Linear approximation

In calculus, linear approximation a method of approximating non-linear differentiable functions by using a linear one.[cite:@boelkins2025active][cite:@openstax2020calculus]

Inbox

• $Lx=(f^{\prime }x)(x-a)+fa$. Then $L$ is called the linearisation of $f$ at $x=a$.

Intuition

The idea of a linear function approximating a curve increasingly well over an increasingly short distance is exactly the thesis of the limit definition of the derivative: the tangent line at a point is the limit of a sequence of secant lines, a sequence which increases in quality of approximation, and decreases in span.

Generally, given a tangent line $L$ at $(a,fa)$, $Lx$ approximates $fx$ increasingly well as $x$ tends to $a$.