Need help plotting a curve, no calculus knowledge....

[yoda-morpheus]

(sorry in advance for not knowing how to use laTex and also not knowing calculus, I thought maybe before I teach myself calculus to figure out this one problem maybe you all could fill in what I'm not figuring out... Thanks in advance...)

I recently found out that I needed to know calculus in order to solve this problem mathematically that I am trying to solve. It's basically a problem involving figuring out the slope of a curve. The problem I have is the curve needs to be able to change according to the properties of the set, so for any set I need to be able to figure out the curve which leads me to my answer 'y' for any 'x'...

It goes like this:

I start out with a numeric set. The numeric set can be plotted on the x axis, so each item in the set is 'x'.

Out of the set, I get the total sum of all the items in the set, denoted as 't'.

I have any target out of 't' denoted as 'r', which is a random percentage of 't', denoted as 'e', which is also 'r/t'. Also, I can find the flat average of 't' by dividing it by the size of the set 'x', denoted as 'a'.

So I have the set X, items x1, x2, x3, x4, x5... xN. I have t = x1+x2+x3+x4..+xN. I have any r along the x axis, but never more than .3 times t. I then have t/N = a.

The problem gets a little more complex with this condition:

if x = a, y = r/t = e. So we can say a flat line would be 'ae'.

Another straight line at an increasing angle from '0' would be given by y = xr/t = xe.

What I have right now is that for any x <= a, the necessary equation for 'y' is then given by any of the following:

x <= a, y = (x/a * r/t) * x = (xr/at) * x = (x^2r)/(at) = (x/a) * e * x = (x^2e)/a, whichever form is simplest to work with up to that point...

This is what I need, basically, up to x = a. However, I don't know how to finish the curve after x > a, or what the resulting curve is called. I didn't know it was a calculus problem until I started looking for an answer online instead of trying out different versions of the formula, then I plotted it on a graph...

So let's assume that e = .3 and a = 52. So exactly at x = 52, y = 15.6. That is where the flat line I mentioned above would be for any x, if all x = a.. xe = 15.6 as well. But that's not what I'm after...

Then at x = 10, it comes out to y = 5.77. At x = 0, y = 0, and at x = 40, y = 9.23.

So then we should realize that the line starts at 0,0, x -> 0 as a condition, x <- 't' as a condition, and it curves from 0, 0 to 15.6, in this example. There will always be a finite set of 'x', and 't' isn't necessarily important once 'r' is decided, to arrive at 'e'. So when it comes to the slope of the line, for any set 'x' I will always know 'e' and 'a' from the set and those two variables, so all we really should care about is any 'x', 'e', and 'a', where for any set 'x', we can determine that 'e' and 'a' will be constants.

When I plotted on a graph and tried to finish the curve for x > a, I found a curve that starts at 0,0, bends up to the given equasion's x,y, then starts to flatten out after that...

So because I don't know calculus, I can't figure out how to finish my curve...

The reason I am working on this is related to a progressive payment plan. But that's an entirely different story that I don't even want to start touching on.

Thanks if you all can finish this curve or correct it so that it works. In addition, it may be helpful to explain that if you wanted to only arrive at 'e', x/a = e at that exact point in the curve...

Thanks a ton. I've spent three days now trying to finish this curve and I still haven't figured it out. I hope I have explained the problem domain in enough detail. I always end up screwing up the x > a part. What was bouncing my noggin at first was I thought I needed the inverse of x <= a. What I actually needed was to continue the line...

Anyway, that's as much info as I can give. If you can figure out the rest or need additional math or parts to the curve, I can offer that. Sorry in advance for never taking calculus, and since I can't remember what type of curve this is, when I go back to college I should probably re-take all of algebra through trig and then finally take calc. Seems I forgot a ton of everything...

 

[yoda-morpheus]

I guess I should clarify the problem domain after looking back at this to simplify the required solution...

I have three variables derived from any set 'x': x, e, and a.

After the set X1...XN (where all members of X are greater than or equal to zero) is established, I then determine the total of X, t and target of t, r; and r/t gives me 'e'. 'a' is determined by t/N. 'a' and 'e' are then constants at that moment, and so it t and r.

The problem is this: The y-axis must curve sharply enough from x,y = 0,0 to x,y; x = a = x,xe. Then it must flatten out after that all the way to t.

The actual problem and caveat is that for the entire set X, all resulting Y values must sum to total exactly r, with a sufficient curve from 0,0 to x,xe and then flatten for x > a, but the y values must sum to r exactly.

Now you can see that while I found an acceptable starting curve approximately just by doing numbers and theoretical realistic distributions, I still have not found the exact appropriate curve. I thought I needed to invert it for the rest, but I actually needed to continue it. Then I realized that I may not have the correct equation even for the start of the curve, but it is approximately what I need...

The equation that I have for x <= a needs to be something like the following, which determines the curved line from 0,0 to x = a. (x is any point along the x axis, and x is any member of the set), and this gives me 'y' (a point on the y axis):

x <= a, y = x²/a * e, or rather x²e/a, either way works out the same.

Then I just need to find the rest of the curve for x > a. I figure what I have done here is split up the curve into two parts up front, when it really needs to be a continuous curve all the way to 't'...

So what I mean by that is I may not actually have the real equation for the actual curve. If I did, I should be able to calculate x > a in the same manner, or at least I figured I could, but it doesn't work that way either... Now I guess you can see why I'm confused as to where I've been and gone with this thing...

Thanks for the help. It's appreciated.