局部线性与线性化

AP 微积分 AB· AP Calculus AB CED — 微分的情境应用· 14 分钟阅读

1. 局部线性：几何基础★★☆☆☆⏱ 3 min

局部线性是可微函数的核心几何性质：当你将可微函数图像上任意点放大足够倍数后，图像会变得和该点处的切线几乎完全一致。对于任意在 $x = a$ 处可微的函数 $y = f (x)$ ，导数 $f^{'} (a)$ 给出切线的斜率。由于切线和原函数在 $x = a$ 处的函数值和斜率都一致，因此对于非常接近 $a$ 的 $x$ 值，函数的行为几乎和切线完全相同。

📘 定义

局部线性

可微函数的一个性质：当在可微点处放大足够倍数后，该点附近的函数图像与该点处的切线无法区分。

例:

对于 $y = sin x$ 在 $x = 0$ 处，放大到区间 $[- 0.1, 0.1]$ 后，曲线会和它的切线 $y = x$ 完全一致。

📐 例题

通过比较函数值和切线近似在 $x = 4.1$ 和 $x = 6$ 处的结果，验证 $f (x) = x$ 在 $x = 4$ 处是局部线性的。

1
计算 $f (4)$ 和 $x = 4$ 处的导数：
$f(4) = \sqrt{4} = 2, \quad f'(x) = \frac{1}{2\sqrt{x}}, \quad f'(4) = \frac{1}{4} = 0.25$
2
写出 $x = 4$ 处切线的点斜式方程：
$y = f(4) + f'(4)(x-4) = 2 + 0.25(x-4) = 0.25x + 1$
3
实际函数值： $f (4.1) = 4.1 \approx 2.0248$ 。切线近似： $y = 2 + 0.25 (0.1) = 2.025$ 。差值仅约0.0002，可忽略不计。
4
实际函数值： $f (6) = 6 \approx 2.4495$ 。切线近似： $y = 2 + 0.25 (2) = 2.5$ 。差值约0.05，大得多。
5
结论：在 $x = 4$ 附近，函数几乎和切线完全一致，因此它在 $x = 4$ 处是局部线性的。近似在远离切点处失效。

Exam tip:

当 AP 考试要求你解释为什么函数在某点是局部线性时，一定要提到该点函数可微，且对于点附近的输入，切线与函数的函数值和斜率都一致。

2. 线性化：公式与计算★★★☆☆⏱ 4 min

线性化（也称为切线近似）是局部线性的实际应用：我们用简单、易于计算的切线来近似切点附近复杂非线性函数的值。在 AP 考试中，线性化的求解和应用是本考点考查频率最高的技能。

📘 定义

$f(x)$在$x=a$处的线性化

$L (x)$

$f (x)$ 在 $x = a$ 处的切线，用于近似 $x = a$ 附近 $f (x)$ 的值。我们总是选择 $f (a)$ 和 $f^{'} (a)$ 都能精确计算的点作为 $a$ 。

例:

To approximate $15$ , we choose $a = 16$ , since $16 = 4$ is known exactly.

L(x) = f(a) + f'(a)(x - a)

📐 例题

求 $f (x) = x^{3} - 2 x + 1$ 在 $a = 2$ 处的线性化，再用它近似 $f (1.98)$ 。

1
计算 $a = 2$ 处的 $f (a)$ ：
$f(2) = (2)^3 - 2(2) + 1 = 8 - 4 + 1 = 5$
2
计算导数并在 $a = 2$ 处求值：
$f'(x) = 3x^2 - 2, \quad f'(2) = 3(4) - 2 = 10$
3
代入线性化公式并化简：
$L(x) = 5 + 10(x - 2) = 10x - 15$
4
计算 $L (1.98)$ 得到近似值：
$L(1.98) = 10(1.98) - 15 = 4.8$
5
The actual value of $f (1.98) \approx 4.801$ , so the approximation is extremely accurate.

