Approximating Values Using Local Linearity and Linearization · 利用局部线性和线性化近似数值
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| tangent line/ˈtændʒənt laɪn/ | 切线 | qiè xiàn |
| local linearity/ˈləʊkl lɪˈnɪərɪti/ | 局部线性性 | jú bù xiàn xìng xìng |
| linearization/ˌlɪnɪəraɪˈzeɪʃn/ | 线性化 | xiàn xìng huà |
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.
放大到足够近,曲线看起来是直的
- 取任意一条光滑曲线,放大到某一点,它会摊平成(几乎)一条直线。
- 那条线是切线,在该点附近它紧贴曲线。
- 所以切线是附近函数的极好近似——这就是局部线性。
- 它让你只用一个斜率和一个点就能估计难算的函数值。
The tangent hugs the curve · 切线与曲线相贴
y = √x
Near the point of tangency the line and curve nearly coincide — that closeness is why $L(x)$ approximates $f(x)$. · 在切点附近，直线与曲线几乎重合——这种接近性使得 $L(x)$ 能够近似 $f(x)$。
The idea that a smooth curve looks straight up close is called local ____. · 光滑曲线在近距离内看起来像直线的概念称为局部 ____。
Local linearity underlies the tangent-line approximation. · 局部线性化是切线近似的基础。
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.
构造切线
- $x=a$ 处的切线过点 $\big(a,f(a)\big)$,斜率为 $f'(a)$。
- 点斜式给出线性化:
-
$$L(x)=f(a)+f'(a)(x-a)$$
- 这个 $L(x)$ 就是切线,换个名字当作近似公式。
The linearization of $f$ at $a$ is $L(x)=$ · $f$ 在 $a$ 处的线性化为 $L(x)=$
Point-slope of the tangent: $f(a)+f'(a)(x-a)$. · 切线的点斜式：$f(a)+f'(a)(x-a)$。
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.
近似一个邻近值
- 要估计 $x$ 接近 $a$ 时的 $f(x)$,改算 $L(x)$——通常是简单算术。
- 选 $a$ 为一个邻近点,使 $f(a)$ 与 $f'(a)$ 简单(一个"好"数)。
- $x$ 越接近 $a$,近似越好。
- 例:要估计 $\sqrt{4.1}$,用 $f(x)=\sqrt x$ 在 $a=4$ 处,那里 $f(4)=2$ 很干净。
Using $L(x)=2+\tfrac14(x-4)$, estimate $\sqrt{4.1}=L(4.1)$. · 使用 $L(x)=2+\tfrac14(x-4)$ 估计 $\sqrt{4.1}=L(4.1)$。
$2+\tfrac14(0.1)=2.025$.
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.
高估还是低估?看凹凸
- 切线落在弯曲图像的一侧,所以 $L(x)$ 偏向一边。
- 凹向上(曲线向上弯,$f''>0$):切线在曲线下方 → $L$ 是低估。
- 凹向下($f''<0$):切线在曲线上方 → $L$ 是高估。
- 检查 $f''$ 的符号就知道你的近似偏向哪边。
If $f$ is concave down near $a$, the tangent-line estimate $L(x)$ is an... · 如果 $f$ 在 $a$ 附近向下凹，则切线估计值 $L(x)$ 是一个...
Concave down → tangent above the curve → overestimate. · 向下凹 → 切线位于曲线上方 → 高估。
A linearization is most accurate for inputs far from the point of tangency. · 线性化对于远离切点的输入最为准确。
It is most accurate near the point; accuracy drops as you move away. · 它在切点 附近 最准确；随着距离增加，准确性下降。
Select all · 所有 correct over/under conclusions. · 选择 所有 正确的过高/过低结论。
Up → tangent below → under; down → tangent above → over; both set by $f''$. · 上凸 → 切线在下 → 低估；下凸 → 切线在上 → 高估；两者均由 $f''$ 决定。
Linearization is only accurate near the point of tangency — far from $a$ the straight line drifts away from the curve. And the over/under call depends on concavity: concave up → tangent below → underestimate; concave down → overestimate. Don't guess the direction; check the sign of $f''$.
线性化只在切点附近准确——离 $a$ 远,直线就偏离曲线。而高估/低估取决于凹凸性:凹向上 → 切线在下 → 低估;凹向下 → 高估。别猜方向;检查 $f''$ 的符号。
Estimate $\sqrt{4.1}$ using linearization at $a=4$.
- $f(x)=\sqrt x=x^{1/2}$, so $f'(x)=\tfrac{1}{2\sqrt x}$; $\;f(4)=2$, $\;f'(4)=\tfrac14$.
- $L(x)=2+\tfrac14(x-4)$, so $L(4.1)=2+\tfrac14(0.1)=2.025$.
- Since $\sqrt x$ is concave down ($f''<0$), this is a slight overestimate (true value $\approx2.0248$).
用 $a=4$ 处的线性化估计 $\sqrt{4.1}$。
- $f(x)=\sqrt x=x^{1/2}$,所以 $f'(x)=\tfrac{1}{2\sqrt x}$;$\;f(4)=2$,$\;f'(4)=\tfrac14$。
- $L(x)=2+\tfrac14(x-4)$,所以 $L(4.1)=2+\tfrac14(0.1)=2.025$。
- 因为 $\sqrt x$ 凹向下($f''<0$),这是略微的高估(真值 $\approx2.0248$)。
Local linearity lets the tangent line approximate a function near a point. The linearization is $L(x)=f(a)+f'(a)(x-a)$; use it to estimate $f(x)$ for $x$ near $a$. Concavity sets the direction of error: concave up → underestimate, concave down → overestimate.
局部线性让切线在一点附近近似函数。线性化是 $L(x)=f(a)+f'(a)(x-a)$;用它估计 $x$ 接近 $a$ 时的 $f(x)$。凹凸性决定误差方向:凹向上 → 低估,凹向下 → 高估。