000	10,000
$\left(1+\frac{1}{n}\right)^n$	2.5937	2.7048	2.7169	2.7183

 

If you try very large values of n on your calculator, you will get closer to the value of e. Like `pi`, e is an irrational number. A decimal approximation of e is ≈ 2.718281828459045 …. e^x is called an exponential to base e.

 

Since e > 1, the function $f(x)=e^x$ is an increasing function. It has the same characteristics as all exponential functions with base b>1, such as the doubling function, $f(x)=2^x$

 

One of the most familiar applications of exponential growth is that of an investment earning compounded interest. Using the formula $A=P{\left(1+\frac{r}{n}\right)}^{nt}$, where A represents the value of the investment at time t, P represents the initial principal, r represents the interest rate expressed in decimal form, t represents the number of years P is invested, and n represents the number of compoundings per year.

When we are looking at continuous compounding we use the formula: $A=P{e}^{rt}$

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Math 1A/1B: Pre-Calculus by Dr. Sarah Eichorn and Dr. Rachel Lehman is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License Links to an external site..