Use of the term first order dependency In a question I am doing it says:

Show explicitly that the function $$y(t)=\frac{-gt^2}{2}+\epsilon t(t-1)$$ yields an action that has no first order dependency on $\epsilon$.

Also my textbook says that

[...] if a certain function $x_0(t)$ yields a stationary value of $S$ then any other function very close to $x_0(t)$ (with the same endpoint values) yields essentially the same $S$, up to first order of any deviations.

I am confused about the first order bit? In the first case does it mean that $\frac{\partial S}{\partial \epsilon}=0$ or that it does not depend of $\epsilon$ but may take some other constant value. In the second case does it mean likewise or something different, please explain?
 A: Hints:


*

*The action is 
$$\tag{A} S[y]:=\int_0^1 \! dt ~L(y,\dot{y}), \qquad L(y,\dot{y})~:=~\frac{m}{2}\dot{y}^2 -mgy,  $$
with Dirichlet boundary conditions 
$$\tag{B}  y(0)~=~0 \quad\text{and}\quad y(1)~=~-\frac{g}{2}. $$

*Calculate explicitly the composed function
$$\tag{C} s(\epsilon)~:=~ S[y_{\epsilon}] , $$
where 
$$\tag{D} y_{\epsilon}(t)~:=~-\frac{gt^2}{2}+\epsilon t(t-1).$$

*Check that the virtual paths (D) satisfy the Dirichlet boundary conditions (B). Why do we need to check that?

*Show explicitly that the function $s(\epsilon)$ has no first order dependence on $\epsilon$. What is the physical significance of this fact? 
References:


*

*David Morin, The Lagrangian Method,  Chap 6, Lecture notes, 2007; Exercise 6.30.

A: First-Order Dependency – A dependency that represents a direct causal relationship (e.g. if agent x produces event z, we say that "x is a first-order dependency of z").
Reference:
http://www.testability.com/Reference/Glossaries.aspx?Glossary=DependencyModeling
