Related rates of change
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Flip to reveal answersChain rule for related rates?
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All 8 Flashcards — Related rates of change
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Question
Chain rule for related rates?
Answer
dy/dt = (dy/dx)(dx/dt) — linked quantities have linked rates.
Question
Circle: how does area rate link to radius rate?
Answer
A = πr² ⇒ dA/dt = 2πr · dr/dt.
Question
Sphere: how does volume rate link to radius rate?
Answer
V = (4/3)πr³ ⇒ dV/dt = 4πr² · dr/dt.
Question
The 4-step related-rates recipe?
Answer
Write the link → differentiate it → apply the chain rule with the known rate → substitute the value and solve.
Question
When do you substitute the given value (e.g. r = 5)?
Answer
LAST — after differentiating. Substituting early removes the variable you need to differentiate.
Question
A balloon's volume grows at 100 cm³/s; find dr/dt at r = 5.
Answer
100 = 4π(25)dr/dt ⇒ dr/dt = 1/π ≈ 0.318 cm/s.
Question
What does a NEGATIVE rate mean in a related-rates problem?
Answer
The quantity is decreasing (e.g. melting, draining) — keep the minus sign.
Question
If dr/dt = 0 at an instant, what is dA/dt for A = πr²?
Answer
dA/dt = 2πr·dr/dt = 0 — the area is momentarily not changing.
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Full study notes for Related rates of change
Topic 5.9 hub
Differentiation rules (HL only)
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