--- name: math-thumb-calculus description: "Apply Chapter 4 (Calculus: Turn Local Change into Global Control) of Mathematical Rules of Thumb to solve, check, or teach problems. Use it to turn a local model into a checked estimate and distinguish an error scale from a proved bound." --- # Calculus: Turn Local Change into Global Control Use this chapter to help the reader make a checked mathematical decision. All 17 numbered rules are available in [the chapter source](references/chapter.md). [The workbook](references/notebook.md) contains a lab, exercises, solutions, and the full rule checklist. [The local rule index](references/rules.json) supplies discovery metadata. ## Start from the reader's task Infer solve, learn, or audit mode from the request. In solve mode, use their supplied numbers and target; in learn mode, use the workbook or their chosen rule; in audit mode, inspect their actual calculation before replacing it. Gather only missing information that changes the choice: Function and domain; expansion point; perturbation; derivative information; desired bound or estimate. If the question falls outside this chapter, say which mathematical operation is missing and suggest a relevant chapter. If the whole-book skill is available, it can carry the task onward, but this chapter works independently. ## Select and apply a rule - **Is the input near a value you understand?** Linearize. If you are solving $f(x)=0$, interpret that linear model’s root as a Newton step. - **Is the question about sensitivity or guaranteed change?** Use relative differentials for small percentage response; use a derivative supremum when a hard interval-wide bound is required. - **Is the question about graph shape?** Partition at critical, nondifferentiable, and endpoint values. Use derivative signs for monotonicity and convexity for tangent–secant bounds. - **Is a thin layer being added?** Multiply boundary measure by thickness, then compare the curvature correction with the tolerance. - **Before integrating, can structure finish the problem?** Check range, parity, and moving endpoints before seeking an antiderivative. - **Is the issue improper accumulation?** Identify the troublesome endpoint, recall the $p=1$ threshold, compare positive integrands in the correct direction, and use adjacent integrals when a decreasing series remainder is needed. - **Do you need an integration method?** Search first for an inner function with its derivative, then consider whether parts transfers complexity profitably. - **Do you need a limit or approximation?** Verify an indeterminate quotient before L’Hôpital, and verify a remainder condition before treating the next Taylor term as a bound. Read the selected complete profile, including its equation, “How to read it,” and “How to use it.” The compact graph assumptions are search cues, not a substitute for the profile. Preserve the numbered citation and role. **Independent**, **Workflow**, and **Specialized** describe the relationship to a calculation; exactness, approximation, bound, diagnostic, and heuristic describe a different dimension. Use verified inputs, show the substitution and units, and interpret the result in the reader's decision. Verify by an appropriate bound, alternative computation, limiting case, residual with conditioning, or sensitivity check. If a required condition fails, reject that use and give the specific missing information or alternative method; do not calculate a plausible-looking answer from an invalid formula. **Essential boundary:** A next Taylor term is not automatically an error bound. Establish an appropriate remainder condition or compare with a verified exact result. For a sufficient independent result, stop with the decision it supports. For a workflow or specialized rule, name the downstream calculation still needed. A numerical demonstration is evidence for that instance, not a universal proof. ## Teach and check understanding Use the [workbook](references/notebook.md) for guided practice. Start with 4.1.1, 4.1.4, 4.3.10 when the reader wants a starting exercise. Ask for an attempt, offer a relevant hint, and reveal the answer when requested or when teaching requires it. Do not force a quiz when the reader asked for a worked solution. Check whether the reader can explain the controlling quantity, apply the rule to a changed input, identify an invalid use, and distinguish a final answer from a preparatory step. Track only demonstrated work. Give a short prerequisite explanation when needed; avoid requiring completion of earlier chapters. ## Return a usable result Include the chosen rule numbers, assumptions that matter, calculation, verification, and next action. For ongoing work, offer this compact record: question; inputs and units; rules; claim type; book role; assumption status; result and error; check; decision; unresolved next step. Write a progress file only when asked or within an already authorized notebook-editing task. The source chapter is a fixed book snapshot. Preserve its mathematical qualifications and historical evidence gaps. Use outside material only when the reader's task needs it, verify material facts appropriately, and identify that material separately from the book. ## Illustrated exploration Open [the browser reader](assets/reader.html) or [the saved illustrated notebook](assets/notebook.ipynb). [Equation cards](references/equations.json) record the book rule, formula, fixed inputs, supported choices, assumptions and executed default results. - **C04-D01: Check a tangent estimate against a remainder**: rule 4.1.1. - **C04-D02: Read a Newton step from its tangent**: rule 4.1.2. - **C04-D03: Distinguish a finite sum from an infinite tail**: rule 4.3.6. - **C04-D04: Use the first omitted Taylor term as an error scale**: rule 4.3.10. - **C04-D05: Confirm 0/0 before using L'Hôpital**: rule 4.3.9. Use a saved illustration only when its conditions fit. Browser controls select finite precomputed choices; they do not calculate arbitrary reader inputs. For different inputs, make a checked calculation using the selected rule. Explain what changes, and never claim the notebook ran or the browser was viewed unless it did. Offer prediction questions for learning; answer direct requests without a mandatory quiz.