Which function grows faster?

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In summary, "function grows faster" means that a mathematical function increases at a higher rate as its input value increases. To compare the growth rates of two functions, you can use methods such as taking the limit of their ratio, analyzing their derivatives, or using graphical representations. Linear growth refers to a constant increase, while exponential growth refers to an increasing increase. Some functions can grow faster than exponential, such as the factorial function (n!). The growth rate of a function can be determined by analyzing its equation, with the power of the highest term for polynomial functions and the base of the exponential term for exponential functions.
  • #1
Dragonfall
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[tex]n^n[/tex]

or

[tex]\prod_{r=0}^{n-1}\left(\begin{array}{c}\sum_{j\leq r}\left(\begin{array}{c}n\\ j\end{array}\right)\\2^r\end{array}\right)[/tex]

Asymptotically, I mean.
 
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  • #2
Since
[tex]\prod_{r=0}^{n-1}{{\sum_{j<r}{n\choose j}}\choose{2^r}}={0\choose1}\prod_{r=1}^{n-1}{{\sum_{j<r}{n\choose j}}\choose{2^r}}=0[/tex]
I'd have to go with [itex]n^n[/itex].
 
  • #3
How about now?
 
  • #4
It seems that the product grows faster.
 

What is the meaning of "function grows faster"?

The phrase "function grows faster" refers to the rate at which a mathematical function increases as its input value increases. A function that grows faster will have a steeper slope or a larger increase in output for a given change in input compared to a function that grows slower.

How do you compare the growth rates of two functions?

To compare the growth rates of two functions, you can use several methods, including taking the limit of the ratio of the two functions, analyzing their derivatives, or using graphical representations such as slope or area under the curve.

What is the difference between linear and exponential growth?

Linear growth refers to a function that increases at a constant rate, while exponential growth refers to a function that increases at an increasing rate. In other words, in exponential growth, the output value increases by a larger amount for each unit increase in the input value, resulting in a steeply increasing curve.

Can a function grow faster than exponential?

Yes, some functions can grow faster than exponential functions. For example, the factorial function (n!) grows faster than any exponential function. Other examples include functions with a power greater than 1, such as n^2 or n^3.

How do I determine the growth rate of a function from its equation?

The growth rate of a function can be determined by analyzing its equation. For polynomial functions, the power of the highest term in the equation indicates the growth rate. For exponential functions, the base of the exponential term represents the growth rate. Other types of functions may require more advanced methods to determine their growth rates.

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