Related rates and removable quotient limits
| English | 中文 | Pinyin |
|---|---|---|
| removable limit/rɪˈmuːvəbl ˈlɪmɪt/ | 可去极限 | kě qù jí xiàn |
| related rate/rɪˈleɪtɪd reɪt/ | 相关变化率 | xiāng guān biàn huà lǜ |
A decision before an answer
- A water-depth rate is not a volume-flow rate: the tank’s cross-sectional area changes with depth. A quotient limit can likewise have a finite value without giving a differentiable extended function.
- Your goal: Translate a geometric rate into a derivative of the relevant quantity.
Read the relationship
- For a quantity V depending on a changing depth h(t), the chain rule gives dV/dt=V′(h) dh/dt. A draining tank has dh/dt<0, so signed dV/dt is negative; an outflow magnitude is its negative. Draw the geometry and name the instantaneous depth before substituting numbers. Do not treat the rate as a static volume divided by elapsed time unless the situation actually specifies a constant average rate.
- Derive and differentiate a spherical-cap volume formula.
A radius-3 spherical tank has depth 1 decreasing at 1/4 per time unit. What is the positive outflow magnitude?
Cross-section area is π(2Rh−h²)=5π. Multiply by depth-decrease magnitude 1/4.
Use the defining rule
- A spherical tank of radius R has cross-section radius squared R²−(R−h)²=2Rh−h² at depth h from the bottom. Integrating these circular areas gives the cap volume V(h)=π(Rh²−h³/3), for 0≤h≤2R. Its derivative π(2Rh−h²) is the current cross-sectional area. At R=3,h=1,dh/dt=−1/4, signed volume rate is −5π/4 and outflow magnitude 5π/4. The formula works for both shallow and deep caps within the stated range.
- Separate continuous quotient extension from differentiable extension.
For x≠0, f(x)=x|x|,g(x)=x. What happens when f/g is filled in at zero?
The quotient is |x|. Filling in zero gives continuity, while the left and right derivatives are −1 and 1.
Check the conditions
- If f and g are continuously differentiable near zero, f(0)=g(0)=0 and g′(0)≠0, then f(x)/g(x) tends to f′(0)/g′(0). This follows from f(x)=f′(0)x+o(x) and g(x)=g′(0)x+o(x); continuity of g′ keeps the quotient defined nearby except at zero. Filling in this limit gives a continuous extension. The argument does not require f′(0) nonzero, and it does not prove the extended quotient differentiable.
- Separate continuous quotient extension from differentiable extension.
For a sphere with R=3 and depth h=1, cross-sectional area is π(6−1)=5π. A depth decrease of 1/4 per time unit therefore removes volume at magnitude 5π/4 per time unit. For f(x)=sin(2x),g(x)=3x, the quotient extends at zero with value 2/3. This limit says nothing about the size of a tank and should not be substituted as a derivative rate without identifying the dependent quantities.
For f(x)=sin(2x) and g(x)=3x, express the removable quotient value as a fraction: ____.
The derivatives at zero are 2 and 3, so their quotient gives the limit 2/3.
Apply the task format
- For f(x)=x|x| and g(x)=x, both are continuously differentiable, f(0)=g(0)=0 and g′(0)=1. Their quotient for x≠0 is |x|, which extends continuously at zero but has unequal one-sided derivatives there. For (f²−f)/(2g−g³), factor to (f/g)(f−1)/(2−g²); its removable limit is −f′(0)/(2g′(0)). Additional factors approach −1 and 2. Distinguish the limit of the quotient from the derivative of its extension.
- Separate continuous quotient extension from differentiable extension.
Use instantaneous cross-section area, not total tank surface area. Report a signed volume change or a positive outflow as requested. A finite removable quotient limit establishes continuity, not differentiability of the filled-in quotient.
Which answer fits this case?
Translate a geometric rate into a derivative of the relevant quantity
A continuously differentiable numerator and denominator with simple denominator zero always give a differentiable quotient extension.
The numerator x|x| and denominator x meet the assumptions, but their filled-in quotient |x| is not differentiable at zero.
Keep the distinctions
- related rate 相关变化率 — A rate obtained by differentiating the relationship between changing quantities.
- removable limit 可去极限 — A finite nearby limit used to fill in a missing function value continuously.
- Translate a geometric rate into a derivative of the relevant quantity.
- Derive and differentiate a spherical-cap volume formula.
- Separate continuous quotient extension from differentiable extension.
Match each term with its precise meaning in this lesson.
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
The order follows the stated process; check each stage before the next.