第 28103 題：複合梯形法

原題與已知條件

嚴格遞增partition、n+1 nodes與逐subinterval寬度。操作=trap-affine-exactness；明示資料={"intercept":3,"nodes":[0,1,5],"slope":-1}；請依 exact_ordered_tuple_declared_by_interface 回答。

• G001 — 嚴格遞增partition、n+1 nodes與逐subinterval寬度。操作=trap-affine-exactness；明示資料={"intercept":3,"nodes":[0,1,5],"slope":-1}；請依 exact_ordered_tuple_declared_by_interface 回答。

定義與適用範圍

沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。

• mko-trapezoidal-quadrature

逐步推導

source roles = [qf-mod50-trap-affine-exactness, trap-affine-exactness]

derived = [[slope, intercept, nodes, endpoint_values, trapezoid_value, exact_integral, exact_on_this_partition, scope]]

Trapezoid affine ORIGINAL value product0 := mul(-1, 0)

Trapezoid affine ORIGINAL value product0 = 0

Trapezoid affine ORIGINAL value plus intercept0 := sum(0, 3)

Trapezoid affine ORIGINAL value plus intercept0 = 3

Trapezoid affine ORIGINAL value product1 := mul(-1, 1)

Trapezoid affine ORIGINAL value product1 = -1

Trapezoid affine ORIGINAL value plus intercept1 := sum(-1, 3)

Trapezoid affine ORIGINAL value plus intercept1 = 2

Trapezoid affine ORIGINAL value product2 := mul(-1, 5)

Trapezoid affine ORIGINAL value product2 = -5

Trapezoid affine ORIGINAL value plus intercept2 := sum(-5, 3)

Trapezoid affine ORIGINAL value plus intercept2 = -2

Trapezoid affine partition ORIGINAL width0 := sub(1, 0)

Trapezoid affine partition ORIGINAL width0 = 1

Trapezoid affine partition ORIGINAL width1 := sub(5, 1)

Trapezoid affine partition ORIGINAL width1 = 4

Trapezoid affine partition ORIGINAL width over2 interval0 := div(1, 2)

Trapezoid affine partition ORIGINAL width over2 interval0 = 1/2

Trapezoid affine partition ORIGINAL endpoint sum interval0 := sum(3, 2)

Trapezoid affine partition ORIGINAL endpoint sum interval0 = 5

Trapezoid affine partition ORIGINAL individual trapezoid interval0 := mul(1/2, 5)

Trapezoid affine partition ORIGINAL individual trapezoid interval0 = 5/2

source roles = [Trapezoid affine partition, 0, 0, 1, 3, 2]

derived = [1, 1/2, 5, 5/2]

Trapezoid affine partition ORIGINAL width over2 interval1 := div(4, 2)

Trapezoid affine partition ORIGINAL width over2 interval1 = 2

Trapezoid affine partition ORIGINAL endpoint sum interval1 := sum(2, -2)

Trapezoid affine partition ORIGINAL endpoint sum interval1 = 0

Trapezoid affine partition ORIGINAL individual trapezoid interval1 := mul(2, 0)

Trapezoid affine partition ORIGINAL individual trapezoid interval1 = 0

source roles = [Trapezoid affine partition, 1, 1, 5, 2, -2]

derived = [4, 2, 0, 0]

Trapezoid affine partition ORIGINAL trapezoid running total0 := sum(0, 5/2)

Trapezoid affine partition ORIGINAL trapezoid running total0 = 5/2

Trapezoid affine partition ORIGINAL trapezoid running total1 := sum(5/2, 0)

Trapezoid affine partition ORIGINAL trapezoid running total1 = 5/2

source roles = [Trapezoid affine partition, [[0, 1], [1, 1], [5, 1]], [[3, 1], [2, 1], [-2, 1]]]

derived = [[0, 1, 5], [3, 2, -2], [1, 4], [5/2, 0], 5/2]

Trapezoid affine ORIGINAL slope over2 := div(-1, 2)

Trapezoid affine ORIGINAL slope over2 = -1/2

Trapezoid affine ORIGINAL right square := mul(5, 5)