Exam tip:

计算近似前一定要化简线性化表达式，减少算术错误。如果你将线性化保留为因式形式，一定要再次检查斜率项的展开是否正确。

3. 微分与变化量近似★★★☆☆⏱ 4 min

微分是线性近似的另一种记号，它关注量的变化而非绝对值。这种记号在实际问题中特别有用，我们可以用它近似量的变化大小，或估计测量产生的误差。

📘 定义

微分

$d y$ (输出变化量), $d x$ (输入变化量)

对于函数 $y = f (x)$ ， $d x = Δ x = x - a$ 是输入的微小变化量。 $d y = f^{'} (x) d x$ 是实际变化量 $Δ y = f (x + d x) - f (x)$ 的线性近似。当 $d x$ 很小时， $Δ y \approx d y$ 。

例:

该记号和线性化直接等价，只是重新整理后突出了变化量。

dy = f'(x) dx

📐 例题

一个球形生日气球的半径测量值为10 cm，最大测量误差为 $\pm 0.1$ cm。利用微分近似计算气球表面积计算值的最大可能误差。

1
回忆表面积公式并标注已知量：
$S(r) = 4\pi r^2, \quad r = 10 \text{ cm}, \quad |dr| = 0.1 \text{ cm}$
2
计算 $S (r)$ 的导数：
$S'(r) = 8\pi r$
3
表面积的最大误差近似为 $∣ d S ∣$ ：
$|dS| = |S'(r) dr|$
4
代入数值得到最终结果：
$|dS| = |8\pi (10)(0.1)| = 8\pi \approx 25.13 \text{ cm}^2$
5
The maximum possible error in the calculated surface area is approximately $8 π$ cm².

Exam tip:

一定要仔细读题：如果题目要求绝对误差，你只需要计算 $∣ d y ∣$ ；如果要求相对误差，你需要计算 $\frac{∣ d y ∣}{y}$ ，百分比误差就是该值乘以100。

4. AP-Style 练习题★★★★☆⏱ 3 min

✓ 快速检测

用这些 AP 风格题目测试你的理解：

What is the linearization of $f (x) = ln (3 x + 1)$ at $a = 0$ ?
- A) $L (x) = 3 x$
- B) $L (x) = 1 + 3 x$
- C) $L (x) = 3 x + ln 2$
- D) $L (x) = x$

📐 例题

Let $f (x) = x^{2} + 9$ : (a) Find the linearization $L (x)$ at $a = 4$ . (b) Approximate $f (4.2)$ and state if the approximation is an overestimate or underestimate. (c) Use differentials to approximate the change in $f (x)$ when $x$ increases from 4 to 4.2.

1
Part (a): Calculate $f (4)$ and $f^{'} (4)$ :
$f(4) = \sqrt{4^2 + 9} = 5, \quad f'(x) = \frac{x}{\sqrt{x^2 + 9}}, \quad f'(4) = \frac{4}{5} = 0.8$
2
Substitute into the linearization formula:
$L(x) = 5 + 0.8(x-4) = 0.8x + 1.8$
3
$L (4.2) = 5.16$ . The second derivative $f^{''} (x) > 0$ for all $x$ , so $f (x)$ is concave up near $x = 4$ . A tangent line to a concave up function lies below the curve, so this is an underestimate.
4
Part (c): Approximate the change with differentials:
$dx = 0.2, \quad dy = f'(4)dx = 0.8(0.2) = 0.16$

📐 例题

The monthly profit of a small coffee shop is modeled by $P (x) = 12 x - 0.005 x^{2} - 2000$ dollars, where $x$ is the number of lattes sold per month. The shop currently sells 1500 lattes per month, and projects sales will increase by 60 lattes next month. Use linearization to approximate the change in profit, and the approximate total profit for next month.

1
Calculate the derivative and evaluate at $x = 1500$ :
$P'(x) = 12 - 0.01x, \quad P'(1500) = -3 \text{ dollars per latte}$
2
Approximate the change in profit:
$dP = P'(1500)dx = -3(60) = -180$
3
Calculate current monthly profit at 1500 lattes:
$P(1500) = 12(1500) - 0.005(1500)^2 - 2000 = 4750$
4
Approximate total profit = $$4750 - $180 = $4570$ . In context: a 60-latte sales increase will decrease total monthly profit by approximately $180.

