第 26447 題：Taylor餘項定理

原題與已知條件

center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-lagrange-witness；資料={"coefficients":[6,11,16,23],"center":0,"x":[1,2],"order":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-lagrange-witness；資料={"coefficients":[6,11,16,23],"center":0,"x":[1,2],"order":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

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-taylor-remainder-theorem

逐步推導

source roles = [taylor-lagrange-witness]

derived = [[degree, order, constant_next_derivative, intermediate_point, strictly_between, Lagrange_remainder, actual_remainder, remainders_equal, certificate]]

Taylor ORIGINAL x minus declared center := sub(1/2, 0)

Taylor ORIGINAL x minus declared center = 1/2

(0)^0

Taylor ORIGINAL derivative0 center power0 term0 = 1

Taylor ORIGINAL derivative0 falling factor times power term0 := mul(1, 1)

Taylor ORIGINAL derivative0 falling factor times power term0 = 1

Taylor ORIGINAL derivative0 ORIGINAL coefficient term0 := mul(6, 1)

Taylor ORIGINAL derivative0 ORIGINAL coefficient term0 = 6

Taylor ORIGINAL derivative0 ascending derivative sum term0 := sum(0, 6)

Taylor ORIGINAL derivative0 ascending derivative sum term0 = 6

(0)^1

Taylor ORIGINAL derivative0 center power1 term1 = 0

Taylor ORIGINAL derivative0 falling factor times power term1 := mul(1, 0)

Taylor ORIGINAL derivative0 falling factor times power term1 = 0

Taylor ORIGINAL derivative0 ORIGINAL coefficient term1 := mul(11, 0)

Taylor ORIGINAL derivative0 ORIGINAL coefficient term1 = 0

Taylor ORIGINAL derivative0 ascending derivative sum term1 := sum(6, 0)

Taylor ORIGINAL derivative0 ascending derivative sum term1 = 6

(0)^2

Taylor ORIGINAL derivative0 center power2 term2 = 0

Taylor ORIGINAL derivative0 falling factor times power term2 := mul(1, 0)

Taylor ORIGINAL derivative0 falling factor times power term2 = 0

Taylor ORIGINAL derivative0 ORIGINAL coefficient term2 := mul(16, 0)

Taylor ORIGINAL derivative0 ORIGINAL coefficient term2 = 0

Taylor ORIGINAL derivative0 ascending derivative sum term2 := sum(6, 0)

Taylor ORIGINAL derivative0 ascending derivative sum term2 = 6

(0)^3

Taylor ORIGINAL derivative0 center power3 term3 = 0

Taylor ORIGINAL derivative0 falling factor times power term3 := mul(1, 0)

Taylor ORIGINAL derivative0 falling factor times power term3 = 0

Taylor ORIGINAL derivative0 ORIGINAL coefficient term3 := mul(23, 0)

Taylor ORIGINAL derivative0 ORIGINAL coefficient term3 = 0

Taylor ORIGINAL derivative0 ascending derivative sum term3 := sum(6, 0)

Taylor ORIGINAL derivative0 ascending derivative sum term3 = 6

source roles = [0, 0, [6, 11, 16, 23]]

derived = [[[0, 6, 1, 1, 6], [1, 11, 1, 0, 0], [2, 16, 1, 0, 0], [3, 23, 1, 0, 0]], 6]

(1/2)^0

Taylor ORIGINAL displacement power0 = 1

Taylor ORIGINAL derivative times displacement0 := mul(6, 1)

Taylor ORIGINAL derivative times displacement0 = 6

Taylor ORIGINAL term0 divide factorial := div(6, 1)

Taylor ORIGINAL term0 divide factorial = 6

(0)^0

Taylor ORIGINAL derivative1 center power0 term1 = 1

Taylor ORIGINAL derivative1 falling factor times power term1 := mul(1, 1)

