thanks everyone. I think my guess is right that the initial value is used to determine dimensionality. it makes even less sense in od.df, where the second argument tells it, but then this function is not supposed to be directly used by an enduser anyway.
I am looking at ForwardDiff. what had confused me is that I had thought it would not be able to decompile the function being passed in, but now I realize that it overloads all known elementary julia math functions to basically keep a symbolic copy, so when the function (not a string containing an unevaluated function) is passed into it, it can reconstruct it and apply the chain rule. clever.
I think there is a small math mistake with abs (and kinks) in it:
julia> f(x)= sum(abs.(x))
f (generic function with 1 method)
julia> ForwardDiff.gradient(f, [0.0])
1-element Array{Float64,1}:
1.0
should probably give NaN or Inf. But this is minor. thanks everyone.
Clarification: Julia has -0.0 and +0.0. In this case, it recognizes -0.0. maybe this is desired behavior??