Trapezoid affine ORIGINAL right square = 25

Trapezoid affine ORIGINAL left square := mul(0, 0)

Trapezoid affine ORIGINAL left square = 0

Trapezoid affine ORIGINAL square difference := sub(25, 0)

Trapezoid affine ORIGINAL square difference = 25

Trapezoid affine ORIGINAL quadratic integral part := mul(-1/2, 25)

Trapezoid affine ORIGINAL quadratic integral part = -25/2

Trapezoid affine ORIGINAL whole width := sub(5, 0)

Trapezoid affine ORIGINAL whole width = 5

Trapezoid affine ORIGINAL intercept integral part := mul(3, 5)

Trapezoid affine ORIGINAL intercept integral part = 15

Trapezoid affine ORIGINAL exact integral := sum(-25/2, 15)

Trapezoid affine ORIGINAL exact integral = 5/2

source roles = [-1, 3, [0, 1, 5], [3, 2, -2]]

derived = [5/2, 5/2, 是]

逐步反向核對

• S029 · premise_and_result_audit — 回查完整來源資料、原操作數及本步條件；有限核對、條件式理由、形式證明與發布資格分開。
• S028 · premise_and_result_audit — 以精確有理數重算 Trapezoid affine ORIGINAL exact integral，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 Trapezoid affine ORIGINAL intercept integral part，不以小數近似替代。
• S026 · premise_and_result_audit — 以精確有理數重算 Trapezoid affine ORIGINAL whole width，不以小數近似替代。
• S025 · premise_and_result_audit — 以精確有理數重算 Trapezoid affine ORIGINAL quadratic integral part，不以小數近似替代。
• S024 · premise_and_result_audit — 以精確有理數重算 Trapezoid affine ORIGINAL square difference，不以小數近似替代。
• S023 · premise_and_result_audit — 以精確有理數重算 Trapezoid affine ORIGINAL left square，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 Trapezoid affine ORIGINAL right square，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 Trapezoid affine ORIGINAL slope over2，不以小數近似替代。
• S020 · premise_and_result_audit — 回查完整來源資料、原操作數及本步條件；有限核對、條件式理由、形式證明與發布資格分開。
• S019 · premise_and_result_audit — 以精確有理數重算 Trapezoid affine partition ORIGINAL trapezoid running total1，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 Trapezoid affine partition ORIGINAL trapezoid running total0，不以小數近似替代。
• S017 · premise_and_result_audit — 回查完整來源資料、原操作數及本步條件；有限核對、條件式理由、形式證明與發布資格分開。
• S016 · premise_and_result_audit — 以精確有理數重算 Trapezoid affine partition ORIGINAL individual trapezoid interval1，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 Trapezoid affine partition ORIGINAL endpoint sum interval1，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 Trapezoid affine partition ORIGINAL width over2 interval1，不以小數近似替代。
• S013 · premise_and_result_audit — 回查完整來源資料、原操作數及本步條件；有限核對、條件式理由、形式證明與發布資格分開。
• S012 · premise_and_result_audit — 以精確有理數重算 Trapezoid affine partition ORIGINAL individual trapezoid interval0，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 Trapezoid affine partition ORIGINAL endpoint sum interval0，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 Trapezoid affine partition ORIGINAL width over2 interval0，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 Trapezoid affine partition ORIGINAL width1，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 Trapezoid affine partition ORIGINAL width0，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 Trapezoid affine ORIGINAL value plus intercept2，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 Trapezoid affine ORIGINAL value product2，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 Trapezoid affine ORIGINAL value plus intercept1，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 Trapezoid affine ORIGINAL value product1，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 Trapezoid affine ORIGINAL value plus intercept0，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 Trapezoid affine ORIGINAL value product0，不以小數近似替代。
• S001 · premise_and_result_audit — 回查完整來源資料、原操作數及本步條件；有限核對、條件式理由、形式證明與發布資格分開。

結論

(-1, 3, [0, 1, 5], [3, 2, -2], 5/2, 5/2, 是, affine_exactness_on_this_partition_does_not_imply_all_functions_are_exact)

依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。