  Finite differences have been widely used in mathematical theory as well as in
scientific and engineering computations. These concepts are constantly
mentioned in calculus. Most frequently-used difference formulas provide
excellent approximations to various derivative functions, including those used
in modeling important physical processes on uniform grids. However, our
research reveals that difference approximations on uniform grids cannot be
applied blindly on nonuniform grids, nor can difference formulas to form
consistent approximations to second derivatives. At best, they may lose
accuracy; at worst they are inconsistent. Detailed consistency and error
analysis, together with simulated examples, will be given.