Taylor ORIGINAL derivative1 falling factor times power term1 = 1

Taylor ORIGINAL derivative1 ORIGINAL coefficient term1 := mul(11, 1)

Taylor ORIGINAL derivative1 ORIGINAL coefficient term1 = 11

Taylor ORIGINAL derivative1 ascending derivative sum term1 := sum(0, 11)

Taylor ORIGINAL derivative1 ascending derivative sum term1 = 11

(0)^1

Taylor ORIGINAL derivative1 center power1 term2 = 0

Taylor ORIGINAL derivative1 falling factor times power term2 := mul(2, 0)

Taylor ORIGINAL derivative1 falling factor times power term2 = 0

Taylor ORIGINAL derivative1 ORIGINAL coefficient term2 := mul(16, 0)

Taylor ORIGINAL derivative1 ORIGINAL coefficient term2 = 0

Taylor ORIGINAL derivative1 ascending derivative sum term2 := sum(11, 0)

Taylor ORIGINAL derivative1 ascending derivative sum term2 = 11

(0)^2

Taylor ORIGINAL derivative1 center power2 term3 = 0

Taylor ORIGINAL derivative1 falling factor times power term3 := mul(3, 0)

Taylor ORIGINAL derivative1 falling factor times power term3 = 0

Taylor ORIGINAL derivative1 ORIGINAL coefficient term3 := mul(23, 0)

Taylor ORIGINAL derivative1 ORIGINAL coefficient term3 = 0

Taylor ORIGINAL derivative1 ascending derivative sum term3 := sum(11, 0)

Taylor ORIGINAL derivative1 ascending derivative sum term3 = 11

source roles = [1, 0, [6, 11, 16, 23]]

derived = [[[1, 11, 1, 1, 11], [2, 16, 2, 0, 0], [3, 23, 3, 0, 0]], 11]

(1/2)^1

Taylor ORIGINAL displacement power1 = 1/2

Taylor ORIGINAL derivative times displacement1 := mul(11, 1/2)

Taylor ORIGINAL derivative times displacement1 = 11/2

Taylor ORIGINAL term1 divide factorial := div(11/2, 1)

Taylor ORIGINAL term1 divide factorial = 11/2

(0)^0

Taylor ORIGINAL derivative2 center power0 term2 = 1

Taylor ORIGINAL derivative2 falling factor times power term2 := mul(2, 1)

Taylor ORIGINAL derivative2 falling factor times power term2 = 2

Taylor ORIGINAL derivative2 ORIGINAL coefficient term2 := mul(16, 2)

Taylor ORIGINAL derivative2 ORIGINAL coefficient term2 = 32

Taylor ORIGINAL derivative2 ascending derivative sum term2 := sum(0, 32)

Taylor ORIGINAL derivative2 ascending derivative sum term2 = 32

(0)^1

Taylor ORIGINAL derivative2 center power1 term3 = 0

Taylor ORIGINAL derivative2 falling factor times power term3 := mul(6, 0)

Taylor ORIGINAL derivative2 falling factor times power term3 = 0

Taylor ORIGINAL derivative2 ORIGINAL coefficient term3 := mul(23, 0)

Taylor ORIGINAL derivative2 ORIGINAL coefficient term3 = 0

Taylor ORIGINAL derivative2 ascending derivative sum term3 := sum(32, 0)

Taylor ORIGINAL derivative2 ascending derivative sum term3 = 32

source roles = [2, 0, [6, 11, 16, 23]]

derived = [[[2, 16, 2, 1, 32], [3, 23, 6, 0, 0]], 32]

(1/2)^2

Taylor ORIGINAL displacement power2 = 1/4

Taylor ORIGINAL derivative times displacement2 := mul(32, 1/4)

Taylor ORIGINAL derivative times displacement2 = 8

Taylor ORIGINAL term2 divide factorial := div(8, 2)

Taylor ORIGINAL term2 divide factorial = 4

Taylor ORIGINAL partial sum0 := sum(0, 6)

