# Brainstorm and confusion of concepts

1. Jan 13, 2016

### Bruno Tolentino

I know several math formulas, like which I will write below.
$$\int_{x_0}^{x_1} f(x) dx$$
$$\frac{\int_{x_0}^{x_1} f(x) dx}{x_1-x_0}$$
$$\frac{\int_{x_0}^{x_1} f(x) dx}{2}$$
$$f(x_1) - f(x_0)$$
$$\frac{f(x_1) - f(x_0)}{x_1-x_0}$$
$$\frac{f(x_1) - f(x_0)}{2}$$
$$\frac{f(x_1) + f(x_0)}{2}$$
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?

2. Jan 13, 2016

### Staff: Mentor

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
None that I'm aware of.
There's no such concept, as far as I know.
There is none.
Again, none of the things you posted is an equation. An equation states that two expressions are equal (i.e., has = in it).
The question is meaningless.