Brainstorm and confusion of concepts

[Bruno Tolentino]

I know several math formulas, like which I will write below.
[tex] \int_{x_0}^{x_1} f(x) dx [/tex]
[tex] \frac{\int_{x_0}^{x_1} f(x) dx}{x_1-x_0} [/tex]
[tex] \frac{\int_{x_0}^{x_1} f(x) dx}{2} [/tex]
[tex] f(x_1) - f(x_0) [/tex]
[tex] \frac{f(x_1) - f(x_0)}{x_1-x_0} [/tex]
[tex] \frac{f(x_1) - f(x_0)}{2} [/tex]
[tex] \frac{f(x_1) + f(x_0)}{2} [/tex]
And I know too that all equations above are importants, appears with very often. But my doubt is the following:
My head is confused, are to much equations and concepts disconnected. I know that the inverse analog of derivative is the primitive, but what's the inverse analog of the arithmetic mean? If the AM of two numbers is (a+b)/2, so the analog inverse is (a-b)/2? And what's the inverse analog of (a+b+c)/3? What's the difference between the equations that I posted above!? What's the inverse analogo of each equation that I wrote above?

 

[Mark44]

I know several math formulas, like which I will write below.
[tex] \int_{x_0}^{x_1} f(x) dx [/tex]
[tex] \frac{\int_{x_0}^{x_1} f(x) dx}{x_1-x_0} [/tex]
[tex] \frac{\int_{x_0}^{x_1} f(x) dx}{2} [/tex]
[tex] f(x_1) - f(x_0) [/tex]
[tex] \frac{f(x_1) - f(x_0)}{x_1-x_0} [/tex]
[tex] \frac{f(x_1) - f(x_0)}{2} [/tex]
[tex] \frac{f(x_1) + f(x_0)}{2} [/tex]
And I know too that all equations above are importants,

None of the above is an equation, so none would be considered a formula. They are all expressions.
Writing them as you have above is a meaningless exercise if you don't know what they represent.
##\int_{x_0}^{x_1} f(x) dx ## -- Could be the area under the graph of y = f(x) between x = 0 and x = 1 (depending on what the function is)
##\frac{\int_{x_0}^{x_1} f(x) dx}{x_1-x_0} ## -- Average value of f on the interval [0, 1]
##\frac{\int_{x_0}^{x_1} f(x) dx}{2} ## -- No significant meaning
## f(x_1) - f(x_0)## -- Vertical distance between two points on the graph of f
## \frac{f(x_1) - f(x_0)}{x_1-x_0}## -- Slope of the secant line between the points ##(x_0, f(x_0))## and ##(x_1, f(x_1))##
## \frac{f(x_1) - f(x_0)}{2}## -- No significant meaning
## \frac{f(x_1) + f(x_0)}{2} ## -- Average (or mean) of two function values

appears with very often. But my doubt is the following:
My head is confused, are to much equations and concepts disconnected. I know that the inverse analog of derivative is the primitive, but what's the inverse analog of the arithmetic mean?

None that I'm aware of.

If the AM of two numbers is (a+b)/2, so the analog inverse is (a-b)/2?

There's no such concept, as far as I know.

And what's the inverse analog of (a+b+c)/3?

There is none.

What's the difference between the equations that I posted above!?

Again, none of the things you posted is an equation. An equation states that two expressions are equal (i.e., has = in it).

What's the inverse analogo of each equation that I wrote above?

The question is meaningless.