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Stuff about e

  1. Apr 15, 2013 #1
    Okay, so I am in grade 12 calc and I was learning about e today, how the slope of the tangent at any point is also the y value at that point. What I was wondering is if there is a function that has the x value equal to the slope at any given point. I think it would look something like a parabola.

    I tried to plug x in for slope to y=mx+b which results with y+x^2, but that does not have the property I am looking for. I also tried y=x^e, it doesn't work out.

    Any thoughts?
  2. jcsd
  3. Apr 15, 2013 #2


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    welcome to pf!

    hi e.mathstudent! welcome to pf! :smile:
    you mean dy/dx = x ?

    hint: what is the derivative (the slope) of a polynomial? :wink:
  4. Apr 15, 2013 #3
    more stuff about e

    does f(x)=(x^2)/2 work?
    it has a derivative of x, so I am going to try that.
  5. Apr 15, 2013 #4
    Oh cool, it works. That was a way simpler answer than I expected.
  6. Apr 15, 2013 #5
    Are there any other cool properties about e or fun things to know?
  7. Apr 15, 2013 #6


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  8. Apr 15, 2013 #7
    Yes. There's a whooole book about it. I happen to be reading it right now: https://www.amazon.com/Story-Number-Princeton-Science-Library/dp/0691141347

    I like e, perhaps moreso than even pi! It's quite easy to calculate the value of e on one's own, as it is the limit as n goes to infinity of (1+1/n)^n

    You can plug in large values of n and calculate to whatever degree you'd like.

    You can approximate it with a taylor series.

    You can use the binomial theorem.

    Have a few more things that I"ll share later, but my wife is home. lol
  9. Apr 15, 2013 #8


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    One way of looking at e ( as e^1 ) is this:

    Assume you have an account of D dollars at a yearly interest rate of 100% , i.e., your account

    doubles every year.

    Now, say you can also compound the interest , e.g., instead of getting 100% yearly, you can

    get 50% after 6 months, and then compound again by 50% six months after that . Then your

    have (1.5)*(1.5)*D =2.25*D dollars, instead of 2*D dollars, by compounding twice. Now, you can

    compound your money, not just twice yearly, but 3-, 4- or more times. If you compounded

    infinitely-often (in the limit), your money will be multiplied, in the limit, by a factor of e, meaning

    you will have e*D dollars at the end of a year by doing this continuous compounding.

    In general, if your interest rate is x (as a fraction ) and you compound your account

    continuously, your D dollars will be worth e^x dollars at the end of the year.
  10. Apr 15, 2013 #9
    The great e vs. pi debate (you will learn a bit, and actually quite funny).

    In five parts:

    Last edited by a moderator: Sep 25, 2014
  11. Apr 15, 2013 #10
    The closest rational approximation of e using integers below 1000 is 878/323. (Source: Eli Maor's book mentioned above).
  12. Apr 16, 2013 #11
    Oh, e. Such an awesome constant. Leonhard Euler is one interesting man.
  13. Apr 16, 2013 #12
    Those videos are epic. :wink:
    Last edited by a moderator: Sep 25, 2014
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