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I have to make a presentation to business directors and I want to explain that for a particular item, the function of cost in effort is beyond "linear growth".

Here's where it gets hairy. My function is

effort = $ax^b$

or said in English "Effort is $x$ to the power of a constant $b$ times a constant $a$". In this context we expect the constant $b$ to be around $1.2$, so effort will increase faster and faster, as you increase $x$. I know I cannot call this "exponential growth" because $x$ my variable is not in the exponent. So what is the correct way to describe this, what sits between "linear growth" and "exponential growth". Can I call this "polynomic growth"?? "polynomial growth"?? This sounds odd for a business context, and odd for an exponent of $1.2$...

Clarification: My $x$ gets up to several million, so the emphasis on the difference with linear growth is very real.

Alraxite
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Mishax
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    You could say "superlinear" growth. – yoknapatawpha Feb 06 '14 at 17:34
  • I don't find it precise. "Superlinear" means "anything more than linear". A 2nd degree polynomial is superlinear, but also exponential is superlinear. – frabala Feb 06 '14 at 17:44
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    Perhaps a graph like this will be more convincing in a presentation than merely finding the correct terminology. – Austin Mohr Feb 06 '14 at 17:56
  • In English it is called a power function http://en.wikipedia.org/wiki/Power_function or otherwise a power law http://en.wikipedia.org/wiki/Power_law – mkl314 Feb 06 '14 at 18:14

2 Answers2

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There is a matrix multiplication algorithm that is about $x^{2.7}$ fast (where $x$ is the size of the input). We say that this algorithm solves the problem in polynomial time, although 2.7 is not an integer. So, I guess "polynomial growth" is ok in a mathematical context. I don't know about business...

frabala
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    You don't have to guess, it's perfectly ok if 2.7 isn't an integer because it's $\mathcal O(x^3)$, which is polynomial. – GPerez Feb 06 '14 at 17:53
  • I agree. I'm from a computer science background and have had "polynomial is good" drilled into me. It's a bit of a shift in mindset to the point where an $x^{0.2}$ ratio in effort to linear is thought of as significant :) – TooTone Feb 06 '14 at 20:12
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Technically $ax^{1.2}$ sits between $ax^1$ linear growth and $ax^2$ quadratic growth. Quadratic growth is a lot less painful than exponential growth. Depending on what you want to emphasise, I would say it's close to linear growth, or greater than linear growth (either could be the case depending on the context). If you plot a graph with the x-axis related to something meaningful to your business you can actually illustrate the difference. A couple of rough plots below to show the idea.

The first just plots both your function and $y=x$ on the same plot: you could add annotation to show the extra costs you're incurring as a result.

enter image description here

The second plots the ratio.

enter image description here

TooTone
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