why the exponential value is considered as 2.718281828whats the history behind exponential funtion..?????

nathanshields | High School Teacher | (Level 1) Associate Educator

Posted on

Imagine you put \$1 in the bank, and it earns 100% interest over the course of the year... cha-ching! \$2.

But if the interest is compounded monthly, then you get to earn "interest on the interest" and get a little more:

`(1+1/12)^12=\$2.613...`

If you compounded every day, you'd get more.  Every second would get you even more, but what about every instant?  Infinite money?

Nope: `lim_(x->infty)(1+1/x)^x` = \$2.71828...

The number e is important in calculus, for determining the rate at which exponential functions increase.  It has the critical property that

`d/(dx)e^x=e^x`

(There's no other number like this).

This allows us to find the rate that any exponential function increases:

`y = 3^x = (e^(ln3))^x=e^(ln3*x)`

Since ln3 is just a constant, then chain rule gives us `y'=e^(ln3*x)*ln3` .

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najm1947 | Elementary School Teacher | (Level 1) Valedictorian

Posted on

There is a long history as to how the number e evolved as given in the referred link and qouted below:

The first time the number e appears in its own right is in 1690. ......the notation e made its first appearance in a letter Euler wrote to Goldbach in 1731. He made various discoveries regarding e in the following years, but it was not until 1748 when Euler published Introductio in Analysin infinitorum that he gave a full treatment of the ideas surrounding e. He showed that

e = 1 + 1/1! +1/2! +1/3! + ...

and that e is the limit of (1 + 1/n)n as n tends to infinity. Euler gave an approximation for e to 18 decimal places,

e = 2.718281828459045235

For further details, go to the link

http://www.gap-system.org/~history/HistTopics/e.html

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