Taylor多項式基礎 · 脈衝 R02
本批已通過 Schema、去重、來源快照與獨立答案重算；仍未標記為人工審定。
本題庫已有對應的中英逐步詳解。原題的 candidate 與英文 ai_candidate 狀態保留；AI 覆核、有限計算與 Lean 形式證明分開標示。
batch-20261014-advance-20261112-r02-taylor-polynomial-foundations · 2026-10-14 · 100 題
欄序為來源綁定的呈現補充；原始 JSONL 與數學答案不變。
欄序與來源綁定資料
1. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-jet-qualification；資料={"derivatives":[1,2,3],"center":-12,"order":2}。請依 [center, order, derivatives, jet_length, complete_jet, certificate] 欄序回答。
查看答案與解釋
(-12, 2, [1, 2, 3], 3, 是, complete_zero_through_n_derivative_jet)
Wave6第六層核對=(-12, 2, [1, 2, 3], 3, 是, complete_zero_through_n_derivative_jet)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-jet-qualification · candidate
2. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-jet-qualification；資料={"derivatives":[2,4,6,8],"center":-11,"order":3}。請依 [center, order, derivatives, jet_length, complete_jet, certificate] 欄序回答。
查看答案與解釋
(-11, 3, [2, 4, 6, 8], 4, 是, complete_zero_through_n_derivative_jet)
Wave6第六層核對=(-11, 3, [2, 4, 6, 8], 4, 是, complete_zero_through_n_derivative_jet)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-jet-qualification · candidate
3. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-jet-qualification；資料={"derivatives":[3,6,9,12,15],"center":-10,"order":4}。請依 [center, order, derivatives, jet_length, complete_jet, certificate] 欄序回答。
查看答案與解釋
(-10, 4, [3, 6, 9, 12, 15], 5, 是, complete_zero_through_n_derivative_jet)
Wave6第六層核對=(-10, 4, [3, 6, 9, 12, 15], 5, 是, complete_zero_through_n_derivative_jet)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-jet-qualification · candidate
4. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-jet-qualification；資料={"derivatives":[4,8,12,16,20,24],"center":-9,"order":5}。請依 [center, order, derivatives, jet_length, complete_jet, certificate] 欄序回答。
查看答案與解釋
(-9, 5, [4, 8, 12, 16, 20, 24], 6, 是, complete_zero_through_n_derivative_jet)
Wave6第六層核對=(-9, 5, [4, 8, 12, 16, 20, 24], 6, 是, complete_zero_through_n_derivative_jet)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-jet-qualification · candidate
5. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-jet-qualification；資料={"derivatives":[5,10,15],"center":-8,"order":2}。請依 [center, order, derivatives, jet_length, complete_jet, certificate] 欄序回答。
查看答案與解釋
(-8, 2, [5, 10, 15], 3, 是, complete_zero_through_n_derivative_jet)
Wave6第六層核對=(-8, 2, [5, 10, 15], 3, 是, complete_zero_through_n_derivative_jet)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-jet-qualification · candidate
6. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-jet-qualification；資料={"derivatives":[6,12,18,24],"center":-7,"order":3}。請依 [center, order, derivatives, jet_length, complete_jet, certificate] 欄序回答。
查看答案與解釋
(-7, 3, [6, 12, 18, 24], 4, 是, complete_zero_through_n_derivative_jet)
Wave6第六層核對=(-7, 3, [6, 12, 18, 24], 4, 是, complete_zero_through_n_derivative_jet)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-jet-qualification · candidate
7. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-jet-qualification；資料={"derivatives":[7,14,21,28,35],"center":-6,"order":4}。請依 [center, order, derivatives, jet_length, complete_jet, certificate] 欄序回答。
查看答案與解釋
(-6, 4, [7, 14, 21, 28, 35], 5, 是, complete_zero_through_n_derivative_jet)
Wave6第六層核對=(-6, 4, [7, 14, 21, 28, 35], 5, 是, complete_zero_through_n_derivative_jet)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-jet-qualification · candidate
8. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-jet-qualification；資料={"derivatives":[8,16,24,32,40,48],"center":-5,"order":5}。請依 [center, order, derivatives, jet_length, complete_jet, certificate] 欄序回答。
查看答案與解釋
(-5, 5, [8, 16, 24, 32, 40, 48], 6, 是, complete_zero_through_n_derivative_jet)
Wave6第六層核對=(-5, 5, [8, 16, 24, 32, 40, 48], 6, 是, complete_zero_through_n_derivative_jet)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-jet-qualification · candidate
9. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-jet-qualification；資料={"derivatives":[9,18,27],"center":-4,"order":2}。請依 [center, order, derivatives, jet_length, complete_jet, certificate] 欄序回答。
查看答案與解釋
(-4, 2, [9, 18, 27], 3, 是, complete_zero_through_n_derivative_jet)
