A way to organize functions by their speed of growth ?

In summary, the conversation discusses organizing functions based on their speed of growth. The participants also mention different types of functions and their derivatives to illustrate this concept. They also suggest checking out Big O notation for further understanding.
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lolgarithms
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A way to organize functions by their "speed of growth"?

How does one say formally in math that a certain function grows "faster" than another?
Doens't really work for trig functions, i know.
you knotice that the exponential function is the function dividing d/dx slower than itself and d/dx faster than itself functions

In order from slower to faster:

Derivative is slower than itself:
Constants

rational function in which quotient is non-constant

Logarithms

Roots

Non-constant polynomials:

b^x, b>1: derivative is proportional to itself

Derivative is faster than itself:

Self-power:x^x

Gamma(x)

Tetrational function [tex]{}^xb=b[4]x[/tex]; b[4]1=b, b[4]2=b^b, b[4]3=b^(b^b), etc. (note the grouping)
 
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What is the purpose of organizing functions by their speed of growth?

The purpose of organizing functions by their speed of growth is to categorize and compare different functions based on how quickly their output value increases as the input value increases. This can help determine the efficiency and performance of a function.

How do you determine the speed of growth of a function?

The speed of growth of a function is determined by looking at its rate of change. This can be calculated by taking the derivative of the function, which represents the slope of the function at any given point.

What are the different categories of function growth?

The different categories of function growth are constant, logarithmic, linear, polynomial, exponential, and factorial. These categories represent different rates of growth, from slowest to fastest.

Why is it important to consider the speed of growth when organizing functions?

Considering the speed of growth when organizing functions is important because it can help determine the most efficient algorithm or approach for solving a problem. Functions with slower growth rates may be more suitable for smaller inputs, while functions with faster growth rates may be better for larger inputs.

Can a function's speed of growth change?

Yes, a function's speed of growth can change depending on the input values. For example, a polynomial function may have a slower growth rate for small input values, but as the input values increase, the growth rate may become much faster. This is why it is important to analyze the speed of growth for different ranges of input values.

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