Приближение значений с использованием локальной линейности и линеаризации

Zoom in far enough and a curve looks straight

• Pick any smooth curve, zoom in on one point, and it flattens into (almost) a straight line.
• That line is the tangent line 切线, and near the point it hugs the curve tightly.
• So the tangent is a great approximation to the function nearby — this is local linearity 局部线性性.
• It lets you estimate hard function values with just a slope and a point.

Build the tangent line

• The tangent at $x=a$ passes through $\big(a,f(a)\big)$ with slope $f'(a)$.
• Point-slope form gives the linearization 线性化:
• $$L(x)=f(a)+f'(a)(x-a)$$
• This $L(x)$ is just the tangent line, renamed as an approximation formula.

Near x = a the tangent line L(x) approximates f(x)

Approximate a nearby value

• To estimate $f(x)$ for $x$ near $a$, compute $L(x)$ instead — it's usually easy arithmetic.
• Choose $a$ to be a nearby point where $f(a)$ and $f'(a)$ are simple (a "nice" number).
• The closer $x$ is to $a$, the better the approximation.
• Example: to estimate $\sqrt{4.1}$, use $f(x)=\sqrt x$ at $a=4$, where $f(4)=2$ is clean.

Over- or underestimate? Ask concavity

• The tangent line lies on one side of a curved graph, so $L(x)$ leans one way.
• Concave up (curve bends upward, $f''>0$): the tangent is below the curve → $L$ is an underestimate.
• Concave down ($f''<0$): the tangent is above the curve → $L$ is an overestimate.
• Checking the sign of $f''$ tells you which way your approximation errs.