Is there an equation that will graph the shape or predict the maximum amplitude of a single sample impulse fed through a One-Pole LPF? If you feed a single sample impulse of a given amplitude through a One-Pole LPF you get an envelope that looks like this:

The output of the LPF from its peak will settle into an exponential decay where time to reach $1/e$ amplitude is given by $1/(2πf)$ where $f$ is the cutoff frequency of the LPF.
I am wondering if there's any way to predict the maximum amplitude of the envelope from the sample rate, LPF cutoff freq, and amplitude of the single sample impulse. In general, is there a non-recursive equation that can describe or approximate the curve that results from these three factors?
If such an equation can be developed or exists, then one can take the derivative and solve where it equals zero to find the max.
Is this possible?
Alternatively, I can solve it numerically by just iterating through a temporary setup of a single sample impulse into an LPF of the given parameters until (output_1 &lt; output). This will find the max as well but this is tedious and computationally consuming.
I would hope some equation can do this faster.
Thanks for any thoughts.
As mentioned in the comments, an impulse response with the shape shown in your question can only be obtained by a system with two real-valued poles. In continuous time, with two distinct poles, the total impulse response is
$$h(t)=\frac{e^{-\alpha_1t}-e^{-\alpha_2t}}{\alpha_2-\alpha_1}u(t)\tag{1}$$
It's a basic exercise to determine the location of the maximum of $(1)$, and the result is
$$t_{max}=\frac{\log\left(\frac{\alpha_1}{\alpha_2}\right)}{\alpha_1-\alpha_2},\qquad\alpha_1\neq\alpha_2\tag{2}$$
For $\alpha_1=\alpha_2=\alpha$, the total impulse response is
$$h(t)=te^{-\alpha t}u(t)\tag{3}$$
and the location of the maximum is
$$t_{max}=\frac{1}{\alpha}\tag{4}$$
The figure below shows an example of an impulse response $(1)$ with $\alpha_1=0.4$ and $\alpha_2=0.6$, and with the location of the maximum predicted by $(2)$.