Calculus I · Unit 2B · review

Related Rates Review

Related Rates Review

Summary

Define variables as functions of time, write a relation, differentiate before inserting values, preserve rate signs and units, and interpret the result in words. Similar triangles are often used to eliminate an extra geometric variable.

Exercise

An oil slick remains circular and area grows at $100$ m $^2$ /min. Find $dr/dt$ at $r=20$ .

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Exercise 1 answer

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Exercise

A cube's volume grows at $12$ cm $^3$ /s. Find $ds/dt$ when $s=2$ .

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Exercise 2 answer

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Exercise

Sand forms a cone whose height always equals its radius. If volume grows at $6$ m $^3$ /min, find height rate at $h=3$ .

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Exercise 3 answer

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Exercise

A spotlight tracks a plane at constant altitude. Derive an angle-rate formula.

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Exercise 4 answer

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Exercise

Explain why substituting $r=10$ into $V=4\pi r^3/3$ before differentiating destroys the related-rate information.

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Exercise 5 answer

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After the explanation

Reading lens

Freeze the geometry at one instant, but differentiate the relationship while every changing quantity is still a function of time.

Mental model

The picture supplies a constraint; implicit differentiation transmits known rates through that constraint to the unknown rate.

Decision

Draw and label first, write one relationship, differentiate with time, then substitute the snapshot measurements and rates.

Common trap

Substituting numerical dimensions before differentiating and thereby erasing the very change the problem asks about.

Check yourself

Does your final rate have the predicted sign, the correct units, and a magnitude compatible with the diagram?

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