Defining Average and Instantaneous Rates of Change at a Point · 定义平均变化率和点在瞬时变化率
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| average rate of change/ˈævrɪdʒ reɪt ɒv tʃeɪndʒ/ | 平均变化率 | píng jūn biàn huà lǜ |
| instantaneous rate of change/ˌɪnstənˈteɪnɪəs reɪt ɒv tʃeɪndʒ/ | 瞬时变化率 | shùn shí biàn huà lǜ |
| difference quotient/ˈdɪfrəns ˈkwəʊʃənt/ | 差商 | chà shāng |
| secant line/ˈsiːkənt laɪn/ | 割线 | gē xiàn |
| tangent line/ˈtændʒənt laɪn/ | 切线 | qiè xiàn |
From "over an interval" to "at an instant"
- Unit 1 built the idea of a limit. Now we use it to measure change.
- Over an interval, the average rate of change 平均变化率 is total change ÷ interval width.
- At a single instant, the instantaneous rate of change 瞬时变化率 is what a speedometer shows.
- The bridge between them is a limit — and that bridge is the whole point of Unit 2.
从"一段区间上"到"一瞬间"
- 第 1 单元建立了极限的想法。现在我们用它来度量变化。
- 在一段区间上,平均变化率是总变化 ÷ 区间宽度。
- 在单一瞬间,瞬时变化率是速度表所显示的。
- 连接两者的桥梁是一个极限——而这座桥正是第 2 单元的全部要点。
Average rate = slope of a secant
- The average rate of change of $f$ from $x=a$ to $x=b$ is the difference quotient 差商:
-
$$\frac{f(b)-f(a)}{b-a}$$
- Geometrically this is the slope of the secant line 割线 joining $\big(a,f(a)\big)$ and $\big(b,f(b)\big)$.
- It is an average: it smooths over everything that happened between $a$ and $b$.
平均变化率 = 割线的斜率
- $f$ 从 $x=a$ 到 $x=b$ 的平均变化率是差商:
-
$$\frac{f(b)-f(a)}{b-a}$$
- 几何上这是连接 $\big(a,f(a)\big)$ 与 $\big(b,f(b)\big)$ 的割线的斜率。
- 它是一个平均:它把 $a$ 与 $b$ 之间发生的一切都抹平了。
Find the average rate of change of $f(x)=x^2$ on $[1,3]$. · 求$f(x)=x^2$在$[1,3]$上的平均变化率。
$\dfrac{9-1}{3-1}=\dfrac{8}{2}=4$.
The average rate of change over an interval equals the slope of the... · 区间上的平均变化率等于...的斜率
Average rate = rise over run between the two endpoints = secant slope. · 平均变化率 = 两个端点之间的升程除以行程 = 割线斜率。
Shrink the interval
- To get the rate at $x=a$, slide $b$ toward $a$ and watch the secant slope.
- As the interval shrinks, the secant pivots toward the tangent line 切线 at $a$.
- The limiting slope is the instantaneous rate of change at $a$.
- No time actually passes at an instant — the limit is what makes "rate at a point" meaningful.
缩小区间
- 要得到在 $x=a$ 处的变化率,让 $b$ 滑向 $a$,盯住割线斜率。
- 当区间缩小,割线转向 $a$ 处的切线。
- 那个极限斜率就是 $a$ 处的瞬时变化率。
- 一瞬间其实没有时间流逝——是极限让"某点处的变化率"变得有意义。
Slide the secant into a tangent · 将割线滑入切线
y = x²
Move the point along $y=x^2$ — the tangent slope you read off is the instantaneous rate, the limit of the secant slopes. · 沿$y=x^2$移动点——你读出的切线斜率即为瞬时变化率，也就是割线斜率的极限。
The instantaneous rate of change at a point equals the slope of the ____ line there. · 点在瞬时变化率等于该处____线的斜率。
It is the limit of secant slopes as the interval shrinks — the tangent slope. · 它是割线斜率在区间缩小时取的极限——即切线斜率。
Reading the rate in context
- The sign tells direction: positive rate → increasing, negative → decreasing.
