# Chapter 11 notebook: Asymptotics: Finding What Matters at Scale **Goal:** Choose a scale estimate that survives the requested algebra and finite-input tolerance. **Start with:** Rules 11.1.3, 11.2.1, 11.3.1. 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 11](../skills/math-thumb-asymptotics/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: Limiting variable and direction; fixed versus varying parameters; dimensionless small parameter; required order; possible cancellation. - 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 main question about relative growth?** State the limit and compare ratios. Use the hierarchy only when exponents, bases, and other parameters are fixed. - **Do two terms appear capable of competing?** Equate their magnitudes, rescale at the resulting transition, and verify every omitted term afterward. - **Does “small” have physical units?** Divide by a reference scale and list all independent dimensionless groups. - **Is the object a factorial, harmonic sum, monotone sum, shrinking tail, or product?** Use the corresponding portable estimate before reaching for specialized machinery. - **Are expansions being combined?** Decide the requested order first, then propagate Big-O remainders only to that order. - **Will leading terms cancel?** Reformulate or compute another term before subtracting. - **Is a series improving and then worsening?** Treat it as potentially asymptotic and compare truncations around the least term. - **Is a smooth sum or concentrated integral still too difficult?** Escalate to Euler–Maclaurin, Laplace’s method, or steepest descent only after checking their specific regularity and geometry. **Chapter-specific stop check:** State what tends to a limit and what stays fixed. Cancellation can destroy the information retained in a leading-order estimate. ## 3. Work the lab Approximate 10! with Stirling's formula sqrt(2 × pi × n) × (n/e)^n. The first approximation is about 3,598,696 compared with 3,628,800, about 0.8296% low. Multiplying by 1+1/(12n) gives about 3,628,685, much closer. Comparing with the exact factorial verifies these finite-n errors; an asymptotic expression alone does not certify a requested tolerance. 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 n = 10 assert isinstance(n, int) and 1 <= n <= 150 exact = math.factorial(n) estimate = math.sqrt(2*math.pi*n)*(n/math.e)**n corrected = estimate*(1+1/(12*n)) print(f"Exact: {exact}; Stirling: {estimate:.3f}; corrected: {corrected:.3f}") print(f"Relative errors: {abs(estimate/exact-1):.6%}, {abs(corrected/exact-1):.6%}") ``` **My prediction, observed result, and explanation:** _Record your work here._ ## 4. Practise without the answers ### Exercise 1 Balance x^2 and epsilon × x^3 for epsilon>0. Where do the terms compete? **My approach, assumptions, calculation, and check:** _Write your attempt here._ ### Exercise 2 Estimate the scale of H_100 using log(n)+Euler's constant, then add the first endpoint correction. **My approach, assumptions, calculation, and check:** _Write your attempt here._ ### Exercise 3 Both sqrt(x^2+x) and x are asymptotic to x as x tends to positive infinity. Does their difference tend to zero? **My approach, assumptions, calculation, and check:** _Write your attempt here._ **Coaching prompt:** “Use the Chapter 11 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 For nonzero x, epsilon × |x|≈1, so |x|≈1/epsilon. The relevant dimensionless ratio is epsilon × x, not epsilon alone. ### Answer 2 log(100)+0.5772156649≈5.182385851. Adding 1/(2 × 100)=0.005 gives 5.187385851, close to H_100≈5.187377518. ### Answer 3 No. Rationalization gives x/(sqrt(x^2+x)+x)=1/(sqrt(1+1/x)+1), which tends to 1/2. Leading equivalents cannot safely be subtracted. ## 6. Build the complete chapter toolkit The new lab samples the chapter; the following checklist covers all 14 rules. Study one thematic group at a time. A large group can take several sessions. - **Regimes and Dominant Behavior:** work with rules 11.1.1, 11.1.2, 11.1.3. - **Portable Large-Scale Estimates:** work with rules 11.2.1, 11.2.2, 11.2.3, 11.2.4, 11.2.5, 11.2.6. - **Guardrails and Escalation:** work with rules 11.3.1, 11.3.2, 11.3.3, 11.3.4, 11.3.5. 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-asymptotics/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 | |---|---|---|---| | 11.1.1: Use the log-power-exponential growth hierarchy | Independent | argument tends to infinity; fixed positive exponents | new | | 11.1.2: Balance competing terms to find the transition scale | Independent | two identifiable competing terms; monotone relative magnitude near crossover | new | | 11.1.3: Express the approximation in a dimensionless small parameter | Workflow | meaningful reference scales; dimensionally homogeneous model | new | | 11.2.1: Use Stirling's formula for factorial scale | Independent | large positive integer | new | | 11.2.2: Estimate harmonic sums with a logarithm and Euler's constant | Independent | large positive integer | new | | 11.2.3: Sandwich a monotone sum with neighboring integrals | Independent | positive monotone function; matching integer samples | new | | 11.2.4: Treat a rapidly shrinking tail as its first term times a constant | Independent | eventual geometric decay; ratio magnitude below one | new | | 11.2.5: Take logs before estimating products | Workflow | positive product or consistent complex branch; logarithmic remainder control | new | | 11.2.6: Use Big-O arithmetic to keep only needed precision | Workflow | common limiting regime; valid component bounds | new | | 11.3.1: Do not subtract asymptotic equivalents blindly | Workflow | known asymptotic equivalents | new | | 11.3.2: Stop a divergent asymptotic series near its least term | Workflow | asymptotic expansion; terms initially decrease | new | | 11.3.3: Improve a sum by adding Euler-Maclaurin endpoint corrections | Specialized | sufficient smoothness; controlled endpoint derivatives | new | | 11.3.4: Find the dominant maximum in large-parameter integrals | Specialized | large parameter; isolated nondegenerate maximum | new | | 11.3.5: Follow steepest descent through a complex saddle point | Specialized | analytic phase; deformable contour | 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 1: algebra; Chapter 4: calculus; 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.