idku
  • idku
I have a question about growth ranking
Mathematics
katieb
  • katieb
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idku
  • idku
So, we are taking about different growths. From smallest to greatest 1) Polynomials - linear growth - quadratic polynomial (with positive coefficient) - cubic polynomial (with positive coefficient) - etc... (some nth degree polynomial) 2) Exponential function f(x)=a(b)\(^x\) - this exponential eventually exceeds the polynomial 3) factorial 4) n\(^n\)
idku
  • idku
So, would I put hyperbolic functions in the exponential? what I want to propose is that a hyperbolic function (as lim x→∞) is smaller than or equivalent to e\(^x\), and larger than any other exponential function if the base of this exponential function is less than e.
nincompoop
  • nincompoop
yes

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idku
  • idku
(and of course is base is bigger than e, this exponential is certainly exceeds hyperbolic)
idku
  • idku
oh yes? nice
nincompoop
  • nincompoop
define hyperbolic function
idku
  • idku
well, it consists of some e^x componenets
idku
  • idku
cosh(x) for example is the average between e^X AND E^-X
idku
  • idku
So... `hyperbolic function` ≥ `(a)^x; where a≤e` `hyperbolic function` < `(a)^x; where a>e`
nincompoop
  • nincompoop
if the difference is the base, then yes
idku
  • idku
yes, with same x exponent... tnx (I was just thinking of different growths)
nincompoop
  • nincompoop
you can wolfram alpha it to make sure
nincompoop
  • nincompoop
test it with one concrete condition where a>e
idku
  • idku
I started this proposal when I proved that e^x is greater than cosh(x)
idku
  • idku
|dw:1435300335244:dw|
idku
  • idku
saying, as x-> infinity
nincompoop
  • nincompoop
correct
idku
  • idku
then sinh(x) is certainly smaller
idku
  • idku
-e^-x
idku
  • idku
-e^x is always >0 e^x is always <0 :)
idku
  • idku
-e^-x is always >0, (correction). but that is always >0 as well.
idku
  • idku
wwhat wouldn't have?
idku
  • idku
other functions are also obviously less than e^x.
idku
  • idku
So I want to change my statment a bit
idku
  • idku
`hyperbolic function` < `(a)^x; where a≥e ` `hyperbolic function` > `(a)^x; where a

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