  The polarization of graphene is calculated exactly within the random phase
approximation for arbitrary frequency, wave vector, and doping. At finite
doping, the static susceptibility saturates to a constant value for low
momenta. At $q=2 k_{F}$ it has a discontinuity only in the second derivative.
In the presence of a charged impurity this results in Friedel oscillations
which decay with the same power law as the Thomas Fermi contribution, the
latter being always dominant. The spin density oscillations in the presence of
a magnetic impurity are also calculated. The dynamical polarization for low $q$
and arbitrary $\omega $ is employed to calculate the dispersion relation and
the decay rate of plasmons and acoustic phonons as a function of doping. The
low screening of graphene, combined with the absence of a gap, leads to a
significant stiffening of the longitudinal acoustic lattice vibrations.
