# Question 28109: An Affine Integrand Is Integrated Exactly on Any Qualified Partition [Return to catalogue page 141](https://ocme.evemisslab.com/en/questions/page/141/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations/#q-20261015-pulse-20261119-r02-0059) 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. R0055-28109 · q-20261015-pulse-20261119-r02-0059 ## Original question and givens Use f(x)=m*x+b with m=-2, b=9 and strictly increasing nodes=[0, 1, 11]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope). - G001 — Use f(x)=m*x+b with m=-2, b=9 and strictly increasing nodes=[0, 1, 11]. Give (slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope). ## 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-trapezoidal-quadrature](https://ocme.evemisslab.com/en/mko/mko-trapezoidal-quadrature/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-mod50-trap-affine-exactness, trap-affine-exactness] After derived = [[slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope]] Conditions: none List this scalar/label/set/point answer role or mode-specific ordered fields,not an imposed universal tuple.Retain legacy domain/underscore operation and integer fractions with/1;original bilingual snapshots stay unchanged. Step identifiers and dependenciesRule: source_answer_roles_or_ordered_fields_without_JS · Uses: G001 S002 · calculationBefore Trapezoid affine ORIGINAL value product0 := mul(-2, 0) After Trapezoid affine ORIGINAL value product0 = 0 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore Trapezoid affine ORIGINAL value plus intercept0 := sum(0, 9) After Trapezoid affine ORIGINAL value plus intercept0 = 9 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Trapezoid affine ORIGINAL value product1 := mul(-2, 1) After Trapezoid affine ORIGINAL value product1 = -2 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Trapezoid affine ORIGINAL value plus intercept1 := sum(-2, 9) After Trapezoid affine ORIGINAL value plus intercept1 = 7 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Trapezoid affine ORIGINAL value product2 := mul(-2, 11) After Trapezoid affine ORIGINAL value product2 = -22 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore Trapezoid affine ORIGINAL value plus intercept2 := sum(-22, 9) After Trapezoid affine ORIGINAL value plus intercept2 = -13 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore Trapezoid affine partition ORIGINAL width0 := sub(1, 0) After Trapezoid affine partition ORIGINAL width0 = 1 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore Trapezoid affine partition ORIGINAL width1 := sub(11, 1) After Trapezoid affine partition ORIGINAL width1 = 10 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore Trapezoid affine partition ORIGINAL width over2 interval0 := div(1, 2) After Trapezoid affine partition ORIGINAL width over2 interval0 = 1/2 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore Trapezoid affine partition ORIGINAL endpoint sum interval0 := sum(9, 7) After Trapezoid affine partition ORIGINAL endpoint sum interval0 = 16 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore Trapezoid affine partition ORIGINAL individual trapezoid interval0 := mul(1/2, 16) After Trapezoid affine partition ORIGINAL individual trapezoid interval0 = 8 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · impliesBefore source roles = [Trapezoid affine partition, 0, 0, 1, 9, 7] After derived = [1, 1/2, 16, 8] Conditions: none Use the own width and aligned endpoints of this ORIGINAL interval times the endpoint average.Widths may differ and values may be negative;do not sort or replace all widths with the first. Step identifiers and dependenciesRule: Trapezoid_EACH_original_interval_own_width_and_aligned_endpoints · Uses: G001, S012 S014 · calculationBefore Trapezoid affine partition ORIGINAL width over2 interval1 := div(10, 2) After Trapezoid affine partition ORIGINAL width over2 interval1 = 5 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore Trapezoid affine partition ORIGINAL endpoint sum interval1 := sum(7, -13) After Trapezoid affine partition ORIGINAL endpoint sum interval1 = -6 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore Trapezoid affine partition ORIGINAL individual trapezoid interval1 := mul(5, -6) After Trapezoid affine partition ORIGINAL individual trapezoid interval1 = -30 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · impliesBefore source roles = [Trapezoid affine partition, 1, 1, 11, 7, -13] After derived = [10, 5, -6, -30] Conditions: none Use the own width and aligned endpoints of this ORIGINAL interval times the endpoint average.Widths may differ and values may be negative;do not sort or replace all widths with the first. Step identifiers and dependenciesRule: Trapezoid_EACH_original_interval_own_width_and_aligned_endpoints · Uses: G001, S016 S018 · calculationBefore Trapezoid affine partition ORIGINAL trapezoid running total0 := sum(0, 8) After Trapezoid affine partition ORIGINAL trapezoid running total0 = 8 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore Trapezoid affine partition ORIGINAL trapezoid running total1 := sum(8, -30) After Trapezoid affine partition ORIGINAL trapezoid running total1 = -22 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · impliesBefore source roles = [Trapezoid affine partition, [[0, 1], [1, 1], [11, 1]], [[9, 1], [7, 1], [-13, 1]]] After derived = [[0, 1, 11], [9, 7, -13], [1, 10], [8, -30], -22] Conditions: none A complete partition is STRICTLY increasing in original order,with n+1 aligned nodes/values for n intervals;the24-node cap is not the vector8 cap.The