  Response functions are central objects in physics. They provide crucial
information about the behavior of physical systems, and they can be directly
compared with scattering experiments involving particles like neutrons, or
photons. Calculations of such functions starting from the many-body Hamiltonian
of a physical system are challenging, and extremely valuable. In this paper we
focus on the two-dimensional (2D) ultracold Fermi atomic gas which has been
realized experimentally. We present an application of the dynamical BCS theory
to obtain response functions for different regimes of interaction strengths in
the 2D gas with zero-range attractive interaction. We also discuss
auxiliary-field quantum Monte Carlo (AFQMC) methods for the calculation of
imaginary-time correlations in these dilute Fermi gas systems. Illustrative
results are given and comparisons are made between AFQMC and dynamical BCS
theory results to assess the accuracy of the latter.
