Approximations

Abstract

Approximations are at the heart of calculus. In Chapter 1 we saw that the transformation of differentials dx = φ'(s)ds can be traced back to the linear approximation Δx ≈ φ'(s)Δs (the microscope equation), and that the factor φ'(s) represented a local lengthmultiplier.We also suggested there that the transformationdxdy = rdrdθ of differentialsfrom Cartesian to polar coordinates has the same explanation: the polar coordinate change map has a linearapproximation (a twovariable “microscope” equation) and the factor r is the local area multiplier for that map. In this chapter we construct a variety of useful approximations to nonlinear functions of one or more variables. However, we save for the following chapter a discussion of the most important approximation, the derivative of a map.