While the term "first order function" isn't standard mathematical terminology, it's likely referring to either a first-order differential equation or something related to first-order logic. Let's address both possibilities:
1. First-Order Differential Equation
A first-order differential equation is a differential equation where the highest order derivative that appears is the first derivative. This means you only have the function itself (y), its first derivative (y' or dy/dx), and possibly the independent variable (x) in the equation. No higher-order derivatives like y'' (second derivative) are present.
-
General Form: A general form of a first-order differential equation is:
F(x, y, y') = 0
Where:
- x is the independent variable.
- y is the dependent variable (a function of x).
- y' is the first derivative of y with respect to x (dy/dx).
- F is a function.
-
Example:
dy/dx + 2y = x
This is a first-order differential equation because the highest derivative present is dy/dx (the first derivative).
-
Another Example:
y' = y2 + sin(x)
Again, only the first derivative, y', appears.
2. Connection to First-Order Logic (Likely Not the Intended Meaning)
Less commonly, the phrasing might allude to first-order logic, a system of formal logic. While not directly a "function," first-order logic deals with predicates and quantifiers applied to individual variables. Predicates can be thought of as functions that return a boolean value (true or false). If the context were logic, one might vaguely refer to a predicate that takes only individual variables (not sets or functions themselves) as somehow related to the idea of "first order." This is, however, an unusual and less precise usage. This is almost certainly not what the question intended.
Summary
If you're encountering "first order function" in a mathematical context, it most likely refers to a first-order differential equation, characterized by only containing the first derivative of the unknown function. It very rarely could refer to related concepts in first-order logic, but this interpretation is unlikely without explicit context.
