On damped, driven harmonic oscillators: Reviewed particular solution
found last time for a driving force 
. We found 
, where a and 
are complicated
functions of 
, and 
. Discussed why we can
obtain the solutions for real driving forces, 
and
, by taking real and imaginary parts respectively
of our solution for the driving force . Then gave
physical interpretation of these solutions: the driving force induces
a response proportional to itself, but with a time lag 
which vanishes in the limit of no friction. The amplitude of
the sinusoidal response depends on the driving frequency 
,
being maximal when 
(this maximal
response is called resonance). Without friction, the resonant frequency
just reduces to the natural frequency of the oscillator. Away from
resonance the amplitude of the response diminishes.
Then had mathematica demo by Chris Goodridge on approximating one-dimensional functions by Taylor expansions. Emphasized approximating functions near extrema by quadratic Taylor expansions, showing that a spring potential is a good approximation to an arbitrary potential function near equilibrium.
-- KB