- The size tells speed: a bigger magnitude means faster change.
- Units matter: for position in metres over time in seconds, the rate is in $\tfrac{\text{m}}{\text{s}}$ — a velocity.
- So average rate = secant slope (over an interval); instantaneous rate = tangent slope (at a point).
在情境中读出变化率
- 符号指明方向:正的变化率 → 递增,负的 → 递减。
- 大小指明快慢:数值越大,变化越快。
- 单位很重要:位置以米、时间以秒计,变化率就以 $\tfrac{\text{m}}{\text{s}}$ 计——一个速度。
- 所以平均变化率 = 割线斜率(区间上);瞬时变化率 = 切线斜率(某点处)。
If the average rate of change of $f$ on $[0,4]$ is $0$, then $f$ must be constant on $[0,4]$. · 如果 $f$ 在 $[0,4]$ 上的平均变化率为 $0$，那么 $f$ 在 $[0,4]$ 上必须是常数。
It only means $f(0)=f(4)$; $f$ could rise then fall (e.g. go out and come back). · 这只意味着$f(0)=f(4)$；$f$可能先上升后下降（例如出去又回来）。
For · 支持 $f(x)=x^2$, average rates on $[3,3.1]$ and $[3,3.01]$ are $6.1$ and $6.01$. Estimate the instantaneous rate at $x=3$. · 对于$f(x)=x^2$，在$[3,3.1]$和$[3,3.01]$上的平均速率分别为$6.1$和$6.01$。估算在$x=3$处的瞬时速率。
The averages close in on $6$. · 平均值收敛于$6$。
A position function has instantaneous rate $-5\,\tfrac{\text{m}}{\text{s}}$ at time $t$. Select all · 所有 correct readings. · 位置函数具有瞬时变化率 $-5\,\tfrac{\text{m}}{\text{s}}$ 在时间 $t$。选择 所有 正确的读数。
Negative rate = moving in the negative direction (so position decreases); its magnitude is the speed; and it is the tangent slope. · 负速率 = 向负方向移动（所以位置减少）；其大小是速率；并且它是切线斜率。
Average and instantaneous rates are not interchangeable. The average rate over $[a,b]$ can be zero while the instantaneous rate is large at every interior point (imagine going out and coming back). Only in the limit of a shrinking interval does the average rate become the instantaneous rate.
平均变化率与瞬时变化率不能互换。$[a,b]$ 上的平均变化率可以为零,而每个内部点的瞬时变化率都很大(想象出去又回来)。只有在区间缩小的极限里,平均变化率才变成瞬时变化率。
For $f(x)=x^2$, find the average rate on $[2,4]$ and estimate the instantaneous rate at $x=2$.
- Average on $[2,4]$: $\dfrac{f(4)-f(2)}{4-2}=\dfrac{16-4}{2}=6$.
- Shrink toward $2$: on $[2,2.1]$, $\dfrac{4.41-4}{0.1}=4.1$; on $[2,2.01]$, $4.01$.
- The average rates close in on $4$ — the instantaneous rate at $x=2$ is $4$.
对 $f(x)=x^2$,求 $[2,4]$ 上的平均变化率,并估计 $x=2$ 处的瞬时变化率。
- $[2,4]$ 上的平均:$\dfrac{f(4)-f(2)}{4-2}=\dfrac{16-4}{2}=6$。
- 向 $2$ 缩小:在 $[2,2.1]$,$\dfrac{4.41-4}{0.1}=4.1$;在 $[2,2.01]$,$4.01$。
- 平均变化率逼近 $4$——$x=2$ 处的瞬时变化率是 $4$。
The average rate of change over $[a,b]$ is the difference quotient $\frac{f(b)-f(a)}{b-a}$ — the slope of a secant line. Shrinking the interval, the secant approaches the tangent line, whose slope is the instantaneous rate of change at that point.
$[a,b]$ 上的平均变化率是差商 $\frac{f(b)-f(a)}{b-a}$——一条割线的斜率。缩小区间,割线逼近切线,其斜率就是该点的瞬时变化率。