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  "body": "\n## Summary\n\nA 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 $a$.\nFor 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|0FNKFT8H6E]].\nThe error is small compared with $\\|x - a\\|$ as $x \\to a$.\n\n## Remarks\n\nFor 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|035V802J58]].\nFor $f(x) = \\sqrt{x}$ near $a = 4$, the approximation $\\sqrt{x} \\approx 2 + (x - 4)/4$ gives $\\sqrt{4.1} \\approx 2.025$, and the true value is $2.0248\\ldots$.\n\n## Explanations\n\n### L1 human\n\nNear a point, a smooth function looks like a straight line, or a flat plane in more variables.\nSo you can estimate values of the function near the point with simple linear arithmetic.\n",
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