Calculus I · Unit 2B · reference

Unit 2B Formula and Strategy Reference

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.

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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