# Chapter 4 notebook: Calculus: Turn Local Change into Global Control **Goal:** Turn a local model into a checked estimate and distinguish an error scale from a proved bound. **Start with:** Rules 4.1.1, 4.1.4, 4.3.10. Use a calculator or the optional Python lab. Programming is optional for the workbook; the Jupyter version requires a Python 3 kernel. All lab numbers are constructed practice inputs, not observed data. **Source:** [Finished Chapter 4](../skills/math-thumb-calculus/references/chapter.md) from *Mathematical Rules of Thumb*. Numbered rules, classifications, and the decision path come from the book. The lab, exercises, answer key, and worksheet prompts are companion additions. ## 1. Frame your decision Write a question in which an answer would change something you do. Gather: Function and domain; expansion point; perturbation; derivative information; desired bound or estimate. - My question and intended decision: - Known inputs and units: - Required accuracy or threshold: - What I expect before calculating: - What I still need to find out: Use the lab as a worked starting point if you do not yet have your own problem. For notation or prerequisites, ask the chapter skill to explain only the concept blocking the next step. ## 2. Choose a route - **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. **Chapter-specific stop check:** A next Taylor term is not automatically an error bound. Establish an appropriate remainder condition or compare with a verified exact result. ## 3. Work the lab Estimate sqrt(101) from the known point 100. The tangent estimate is 10+(1/20)(1)=10.05. On [100,101], |f''(x)|=1/(4 × x^(3/2)) <= 1/4000, so the first-order Taylor error is at most (1/2)(1/4000)(1)^2=0.000125. Exact evaluation gives about 10.04987562, within this bound. Predict the sign and scale before running the code. Then change one input and explain why the result moves. The code checks the constructed example; it does not prove the rule for every possible input. Code assertions may describe the example's chosen regime, so inspect them before changing that regime. ```python import math base, change = 100.0, 1.0 assert base > 0 and base+change > 0 estimate = math.sqrt(base)+change/(2*math.sqrt(base)) exact = math.sqrt(base+change) lower_endpoint = min(base, base+change) bound = change**2/(8*lower_endpoint**1.5) print(f"Estimate {estimate:.8f}; exact {exact:.8f}; bound {bound:.8f}") assert abs(exact-estimate) <= bound+1e-14 ``` **My prediction, observed result, and explanation:** _Record your work here._ ## 4. Practise without the answers ### Exercise 1 Volume is proportional to r^3. Estimate the effect of a 1% radius increase, then check exactly. **My approach, assumptions, calculation, and check:** _Write your attempt here._ ### Exercise 2 Evaluate the integral of x^3 from -2 to 2 without finding an antiderivative. **My approach, assumptions, calculation, and check:** _Write your attempt here._ ### Exercise 3 A solver applies L'Hopital to (x+1)/(x+2) as x tends to 0 and obtains 1. Diagnose the mistake. **My approach, assumptions, calculation, and check:** _Write your attempt here._ **Coaching prompt:** “Use the Chapter 4 skill to help me with Exercise 2. Ask for my attempt, give one useful hint if I need it, and help me check the assumptions before showing the answer.” ## 5. Answer key and reasoning Read this after attempting the exercises, or use it immediately if you prefer a complete walkthrough. An answer is complete only when its assumptions and stopping point are clear. ### Answer 1 Relative differential gives 3%. Exact increase is 1.01^3-1=0.030301, or 3.0301%; the difference is 0.0301 percentage points. ### Answer 2 The integrand is odd and the domain symmetric, so the integral is exactly zero. ### Answer 3 The original limit is 1/2 by continuity; the expression has neither 0/0 nor infinity/infinity form. Differentiating numerator and denominator is not a valid transformation here. ## 6. Build the complete chapter toolkit The new lab samples the chapter; the following checklist covers all 17 rules. Study one thematic group at a time. A large group can take several sessions. - **Build Useful Local Models:** work with rules 4.1.1, 4.1.2, 4.1.3, 4.1.4. - **Read Shape from Derivatives and Convexity:** work with rules 4.2.1, 4.2.2, 4.2.3. - **Control Integrals, Series, and Method Choice:** work with rules 4.3.1, 4.3.2, 4.3.3, 4.3.4, 4.3.5, 4.3.6, 4.3.7, 4.3.8, 4.3.9, 4.3.10. For each selected