Taylor ORIGINAL partial sum0 = 6

Taylor ORIGINAL partial sum1 := sum(6, 11/2)

Taylor ORIGINAL partial sum1 = 23/2

Taylor ORIGINAL partial sum2 := sum(23/2, 4)

Taylor ORIGINAL partial sum2 = 31/2

(1/2)^0

Taylor ORIGINAL full function at x ORIGINAL argument power0 = 1

Taylor ORIGINAL full function at x coefficient term0 := mul(6, 1)

Taylor ORIGINAL full function at x coefficient term0 = 6

Taylor ORIGINAL full function at x ascending partial sum0 := sum(0, 6)

Taylor ORIGINAL full function at x ascending partial sum0 = 6

(1/2)^1

Taylor ORIGINAL full function at x ORIGINAL argument power1 = 1/2

Taylor ORIGINAL full function at x coefficient term1 := mul(11, 1/2)

Taylor ORIGINAL full function at x coefficient term1 = 11/2

Taylor ORIGINAL full function at x ascending partial sum1 := sum(6, 11/2)

Taylor ORIGINAL full function at x ascending partial sum1 = 23/2

(1/2)^2

Taylor ORIGINAL full function at x ORIGINAL argument power2 = 1/4

Taylor ORIGINAL full function at x coefficient term2 := mul(16, 1/4)

Taylor ORIGINAL full function at x coefficient term2 = 4

Taylor ORIGINAL full function at x ascending partial sum2 := sum(23/2, 4)

Taylor ORIGINAL full function at x ascending partial sum2 = 31/2

(1/2)^3

Taylor ORIGINAL full function at x ORIGINAL argument power3 = 1/8

Taylor ORIGINAL full function at x coefficient term3 := mul(23, 1/8)

Taylor ORIGINAL full function at x coefficient term3 = 23/8

Taylor ORIGINAL full function at x ascending partial sum3 := sum(31/2, 23/8)

Taylor ORIGINAL full function at x ascending partial sum3 = 147/8

Taylor ORIGINAL f(x) minus Taylor value := sub(147/8, 31/2)

Taylor ORIGINAL f(x) minus Taylor value = 23/8

source roles = [[6, 11, 16, 23], 0, 1/2, 2, [6, 11, 32], [6, 11/2, 4]]

derived = [31/2, 147/8, 23/8, 3]

(0)^0

Taylor ORIGINAL next derivative center power0 term3 = 1

Taylor ORIGINAL next derivative falling factor times power term3 := mul(6, 1)

Taylor ORIGINAL next derivative falling factor times power term3 = 6

Taylor ORIGINAL next derivative ORIGINAL coefficient term3 := mul(23, 6)

Taylor ORIGINAL next derivative ORIGINAL coefficient term3 = 138

Taylor ORIGINAL next derivative ascending derivative sum term3 := sum(0, 138)

Taylor ORIGINAL next derivative ascending derivative sum term3 = 138

source roles = [3, 0, [6, 11, 16, 23]]

derived = [[[3, 23, 6, 1, 138]], 138]

Taylor witness ORIGINAL center plus x := sum(0, 1/2)

Taylor witness ORIGINAL center plus x = 1/2

Taylor witness ORIGINAL midpoint := div(1/2, 2)

Taylor witness ORIGINAL midpoint = 1/4

(1/2)^3

Taylor witness ORIGINAL displacement power = 1/8

Taylor witness constant derivative times power := mul(138, 1/8)

Taylor witness constant derivative times power = 69/4

Taylor witness divide next factorial := div(69/4, 6)

Taylor witness divide next factorial = 23/8

source roles = [3, 2, 0, 1/2, 138, 1/4]

derived = [是, 23/8, 23/8, 是]

