Linear approximation

Summary

A linear approximation of a function $f$ near a point $a$ replaces $f$ by an affine function. This function has the same value and the same derivative at $a$. For a differentiable map $f$ from an open subset of $\mathbb{R}^n$ to $\mathbb{R}^m$, it is $f(x) \approx f(a) + Df(a)(x - a)$, where $Df(a)$ is the total derivative. The error is small compared with $\|x - a\|$ as $x \to a$.

Remarks

For a function of one variable, the graph of the linear approximation is the tangent line. For a function of two variables, it is the tangent plane. For $f(x) = \sqrt{x}$ near $a = 4$, the approximation $\sqrt{x} \approx 2 + (x - 4)/4$ gives $\sqrt{4.1} \approx 2.025$, and the true value is $2.0248\ldots$.

Explanations

L1 human

Near a point, a smooth function looks like a straight line, or a flat plane in more variables. So you can estimate values of the function near the point with simple linear arithmetic.