Linear Approximation and Differentials

Read this section to learn how linear approximation and differentials are connected. Work through practice problems 1-10.

Introduction

Newton's method used tangent lines to "point toward" a root of the function. In this section we examine and use another geometric characteristic of tangent lines:

If $\quad \mathrm{f}$ is differentiable at a and $\mathrm{x}$ is close to $\mathrm{a}$ ,

then the tangent line $\mathrm{L}(\mathrm{x})$ is close to $\mathrm{f}(\mathrm{x})$ . (Fig. 1)

This idea is used to approximate the values of some commonly used functions and to predict the "error" or uncertainty in a final calculation if we know the "error" or uncertainty in our original data. Finally, we define and give some examples of a related concept called the differential of a function.

Source: Dale Hoffman, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2012/12/MA005-3.9-Linear-Approximation.pdf
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