Symmetric Difference Quotient

      f(x + h) - f(x - h)
              2h

The symmetric difference quotient is simply the average of the two difference quotients

     f(x + h) - f(x)                           f(x) - f(x - h)
            h                                               h

and can often give a better estimate than a one-sided difference quotient.

        f(2 + .001) - f(2 - .001)
                    .002

       =  2.001³ - 1.999³
                 .002

       = 12.000001

b.  using a right sided difference quotient

     f(2 + .001) - f(2)
               .001

    =   2.001³ - 2³
               .001

    =  12.006001

c.  Find the exact value of f ' (2) and compare to the answers in a and b.

     f ' (x) = 3x²

     so f ' (2) = 12