Wave6第六層核對=(-4, 2, [9, 18, 27], 3, 是, complete_zero_through_n_derivative_jet)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-jet-qualification · candidate
10. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-jet-qualification；資料={"derivatives":[10,20,30,40],"center":-3,"order":3}。請依 [center, order, derivatives, jet_length, complete_jet, certificate] 欄序回答。
查看答案與解釋
(-3, 3, [10, 20, 30, 40], 4, 是, complete_zero_through_n_derivative_jet)
Wave6第六層核對=(-3, 3, [10, 20, 30, 40], 4, 是, complete_zero_through_n_derivative_jet)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-jet-qualification · candidate
11. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-jet-qualification；資料={"derivatives":[11,22,33,44,55],"center":-2,"order":4}。請依 [center, order, derivatives, jet_length, complete_jet, certificate] 欄序回答。
查看答案與解釋
(-2, 4, [11, 22, 33, 44, 55], 5, 是, complete_zero_through_n_derivative_jet)
Wave6第六層核對=(-2, 4, [11, 22, 33, 44, 55], 5, 是, complete_zero_through_n_derivative_jet)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-jet-qualification · candidate
12. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-jet-qualification；資料={"derivatives":[12,24,36,48,60,72],"center":-1,"order":5}。請依 [center, order, derivatives, jet_length, complete_jet, certificate] 欄序回答。
查看答案與解釋
(-1, 5, [12, 24, 36, 48, 60, 72], 6, 是, complete_zero_through_n_derivative_jet)
Wave6第六層核對=(-1, 5, [12, 24, 36, 48, 60, 72], 6, 是, complete_zero_through_n_derivative_jet)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-jet-qualification · candidate
13. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-jet-qualification；資料={"derivatives":[13,26,39],"center":0,"order":2}。請依 [center, order, derivatives, jet_length, complete_jet, certificate] 欄序回答。
查看答案與解釋
(0, 2, [13, 26, 39], 3, 是, complete_zero_through_n_derivative_jet)
Wave6第六層核對=(0, 2, [13, 26, 39], 3, 是, complete_zero_through_n_derivative_jet)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-jet-qualification · candidate
14. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-jet-qualification；資料={"derivatives":[14,28,42,56],"center":1,"order":3}。請依 [center, order, derivatives, jet_length, complete_jet, certificate] 欄序回答。
查看答案與解釋
(1, 3, [14, 28, 42, 56], 4, 是, complete_zero_through_n_derivative_jet)
Wave6第六層核對=(1, 3, [14, 28, 42, 56], 4, 是, complete_zero_through_n_derivative_jet)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-jet-qualification · candidate
15. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-jet-qualification；資料={"derivatives":[15,30,45,60,75],"center":2,"order":4}。請依 [center, order, derivatives, jet_length, complete_jet, certificate] 欄序回答。
查看答案與解釋
(2, 4, [15, 30, 45, 60, 75], 5, 是, complete_zero_through_n_derivative_jet)
Wave6第六層核對=(2, 4, [15, 30, 45, 60, 75], 5, 是, complete_zero_through_n_derivative_jet)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-jet-qualification · candidate
16. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-jet-qualification；資料={"derivatives":[16,32,48,64,80,96],"center":3,"order":5}。請依 [center, order, derivatives, jet_length, complete_jet, certificate] 欄序回答。
查看答案與解釋
(3, 5, [16, 32, 48, 64, 80, 96], 6, 是, complete_zero_through_n_derivative_jet)
Wave6第六層核對=(3, 5, [16, 32, 48, 64, 80, 96], 6, 是, complete_zero_through_n_derivative_jet)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-jet-qualification · candidate
17. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-jet-qualification；資料={"derivatives":[17,34,51],"center":4,"order":2}。請依 [center, order, derivatives, jet_length, complete_jet, certificate] 欄序回答。
查看答案與解釋
(4, 2, [17, 34, 51], 3, 是, complete_zero_through_n_derivative_jet)
Wave6第六層核對=(4, 2, [17, 34, 51], 3, 是, complete_zero_through_n_derivative_jet)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-jet-qualification · candidate
18. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-jet-qualification；資料={"derivatives":[18,36,54,72],"center":5,"order":3}。請依 [center, order, derivatives, jet_length, complete_jet, certificate] 欄序回答。
查看答案與解釋
(5, 3, [18, 36, 54, 72], 4, 是, complete_zero_through_n_derivative_jet)
Wave6第六層核對=(5, 3, [18, 36, 54, 72], 4, 是, complete_zero_through_n_derivative_jet)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-jet-qualification · candidate
19. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-jet-qualification；資料={"derivatives":[19,38,57,76,95],"center":6,"order":4}。請依 [center, order, derivatives, jet_length, complete_jet, certificate] 欄序回答。
查看答案與解釋
(6, 4, [19, 38, 57, 76, 95], 5, 是, complete_zero_through_n_derivative_jet)
Wave6第六層核對=(6, 4, [19, 38, 57, 76, 95], 5, 是, complete_zero_through_n_derivative_jet)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-jet-qualification · candidate
20. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-jet-qualification；資料={"derivatives":[20,40,60,80,100,120],"center":7,"order":5}。請依 [center, order, derivatives, jet_length, complete_jet, certificate] 欄序回答。
查看答案與解釋
(7, 5, [20, 40, 60, 80, 100, 120], 6, 是, complete_zero_through_n_derivative_jet)
Wave6第六層核對=(7, 5, [20, 40, 60, 80, 100, 120], 6, 是, complete_zero_through_n_derivative_jet)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-jet-qualification · candidate
21. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-jet-qualification；資料={"derivatives":[21,42,63],"center":8,"order":2}。請依 [center, order, derivatives, jet_length, complete_jet, certificate] 欄序回答。
查看答案與解釋
(8, 2, [21, 42, 63], 3, 是, complete_zero_through_n_derivative_jet)
Wave6第六層核對=(8, 2, [21, 42, 63], 3, 是, complete_zero_through_n_derivative_jet)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-jet-qualification · candidate
22. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-jet-qualification；資料={"derivatives":[22,44,66,88],"center":9,"order":3}。請依 [center, order, derivatives, jet_length, complete_jet, certificate] 欄序回答。
查看答案與解釋
(9, 3, [22, 44, 66, 88], 4, 是, complete_zero_through_n_derivative_jet)
Wave6第六層核對=(9, 3, [22, 44, 66, 88], 4, 是, complete_zero_through_n_derivative_jet)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-jet-qualification · candidate
23. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-jet-qualification；資料={"derivatives":[23,46,69,92,115],"center":10,"order":4}。請依 [center, order, derivatives, jet_length, complete_jet, certificate] 欄序回答。
查看答案與解釋
(10, 4, [23, 46, 69, 92, 115], 5, 是, complete_zero_through_n_derivative_jet)
Wave6第六層核對=(10, 4, [23, 46, 69, 92, 115], 5, 是, complete_zero_through_n_derivative_jet)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-jet-qualification · candidate
24. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-jet-qualification；資料={"derivatives":[24,48,72,96,120,144],"center":11,"order":5}。請依 [center, order, derivatives, jet_length, complete_jet, certificate] 欄序回答。
查看答案與解釋
(11, 5, [24, 48, 72, 96, 120, 144], 6, 是, complete_zero_through_n_derivative_jet)
Wave6第六層核對=(11, 5, [24, 48, 72, 96, 120, 144], 6, 是, complete_zero_through_n_derivative_jet)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-jet-qualification · candidate
25. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-jet-qualification；資料={"derivatives":[25,50,75],"center":12,"order":2}。請依 [center, order, derivatives, jet_length, complete_jet, certificate] 欄序回答。
查看答案與解釋
(12, 2, [25, 50, 75], 3, 是, complete_zero_through_n_derivative_jet)
Wave6第六層核對=(12, 2, [25, 50, 75], 3, 是, complete_zero_through_n_derivative_jet)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-jet-qualification · candidate
26. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={"derivatives":[1,2,3],"center":-12,"order":2}。請依 [center, order, shifted_coefficients, indexed_coefficients, certificate] 欄序回答。
查看答案與解釋
(-12, 2, [1, 2, 3/2], [[0, 1], [1, 2], [2, 3/2]], coefficient_k_equals_derivative_k_over_k_factorial)
Wave6第六層核對=(-12, 2, [1, 2, 3/2], [[0, 1], [1, 2], [2, 3/2]], coefficient_k_equals_derivative_k_over_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-coefficient-construction · candidate
27. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={"derivatives":[2,4,6,8],"center":-11,"order":3}。請依 [center, order, shifted_coefficients, indexed_coefficients, certificate] 欄序回答。
查看答案與解釋
(-11, 3, [2, 4, 3, 4/3], [[0, 2], [1, 4], [2, 3], [3, 4/3]], coefficient_k_equals_derivative_k_over_k_factorial)
