Calculus I · Unit 2B · review

Unit 2B Review: Applications of Derivatives

Unit 2B Review and Exam Preparation

Application Decision Guide

Decision

Match the question to the derivative tool

Calculus reference table
Question type	Primary tools
position, velocity, acceleration	first and second time derivatives; sign charts
small-change estimate	tangent-line linearization or differentials
measurement uncertainty	absolute and relative differentials
changing geometric quantities	related-rates equation differentiated in time
absolute extrema on a closed interval	endpoints plus interior critical numbers
local maxima or minima	first-derivative sign change or second-derivative test
increasing/decreasing behavior	sign of $f'$
concavity and inflection	sign of $f"$ plus actual sign change
complete curve sketch	domain, limits, intercepts, $f'$ , and $f"$
maximum or minimum design	objective, constraint, feasible domain, verification
$0/0$ or $\infty/\infty$ limit	algebra first when simpler; otherwise L'Hopital under hypotheses
root approximation	Newton iteration and convergence check

In ordinary language

The final sentence test

A complete applied answer should name the quantity, direction or sign, units, reference point, and practical interpretation. If your final line is only $x=3$ , you may have found a candidate while failing to answer the actual question.

After the explanation

Use the section idea

Reading lens

Mixed applications remove the method label; classify the model, commit to a complete attempt, and diagnose the first wrong decision.

Mental model

A complete response includes setup, calculus, candidates or rates, units, interpretation, and a reasonableness check.

Decision

Choose the governing relationship before calculating, then use the attempt-gated answer as feedback rather than as a shortcut.

Common trap

Reading an answer before modeling the problem or treating every miss as algebra when the first error was conceptual.

Check yourself

Can you defend the model and domain even before checking whether the final number matches?

Source & rights

Original instruction with traceable references.

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Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.