Change in Tandem · 同步变化
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| input/ˈɪnpʊt/ | 输入 | shū rù |
| output/ˈaʊtpʊt/ | 输出 | shū chū |
| increasing/ɪnˈkriːsɪŋ/ | 递增 | dì zēng |
| decreasing/ˈdiːkriːsɪŋ/ | 递减 | dì jiǎn |
| constant/ˈkɒnstənt/ | 常数 | cháng shù |
| interval/ˈɪntəvl/ | 区间 | qū jiān |
| concavity/kənˈkævɪti/ | 凹凸性 | āo tū xìng |
| rate of change/reɪt ɒv tʃeɪndʒ/ | 变化率 | biàn huà lǜ |
| concave up/kɒnˈkeɪv ʌp/ | 凹 | āo |
| concave down/kɒnˈkeɪv daʊn/ | 凸 | tū |
| turning point/ˈtɜːnɪŋ pɔɪnt/ | 转折点 | zhuǎn zhé diǎn |
| local maximum/ˈləʊkl ˈmæksɪməm/ | 极大值 | jí dà zhí |
| local minimum/ˈləʊkl ˈmɪnɪməm/ | 极小值 | jí xiǎo zhí |
Watch the water climb
- Pour water into a glass at a steady rate and watch the level rise.
- In a straight glass the level climbs steadily; in a curvy vase it speeds up and slows down.
- Two quantities — the water poured and the height — change together, in tandem.
- Precalculus starts by describing exactly how one quantity changes as another one does.
看水位上升
- 以稳定的速度往杯子里倒水,看着水位上升。
- 在笔直的杯子里,水位稳稳地爬升;在弯曲的花瓶里,它时快时慢。
- 两个量——倒入的水和高度——在一起变化,也就是同步变化。
- 微积分预备(Precalculus)的起点,就是精确地描述一个量如何随另一个量而变化。
A function links an input to an output
- A function takes an input 输入 (a value of $x$) and returns exactly one output 输出 (written $f(x)$).
- As the input changes, the output changes with it — that is change in tandem.
- We write $f(2)$ to mean "the output when the input is $2$".
- So the whole story of a function is one question: as $x$ moves, where does $f(x)$ go?
函数把输入连到输出
- 函数接收一个输入(input)(一个 $x$ 的值),并返回唯一一个输出(output)(记作 $f(x)$)。
- 当输入变化时,输出随之变化——这就是同步变化。
- 我们写 $f(2)$ 表示"输入为 $2$ 时的输出"。
- 所以一个函数的全部故事就是一个问题:当 $x$ 移动时,$f(x)$ 去了哪里?
Increasing, decreasing, or constant
- Pick an interval 区间 — a stretch of the input axis — and follow the output from left to right.
- The function is increasing 递增 where the output rises as the input moves right.
- It is decreasing 递减 where the output falls as the input moves right.
- It is constant 常数 where the output stays level.
- You can read this straight off a graph: uphill is increasing, downhill is decreasing.
递增、递减,还是恒定
- 选一段区间(interval)——输入轴上的一段——然后从左到右跟踪输出。
- 输出随输入右移而上升的地方,函数是递增(increasing)的。
- 输出随输入右移而下降的地方,函数是递减(decreasing)的。
- 输出保持水平的地方,函数是恒定(constant)的。
- 这些都能直接从图像上读出:上坡是递增,下坡是递减。
A function is increasing on an interval when… · 一个函数在某区间上递增,是指……
Increasing is about direction, not size: move right along the input and the output climbs. It can be increasing while its values are still negative. · 递增讲的是方向,不是大小:沿输入向右移动,输出随之上升。即使输出仍是负数,函数也可以是递增的。
If a graph is perfectly flat (horizontal) over an interval, the function is constant · 恒定的 there. · 如果图像在某区间上是完全水平的,那么函数在那里是恒定的。
A flat graph means the output does not change as the input changes — that is exactly what constant means. · 水平的图像意味着输出不随输入改变——这正是恒定的含义。
Concavity: is the change speeding up or slowing down?
- Even while a curve keeps increasing, its rate of change 变化率 (how fast the output moves) can itself change.
- Concavity 凹凸性 describes which way the curve bends.
- Concave up 凹 — the curve bends upward like a cup, so the rate of change is increasing.
- Concave down 凸 — the curve bends downward like a dome, so the rate of change is decreasing.
- The curvy vase again: where it narrows, the height climbs faster and faster (concave up).
凹凸性:变化在加快还是变慢?
