Connecting a Function, Its First Derivative, and Its Second Derivative · 连接一个函数、其一阶导数和其二阶导数
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| zero/ˈzɪərəʊ/ | 零点 | líng diǎn |
Three graphs telling one story
- $f$, $f'$, and $f''$ are three views of the same function — and they must all agree.
- A feature of $f$ shows up as a specific feature of $f'$ or $f''$.
- The exam loves handing you one of the three and asking about another.
- Master the translation table and these questions become quick.
三张图讲同一个故事
- $f$、$f'$、$f''$ 是同一个函数的三个视角——它们必须全部一致。
- $f$ 的一个特征会以 $f'$ 或 $f''$ 的某个特征出现。
- 考试喜欢给你三者之一,却问另一个。
- 掌握这张翻译表,这些题就变快。
What the zeros of $f'$ say
- A zero 零点 of $f'$ (where $f'=0$) is a critical point of $f$ — a possible extremum.
- $f'$ goes $+\to-$ at that zero → $f$ has a local max; $-\to+$ → local min.
- Where $f'>0$, $f$ increases; where $f'<0$, $f$ decreases.
- So the sign of $f'$ gives increase/decrease, and its zeros give extrema.
$f'$ 的零点说了什么
- $f'$ 的一个零点($f'=0$ 处)是 $f$ 的临界点——一个可能极值。
- $f'$ 在该零点从 $+\to-$ → $f$ 有局部最大;$-\to+$ → 局部最小。
- $f'>0$ 之处 $f$ 递增;$f'<0$ 之处 $f$ 递减。
- 所以 $f'$ 的符号给递增/递减,它的零点给极值。
One function, three readings · 一个函数，三种读数
y = ax³ + cx
On this curve, the zeros of $f'$ mark the peak and valley, and the zero of $f''$ marks the inflection. · 在这条曲线上，$f'$的零点标记了峰和谷，而$f''$的零点标记了拐点。
A sign-changing zero of $f'$ marks, on the graph of $f$, a... · $f'$的变号零点在$f$的图像上标记了一个...
Zeros of $f'$ (sign-changing) are extrema of $f$. · $f'$的零点（变号）是$f$的极值。
On an interval where the graph of $f'$ is positive, $f$ is increasing. · 在$f'$的图像为正的区间上，$f$是递增的。
$f'>0$ ⇔ $f$ increasing. · $f'>0$ ⇔ $f$递增。
What the zeros of $f''$ say
- A sign-changing zero of $f''$ is a point of inflection of $f$.
- Where $f''>0$, $f$ is concave up; where $f''<0$, concave down.
- Note: $f''$ is the derivative of $f'$, so a zero of $f''$ is where $f'$ has a max or min (its steepest slope on $f$).
- The sign of $f''$ gives concavity; its sign-changing zeros give inflections.
$f''$ 的零点说了什么
- $f''$ 变号的零点是 $f$ 的拐点。
- $f''>0$ 之处 $f$ 上凹;$f''<0$ 之处下凹。
- 注意:$f''$ 是*$f'$ 的导数*,所以 $f''$ 的零点是 $f'$ 取最大或最小之处($f$ 上最陡的斜率)。
- $f''$ 的符号给凹凸性;它变号的零点给拐点。
A sign-changing zero of $f''$ marks, on the graph of $f$, a... · $f''$的变号零点在$f$的图像上标记了一个...
Sign-changing zeros of $f''$ are inflection points of $f$. · $f''$的变号零点是$f$的拐点。
From the graph of $f''$ you can read which features of $f$? · 从$f''$的图像中可以读出$f$的哪些特征？
Concavity + inflections come from $f''$; exact values need $f$ itself. · 凹凸性和拐点来自$f''$；精确值需要$f$本身。
Reading across representations
- Given a graph of $f'$: its zeros → extrema of $f$; its sign → increase/decrease of $f$; its own increase/decrease → concavity of $f$.
- Given a table of values: estimate $f'$ and $f''$ with difference quotients, then apply the same rules.
- Given a formula: differentiate and use sign charts.
- Same logic, three languages — translate freely.
跨表示阅读
- 给定 $f'$ 的图:它的零点 → $f$ 的极值;它的符号 → $f$ 的递增/递减;它自身的递增/递减 → $f$ 的凹凸性。
- 给定一张值表:用差商估计 $f'$ 与 $f''$,再套用同样的规则。
- 给定一个公式:求导并用符号表。
- 同一逻辑,三种语言——自由翻译。
The graph of $f'$ is positive then negative, crossing zero at $x=2$. At $x=2$, $f$ has a... · $f'$的图像先正后负，在$x=2$处穿过零点。在$x=2$处，$f$有一个...
$f'$ goes $+\to-$ → local max of $f$. · $f'$从$+\to-$变化 → $f$的局部最大值。
On a graph of $f'$, it is the ____ (not the peaks) that mark the extrema of $f$. · 在$f'$的图像上，它是____（而不是峰值）标记了$f$的极值。
Zeros of $f'$ = extrema of $f$; peaks of $f'$ relate to inflections of $f$. · $f'$的零点 = $f$的极值；$f'$的峰值与$f$的拐点相关。
Keep the levels straight. A zero of $f'$ is an extremum of $f$ (not of $f'$). A zero of $f''$ is an inflection of $f$ and an extremum of $f'$. Reading a graph of $f'$ as if it were $f$ — treating its peaks as maxima of $f$ — is a classic mix-up. Its zeros, not its peaks, mark $f$'s extrema.
把层次分清。$f'$ 的零点是 $f$ 的极值(不是 $f'$ 的)。$f''$ 的零点是 $f$ 的拐点且是 $f'$ 的极值。把 $f'$ 的图当作 $f$ 来读——把它的峰当作 $f$ 的最大值——是经典的混淆。是它的零点、而非峰,标记 $f$ 的极值。
You are given the graph of $f'$, which is positive on $(-\infty,2)$, zero at $x=2$, and negative after. What happens to $f$ at $x=2$?
- $f'$ changes $+\to-$ at $x=2$ → $f$ has a local maximum there.
- $f$ is increasing before $x=2$ and decreasing after (matching $f'>0$ then $f'<0$).
- The value $f'(2)=0$ means a horizontal tangent on $f$.
给你 $f'$ 的图:在 $(-\infty,2)$ 为正,在 $x=2$ 为零,之后为负。$f$ 在 $x=2$ 处会怎样?
- $f'$ 在 $x=2$ 从 $+\to-$ → $f$ 在那里有局部最大值。
- $f$ 在 $x=2$ 前递增、之后递减(与 $f'>0$ 然后 $f'<0$ 相符)。
- 值 $f'(2)=0$ 意味着 $f$ 上有一条水平切线。
$f$, $f'$, $f''$ tell one consistent story. Zeros of $f'$ (sign-changing) are $f$'s extrema; the sign of $f'$ gives increase/decrease. Zeros of $f''$ (sign-changing) are $f$'s inflection points; the sign of $f''$ gives concavity. Translate this across graphs, tables, and formulas.
$f$、$f'$、$f''$ 讲一个一致的故事。$f'$ 的(变号)零点是 $f$ 的极值;$f'$ 的符号给递增/递减。$f''$ 的(变号)零点是 $f$ 的拐点;$f''$ 的符号给凹凸性。把这跨图、表、公式翻译。