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November 6th, 2016, 07:54 PM   #1
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looking for an equation of a curve

While reading about Riemann, I came across a relation between random numbers and e. I am not sure if Riemann discovered the relation.

This led to an empirically calculated curve that connects the points 0,1 and 1,e, in which the slope at x=0 is 1 and the slope at x=1 is 2.

Has anyone seen such a curve and might there be a formula for it?

Last edited by skipjack; November 7th, 2016 at 12:16 AM.
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November 6th, 2016, 08:18 PM   #2
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$f(x) = x\left(\dfrac x 2 +1\right)$ satisfies

$f^\prime(0)=1,~f^\prime(1)=1$
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November 6th, 2016, 11:33 PM   #3
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sorry, I gave wrong boundary condition

I used Excel trendline and requested an exponential curve.
it gave a curve almost identical to y=e^x.
this means the slope at x=1 is e, not 2 as I stated before.
So my question is answered.
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