Rates of Change · 变化率
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| rate of change/reɪt ɒv tʃeɪndʒ/ | 变化率 | biàn huà lǜ |
| average rate of change/ˈævrɪdʒ reɪt ɒv tʃeɪndʒ/ | 平均变化率 | píng jūn biàn huà lǜ |
| interval/ˈɪntəvl/ | 区间 | qū jiān |
| slope/sləʊp/ | 斜率 | xié lǜ |
| secant line/ˈsiːkənt laɪn/ | 割线 | gē xiàn |
| constant/ˈkɒnstənt/ | 常数 | cháng shù |
How fast, exactly?
- A car's speedometer answers a question the odometer cannot: not how far, but how fast right now.
- "How fast is the output changing?" is the central question of this whole course.
- The answer is a number called a rate of change 变化率.
- This lesson measures it two ways: over a stretch, and at a single instant.
到底有多快?
- 汽车的速度表回答了里程表答不了的问题:不是跑了多远,而是此刻多快。
- "输出变化得有多快?"是整门课程的核心问题。
- 答案是一个数,叫做变化率(rate of change)。
- 这节课用两种方式来测量它:在一段区间上,以及在某一瞬间。
Average rate of change
- Over an interval 区间 from $x = a$ to $x = b$, compare how much the output changed to how much the input changed.
- The average rate of change 平均变化率 is $\dfrac{f(b) - f(a)}{b - a}$ — change in output over change in input.
- It is a ratio, so it always carries units: metres per second, dollars per year, °C per minute.
- Example in words: if temperature rose 12°C over 4 hours, the average rate was 3°C per hour.
平均变化率
- 在从 $x = a$ 到 $x = b$ 的区间(interval)上,比较输出变了多少与输入变了多少。
- 平均变化率(average rate of change)是 $\dfrac{f(b) - f(a)}{b - a}$——输出的变化除以输入的变化。
- 它是一个比值,所以总是带有单位:米每秒、美元每年、°C 每分钟。
- 用话来说:如果温度在 4 小时内上升了 12°C,那么平均变化率是每小时 3°C。
The average rate of change of $f$ over the interval $[a, b]$ equals… · $f$ 在区间 $[a, b]$ 上的平均变化率等于……
It is the change in output divided by the change in input — a ratio, so it carries units of output per unit of input. · 它是输出的变化除以输入的变化——一个比值,所以它的单位是每单位输入对应的输出。
It is the slope of a secant line
- Draw the straight line joining the two endpoints $(a, f(a))$ and $(b, f(b))$: that is the secant line 割线.
- Its slope 斜率 — rise over run — is exactly the average rate of change.
- Steeper secant → faster average change; a downhill secant → a negative rate.
它是割线的斜率
- 把连接两个端点 $(a, f(a))$ 和 $(b, f(b))$ 的直线画出来:那就是割线(secant line)。
- 它的斜率(slope)——纵向变化除以横向变化——正是平均变化率。
- 割线越陡,平均变化越快;向下倾斜的割线,则代表负的变化率。
A runner's distance is $f(t)$ metres. If $f(2) = 10$ and $f(5) = 40$, what is the average speed over $[2, 5]$? · 一名跑步者跑过的距离是 $f(t)$ 米。若 $f(2) = 10$ 且 $f(5) = 40$,那么在 $[2, 5]$ 上的平均速度是多少?
Average speed $= \dfrac{40 - 10}{5 - 2} = \dfrac{30}{3} = 10$ m/s — the secant slope over that interval. · 平均速度 $= \dfrac{40 - 10}{5 - 2} = \dfrac{30}{3} = 10$ m/s——就是这段区间上的割线斜率。
On a graph, the average rate of change over $[a, b]$ is the slope of the… · 在图像上,$[a, b]$ 上的平均变化率是这条线的斜率……
The secant line joins $(a, f(a))$ and $(b, f(b))$; its slope is exactly the average rate of change. · 割线连接 $(a, f(a))$ 和 $(b, f(b))$;它的斜率正是平均变化率。
The rate at a single point
- A secant needs two points. To get the rate at just one point, shrink the interval.
