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Re: new tutor here
Posted:
Jun 24, 2013 5:29 PM
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Joe N says:
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[1] Draw the line (y=x/e) from 0 through (e,1) and (pi,pi/e). Draw the log curve through the points (e,1) and (pi,log(pi)). The first line is tangent to the log curve at x=e (calculus fact#1). If you've drawn correctly, mentally or on paper or computer, you will easily see that pi/e > log(pi)* (These are the y values of the intersection of vertical line x=pi with the log curve, and the line 1/e, respectively.)
pi/e > log(pi)*
pi > e log(pi)
e^pi > pi^e
*: Analytically, you can use the mean value theorem (calculus fact#2) to *prove* that it is so. Probably can do so just based on the fact that log is continuous with monotonically decreasing derivative, but, eh, life is short.
[2] As I said, there is an even more direct way of seeing this (without using log), but also uses a couple calculus facts.
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Here's a link to the desmos graph of the first pic:
[1] https://www.desmos.com/calculator/kahmpfofbx
Here's a link to the visualization mentioned for the more "direct" way of seeing:
[2] https://www.desmos.com/calculator/8ojs1gfojk
Anyone care to tease the meaning out of this pic?
Cheers,
Joe N
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