Calculus I · Unit 2B · diagnostic

Derivative Applications Bridge Diagnostic

Bridge Diagnostic

Before beginning, calculate or explain each item.

Exercise

Differentiate $x^4-3x^2+1$ .

Answer reveal

Exercise 1 answer

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Exercise

Differentiate $(x^2+1)^5$ .

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Exercise 2 answer

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Exercise

Differentiate $x^2e^x$ .

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Exercise 3 answer

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Exercise

Find $dy/dx$ for $x^2+y^2=25$ .

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Exercise 4 answer

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Exercise

If $s(t)=t^3-6t$ , find $s'(t)$ and $s"(t)$ .

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Exercise 5 answer

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Exercise

State the units of $C'(q)$ if $C$ is dollars and $q$ is items.

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Exercise 6 answer

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Exercise

Solve $3x^2-12x+9=0$ .

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Exercise 7 answer

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Exercise

Explain why a solution of $f'(x)=0$ is only a candidate for an extremum.

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Exercise 8 answer

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Worked solution

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Missing more than two computational items suggests a short return to the corresponding Unit 2A review page. Application work becomes much easier when differentiation itself is no longer consuming all available attention.

After the explanation

Use the section idea

Reading lens

Begin with the modeled quantity and the question it must answer; differentiation belongs in the middle of the translation, not at the beginning.

Mental model

An application moves from situation to variables, units, relationship, derivative objective, feasible domain, and finally a contextual claim.

Decision

Name what changes, with respect to what, and in which units before selecting a derivative method.

Common trap

Calculating a derivative of an unexamined formula and reporting a bare number that does not answer the situation.

Check yourself

Can you state the variables, units, assumptions, relationship, and requested rate in complete sentences?

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.