- Take average rates over smaller and smaller intervals around the point.
- As the interval closes toward zero, the secant tips over into a tangent — its slope is the rate at that exact point.
- This "rate at an instant" is the idea that grows into calculus next year.
某一点处的变化率
- 割线需要两个点。要得到单独一个点处的变化率,就把区间缩小。
- 在这个点周围,取越来越小的区间上的平均变化率。
- 当区间收缩趋向零时,割线就倾斜成一条切线——它的斜率就是那一点处的变化率。
- 这个"瞬间变化率"的想法,明年会成长为微积分。
Shrink the interval until the secant becomes a tangent · 让区间不断缩小,割线就变成了切线
f(x) = 0.5x²
Drag the point to read the slope at a single spot — this is the limit of the average rate of change as the interval shrinks to zero. · 拖动这个点,读出某一点处的斜率——这就是区间缩小到零时平均变化率的极限。
To estimate the rate of change at a single point, average rates are taken over very ____ intervals around it. · 要估计某一点处的变化率,就在它周围取很____的区间上的平均变化率。
As the interval shrinks toward zero, the secant slope approaches the slope at that exact point. · 当区间缩小趋向零时,割线的斜率就趋近于那一点处的斜率。
Sign, size, and units
- A positive rate means the output is rising; a negative rate means it is falling; zero means no change.
- A bigger number (ignoring sign) means faster change.
- For a straight line the rate is the same everywhere — it is constant 常数.
- Always state the units: a rate of change with no units is only half an answer.
正负、大小与单位
- 正的变化率表示输出在上升;负的变化率表示在下降;零表示没有变化。
- 数字越大(不看正负),变化越快。
- 对一条直线来说,变化率处处相同——它是常数(constant)。
- 一定要写出单位:没有单位的变化率只答对了一半。
A negative · 负形 average rate of change means the output decreased over the interval. · 负的平均变化率意味着输出在这段区间内减小了。
A negative ratio means the output ended lower than it started — the function fell, on average, across the interval. · 负的比值意味着输出结束时比开始时更低——函数在这段区间内平均是下降的。
Select all · 所有 true statements about average rate of change. · 选出关于平均变化率的所有正确说法。
It can be positive, negative, or zero. Everything else is true: it is a slope, it carries units, and it approximates a point rate over small intervals. · 它可以是正、负或零。其余都对:它是斜率,带有单位,并在很小的区间上近似某一点的变化率。
Average rate of change is about the two endpoints only — it ignores everything in between. A hiker who climbs then descends back to the same height has an average rate of zero, even though they were moving the whole time.
平均变化率只关乎两个端点——它忽略中间发生的一切。一个先爬升再下降回到同样高度的登山者,平均变化率是零,尽管他全程都在移动。
A car travels 150 km between 1:00 and 3:00.
- Average speed $= \dfrac{150 \text{ km}}{2 \text{ h}} = 75$ km/h.
- That is the secant slope of the distance–time graph over those two hours.
- The car's actual speed changed constantly — 75 km/h is just the average.
一辆车在 1:00 到 3:00 之间行驶了 150 km。
- 平均速度 $= \dfrac{150 \text{ km}}{2 \text{ h}} = 75$ km/h。
- 那就是这两小时里距离–时间图上割线的斜率。
- 车的实际速度一直在变——75 km/h 只是平均值。
The average rate of change over $[a, b]$ is $\dfrac{f(b) - f(a)}{b - a}$, the slope of the secant line through the endpoints. Shrink the interval toward zero and it becomes the rate at a single point. Always report the units.
$[a, b]$ 上的平均变化率是 $\dfrac{f(b) - f(a)}{b - a}$,也就是穿过端点的割线的斜率。把区间缩小趋向零,它就变成某一点处的变化率。永远要写出单位。