Exponential Functions
Posted by Taylor
Today in precalc, we spent most of the time working on problems out of a packet that deals with exponential functions. Because we learned the basics about exponential functions yesterday, that means that today we applied these concepts.
The problem we had to solve today was to determine if there was a limit to the amount of money that would be earned in a savings account if we continually added a compound interest of 100%. The expression given to us (a form of an exponential function) was (1+1/n)^n.
First we had to make a table.
| y | y=1/n | y=1+1/n | y=(1+1/n)^n |
| 1 | 1 | 2 | 2 |
|
2 |
.5 | 1.5 | 2.25 |
| 5 | .2 | 1.2 | 2.49 |
| 10 | .1 | 1.1 | 2.59 |
| 50 | .02 | 1.02 | 2.69 |
| 100 | .01 | 1.01 | 2.70 |
| 500 | .002 | 1.002 | 2.72 |
If you look at the last few values, you might notice that they have seemed to reach a peak at about 2.72. This value is actually an irrational number (like pi) that is called Euler’s number. The symbol for this value is e. The exact value (or as close as I can get) is 2.718281828… The formal definition of e is (1+1/n)^n.
So when applying this concept to banking, this means that if you want to make the most amount of money by having the interest compiled more frequently, there is actually a limit at which the final amount will not increase at a significant rate.
Basically, today was a day where we worked on a few problems and continued our study into exponential functions. The next scribe will be…Shannon. Be happy. You have the entire weekend.
January 29th, 2010 at 10:23 am
Just wanted to clarify that the definition of e is the limit of the expression that Taylor gave as n approaches infinity.