Introducing Calculus: Can Change Occur at an Instant? · 微积分导论：变化能否在瞬间发生？
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| average rate of change/ˈævrɪdʒ reɪt ɒv tʃeɪndʒ/ | 平均变化率 | píng jūn biàn huà lǜ |
| slope/sləʊp/ | 斜率 | xié lǜ |
| secant line/ˈsiːkənt laɪn/ | 割线 | gē xiàn |
| tangent line/ˈtændʒənt laɪn/ | 切线 | qiè xiàn |
| instantaneous rate of change/ˌɪnstənˈteɪnɪəs reɪt ɒv tʃeɪndʒ/ | 瞬时变化率 | shùn shí biàn huà lǜ |
| limit/ˈlɪmɪt/ | 极限 | jí xiàn |
| derivative/dɪˈrɪvətɪv/ | 导数 | dǎo shù |
How fast, right now?
- Your car's speedometer reads $60\ \tfrac{\text{km}}{\text{h}}$ at a single instant.
- But speed is distance ÷ time — and at an instant, no time passes and no distance is covered.
- So how can you have a speed at a single moment?
- Answering this question is where calculus begins.
此刻到底有多快?
- 你的车速表在某一瞬间显示 $60\ \tfrac{\text{km}}{\text{h}}$。
- 但速度是距离 ÷ 时间——而在一个瞬间,没有时间流逝,也没有走过距离。
- 那么,一个瞬间怎么会有速度呢?
- 回答这个问题,正是微积分(calculus)的起点。
Average rate of change
- Over an interval, change is easy: it's the total change divided by the time taken.
- This is the average rate of change 平均变化率 of a function.
- On a graph it is the slope 斜率 of the straight line joining two points — a secant line 割线.
- Example: from $x=1$ to $x=3$, the average rate is $\dfrac{f(3)-f(1)}{3-1}$.
平均变化率
- 在一段区间上,变化很好算:就是总变化除以所花的时间。
- 这就是一个函数的平均变化率(average rate of change)。
- 在图上,它是连接两点的直线的斜率(slope)——一条割线(secant line)。
- 例子:从 $x=1$ 到 $x=3$,平均变化率是 $\dfrac{f(3)-f(1)}{3-1}$。
The slope of a secant line between two points on a curve gives the... · 曲线上两点间割线的斜率给出……
A secant joins two points, so its slope is the average rate of change over that interval. · 割线连接两点，因此其斜率是该区间上的平均变化率。
Squeeze the two points together
- Now slide the second point closer and closer to the first.
- The secant line pivots and gets closer to just touching the curve at one point.
- That touching line is the tangent line 切线.
把两个点挤到一起
- 现在让第二个点越来越靠近第一个点。
- 割线会转动,越来越接近只在一点轻触曲线。
- 那条轻触的直线就是切线(tangent line)。
As the second point slides toward the first, the secant line approaches the... · 当第二点滑向第一点时，割线趋近于……
The secant pivots until it just touches the curve — it approaches the tangent line. · 割线旋转直至仅接触曲线——它趋近于切线。
The instantaneous rate of change
- The slope of the tangent line is the instantaneous rate of change 瞬时变化率 at that point.
- That is the speedometer reading — the rate at one exact instant.
- For a position-time graph, it is the instantaneous velocity.
瞬时变化率
- 切线的斜率就是该点的瞬时变化率(instantaneous rate of change)。
- 那就是速度表的读数——某一确切瞬间的变化率。
- 对于位置-时间图,它就是瞬时速度。
Average rate or instantaneous rate? · 平均速率还是瞬时速率？
Decide whether each quantity is measured over an interval or at a single instant. · 判断每个量是在区间上测量还是在单点测量。
The slope of the tangent line is the ____ rate of change. · 切线的斜率是____变化率。
The tangent line touches at one point, so its slope is the instantaneous rate of change there. · 切线在某一点相切，因此其斜率是该点的瞬时变化率。
You cannot find the instantaneous rate by plugging in one point — that gives $\tfrac{0}{0}$, which is undefined. You have to watch what the average rate approaches as the interval shrinks.
你不能靠代入一个点来求瞬时变化率——那会得到 $\tfrac{0}{0}$,是没有定义的。你必须观察当区间缩小时,平均变化率趋近于什么。
You can find the instantaneous rate of change by plugging a single point into the average-rate formula. · 你可以将一个单点代入平均速率公式来找到瞬时变化率。
That gives $\tfrac{0}{0}$, which is undefined. You must take the limit as the interval shrinks to zero. · 这给出$\tfrac{0}{0}$，这是未定义的。你必须取当区间收缩至零时的极限。
This "approaches" is a limit
- We never actually divide by zero; we ask what value the slope approaches.
- That value is a limit 极限 — the central idea of calculus.
- The limit of the average rate of change, as the interval shrinks to zero, is called the derivative 导数.
- The whole first unit of the course is about making this idea precise.
这个“趋近”就是极限
- 我们从不真的去除以零;我们问斜率趋近于什么值。
- 那个值就是极限(limit)——微积分的核心思想。
- 当区间缩小到零时,平均变化率的极限,就叫做导数(derivative)。
- 本课程的整个第一单元,就是要把这个思想说精确。
Select all · 所有 true statements about the derivative at a point. · 选择关于某点导数的所有正确陈述。
The derivative is the limit of average rates as the interval shrinks — the tangent slope, i.e. the instantaneous rate. · 导数是平均变化率在区间收缩时的极限——即切线斜率，也就是瞬时变化率。
A ball's height is $h(t) = 5t^2$ metres. Estimate its speed at $t=2$ s.
- Average rate from $t=2$ to $t=2.1$: $\dfrac{h(2.1)-h(2)}{0.1} = \dfrac{22.05-20}{0.1} = 20.5\ \tfrac{\text{m}}{\text{s}}$.
- From $t=2$ to $t=2.01$: $\dfrac{20.2005-20}{0.01} = 20.05\ \tfrac{\text{m}}{\text{s}}$.
- The values close in on $20\ \tfrac{\text{m}}{\text{s}}$ — that limit is the instantaneous speed.
一个球的高度是 $h(t) = 5t^2$ 米。估计它在 $t=2$ s 时的速度。
- 从 $t=2$ 到 $t=2.1$ 的平均变化率:$\dfrac{h(2.1)-h(2)}{0.1} = \dfrac{22.05-20}{0.1} = 20.5\ \tfrac{\text{m}}{\text{s}}$。
- 从 $t=2$ 到 $t=2.01$:$\dfrac{20.2005-20}{0.01} = 20.05\ \tfrac{\text{m}}{\text{s}}$。
- 这些值逐渐逼近 $20\ \tfrac{\text{m}}{\text{s}}$——那个极限就是瞬时速度。
A ball's height is $h(t)=5t^2$ m. Estimate its instantaneous speed at $t=2$ s (in m/s). · 球的高度为$h(t)=5t^2$米。估算其在$t=2$秒时的瞬时速度（单位：m/s）。
Shrinking the interval, the average rate closes in on $20\ \tfrac{\text{m}}{\text{s}}$ — that limit is the instantaneous speed. · 缩小区间后，平均速率趋近于$20\ \tfrac{\text{m}}{\text{s}}$——该极限即为瞬时速度。
Average rate of change = slope of a secant line over an interval. Shrink the interval to zero and the secant approaches the tangent line, whose slope is the instantaneous rate of change. That limiting value is the derivative.
平均变化率 = 一段区间上割线的斜率。把区间缩小到零,割线逼近切线,切线的斜率就是瞬时变化率。那个极限值,就是导数。