Searching for a Function with x=Slope of Tangent

[e.mathstudent]

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?

 

[tiny-tim]

welcome to pf!

hi e.mathstudent! welcome to pf!

… 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.

you mean dy/dx = x ?

hint: what is the derivative (the slope) of a polynomial?

 

[e.mathstudent]

more stuff about e

does f(x)=(x^2)/2 work?
it has a derivative of x, so I am going to try that.

 

[e.mathstudent]

Oh cool, it works. That was a way simpler answer than I expected.

 

[e.mathstudent]

Are there any other cool properties about e or fun things to know?

 

[tiny-tim]

start with http://en.wikipedia.org/wiki/E_(mathematical_constant)

 

[dkotschessaa]

Are there any other cool properties about e or fun things to know?

Yes. There's a whooole book about it. I happen to be reading it right now: https://www.amazon.com/dp/0691141347/?tag=pfamazon01-20

I like e, perhaps more so 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

 

[Bacle2]

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.

 

[dkotschessaa]

The great e vs. pi debate (you will learn a bit, and actually quite funny).

In five parts:

 

[dkotschessaa]

The closest rational approximation of e using integers below 1000 is 878/323. (Source: Eli Maor's book mentioned above).

 

[Lyrassia]

Oh, e. Such an awesome constant. Leonhard Euler is one interesting man.

 

[Lyrassia]

The great e vs. pi debate (you will learn a bit, and actually quite funny).

In five parts:

Those videos are epic.