# Flat Function, Maximally Flat

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Informally speaking, a flat function has a slope of zero; They are flat, in the same sense that a floor of a house is flat. A more strict definition is that it is a smooth function with derivatives that vanish to zero at a given point. In other words, the values for the derivative get smaller and the slope flattens as it approaches that point.

An infinitely flat function will have all its derivatives (i.e. an infinite number of them) equal zero. For practical purposes (i.e. having to test all derivatives up to the ∞th), flat functions are defined for being flat to a certain degree. A “third degree flat function” for example, has all the derivatives up to the third, checked as being equal to zero.
All constant functions are flat functions, so do not need to have their derivatives checked. However, for other functions you can’t assume true flatness unless you can prove that every possible derivative is flat. Otherwise, you should note the degree of flatness.

For example, the even function f(x) = x2 is flat at x = 0 (at least to the tenth degree, which is as far as I checked):

Many even functions are flat at x = 0.

Flat functions are important in real analysis because they do not have a meaningful Taylor series expansion around 0; the non-constant part of the function always lies in the series remainder [1].

## Flat Function Examples

The following well-known example of a flat function is flat at the origin and not analytic at any point:

As this particular function is known to have all-zero derivatives at the origin, it is sometimes called The flat function.
Other examples [2]:

• f(x) = exp(-1/x) on (0, 1] and f(0) = 0,
• on [0. 1]

## Maximally Flat Function

A maximally flat function has as many of its derivatives as possible equal to zero [3]. Another way to put this: we want the function to be flat at a certain point and to change from “flatness” as slowly as possible.

Given a function fitting a certain form, the goal is to fit the “best” function to the formula— the one with the maximum number of zero derivatives (first, second,…nth derivative). The following image shows a few functions that fit the form f(x) = ax3 + bx2 + cx. The maximally flat function at x = 5 is the function f(x) = 5x3 – 15x2 + 15x [4], because it’s the one with the slope that “steepens” the slowest as we move away from f(x) = 5:

A maximally flat function at f(x) = 5 (red graph), shown with two other possibilities.

## References

[1] Pan, Y. & Wang, M. When is a Function not Flat? Retrieved May 7, 2021 from: http://www.stat.uchicago.edu/~meiwang/research/continuation.pdf
[2] Stoica, G. When Must a Flat Function be Identically 0? Retrieved May 7, 2021 from: https://www.tandfonline.com/doi/abs/10.1080/00029890.2018.1470415?journalCode=uamm20
[3] Ghausi, M. (1971). Electronic Circuits Devices, Models, Functions, Analysis, and Design. The University of Michigan.
[4] Maximally Flat Transformer Functions.doc. (2020). Retrieved May 7, 2021 from: http://www.ittc.ku.edu/~jstiles/723/handouts/Maximally%20Flat%20Transformer%20Functions.pdf

CITE THIS AS:
Stephanie Glen. "Flat Function, Maximally Flat" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/flat-function-maximally-flat/
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