It is said that different functions grow at different rates. What does a growth rate mean and how is it defined? Are there multiple definitions of a growth rate?
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Let my Wiki that for you. – amWhy Jul 24 '17 at 22:16
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- "It is said [from whom] that different functions grow at different rates. Who said what to whom?
– amWhy Jul 24 '17 at 22:18 -
Possible Duplicate: https://math.stackexchange.com/questions/146912/comparing-the-growth-rates?rq=1 – amWhy Jul 24 '17 at 22:20
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First, thanks for the rude reply. Second, about your second comment: here in this article, page 1: http://www.math.uconn.edu/~kconrad/blurbs/analysis/growth.pdf . Thank you. – LearningMath Jul 24 '17 at 22:21
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3My question is what is a growth rate, are there multiple definitions of it, and what does it mean for one function to grow faster than another? – LearningMath Jul 24 '17 at 22:23
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3Nothing whatsoever that I said was rude. I provided you with an overview of various function's growth rate. You failed to cite the source of the first sentence, and I pointed this out. You need to learn quite a bit more about this site. Thank you. – amWhy Jul 24 '17 at 22:25
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You might also want to [compare growth rate of functions(https://math.stackexchange.com/questions/86116/compare-growth-rate-of-functions?rq=1) – amWhy Jul 24 '17 at 22:29
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This may be a starting point: https://en.wikipedia.org/wiki/Big_O_notation – Michael Hardy Jul 25 '17 at 00:18
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An other possible definition for growth rate would be to describe the behavior of a function in the long term. Let me be a little more precise:
We say that $f: \mathbb{R} \to \mathbb{R}$ is of polynomial growth if there exist a polynomial $P$ such that:
$$\lim_{x \to \infty} \dfrac{f(x)}{P(x)}=1.$$
Similarly, one could talk about exponential growth or logarithmic growth, by changing the requirement on $P$ to an exponential function or a logarithmic function. You could also be more specific and describe the growth of the polynomial using it's degree.
Maxime Scott
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How is this related to comparing the derivatives of the two functions like in @Salahamam_ Fatima's answer? And why the limit is equal to one? Thanks. – LearningMath Jul 24 '17 at 22:37
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There are different possible meanings for growth rate. In @Salahaman_Fatima's answer, we look at the growth rate between two points. In my answer, we look at the long term behavior of your function. This is used in many different situation, including computer science when you try to write efficient code. In that situation, your goal would be to have the lowest growth rate so that large examples won't be taking to much time to compute. – Maxime Scott Jul 24 '17 at 22:43
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Why is the limit in your answer set to $1$? And where is the definition in @Salahaman_Fatima's answer used. Thanks again for your answer. – LearningMath Jul 24 '17 at 22:46
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@LearningMath You are being rude. Salahama is answering a question about rate of change. You are asking about growth rate functions. – amWhy Jul 24 '17 at 23:08
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Wait, what? Let me cite @MaximeScott: "In Salahaman_Fatima's answer, we look at the growth rate between two points." So, obviously, this is a growth rate too. I'm not familiar with the term and it's usage, so stop acting like a child, and don't put words in my mouth. The last thing I'm trying to do is being rude. I asked a question, and that means I am not familiar with the contents in it. Peace. And one last thing: Can you explain to me, how on earth am I being rude to Salahaman_Fatima or Maxime Scott? – LearningMath Jul 24 '17 at 23:12
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I apologize for the misunderstanding, but in the other answer, we are indeed looking at the rate of change of a function, which is a speed of growth between two points, but is not the growth of change as I defined it. – Maxime Scott Jul 24 '17 at 23:14
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@MaximeScott So now, if I say that one function grows faster than another, am I referring to derivatives (speed of growth) or growth rate? – LearningMath Jul 24 '17 at 23:24
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Hm, under your definition, $\sqrt x$ does not have polynomial growth. If one wanted to include such functions as polynomial growth, it might suit better to use:$$\exists n\in\mathbb N\text{ s.t. }\lim_{x\to\infty}\frac{f(x)}{x^n}=0$$ – Simply Beautiful Art Jul 24 '17 at 23:54
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Let $f $ be a function from R to R.
The growth rate from $a $ to $b\ne a$ is given by $$\Delta (f,a,b)=\frac {f (b)-f (a)}{b-a} .$$
$f $ grows faster than $g $ from $a $ to $b $ if $$\Delta (f,a,b)>\Delta (g,a,b) $$
hamam_Abdallah
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Is this the only definition of a growth rate? And in this case, it is said that $f$ grows faster than $g$ right? – LearningMath Jul 24 '17 at 22:25
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This is a definition of rate of change (i.e. the derivative). This is not a definition of the growth rate of functions. – amWhy Jul 24 '17 at 23:09
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@amWhy in french, Delta is called : le taux ( rate) d'accroissement ( growth). – hamam_Abdallah Jul 24 '17 at 23:29