Subject: Calculus

E

Calculus.E History

Hide minor edits - Show changes to output

October 31, 2011 by matthew_suan -
Added line 43:
October 31, 2011 by matthew_suan -
Added lines 35-43:

!!!Example #1
Give other names of the mathematical constant {$e$}.

(:toggle id=box1 show='Show Answer with Solution' init=hide button=1:)
>>id=box1 border='1px solid #999' padding=5px bgcolor=#edf<<
1. Euler's number\\
2. Napier's constant\\
>><<
July 03, 2011 by matthew_suan -
Changed lines 36-37 from:
to:
'''NEXT TOPIC:''' [[Exponential function]]
March 28, 2011 by matthew_suan -
Added lines 26-36:

!!Some Cool Stuffs
Mathematicians found many cool stuffs about e. Some of which are numbered below.

1. e has the value of 2.718 281 828 459 045. Notice the successive re-occurrence of the number 1828.

2. Upon taking the derivative of {$e^x$}, you get {$e^x$} back again. It is the only number who could do that.

3. Upon taking the integral of {$e^x$}, you also get back {$e^x$} again. (Of course since integration and differentiation are inverse operations).
March 28, 2011 by matthew_suan -
Changed lines 10-11 from:
{$$e = \sum_{i=1}^{\infty}\frac{i!}$$}
to:
{$$e = \sum_{i=1}^{\infty}\frac{1}{i!}$$}
Changed line 21 from:
{$$= \ln{x}|_{1}^{e} = \ln{e} - \ln{1] =\ln{e} + 0$$}
to:
{$$= \ln{x}|_{1}^{e} = \ln{e} - \ln{1} =\ln{e} + 0$$}
March 28, 2011 by matthew_suan -
March 28, 2011 by matthew_suan -
Added lines 1-26:
'''e''' is a mathematical constant that has a value of 2.71828... (the tail numbers goes on forever). It is one of mathematic's most famous constant due to its wide array of uses and some cool stuff. Some of which are used in describing a nonlinear values such as growth or decay and the bell curve(statistics). It is also even used in some probability, counting and prime number distribution problems and even used to describe certain physical and chemical phenomena.

!!e as a Formula
The constant e can also be expressed by the following formula;

{$$e = \lim_{x \to \infty} \left (1 + \frac{1}{x}\right )^x$$}

e is sometimes called as Euler's number, Eulerian number or Napier's constant and it can also be represented by this formula;

{$$e = \sum_{i=1}^{\infty}\frac{i!}$$}

!!e as an Area
e is also defined as the value that makes the shaded area in the graph of {$y=\frac{1}{x}$} below equal to 1.

%center%Attach:e.png

If we compute the shaded area above from {$x=1$} to some point {$e$}, we get

{$$A = \int_{1}^{e} \frac{1}{x}dx = 1$$}

{$$= \ln{x}|_{1}^{e} = \ln{e} - \ln{1] =\ln{e} + 0$$}

which leads us to some basic property of e which is

{$$\ln{e} = 1$$}