Determining Intervals on Which a Function Is Increasing or Decreasing · 确定函数递增或递减的区间
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| sign of the first derivative/saɪn ɒvðə fɜːst dɪˈrɪvətɪv/ | 一阶导数的符号 | yī jiē dǎo shù de fú hào |
| sign chart/saɪn tʃɑːt/ | 符号表 | fú hào biǎo |
The sign of $f'$ tells the story
- Where a curve rises or falls is written in the sign of the first derivative 一阶导数的符号.
- $f'(x)>0$ on an interval → $f$ is increasing there (going uphill).
- $f'(x)<0$ → $f$ is decreasing (going downhill).
- So to map out where $f$ climbs and drops, you just track the sign of $f'$.
$f'$ 的符号讲出整个故事
- 曲线在何处上升或下降,写在一阶导数符号里。
- 某区间上 $f'(x)>0$ → $f$ 在那里递增(上坡)。
- $f'(x)<0$ → $f$ 递减(下坡)。
- 所以要画出 $f$ 何处爬升、何处下降,你只需追踪 $f'$ 的符号。
Uphill where f′ > 0 · f′ > 0 处上坡
y = ax³ + cx
The curve climbs where its slope is positive and falls where negative — the sign of $f'$ maps every stretch. · 曲线在其斜率为正处上升，斜率为负处下降——$f'$ 的符号映射了每一段。
A function is increasing on an interval exactly when... · 函数在区间上递增当且仅当...
Positive first derivative means increasing. · 正的导数意味着递增。
Build a sign chart
- Start by finding the critical points (where $f'=0$ or is undefined) — these split the number line.
- Draw a sign chart 符号表: mark the critical points, creating open intervals between them.
- Pick a test point in each interval and check the sign of $f'$ there.
- One sign per interval is enough — $f'$ can't change sign without passing through a critical point.
建立符号表
- 先找临界点($f'=0$ 或无定义之处)——它们把数轴分段。
- 画一张符号表:标出临界点,在它们之间形成开区间。
- 在每个区间取一个测试点,检查那里 $f'$ 的符号。
- 每个区间一个符号就够了——$f'$ 不经过临界点就无法变号。
For · 支持 $f'(x)=3(x-1)(x+1)$, evaluate $f'(0)$ to test the middle interval. · 对于 $f'(x)=3(x-1)(x+1)$，评估 $f'(0)$ 以测试中间区间。
$3(-1)(1)=-3<0$, so $f$ is decreasing on $(-1,1)$. · $3(-1)(1)=-3<0$，所以 $f$ 在 $(-1,1)$ 上递减。
The critical points split the number line into intervals for a ____ chart of $f'$. · 临界点将数轴划分为用于绘制 $f'$ 的 ____ 表。
A sign chart tracks where $f'$ is positive or negative. · 符号表追踪 $f'$ 为正或负的区间。
Read off the intervals
- Each interval where $f'>0$ is an interval of increase; each where $f'<0$ is an interval of decrease.
- Write them in interval notation: e.g. increasing on $(-\infty,-1)\cup(1,\infty)$.
- Use the function's domain as the outer boundaries.
- (Whether to include a critical point is a convention detail; the AP usually accepts open intervals.)
读出区间
- 每个 $f'>0$ 的区间是递增区间;每个 $f'<0$ 的区间是递减区间。
- 用区间记号写出:如在 $(-\infty,-1)\cup(1,\infty)$ 上递增。
- 用函数的定义域作为最外边界。
- (是否把临界点包含进去是约定细节;AP 通常接受开区间。)
For · 支持 $f(x)=x^3-3x$ (critical points $\pm1$), select all · 所有 intervals of increase. · 对于 $f(x)=x^3-3x$（临界点 $\pm1$），选择 所有 递增区间。
$f'>0$ outside $[-1,1]$; decreasing in between. · $f'>0$ 在 $[-1,1]$ 外部；在内部递减。
Why it works
- Between two consecutive critical points, $f'$ keeps one sign — no sign flip without a zero or a break.
- So one test value settles the whole interval.
- This sign information is the foundation of the First Derivative Test (next lesson).
- Increasing/decreasing behavior is exactly what a rate function encodes.
为何有效
- 在两个相邻临界点之间,$f'$ 保持一个符号——没有零点或断裂就不会变号。
- 所以一个测试值就能定下整个区间。
- 这个符号信息是一阶导数检验(下一课)的基础。
- 递增/递减行为正是变化率函数所编码的。
A function that is negative (below the $x$-axis) can still be increasing. · 一个为负（在 $x$-轴下方）的函数仍然可以是递增的。
Increasing depends on $f'>0$, not the sign of $f$. · 递增取决于 $f'>0$，而非 $f$ 的符号。
One test point settles a whole interval because $f'$ cannot change sign without... · 一个测试点即可确定整个区间，因为 $f'$ 除非...
A sign change requires a zero or an undefined point of $f'$. · 符号变化需要 $f'$ 为零或未定义的点。
Track the sign of $f'$, not the sign of $f$. A function can be negative yet increasing (e.g. rising from $-5$ toward $-1$). "Increasing" means the outputs are getting larger ($f'>0$), regardless of whether $f$ itself is above or below the $x$-axis.
追踪 $f'$ 的符号,而不是 $f$ 的符号。一个函数可以为负却递增(如从 $-5$ 升向 $-1$)。"递增"意味着输出在变大($f'>0$),无论 $f$ 本身在 $x$ 轴上方还是下方。
Where is $f(x)=x^3-3x$ increasing?
- $f'(x)=3(x-1)(x+1)$; critical points $x=-1,1$.
- Test $x=-2$: $f'>0$. Test $x=0$: $f'=-3<0$. Test $x=2$: $f'>0$.
- Increasing on $(-\infty,-1)$ and $(1,\infty)$; decreasing on $(-1,1)$.
$f(x)=x^3-3x$ 在何处递增?
- $f'(x)=3(x-1)(x+1)$;临界点 $x=-1,1$。
- 测 $x=-2$:$f'>0$。测 $x=0$:$f'=-3<0$。测 $x=2$:$f'>0$。
- 在 $(-\infty,-1)$ 与 $(1,\infty)$ 上递增;在 $(-1,1)$ 上递减。
The sign of $f'$ determines monotonic behavior: $f'>0$ → increasing, $f'<0$ → decreasing. Split the number line at the critical points, build a sign chart with one test point per interval, and read off the intervals of increase and decrease. Track the sign of $f'$, not of $f$.
$f'$ 的符号决定单调行为:$f'>0$ → 递增,$f'<0$ → 递减。在临界点处把数轴分段,用每区间一个测试点建符号表,读出递增与递减区间。追踪 $f'$ 的符号,而非 $f$ 的。