From observations, for quadratic functions, the average rate of change in $[a,b]$ is the arithmetic average of the derivative evaluated at $a$ and $b$. $${\rm{Avg. ROC}_{[a,b]}}=\frac{f'(a)+f'(b)}{2}\qquad f(x)=c_0+c_1x+c_2x^2$$ So, motivated by this video by Michael Penn, I set out to find "which" average works to find the average ROC for the exponential function and others.
Video Overview:
Basically, all averages can be thought of as special cases of $$A(t)=\frac{\int_a^bx^{t}dx}{\int_a^bx^{t-1}dx}\tag{1}$$ For example, $A(1)$ is the arithmetic mean, $A(-0.5)$ is the geometric mean, $A(2)$ is the centroidal mean, etc.
I experimented with values of $t$ in $\rm Eq. (1)$ until a satisfying result was found for respective kinds of functions. Here are my results:
- Exponentials: $0$ ("Logarithmic Mean")
- Quadratic: $1$ ("Arithmetic Mean")
- Cubics: $\frac12$ ("Heronian Mean")
- Quartic: $\frac13$ (?)
- Quintic: $\frac14$ (?)
- (Speculation) $n$-degree polynomial: $\frac{1}{n-1}$
- Logarithms: $-1$ (?)
- (Speculation) $\operatorname{Li}_n$ : $-n$
The last two numbers were tested for small $a$ and $b$, and are very very crude approximations using Desmos. Here is my Desmos (edited and cleaned for others to use!).
I want an explanation of these various phenomena and the pattern behind these numbers.
Edit. I deleted this question because I thought I knew the answer. As it turns out, I didn't. After almost a month, I returned to this question to try it again. Unfortunately, I was stuck. $$\frac{\int_a^bx^{t}dx}{\int_a^bx^{t-1}dx}=\frac{n}{n+1}\frac{[f'(b)]^{n+1}-[f'(a)]^{n+1}}{[f'(b)]^n-[f'(a)]^n}$$ By our condition that it should equal the average rate of change, $$\frac{n}{n+1}\frac{[f'(b)]^{n+1}-[f'(a)]^{n+1}}{[f'(b)]^n-[f'(a)]^n}=\frac{f(b)-f(a)}{b-a}\tag{2}$$ I need to somehow find an equation that relates $n$ and a function $f(x)$ with $\rm Eq. (2)$. I didn't know where to go from here on. I spent almost an hour fiddling around with Laplace transforms and other algebraic manipulations, but I got nowhere.