Calculus I · Unit 2B · review

Derivative Applications Modeling Studio Review

Mixed Modeling Problems

Exercise

For $C(t)=20te^{-0.5t}$ , find the peak time and concentration and state where concentration increases.

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Exercise 1 answer

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Exercise

A sphere radius grows at $0.2$ cm/s. Find the area and volume rates at $r=5$ .

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Exercise 2 answer

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Exercise

A camera is $30$ m from a road. Derive angular rate when a vehicle moves at $15$ m/s.

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Exercise 3 answer

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Exercise

For $D(v)=1.0v+v^2/(2(0.75)(9.8))$ , compare $D'(15)$ and $D'(30)$ .

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Exercise 4 answer

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Exercise

A spherical sensor radius is measured as $6.00\pm0.03$ mm. Estimate relative volume uncertainty.

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Exercise 5 answer

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Exercise

A company's demand is $p(q)=100-0.1q$ and cost is $C(q)=5000+25q$ . Find the profit-maximizing feasible production level.

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Exercise

A response is $R(d)=120d/(d+10)$ . Analyze monotonicity, concavity, and limiting response.

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Exercise 7 answer

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Exercise

Build a derivative model for one changing quantity from your own field and state its units, assumptions, and domain.

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Exercise 8 answer

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After the explanation

Use the section idea

Reading lens

Treat each derivative model as a conditional claim whose variables, units, assumptions, calibration range, and limitations remain visible.

Mental model

A useful model connects a measurable input to a measurable output, while its derivative describes local sensitivity inside a stated domain.

Decision

Define the relationship and objective, differentiate, evaluate candidates or rates, then test sign, scale, units, and assumption sensitivity.

Common trap

Extending a fitted model outside its data range or presenting medication, stopping-distance, or business outputs without the assumptions that shape them.

Check yourself

What observation would falsify the model, and how would the conclusion change if its strongest assumption failed?

Source & rights

Original instruction with traceable references.

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