- 即使一条曲线一直在递增,它的变化率(rate of change)(输出移动得多快)本身也可以改变。
- 凹凸性(concavity)描述的是曲线向哪个方向弯。
- 凹(concave up)——曲线像杯子一样向上弯,所以变化率在增大。
- 凸(concave down)——曲线像圆顶一样向下弯,所以变化率在减小。
- 再看那个弯曲的花瓶:在它变窄的地方,高度越爬越快(凹)。
Saying a curve is concave up tells you that… · 说一条曲线是凹的,告诉你的是……
Concave up bends like a cup: the rate of change itself is rising. A curve can be concave up while it is going down — think of a slide flattening out at the bottom. · 凹的形状像一个杯子:变化率本身在上升。曲线在下降时也可以是凹的——想象滑梯在底部逐渐变平。
Turning points: local maxima and minima
- Where a curve stops increasing and starts decreasing, it reaches a local maximum 极大值 — a peak.
- Where it stops decreasing and starts increasing, it reaches a local minimum 极小值 — a valley.
- Each such peak or valley is a turning point 转折点.
- "Local" means highest or lowest nearby — not necessarily over the whole graph.
转折点:极大值与极小值
- 曲线从递增转为递减的地方,到达一个极大值(local maximum)——一个山峰。
- 曲线从递减转为递增的地方,到达一个极小值(local minimum)——一个山谷。
- 每一个这样的山峰或山谷,都是一个转折点(turning point)。
- "局部"指的是附近最高或最低——不一定是整张图上最高或最低。
The tangent's slope IS the rate of change · 切线的斜率就是变化率
f(x) = x³ − 3x
Drag the point and read the tangent's slope — positive where the curve rises, negative where it falls, and zero at a turning point. · 拖动这个点,读出切线的斜率——曲线上升处为正,下降处为负,在转折点处为零。
A point where a graph stops increasing and starts decreasing is a local . · 图像从递增转为递减的那个点,是一个局部。
At a local maximum the curve turns from uphill to downhill — a peak that is highest compared with nearby points. · 在极大值处,曲线从上坡转为下坡——一个与附近各点相比最高的山峰。
Look at the curve in the figure. Select all · 所有 the true statements. · 看图中的曲线。选出所有正确的说法。
The curve rises, turns at a peak, falls to a valley, then rises again. It is not · 不 concave up everywhere — it bends down near the peak and up near the valley. · 曲线先上升,在山峰处转弯,下降到谷底,然后再次上升。它并非处处都凹——在山峰附近向下弯,在谷底附近向上弯。
"Increasing" and "concave up" are different ideas. Increasing is about the output going up; concave up is about the rate of change going up. A curve can be decreasing and concave up at the same time — falling, but flattening out as it falls.
"递增"和"凹"是不同的概念。递增讲的是输出在上升;凹讲的是变化率在上升。一条曲线可以同时是递减且是凹的——一边下降,一边在下降中逐渐变平。
Comparing two outputs
- To compare a function at two inputs $a$ and $b$, compare their outputs $f(a)$ and $f(b)$.
- If $b > a$ and $f(b) > f(a)$, the output grew as the input grew.
- The net change in output from $a$ to $b$ is simply $f(b) - f(a)$.
- If the function returns to the same value, the net change is zero — even if it rose and fell along the way.
比较两个输出
- 要在两个输入 $a$ 和 $b$ 处比较函数,就比较它们的输出 $f(a)$ 和 $f(b)$。
- 如果 $b > a$ 且 $f(b) > f(a)$,说明输入增大时输出也增大了。
- 从 $a$ 到 $b$ 输出的净变化就是 $f(b) - f(a)$。
- 如果函数回到了同一个值,净变化就是零——即使中途有升有降。
A function has $f(2) = 5$ and $f(6) = 5$. What is the net change in output from $x = 2$ to · 到 $x = 6$? · 某函数满足 $f(2) = 5$ 且 $f(6) = 5$。从 $x = 2$ 到 $x = 6$,输出的净变化是多少?
Net change $= f(6) - f(2) = 5 - 5 = 0$. The output ended where it started — even though it may have risen and fallen in between. · 净变化 $= f(6) - f(2) = 5 - 5 = 0$。输出回到了起点——尽管中间可能有升有降。
Match each term to what it describes. · 把每个术语和它的描述配对。
Increasing/decreasing describe the output's direction; concavity describes how the rate of change itself is changing; a local maximum is where the direction flips. · 递增/递减描述输出的方向;凹凸性描述变化率本身如何变化;极大值是方向翻转的地方。
A cup of tea cools from 90°C to room temperature.
- Temperature (output) is decreasing the whole time — it never goes back up.
- But it falls fast at first, then slowly: the curve is concave up.
- There is no local maximum or minimum in between — the direction never flips.
一杯茶从 90°C 冷却到室温。
- 温度(输出)全程都在递减——它从不回升。
- 但它先降得快,后降得慢:曲线是凹的。
- 中途没有极大值或极小值——方向从未翻转。
A function's behaviour is read from its graph: it is increasing or decreasing on intervals, its concavity tells you whether the rate of change is speeding up or slowing down, and its turning points are the local maxima and minima where the direction flips.
一个函数的行为要从它的图像来读:它在各个区间上递增或递减,它的凹凸性告诉你变化率是在加快还是变慢,而它的转折点就是方向翻转处的极大值和极小值。