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.

Author Bio

As a freshman at in Baylor University's Honor's College, Brianstarted this research project that gave way to his paper in August2005. He worked with his faculty advisor, Professor Qin Sheng, onapplying calculus knowledge to derivative approximations that manypure and applied mathematicians were interested in. Brian's firstinterest in working on such a research project was innocently due tohis AP calculus training, and advanced lectures taught at Baylor.Brian continued his work throughout his first year. Preliminaryresults were reported at MAA Sectional Meeting at Midwestern StateUniversity, Texas, in April 2006. As a sophomore, Brian is nowinvolved in more complex research projects leading to his degree.Brian enjoys science fiction, traveling and visiting his family inSan Antonio, TX.

Andrew is currently a sophomore at the Westwood High School in Austin. He and his family have just moved from Dayton, Ohio. He has been very interested in mathematics and science subjects, in particular computer science and applied mathematics. He enjoys programming in Java and MatLab for solving real problems and for playing games. Andrew studies in an enriched TAG Program. He completed his introductory modern physics in the CTD Program at Northwestern University. Under the guidance of his summer research advisor, Professor Qin Sheng, he not only learned the concepts of calculus, but also understood the importance of approximations via finite differences.Working independently, he developed all MatLab programs required to show either the consistency or inconsistency of different formulas. Andrew also enjoys many activities beyond the sciences, including reading, playing computer games, swimming, and playing his cello with fellows in the school orchestra.