5. 常见陷阱

错误做法:

Writing the linearization as $L (x) = f (a) + f^{'} (a) x$ instead of $L (x) = f (a) + f^{'} (a) (x - a)$ , omitting the $(x - a)$ term.

原因:

Students misremember the formula and forget it is derived from point-slope form of a line.

正确做法:

Always start from point-slope form $(y - f (a)) = f^{'} (a) (x - a)$ , then rearrange to get $L (x)$ .

错误做法:

Choosing $a$ as the input you want to approximate instead of a nearby known point (e.g., approximating $15$ by choosing $a = 15$ instead of $a = 16$ ).

原因:

Students misread the problem and forget the purpose of $a$ is to have an exactly known value of $f (a)$ .

正确做法:

Always pick the closest round number to your unknown input where you can compute $f (a)$ exactly.

错误做法:

Forgetting the chain rule when differentiating composite functions for linearization (e.g., writing the derivative of $e^{3 x}$ as $e^{3 x}$ instead of $3 e^{3 x}$ ).

原因:

Students rush through differentiation after setting up the problem and skip the chain rule step.

正确做法:

After calculating any derivative for linearization, double-check that you applied the chain rule for composite inner functions.

错误做法:

Claiming a linear approximation is accurate for a point far from $a$ .

原因:

Students assume linearization works everywhere, not just locally.

正确做法:

After calculating an approximation, check how far your input is from $a$ , and explicitly note that accuracy decreases as distance from $a$ increases.

错误做法:

Writing $Δ y = d y$ instead of $Δ y \approx d y$ , treating the approximation as exact.

原因:

Students forget that $d y$ is a linear approximation, not the actual change.

正确做法:

Always use an approximation symbol when relating $Δ y$ to $d y$ , and explicitly state the relationship is approximate for small $d x$ .

错误做法:

Confusing the measured value of a quantity with its error in differential error problems (e.g., using $d r = 10$ cm instead of $d r = 0.1$ cm in the balloon example).

原因:

Students mix up which value corresponds to the input and which corresponds to the error.

正确做法:

Label all variables before substituting: $r$ = measured input, $d r$ = error in $r$ .

6. 速查表

类别	公式	说明
局部线性性质	Near $x = a$ , $f (x) \approx$ tangent line	Only applies to differentiable functions at $x = a$
Linearization at $x = a$	$L (x) = f (a) + f^{'} (a) (x - a)$	$a$ must be a point with known $f (a)$ and $f^{'} (a)$
Linear Approximation	$f (x) \approx L (x)$	Only accurate for $x$ very close to $a$
Differential of $y = f (x)$	$d y = f^{'} (x) d x$	$d y$ approximates the change in $y$
Actual vs Approximate Change	$Δ y = f (x + d x) - f (x) \approx d y$	$Δ y$ is exact, $d y$ is approximate
Max Absolute Error	$∣ Error ∣ \approx ∣ f^{'} (x) d x ∣$	$d x$ = maximum error in input
Relative Error	$\frac{∣ d y ∣}{∣ f ( x ) ∣}$	Unitless error proportional to quantity size
Percentage Error	$100 \times \frac{∣ d y ∣}{∣ f ( x ) ∣}$	Error expressed as a percentage

真题中的出现

AI 根据考纲规律估算的考点位置,请对照官方真题核实准确性。仅作复习重点参考。

2023 · MCQ
求给定点处的线性化
2022 · FRQ
利用微分近似误差

下一步

局部线性和线性化是 AP 微积分 AB 第4单元后续核心考点的基础前置知识。下一个直接考点是相关变化率，它依赖微分（微小线性变化）的思想来关联关联量的变化率；如果不理解 $Δ y \approx d y = f^{'} (x) d x$ ，你很难正确建立相关变化率的方程。本考点还介绍了用简单线性模型近似复杂非线性函数的核心思想，这是科学、经济学和工程学中所有应用微积分的基础，也为你后续学习曲线绘图和优化等考点做准备。