Constructing approximations composed of piece-wise elementary functions, such as cubic splines, is an important part an engineer's mathematical skill. In this paper, we show how a computer algebra system can be simply exploited to find local parabolic approximations that, besides making stunning improvements to accuracy (over the linear)encourage students to view approximations, rather than global formulae, as the norm.

Parabolic approximations are developed for a Simpson's rule for indefinite integrals with any number of unevenly spaced sub-divisions through the interval of integration, central difference approximations to the derivative, Muller's root finding method and a generalized Euler method, seen from a unified approach that extends the linear approximation technique.  The techniques are well known in books on numerical analysis, but here the emphasis is not on error analyses per se but on simple applications of the calculus, made feasible for the classroom by the use of a computer algebra system in an elementary, but non-trivial, way.

The rules for differentiating polynomials, the Fundamental Theorem of Calculus and the solution of three linear algebraic equations in three unknowns, are assumed known for this work which leads on naturally to Taylor series approximations. The fact that such, or even better, algorithms might already be programmed into the computer algebra system being used, should not detract from the pedagogic value.