rule, read its equation, explanation, and worked use in the source. Reproduce that example; change one input; then change one assumption so the rule is no longer justified. Record the result and what check catches the failure. Historical examples remain labeled and qualified as in the source. Read each complete numbered profile in the [chapter reference](../skills/math-thumb-calculus/references/chapter.md) before applying it. The cues below abbreviate the graph metadata; they are not complete conditions. Change the status only after doing the practice described below. | Rule | Book role | First assumptions to inspect | Practice status | |---|---|---|---| | 4.1.1: Linearize near a point you already understand | Independent | differentiability; nearby input | new | | 4.1.2: Interpret a Newton step as the tangent-line root | Workflow | nonzero derivative; sufficiently near simple root | new | | 4.1.3: Use relative differentials for power-law sensitivity | Independent | small relative change; nonzero positive reference for real log form | new | | 4.1.4: Turn a derivative bound into a change bound | Independent | derivative exists or function absolutely continuous; valid uniform derivative bound | new | | 4.2.1: Use the derivative sign before solving for extrema | Independent | differentiability on interval | new | | 4.2.2: Bracket a convex function between tangents and secants | Independent | convex function; points within convex domain | new | | 4.2.3: Estimate a thin shell as boundary measure times thickness | Independent | thickness small relative to curvature radius; regular boundary | new | | 4.3.1: Check an integral average against the function's range | Independent | continuous or integrable bounded function; finite interval | new | | 4.3.2: Exploit even and odd symmetry before integrating | Independent | symmetric domain; verified function symmetry | new | | 4.3.3: Differentiate an accumulated integral at its moving endpoint | Independent | continuous integrand near evaluation point; differentiable limit function | new | | 4.3.4: Memorize the p-integral convergence threshold | Independent | positive power law tail; infinite upper limit or zero endpoint as specified | new | | 4.3.5: Compare positive integrands with a known benchmark | Independent | pointwise order on common domain; integrability | new | | 4.3.6: Bracket a decreasing series tail with integrals | Workflow | positive decreasing summand; matching continuous extension | new | | 4.3.7: Look for an inner function and its derivative | Workflow | composition structure; matching derivative factor | new | | 4.3.8: Choose integration by parts to move complexity | Workflow | product structure; differentiation simplifies one factor | new | | 4.3.9: Use L'Hôpital only after confirming an indeterminate form | Workflow | zero over zero or infinity over infinity; differentiability near limit | new | | 4.3.10: Use the first omitted Taylor term as an error scale | Workflow | sufficient smoothness; next derivative control | new | For a completed row, record: **rule number / my new input / mathematical claim type / verified assumptions / calculation / check / valid use / rejected use / next step**. “Practised” means you worked an example. “Demonstrated” means you can explain a valid use, transfer it, and reject a misuse without the answer key. ## 7. Apply it to your own problem Return to your opening question. Choose the smallest rule set that can settle it. Use the book's independent/workflow/specialized classification separately from the claim type (exact, approximate, bound, diagnostic, or heuristic). - Selected rule number(s) and reason: - Assumptions that hold, fail, or remain uncertain: - Substitution with units: - Result and error, uncertainty, or bound: - Independent check or limiting case: - Decision this supports: - Stop here, do a named next calculation, or gather missing information: ## 8. Transfer and continue Useful nearby chapters: Chapter 5: analysis; Chapter 11: asymptotics; Chapter 20: numerical methods. Bring the question, units, assumptions, result type, and uncertainty to the next chapter. Choose a bridge only when it supplies an operation you actually need. **Completion check:** Explain this chapter's lab in your own words; solve one changed-input exercise; reject one invalid use; and produce a decision record for your own problem. If one check fails, revisit that part of the chapter rather than marking every rule complete.