Linear approximation

The linear approximation of a function $f$ at a point $a$, also called the tangent line approximation or local linearization, is the tangent line $L(x) = f(a) + f'(a)(x - a)$. It exists where $f$ is differentiable at $a$. Also written $f(x_0 + \Delta x) \approx f(x_0) + f'(x_0)\,\Delta x$, with $x_0$ for $a$. This page shows it on seven functions: four smooth ones, and three that are not differentiable at 0.

The graph is centred horizontally on the blue point P $= (a, f(a))$, with equal scales on both axes. The red step Δx also sets the zoom: the view shrinks with it, stretching as needed to keep P, the tangent's rise and Q in sight. Over the step the tangent line rises by the blue differential $dy = f'(a)\,\Delta x$, while the curve rises by $\Delta y = f(a + \Delta x) - f(a)$. The yellow arrow is the error:

A function is differentiable at $a$ exactly when there is a finite number $m$ with $f(a + h) - f(a) - m h = o(|h|)$, that is, the error divided by $h$ tends to 0 as $h \to 0$ from both sides; then $m = f'(a)$. The page draws the step to the right only ($h = \Delta x > 0$), so the picture illustrates this rather than proves it; where a tangent exists is read from each function's formula. Because the view shrinks together with $\Delta x$, $|E|/\Delta x$ says roughly how far, in view sizes, the curve has strayed from the line at Q, which is why a differentiable curve looks straighter the further you zoom in. If $f''(a) \ne 0$, then $E \approx \tfrac12 f''(a)\,\Delta x^2$ for small $\Delta x$, so once $\Delta x$ is small each tenfold zoom makes the error about a hundred times smaller. Where $f''(a) = 0$, as for $\sin x$ at 0, it shrinks faster still.