Question

Could anyone explain 2A-11 a bit further starting from the algebra part: c((Vo+dV)^-k and going forward?

Answer 1

the idea is that there is a known quadratic approximation to any function of the the form (1 + x)^r

the equation we have been given is p(v^r) = Constant
and we want it in the form of (1 + x) ^ r

so initially we have p = C * ( v0 ^ -k) and v0 = v0 +dv
or p = C * (v0 + dv)^ -k
to get it in the (1 + x) form first term should be 1.... so lets take the v0 term outside...

p = C * (v0^-k) * (1 + dv/v0)^-k

if A = C * (v0 ^ -k) r = -k and x = dv/d0 (point to note as dv/v0 -> 0, x -> 0)
then the equation is in the terms of
p = A * ( 1 + x ) ^ r
= A ( 1 +rx + (1/2)r(r-1)x^2) ... From quad approximation

just replacing values of A, r, x we get the desired answer