Approximation of functions

In mathematics, a linear approximation is an approximation of a general function using a linear function (more precisely, an affine function). They are widely used in the method of finite differences to produce first order methods for solving or approximating solutions to equations.

Given a twice continuously differentiable function ${\displaystyle f}$ of one real variable, Taylor's theorem for the case ${\displaystyle n=1}$ states that

where ${\displaystyle R_{2}}$ is the remainder term. The linear approximation is obtained by dropping the remainder:

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