Calculus I · 2B · reference

Unit 2B Formula and Strategy Reference

Use this applications of derivatives reference for formulas, notation, domains, units, and problem-solving strategy.

Course progressSupporting resourceReturn to the unit map →

Section overview

Review, practice, exams, and reference

What this section is building

Use this applications of derivatives reference for formulas, notation, domains, units, and problem-solving strategy.

Notice

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

Decide

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

Avoid

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

Formula and Strategy Reference

\begin{aligned} v&=s',& a&=s'',& \text{speed}&=|v|,\\ L(x)&=f(a)+f'(a)(x-a),& dy&=f'(x)dx,\\ x_{n+1}&=x_n-\frac{f(x_n)}{f'(x_n)},& \frac{d}{dt}F(x(t),y(t))&=0\text{ for a constant relation}. \end{aligned}

Closed-interval absolute extrema require evaluation at endpoints and all interior critical numbers. The Mean Value Theorem guarantees some $c$ with

f'(c)=\frac{f(b)-f(a)}{b-a}.

Increasing/decreasing behavior comes from the sign of $f'$ ; concavity from the sign of $f"$ . Optimization requires an objective, constraint, feasible domain, and verification.

L'Hopital's Rule applies to limits of quotients producing $0/0$ or $\infty/\infty$ , under its hypotheses. Other indeterminate forms must first be transformed.

Summary

End of Unit 2B

You should now be able to choose a derivative tool from the question being asked, build the necessary model, carry out the calculus, and interpret the result with units and assumptions. That is the practical core of differential calculus.

Source & rights

Original instruction with traceable references.

BetterGrades-original composition declared by source handoff; owner provenance review required before public release

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

Formula and Strategy Reference

\begin{aligned} v&=s',& a&=s'',& \text{speed}&=|v|,\\ L(x)&=f(a)+f'(a)(x-a),& dy&=f'(x)dx,\\ x_{n+1}&=x_n-\frac{f(x_n)}{f'(x_n)},& \frac{d}{dt}F(x(t),y(t))&=0\text{ for a constant relation}. \end{aligned}

Closed-interval absolute extrema require evaluation at endpoints and all interior critical numbers. The Mean Value Theorem guarantees some

c

with

f'(c)=\frac{f(b)-f(a)}{b-a}.

Increasing/decreasing behavior comes from the sign of

f'

; concavity from the sign of

f"

. Optimization requires an objective, constraint, feasible domain, and verification.

L'Hopital's Rule applies to limits of quotients producing

0/0

\infty/\infty

, under its hypotheses. Other indeterminate forms must first be transformed.