Why study Simpson's Rule, or how to approximate errors? If you have a lot of different calculators, computers and computer programs that can estimate pi, for example, try them. They may not agree in all the decimal places. How do you know which one is correct? Actually we (my husband and I) did this several years ago and were surprised by the descrepancies. Even if your algorithms produce the same numbers to 20 decimal places, they still may be wrong; after all, the harmonic series always converges on a computer or calculator (the divergence is too slow for this finite machines).