table determines this sum,not a unique integral without a full integrand. Step identifiers and dependenciesRule: Trapezoid_COMPLETE_partition_n_intervals_nplus1_nodes_no_unique_integrand · Uses: G001, S019 S021 · calculationBefore Trapezoid affine ORIGINAL slope over2 := div(-2, 2) After Trapezoid affine ORIGINAL slope over2 = -1 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore Trapezoid affine ORIGINAL right square := mul(11, 11) After Trapezoid affine ORIGINAL right square = 121 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore Trapezoid affine ORIGINAL left square := mul(0, 0) After Trapezoid affine ORIGINAL left square = 0 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · calculationBefore Trapezoid affine ORIGINAL square difference := sub(121, 0) After Trapezoid affine ORIGINAL square difference = 121 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · calculationBefore Trapezoid affine ORIGINAL quadratic integral part := mul(-1, 121) After Trapezoid affine ORIGINAL quadratic integral part = -121 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · calculationBefore Trapezoid affine ORIGINAL whole width := sub(11, 0) After Trapezoid affine ORIGINAL whole width = 11 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025 S027 · calculationBefore Trapezoid affine ORIGINAL intercept integral part := mul(9, 11) After Trapezoid affine ORIGINAL intercept integral part = 99 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026 S028 · calculationBefore Trapezoid affine ORIGINAL exact integral := sum(-121, 99) After Trapezoid affine ORIGINAL exact integral = -22 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027 S029 · impliesBefore source roles = [-2, 9, [0, 1, 11], [9, 7, -13]] After derived = [-22, -22, Yes] Conditions: none The WHOLE affine law makes each trapezoid equal to the antiderivative difference over the same interval.Internal endpoints telescope,hence exactness on any qualified partition;not for arbitrary functions. Step identifiers and dependenciesRule: Trapezoid_WHOLE_affine_exactness_each_interval_antiderivative_telescopes · Uses: G001, S028 ## Reverse checks - S029 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct. - S028 · premise_and_result_audit — Recompute Trapezoid affine ORIGINAL exact integral with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Recompute Trapezoid affine ORIGINAL intercept integral part with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Recompute Trapezoid affine ORIGINAL whole width with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute Trapezoid affine ORIGINAL quadratic integral part with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Recompute Trapezoid affine ORIGINAL square difference with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute Trapezoid affine ORIGINAL left square with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute Trapezoid affine ORIGINAL right square with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute Trapezoid affine ORIGINAL slope over2 with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct. - S019 · premise_and_result_audit — Recompute Trapezoid affine partition ORIGINAL trapezoid running total1 with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute Trapezoid affine partition ORIGINAL trapezoid running total0 with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct. - S016 · premise_and_result_audit — Recompute Trapezoid affine partition ORIGINAL individual trapezoid interval1 with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute Trapezoid affine partition ORIGINAL endpoint sum interval1 with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute Trapezoid affine partition ORIGINAL width over2 interval1 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct. - S012 · premise_and_result_audit — Recompute Trapezoid affine partition ORIGINAL individual trapezoid interval0 with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute Trapezoid affine partition ORIGINAL endpoint sum interval0 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute Trapezoid affine partition ORIGINAL width over2 interval0 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute Trapezoid affine partition ORIGINAL width1 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute Trapezoid affine partition ORIGINAL width0 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute Trapezoid affine ORIGINAL value plus intercept2 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute Trapezoid affine ORIGINAL value product2 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute Trapezoid affine ORIGINAL value plus intercept1 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Trapezoid affine ORIGINAL value product1 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Trapezoid affine ORIGINAL value plus intercept0 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute Trapezoid affine ORIGINAL value product0 with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct. ## Result (-2, 9, [0, 1, 11], [9, 7, -13], -22, -22, Yes, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact) 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: 17260127694e41396d234f7eb0dfbc6cdaeb32cebb379aa801a14a36b5e84e8d - candidate record SHA-256: c524d227a6c11e35cb6509bc74825cbd1fb9936f1c80d4faf1b08c88b5a1c03c - semantic review receipt SHA-256: 6af367dae24a584f3596525c4953bbc2115343efd68cc9083693eb5c1ef2e683 - question-first witness SHA-256: 611eb69f3716404b968296a8b0d7d8c61b5ca18129d406f1ff9c7c11e10ffb4e - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-pulse-20261119-r02-0059.json)