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To find the extremal for a functional in the context of quantitative methods, we generally apply the calculus of variations. A functional is an expression of the form: $$ J[x] = \int_{t_0}^{t_f} F(t, x, \dot{x}) \, dt $$ where $x(t)$ is the unknown function to be determined, $\dot{x} = \frac{dx}{dt}$, and $F$ is a given function of _______ ________ _____ ________ ___.
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