You might have seen the famous approximation $\sin(x) \approx x$ for $x$ near 0. We can use derivatives to approximate non-linear functions by simpler linear functions. We can better understand this by looking at the graph a function $f$ under magnification. The blue curve is the graph of $f(x) = \sin(x)$ and the yellow line is its tangent at $x = 0$. We can see that, under magnification of the curve at the point $x = 0$, the tangent line to the curve at $x = 0$ is a close approximation of the function for $x$ near 0. In general, for a function $f$: if $f$ is differentiable at a point $x_0$, then stronger and stronger magnifications at $(x_0,f(x_0))$ eventually make the curve containing $(x_0,f(x_0))$ look more and more like a non-vertical line segment, that line being the tangent line to the graph of $f$ at $(x_0,f(x)_0))$. Thus, a function $f$ that is differentiable at $x_0$ is said to be locally linear at the point $(x_0,f(x_0))$ . By contrast, the graph of a functi...
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