# Question 24024: Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial [Return to catalogue page 121](https://ocme.evemisslab.com/en/questions/page/121/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261112-r02-taylor-polynomial-foundations/#q-20261014-advance-20261112-r02-taylor-polynomial-0049) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0047-24024 · q-20261014-advance-20261112-r02-taylor-polynomial-0049 ## Original question and givens Use the derivative values=[24, 48, 72, 96, 120, 144] at the fixed center=11, through declared order=5≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate). - G001 — Use the derivative values=[24, 48, 72, 96, 120, 144] at the fixed center=11, through declared order=5≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate). ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-taylor-polynomial](https://ocme.evemisslab.com/en/mko/mko-taylor-polynomial/) ## Step-by-step derivation S001 · impliesBefore source roles = [taylor-coefficient-construction] After derived = [[center, order, shifted_coefficients, indexed_coefficients, certificate]] Conditions: none List the source field order without JavaScript.Gradient boundary uses its original mode to choose6,8 or7 fields;no padding or answer-derived mode. Step identifiers and dependenciesRule: source_ordered_fields_without_JS · Uses: G001 S002 · impliesBefore source roles = [11, 5, [24, 48, 72, 96, 120, 144]] After derived = [[0, 1, 2, 3, 4, 5], 6, Yes] Conditions: none The data are assumed derivatives of the same function at the same a.A numeric list does not verify differentiability of an unknown function.Order0 is f(a). Step identifiers and dependenciesRule: Taylor_assumed_complete_derivative_jet_not_differentiability_test · Uses: G001, S001 S003 · impliesBefore source roles = [0] After derived = [1] Conditions: none The empty product0! is1,not0. Step identifiers and dependenciesRule: empty_factorial_is_multiplicative_unit · Uses: G001, S002 S004 · calculationBefore Taylor coefficient0 derivative over factorial := div(24, 1) After Taylor coefficient0 derivative over factorial = 24 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Taylor k1 factorial step1 := mul(1, 1) After Taylor k1 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Taylor coefficient1 derivative over factorial := div(48, 1) After Taylor coefficient1 derivative over factorial = 48 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore Taylor k2 factorial step1 := mul(1, 1) After Taylor k2 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore Taylor k2 factorial step2 := mul(1, 2) After Taylor k2 factorial step2 = 2 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore Taylor coefficient2 derivative over factorial := div(72, 2) After Taylor coefficient2 derivative over factorial = 36 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore Taylor k3 factorial step1 := mul(1, 1) After Taylor k3 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore Taylor k3 factorial step2 := mul(1, 2) After Taylor k3 factorial step2 = 2 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore Taylor k3 factorial step3 := mul(2, 3) After Taylor k3 factorial step3 = 6 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore Taylor coefficient3 derivative over factorial := div(96, 6) After Taylor coefficient3 derivative over factorial = 16 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore Taylor k4 factorial step1 := mul(1, 1) After Taylor k4 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore Taylor k4 factorial step2 := mul(1, 2) After Taylor k4 factorial step2 = 2 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore Taylor k4 factorial step3 := mul(2, 3) After Taylor k4 factorial step3 = 6 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore Taylor k4 factorial step4 := mul(6, 4) After Taylor k4 factorial step4 = 24 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore Taylor coefficient4 derivative over factorial := div(120, 24) After Taylor coefficient4 derivative over factorial = 5 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore Taylor k5 factorial step1 := mul(1, 1) After Taylor k5 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore Taylor k5 factorial step2 := mul(1, 2) After Taylor k5 factorial step2 = 2 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore Taylor k5 factorial step3 := mul(2, 3) After Taylor k5 factorial step3 = 6 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore Taylor k5 factorial step4 := mul(6, 4) After Taylor k5 factorial step4 = 24 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore Taylor k5 factorial step5 := mul(24, 5) After Taylor k5 factorial step5 = 120 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · calculationBefore Taylor coefficient5 derivative over factorial := div(144, 120) After Taylor coefficient5 derivative over factorial = 6/5 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · impliesBefore source roles = [11, 5, [24, 48, 36, 16, 5, 6/5]] After derived = [T_n(x)=sum c_k*(x-center)^k, T_n^(j)(center)=declared derivative[j] for EVERY0<=j<=n, no remainder or infinite-series certificate] Conditions: none For each j,terms kj still contain a positive power at a;only k=j yields j!*c_j.This matches the full jet.Order0 retains f(a),without automatically identifying the original function or bounding a remainder. Step identifiers and dependenciesRule: Taylor_shifted_basis_matches_derivatives_ALL_orders · Uses: G001, S024 ## Reverse checks - S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S024 · premise_and_result_audit — Recompute Taylor coefficient5 derivative over factorial with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute Taylor k5 factorial step5 with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute Taylor k5 factorial step4 with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute Taylor k5 factorial step3 with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute Taylor k5 factorial step2 with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute Taylor k5 factorial step1 with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute Taylor coefficient4 derivative over factorial with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute Taylor k4 factorial step4 with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute Taylor k4 factorial step3 with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute Taylor k4 factorial step2 with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute Taylor k4 factorial step1 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute Taylor coefficient3 derivative over factorial with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute Taylor k3 factorial step3 with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute Taylor k3 factorial step2 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute Taylor k3 factorial step1 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute Taylor coefficient2 derivative over factorial with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute Taylor k2 factorial step2 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute Taylor k2 factorial step1 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute Taylor coefficient1 derivative over factorial with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute Taylor k1 factorial step1 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Taylor coefficient0 derivative over factorial with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result (11, 5, [24, 48, 36, 16, 5, 6/5], [[0, 24], [1, 48], [2, 36], [3, 16], [4, 5], [5, 6/5]], coefficient_k_equals_derivative_k_over_k_factorial) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 527a19cdcca8a94985653c66462b18bdaf611e0b14b37271bdaa38516a7c875b - candidate record SHA-256: 64fd383b30b681bab0bfb182b1fcbbe821142bfb3dd5a2885ecfc8a15ad48034 - semantic review receipt SHA-256: 9b43b9823e9bd2478d6caa5b57ffed161c2c8956af86974e901d5662ad994386 - question-first witness SHA-256: eb97596068e2d5648033bab2b8f6ca9b653bbe8cb77c0f8e1d16d0c40e46f50b - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261112-r02-taylor-polynomial-0049.json)