/sci/ - Science & Math

[Anonymous No.9512923]

Hey /sci/, here's some basic big boy maths that I'm having trouble working through. I just need to know what I'm missing here to complete my model.

Suppose you have an exponential function [math]y=f(x)[/math], and [math]R[/math] is that function's area under its curve from [math](x,x_0)[/math] and we're only interested in the function from range [math](S_L,S_u)[/math]. Essentially the horizontal and vertical boundaries for this function (the following points: [math](x,S_L)[/math] & [math](x_0,S_u)[/math] ) are fixed and what changes as [math]R[/math] changes is the curvature of the function as can be seen in the attached picture.

I fear that I'm lacking the mathematical know-how to make a function that properly reflects this concept as all I've been doing has been dicking around with Desmos and various exponential functions trying to find a model that I could fit some variables to.

[Anonymous No.9512933]

I basically want to combine this with a bell curve and simulate an "income game", just a small personal project

[Anonymous No.9512948]

in basic terms what I'm asking is if there is a simple method for pinning a function down at two points and then transforming it by increasing or decreasing the area under its curve

[Anonymous No.9513000]

>>9512948
a function in general, or the exponential function? because it looks like all you're talking about is take the integral of your function with some algebraic constants (like f(x) = a* e^(bx)) and set that equal to R

[Anonymous No.9513522]

>>9513000

the exponential function, in this case

the problem is that R (being the area under the curve) is a dependent property of the function itself, whereas I want to use it as the value that modifies the shape of the function

[Anonymous No.9513534]

>>9513522
>the problem is that R (being the area under the curve) is a dependent property of the function itself
Of course it is, it's the bloody area under the curve.

[Anonymous No.9513569]

>>9513534
>Of course it is, it's the bloody area under the curve.
I don't understand why you'd even post that, you ignored the most important part here.

I want to modify the area under the curve without moving the exponential function from the fixed points [math](x,S_L)[/math] & [math](x_0,S_u)[/math]. Therefore I need to use a variable(that is essentially on a 1-to-1 scale with R) in some way that modifies the function.

>>9513000
>like f(x) = a* e^(bx)
I have tried setting constants, but the only way to change the area in-so-far is to play with the constants until you get a the fixed points back. I am thinking that I am probably missing a formula or property that greatly simplifies this instead of manually playing with a or b.

[Anonymous No.9513896]

This post would be better off on /wsr/, anon.

Anyway, I think there's only one equation of the form a*e^bx that passes through any two given points. Try reverse engineering the equation to find a and b in terms of the points' co-ordinates.

[Anonymous No.9513923]

>>9513569
Oh, so SL and Su are not the asymptotes of the function?

[Anonymous No.9513969]

>>9513896
>>9513923
Nevermind, i'm an illiterate brainlet. Disregard the second paragraph.

[Anonymous No.9514064]

>>9513896

yeah, I just figured there would be a higher concentration of high-level mathematicians here

>>9513896
>Anyway, I think there's only one equation of the form a*e^bx that passes through any two given points. Try reverse engineering the equation to find a and b in terms of the points' co-ordinates.

I'll try reverse engineering it. If there is only one equation of the form a*e^bx that passes through any two given points that would make this a hundred times more difficult.

>>9513923

No, they are points the function touches, the idea is to modify the function's shape without changing those two points. In fact, you could think of the domain as [math] [x,x_0] [/math] and the range as [math] [S_L, S_u] [/math] for some values of [math] R \geq 0 [/math].

I will post a similar example with a parabola for coherence.

[Anonymous No.9514094]

>>9512923
>>9514064

same concept explained with a parabola, maybe it's more intuitive to grasp this way

[Anonymous No.9514100]

>>9514064
given points (xL,yL), (xU,yU)
assume f(x) = a*e^(bx)+c
a*e^(b*xL)+c = yL
a*e^(b*xU)+c = yU
int(f(x),xL,xU) = (a/b)*e^(b*x)+c*x | x=xU,xL = R

solve for a,b,and c

actually you have 5 knowns and 3 unknowns here so you probably need to look at linear combinations f(x) = a*e^(b*x) + c * e^(d*x) + f

but that's the algebraic procedure

[Anonymous No.9514111]

>>9514100

thank you very much, looks promising

[Anonymous No.9514198]

>>9514111
oh and i forgot you actually want
int(f(x),xL,Xu) - (xu-xL)*sL = R
the way you've defined it