Linear Approximation and Differentials

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


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,
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