MATH.10:4.3 - Calculate the effect of the variation

Form the difference

DeltaJ(h) = J(x(h)) - J(x(0)).

Substitute the whole changed candidate, including dependent values. Simplify enough to determine the sign or magnitude relevant to the question.

For a differentiable comparison with a scalar parameter h near zero, write

DeltaJ(h) = a*h + r(h), with r(h)/h -> 0,

when that expansion is justified. The coefficient a is the first variation along this family. Higher-order terms or a bound on the remainder can be needed when a vanishes, or when choosing a finite step.

A derivative tells how sufficiently small changes behave under its limit assumptions. To select a particular step, check the finite difference or a bound that covers that step. For a discrete family, compare its admissible members directly; differentiability is not an entry requirement.

If the calculation uses a numerical approximation, relate its error to the sign or comparison being used. C.29.2 develops that computational task. A difference interval entirely below zero can establish improvement for minimization; an interval spanning zero leaves that comparison unresolved.