Point-wise calculus

The work of Richard Neidinger implementing automatic differentiation in APL as a vector arithmetic is reformulated and extended. For functions of a single variable, an arithmetic is developed for function samples, nested vectors whose components hold the values, at any number of given sample points, of a function and its derivatives up to any specified order. It is argued that, for teaching purposes, this sampling provides a more intuitive introduction to mathematical functions and the rules of calculus than do algebraic formulae and that for certain calculations (such as the computation of polynomial approximations of high degree) the formulation provides superior algorithms for computation. As such, it offers an alternative approach to the teaching of elementary college mathematics.