逐步反向核對

• S078 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S077 · premise_and_result_audit — 以精確有理數重算 Taylor witness divide next factorial，不以小數近似替代。
• S076 · premise_and_result_audit — 以精確有理數重算 Taylor witness constant derivative times power，不以小數近似替代。
• S075 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S074 · premise_and_result_audit — 以精確有理數重算 Taylor witness ORIGINAL midpoint，不以小數近似替代。
• S073 · premise_and_result_audit — 以精確有理數重算 Taylor witness ORIGINAL center plus x，不以小數近似替代。
• S072 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S071 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL next derivative ascending derivative sum term3，不以小數近似替代。
• S070 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL next derivative ORIGINAL coefficient term3，不以小數近似替代。
• S069 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL next derivative falling factor times power term3，不以小數近似替代。
• S068 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S067 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S066 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL f(x) minus Taylor value，不以小數近似替代。
• S065 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL full function at x ascending partial sum3，不以小數近似替代。
• S064 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL full function at x coefficient term3，不以小數近似替代。
• S063 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S062 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL full function at x ascending partial sum2，不以小數近似替代。
• S061 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL full function at x coefficient term2，不以小數近似替代。
• S060 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S059 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL full function at x ascending partial sum1，不以小數近似替代。
• S058 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL full function at x coefficient term1，不以小數近似替代。
• S057 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S056 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL full function at x ascending partial sum0，不以小數近似替代。
• S055 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL full function at x coefficient term0，不以小數近似替代。
• S054 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S053 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL partial sum2，不以小數近似替代。
• S052 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL partial sum1，不以小數近似替代。
• S051 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL partial sum0，不以小數近似替代。
• S050 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL term2 divide factorial，不以小數近似替代。
• S049 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative times displacement2，不以小數近似替代。
• S048 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S047 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S046 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative2 ascending derivative sum term3，不以小數近似替代。
• S045 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative2 ORIGINAL coefficient term3，不以小數近似替代。
• S044 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative2 falling factor times power term3，不以小數近似替代。
• S043 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S042 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative2 ascending derivative sum term2，不以小數近似替代。
• S041 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative2 ORIGINAL coefficient term2，不以小數近似替代。
• S040 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative2 falling factor times power term2，不以小數近似替代。
• S039 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S038 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL term1 divide factorial，不以小數近似替代。
• S037 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative times displacement1，不以小數近似替代。
• S036 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S035 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S034 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative1 ascending derivative sum term3，不以小數近似替代。
• S033 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative1 ORIGINAL coefficient term3，不以小數近似替代。
• S032 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative1 falling factor times power term3，不以小數近似替代。
• S031 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S030 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative1 ascending derivative sum term2，不以小數近似替代。
• S029 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative1 ORIGINAL coefficient term2，不以小數近似替代。
• S028 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative1 falling factor times power term2，不以小數近似替代。
• S027 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S026 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative1 ascending derivative sum term1，不以小數近似替代。
• S025 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative1 ORIGINAL coefficient term1，不以小數近似替代。
• S024 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative1 falling factor times power term1，不以小數近似替代。
• S023 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S022 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL term0 divide factorial，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative times displacement0，不以小數近似替代。
• S020 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S019 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S018 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative0 ascending derivative sum term3，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative0 ORIGINAL coefficient term3，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative0 falling factor times power term3，不以小數近似替代。
• S015 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S014 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative0 ascending derivative sum term2，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative0 ORIGINAL coefficient term2，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative0 falling factor times power term2，不以小數近似替代。
• S011 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S010 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative0 ascending derivative sum term1，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative0 ORIGINAL coefficient term1，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative0 falling factor times power term1，不以小數近似替代。
• S007 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S006 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative0 ascending derivative sum term0，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative0 ORIGINAL coefficient term0，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative0 falling factor times power term0，不以小數近似替代。
• S003 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S002 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL x minus declared center，不以小數近似替代。
• S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。

結論

(3, 2, 138, 1/4, 是, 23/8, 23/8, 是, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)

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。