Advanced Calculus pp 71-104 | Cite as

# 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 transformation*dxdy = 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.

## Keywords

Homogeneous Polynomial Remainder Function Taylor Polynomial Binomial Expansion Microscope Equation## Preview

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