# Math Help - essay on the number\letter e help!!!!

1. ## essay on the number\letter e help!!!!

i need to write an extra credit essay on e. 3 pages double spaced and all i have is half a page on how it was discovered. i dont know what to write and how i can fill 3 pages because its soo confusing. its due thursday any help is appreciated. im goin nuts

2. I suggest discussing the fact that the derivative of an exponential function is a scalar multiple of the same function. The number $\displaystyle e$ is the base that makes this scalar multiple = 1, in other words, makes the function its own derivative. This will give you a limit definition for $e$.

Then I would suggest using the Taylor series to create a polynomial for $\displaystyle e^x$, and then evaluating the polynomial at $\displaystyle x= 1$ to evaluate $\displaystyle e$ to a number of decimal places.

Does this help?

3. You certainly could say a lot about the fact that $e^{ix}= cos(x)+ i sin(x)$ or, taking $x= \pi$ in that, that $e^{i\pi}= -1$. You could also discuss the "hyperbolic functions", $cosh(x)= \frac{e^x+ e^{-x}}{2}$ and $sinh(x)= \frac{e^x- e^{-x}}{2}$, what differential equations they satisfy, and their relationship to the circular (trig) functions.

Of course, since you have told us nothing about your class, or what level of mathematics you are at we have no idea whether you know what a "derivative" or "differential equation" is, or what you know about complex numbers so we have no idea whether any of this is helpful!