C.12 · 관련 변화율과 제거 가능한 분산 한계
GRE Subject Test · GRE Mathematics 주제 30 3:02 영어 내레이션 · 영어 + 중국어 자막 burned-in
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This unit translates a geometric rate into a derivative, derives and differentiates a spherical-cap volume formula, and separates continuous extension of a quotient from differentiable extension.
本单元把几何上的变化率转化为导数，推导并求导球冠体积公式，并区分商的连续延拓与可微延拓。
For a volume depending on a changing depth, the chain rule gives dV dt equals V prime of h times dh dt.
当体积依赖于变化的深度时，链式法则给出 dV/dt = V′(h)·dh/dt。
A draining tank has negative dh dt, so the signed rate is negative and the outflow is its magnitude.
排水时 dh/dt < 0，所以带符号的变化率为负，流出量是它的绝对值。
Name the instantaneous depth before substituting numbers.
代入数值前先确定瞬时深度。
A spherical tank of radius R has cross-section radius squared 2 R h minus h squared at depth h.
半径为 R 的球形罐在深度 h 处截面半径的平方是 2Rh − h²。
Integrating gives the cap volume pi times R h squared minus h cubed over 3, whose derivative is the cross-section area.
积分得球冠体积 π(Rh² − h³/3)，其导数就是截面面积。
With R equals 3 at depth 1, the cross-sectional area is pi times 5, so a fall of a quarter per unit time drains 5 pi over 4.
R = 3、深度为 1 时，截面面积是 5π，所以每单位时间下降 1/4 就排出 5π/4。
If f and g vanish at zero with g prime of zero nonzero, f over g tends to f prime over g prime of zero; filling in gives a continuous extension.
若 f、g 在零处为零且 g′(0) ≠ 0，则 f/g 趋于 f′(0)/g′(0)；补上这个值得到连续延拓。
It need not be differentiable: for x times the absolute value of x, over x, the quotient is the absolute value of x, which extends continuously at zero but has a corner.
但它不一定可微：对 x|x|/x，商是 |x|，在零处可以连续延拓，却有一个尖角。
With R 3 and depth 1, the cross-sectional area is 5 pi, so a quarter-unit fall removes 5 pi over 4.
R = 3、深度为 1 时，截面面积是 5π，所以下降四分之一单位就排出 5π/4。
Separately, sin 2x over 3x extends at zero with value 2 thirds.
另外，sin 2x/(3x) 在零处可延拓，值为 2/3。
That limit says nothing about a tank.
这个极限与水罐毫无关系。
Use the instantaneous cross-section area, not total tank surface area.
用瞬时截面面积，而不是水罐的总表面积。
Report a signed change or a positive outflow as requested.
按要求报告带符号的变化或正的流出量。
A finite removable quotient limit establishes continuity, not differentiability.
有限的可去商极限只能说明连续，不能说明可微。
Integrating disks gives the cap volume.
对圆盘积分得球冠体积。
Its derivative is pi times 2 R h minus h squared, which is 3 pi at this depth; times minus 0.1 gives minus 0.3 pi cubic metres a minute.
其导数是 π(2Rh − h²)，在此深度为 3π；乘以 −0.1 得每分钟 −0.3π 立方米。
The outflow magnitude is 0.3 pi.
流出量是 0.3π。
Factor as f over g times f minus 1 over 2 minus g squared; the limit is 2 times minus a half, which is minus 1.
分解为 (f/g)·(f − 1)/(2 − g²)；极限是 2 × (−1/2) = −1。
But differentiability does not follow: with f equals 4x plus x times the absolute value of x, the expansion has an absolute-value term, so the one-sided derivatives differ.
但不能推出可微：取 f = 4x + x|x|，展开式中有绝对值项，所以左右导数不同。
Check yourself.
自我检测。
The cap's volume derivative is the cross-section area.
球冠体积的导数是截面面积。
The outflow is 5 pi over 4.
流出量是 5π/4。
sin 2x over 3x tends to 2 thirds.
sin 2x/(3x) 趋于 2/3。
The extended x absolute x over x is not differentiable.
x|x|/x 延拓后不可微。
And the quotient limit is minus 1.
商的极限是 −1。