Central difference quotients are often preferred over forward or backward difference quotients when estimating the derivative of a differentiable function from tabulated data.
(a) Explain qualitatively why using data on both sides of the point of interest tends to improve the accuracy of a numerical derivative estimate.
(b) Describe how the spacing between data points influences the error in a numerical derivative approximation. Comment on both extremely large and extremely small spacings.
(c) Suppose you are given equally spaced data for a smooth function but may choose how many neighboring points to include in your calculation. Discuss one advantage and one potential disadvantage of using a wider stencil (more points) for the estimate.
(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.