Estimating Derivatives of a Function at a Point · 估算函数在一点的导数
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| estimate/ˈestɪmət/ | 估计值 | gū jì zhí |
| difference quotient/ˈdɪfrəns ˈkwəʊʃənt/ | 差商 | chà shāng |
Measuring a slope you can't compute exactly
- Sometimes you have a graph or a table, not a formula — but you still want $f'$ at a point.
- You can estimate the derivative from what you have.
- From a graph: measure the steepness of the tangent line by eye.
- From a table: use a difference quotient over the closest available inputs.
度量一个无法精确算出的斜率
- 有时你手上有一张图或一张表,而非公式——但你仍想要某点的 $f'$。
- 你可以从已有的东西估计导数。
- 从图上:用眼睛量出切线的陡峭程度。
- 从表上:用最接近的可用输入做差商。
Estimate the slope by eye · 目测估算斜率
y = x²
Move the point and watch the tangent — its steepness is the derivative you would estimate from a graph. · 移动点并观察切线——其陡峭程度就是你会从图中估算出的导数。
From a graph: tangent steepness
- Draw (or imagine) the tangent line at the point, then read its slope as rise ÷ run.
- Pick two clear points on the tangent, not on the curve, and compute $\tfrac{\Delta y}{\Delta x}$.
- Steeper tangent → larger $|f'|$; a flat tangent → $f'\approx0$.
- Uphill tangent → positive derivative; downhill → negative.
从图上:切线的陡度
- 在该点画出(或想象)切线,再把它的斜率读作纵变化 ÷ 横变化。
- 取切线上的两个清晰点(不是曲线上的点),计算 $\tfrac{\Delta y}{\Delta x}$。
- 切线越陡 → $|f'|$ 越大;切线水平 → $f'\approx0$。
- 切线上坡 → 导数为正;下坡 → 为负。
To estimate $f'(a)$ from a graph, you should measure the slope of the... · 要从图中估算$f'(a)$，你应该测量...的斜率
The derivative is the tangent slope, read from points on the tangent line. · 导数是切线斜率，从切线上的点读出。
From a table: nearest difference quotient
- With table values, approximate $f'(a)$ by the difference quotient over the closest points:
- A symmetric estimate is best: $f'(a)\approx\dfrac{f(a+h)-f(a-h)}{2h}$, using one point on each side.
- Or use a one-sided quotient $\dfrac{f(a+h)-f(a)}{h}$ if only one neighbor is available.
- Smaller $h$ (closer inputs) generally gives a better estimate 估计值.
从表上:最近的差商
- 有了表值,用最接近的点做差商来近似 $f'(a)$:
- 对称估计最好:$f'(a)\approx\dfrac{f(a+h)-f(a-h)}{2h}$,两侧各取一个点。
- 若只有一侧的邻点,就用单侧差商 $\dfrac{f(a+h)-f(a)}{h}$。
- $h$ 越小(输入越近),估计通常越好。
A table gives $f(1.9)=4.2$ and $f(2.1)=5.8$. Use the symmetric difference quotient to estimate $f'(2)$. · 表格给出了$f(1.9)=4.2$和$f(2.1)=5.8$。使用对称差商来估算$f'(2)$。
$\dfrac{5.8-4.2}{2(0.1)}=\dfrac{1.6}{0.2}=8$.
Using input values closer to the point generally improves a table-based derivative estimate. · 使用更接近该点的输入值通常能改善基于表的导数估算。
A smaller interval better approximates the tangent slope. · 更小的区间更好地近似切线斜率。
A derivative read from a table or graph is an , not an exact value. · 从表或图中读出的导数是一个，而非精确值。
Only algebra on a formula gives the exact derivative. · 只有对公式进行代数运算才能得到精确导数。
From $f(3)=10$ and $f(3.2)=11.4$, estimate $f'(3)$ with a one-sided difference quotient. · 从$f(3)=10$和$f(3.2)=11.4$出发，使用单侧差商估算$f'(3)$。
$\dfrac{11.4-10}{3.2-3}=\dfrac{1.4}{0.2}=7$.
Reading sign and size
- The sign of the estimate says which way $f$ is heading: $+$ rising, $-$ falling.
- The size says how fast: a derivative of $12$ means $f$ changes about $12$ units per unit of $x$ there.
- Always attach units and interpret in context (e.g. "the temperature is rising about $2^\circ$C per minute").
- An estimate is an approximation — say so, and use the closest data you have.
读出符号与大小
- 估计的符号说明 $f$ 朝哪个方向:$+$ 上升,$-$ 下降。
- 大小说明多快:导数为 $12$ 意味着在那里 $f$ 每单位 $x$ 变化约 $12$ 个单位。
- 永远附上单位并在情境中解读(如"温度每分钟上升约 $2^\circ$C")。
- 估计是一个近似——要说明这一点,并使用你手上最接近的数据。
If an estimated derivative at a point is negative, the function there is... · 如果在某点的估算导数为负，则该处的函数是...
A negative rate means the function is falling there. · 负速率意味着函数在那里正在下降。
When estimating from a graph, read the slope of the tangent line, not of a nearby secant, and take your two points off the tangent, not off the curve. From a table, don't reuse the point $a$ itself as both inputs — you need two different inputs bracketing (or approaching) $a$.
从图上估计时,读切线的斜率,而不是附近割线的斜率,并且两个点取自切线而非曲线。从表上做时,别把点 $a$ 本身当作两个输入——你需要两个不同的输入来夹住(或逼近)$a$。
A table gives $f(1.9)=4.2$, $f(2)=5.0$, $f(2.1)=5.9$. Estimate $f'(2)$.
- Symmetric difference quotient: $f'(2)\approx\dfrac{f(2.1)-f(1.9)}{2(0.1)}=\dfrac{5.9-4.2}{0.2}=\dfrac{1.7}{0.2}=8.5$.
- (A one-sided estimate $\dfrac{5.9-5.0}{0.1}=9$ is close but less balanced.)
- So $f'(2)\approx8.5$ — $f$ is rising at about $8.5$ units per unit of $x$.
一张表给出 $f(1.9)=4.2$、$f(2)=5.0$、$f(2.1)=5.9$。估计 $f'(2)$。
- 对称差商:$f'(2)\approx\dfrac{f(2.1)-f(1.9)}{2(0.1)}=\dfrac{5.9-4.2}{0.2}=\dfrac{1.7}{0.2}=8.5$。
- (单侧估计 $\dfrac{5.9-5.0}{0.1}=9$ 接近,但不够均衡。)
- 所以 $f'(2)\approx8.5$——$f$ 每单位 $x$ 上升约 $8.5$ 个单位。
To estimate a derivative: from a graph, read the slope of the tangent line (rise ÷ run using points on the tangent); from a table, use a difference quotient over the closest inputs — the symmetric quotient $\frac{f(a+h)-f(a-h)}{2h}$ is most accurate. The sign gives direction, the size gives rate.
估计导数:从图上,读切线的斜率(用切线上的点算纵 ÷ 横);从表上,用最接近输入的差商——对称差商 $\frac{f(a+h)-f(a-h)}{2h}$ 最准。符号给方向,大小给速率。