Central difference quotients are often preferred over forward or backward difference quotients when estimating the derivative of a differentiable function from tabulated data.
(a) Provide a qualitative explanation for why incorporating data on both sides of a point generally yields a more accurate derivative estimate.
(b) Discuss the effect of data spacing on the magnitude of the approximation error, referring to both very large and very small spacings.
(c) For equally spaced data, analyze one advantage and one disadvantage of using many neighboring points when constructing a numerical derivative estimate.