What would be the function of said scenario?

[Dagl1]

Hi there,

Everyday I am reading articles besides doing experiments in the lab and I was joking with one of my friends that it is going to take me forever to read all 90 articles I still have on my desk, because on average I add 0.7 more articles per article that I read. So then I was wondering what type of mathematical function this would. I have written a quick python program which approaches the limit of whatever this function would be (I think it approaches 300 articles). I however still haven't been able to think of what the actual function would look like. Could anyone help me out, thanks!

-Dagl

PS. the python code (I know this can be written a lot better, but it works so whatever, although of course if you want to give me a tip sure (there is no need for having both B and X));

X = 90
B = 90
Xtotal = 0
while X > 0.0000001:
    X = X *0.7
    Xtotal = Xtotal + X
    print("X = "+ str(X))
    print("Xtotal = "+ str(Xtotal))
print(Xtotal+B)

[taeto]

It comes to 300 articles because \(\frac{90}{1 - 0.7} = 300,\) no?

[Dagl1]

Thank you!

That looks awfully familiar to the type of functions I came across in high school, but now I have two more questions! 

Firstly, if we would make this an Y(x) = function, what would it become (90 - x) / (1-0.7)?

Second, Why is that 1 there in the formula (as in, in highschool I just learned that this was a function, but now I would like to know WHY it is like this, if that makes sense).

Again, thank you very much

-Dagl

[taeto]

8 hours ago, Dagl1 said:

Firstly, if we would make this an Y(x) = function, what would it become (90 - x) / (1-0.7)?

If your dog eats \(x\) of the articles before you begin, and you start from only \(90-x\) articles instead of \(90,\) yes. Then \(Y(x) = \frac{90-x}{1-0.7}\) would be the total number of articles you would have to read. 

Remember that \(Y\) is the name of the function, and \(x\) is the name of its argument. It is \(Y\) that is the function, not \(Y(x),\) which is the value of the function \(Y\) at \(x.\)

8 hours ago, Dagl1 said:

Second, Why is that 1 there in the formula (as in, in highschool I just learned that this was a function, but now I would like to know WHY it is like this, if that makes sense).

If for every 2 articles you have read, you have to add .7 more articles, then the 1 would become a 2, etc. 

Edited February 19, 2019 by taeto

[Dagl1]

@taeto

Thanks! 
For some reason I thought this would be function that looks similar to y = ln(x) (in shape) but then with 300 being the limit, but I suppose that is only the case for the python program I wrote as with that one will just approach the total number of articles read (which has little to do with x in a function).

Hmmmm learn something everyday, I always thought Y(x) is the function with x being the argument, thanks!

Regarding the 1, ye that makes sense.

Thanks! I think I am gonna go and do some higschool math because I hadn't realized that I lost soo much knowledge since I left hahaha.

-Dagl

[taeto]

Yeah. How long have you been stuck with programming?  But seriously, I actually love programming stuff, did a lot of research in pure maths by writing programs, letting them run for a bit, and check out what kind of garbage they would deliver as output. It really works quite well to get papers in pure maths and theoretical computer science.

Your intuition is not unreasonable. I also thought that the answer would not be something linear. Though on reflection, it is the case that as you work through your load, it gets gradually lighter, though not by a whole lot.

 

 

[Dagl1]

Hahaha I am unsure how to answer that question; I have done 2 months of python programming about 1.5 years ago. But writing this program took only about a minute or 2. It's interesting to note that writing this program was extremely easy but I had been been trying to figure out the formula for a whole 45 minutes. 

Now I do have one more question, what would be the function (or at least the structure) of a function which starts at 90 and then gradually approaches 300 (with 300 being the limit)?

-Dagl1

[taeto]

1 hour ago, Dagl1 said:

Now I do have one more question, what would be the function (or at least the structure) of a function which starts at 90 and then gradually approaches 300 (with 300 being the limit)?

I assume you mean a function \(f : [0,\infty) \to \mathbb{R} \) with \(f(0)=90\) and \(\lim_{x\to \infty} f(x) = 300.\) 

The quality of the answer depends on your personal taste in functions. You can have something absolutely natural, but with an intricate look. Or with a very simple design, but properties that are ugly when you look into it.

The function \(f \) given by \(f(x)= 90 + \frac{210x}{x+1}\) is an obvious first choice. It looks pretty enough in a graph. Analytically it is still a little questionable. I do not know how you feel about a function involving a hyperbolic arctangent as an alternative.

[Dagl1]

@taeto

I was definitely thinking of 90 + 210x/(x+1). You have already been answering all my questions so if you don't feel like helping helpless me I can understand; but your answer raises three new questions! 

Firstly, why do we put /(x+1), where does that come from in a more, hmm logical/explanatory sense?
Secondly, why is that analytically questionable;p?
Lastly, well of course I think that the hyperbolic arctangent is a little mundane of course (I had to google what this was hahahah)! Would such a function be less analytically questionable?

Thanks for the amazing answers, if you don't feel like explaining this stuff, that is fine as I already have my original answer!

-Dagl

[taeto]

I suppose that \(x/(x+1)\) is the simplest expression that starts at \(0\) for \(x=0\) and increases to \(1\) in the limit as \(x\to \infty.\) 

It is not completely natural though, in that the \(+1\) may appear a little arbitrary, since you can replace \(x+1\) by \(x+c\) for any choice of \(c > 0\) and have the same effect. Also the function is not defined for \(x=-1\) resp. \(x=-c.\) 

Actually I meant the hyperbolic tangent \(\tanh \) in place of the \(x/(x+1)\) bit, but mistyped out of habit. It is nice throughout the reals and behaves similarly from \(0\) and up. It has no "arbitrary" looking parameters either. 

[Dagl1]

I see, so because when x is infinite it doesn't matter if it is +1 or +20 million because the function will still increase to 1 when x moves towards infinity?

Hyperbolic tangent tanh does look nice, but it does make me notice that these functions don't follow the iterations the program follows (where (0,90) (1,153) (2, 197)). Of course they get to the same limit, but they don't follow the same incremental steps (see below for output of program (I changed the code a little to accurately reflect data)).

increment = 62.99999999999999
Y = 153.0
x = 1
increment = 44.099999999999994
Y = 197.1
x = 2
increment = 30.869999999999994
Y = 227.97
x = 3
increment = 21.608999999999995
Y = 249.579
x = 4
increment = 15.126299999999995
Y = 264.70529999999997
x = 5