I am trying to calculate the values of $a$ and $b$ in the following function: $$ f(x) = -e^{ax} + b + 1. $$ There are a few "rules" in play:
$\int\limits_{0}^{n} f(x)\, \mathrm{d}x= v$
$f(n) = rb$
where
- $v$ is the total area of the subgraph of $f$,
- $n$ is the length of the base of this subgraph
- $r$ is the desired ratio between the start ($v_{0}$) and end ($v_{n}$) points (ex. 0.5)
Is it possible to derive equations to calculate $a$ and $b$, given $v$, $n$ and $r$? If so, would you be willing to show me how?
The constraints are written
$$\begin{cases}\dfrac{1-e^{an}}a+(b+1)n=v,\\-e^{an}+b+1=rb.\end{cases}$$
You can eliminate $b$ using the second equation,
$$b=\frac{1-e^{an}}{r-1},$$
which you plug in the first, giving a nasty nonlinear equation in $an$
$$\dfrac{1-e^{an}}{an}+\frac{1-e^{an}}{r-1}=\frac vn-1.$$
This must be solved by a numerical method. (There is a slight hope that it can be done with Lambert's W function, but I have not investigated.)