Wave6第六層核對=(-11, 3, [2, 4, 3, 4/3], [[0, 2], [1, 4], [2, 3], [3, 4/3]], coefficient_k_equals_derivative_k_over_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-coefficient-construction · candidate
28. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={"derivatives":[3,6,9,12,15],"center":-10,"order":4}。請依 [center, order, shifted_coefficients, indexed_coefficients, certificate] 欄序回答。
查看答案與解釋
(-10, 4, [3, 6, 9/2, 2, 5/8], [[0, 3], [1, 6], [2, 9/2], [3, 2], [4, 5/8]], coefficient_k_equals_derivative_k_over_k_factorial)
Wave6第六層核對=(-10, 4, [3, 6, 9/2, 2, 5/8], [[0, 3], [1, 6], [2, 9/2], [3, 2], [4, 5/8]], coefficient_k_equals_derivative_k_over_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-coefficient-construction · candidate
29. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={"derivatives":[4,8,12,16,20,24],"center":-9,"order":5}。請依 [center, order, shifted_coefficients, indexed_coefficients, certificate] 欄序回答。
查看答案與解釋
(-9, 5, [4, 8, 6, 8/3, 5/6, 1/5], [[0, 4], [1, 8], [2, 6], [3, 8/3], [4, 5/6], [5, 1/5]], coefficient_k_equals_derivative_k_over_k_factorial)
Wave6第六層核對=(-9, 5, [4, 8, 6, 8/3, 5/6, 1/5], [[0, 4], [1, 8], [2, 6], [3, 8/3], [4, 5/6], [5, 1/5]], coefficient_k_equals_derivative_k_over_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-coefficient-construction · candidate
30. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={"derivatives":[5,10,15],"center":-8,"order":2}。請依 [center, order, shifted_coefficients, indexed_coefficients, certificate] 欄序回答。
查看答案與解釋
(-8, 2, [5, 10, 15/2], [[0, 5], [1, 10], [2, 15/2]], coefficient_k_equals_derivative_k_over_k_factorial)
Wave6第六層核對=(-8, 2, [5, 10, 15/2], [[0, 5], [1, 10], [2, 15/2]], coefficient_k_equals_derivative_k_over_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-coefficient-construction · candidate
31. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={"derivatives":[6,12,18,24],"center":-7,"order":3}。請依 [center, order, shifted_coefficients, indexed_coefficients, certificate] 欄序回答。
查看答案與解釋
(-7, 3, [6, 12, 9, 4], [[0, 6], [1, 12], [2, 9], [3, 4]], coefficient_k_equals_derivative_k_over_k_factorial)
Wave6第六層核對=(-7, 3, [6, 12, 9, 4], [[0, 6], [1, 12], [2, 9], [3, 4]], coefficient_k_equals_derivative_k_over_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-coefficient-construction · candidate
32. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={"derivatives":[7,14,21,28,35],"center":-6,"order":4}。請依 [center, order, shifted_coefficients, indexed_coefficients, certificate] 欄序回答。
查看答案與解釋
(-6, 4, [7, 14, 21/2, 14/3, 35/24], [[0, 7], [1, 14], [2, 21/2], [3, 14/3], [4, 35/24]], coefficient_k_equals_derivative_k_over_k_factorial)
Wave6第六層核對=(-6, 4, [7, 14, 21/2, 14/3, 35/24], [[0, 7], [1, 14], [2, 21/2], [3, 14/3], [4, 35/24]], coefficient_k_equals_derivative_k_over_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-coefficient-construction · candidate
33. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={"derivatives":[8,16,24,32,40,48],"center":-5,"order":5}。請依 [center, order, shifted_coefficients, indexed_coefficients, certificate] 欄序回答。
查看答案與解釋
(-5, 5, [8, 16, 12, 16/3, 5/3, 2/5], [[0, 8], [1, 16], [2, 12], [3, 16/3], [4, 5/3], [5, 2/5]], coefficient_k_equals_derivative_k_over_k_factorial)
Wave6第六層核對=(-5, 5, [8, 16, 12, 16/3, 5/3, 2/5], [[0, 8], [1, 16], [2, 12], [3, 16/3], [4, 5/3], [5, 2/5]], coefficient_k_equals_derivative_k_over_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-coefficient-construction · candidate
34. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={"derivatives":[9,18,27],"center":-4,"order":2}。請依 [center, order, shifted_coefficients, indexed_coefficients, certificate] 欄序回答。
查看答案與解釋
(-4, 2, [9, 18, 27/2], [[0, 9], [1, 18], [2, 27/2]], coefficient_k_equals_derivative_k_over_k_factorial)
Wave6第六層核對=(-4, 2, [9, 18, 27/2], [[0, 9], [1, 18], [2, 27/2]], coefficient_k_equals_derivative_k_over_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-coefficient-construction · candidate
35. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={"derivatives":[10,20,30,40],"center":-3,"order":3}。請依 [center, order, shifted_coefficients, indexed_coefficients, certificate] 欄序回答。
查看答案與解釋
(-3, 3, [10, 20, 15, 20/3], [[0, 10], [1, 20], [2, 15], [3, 20/3]], coefficient_k_equals_derivative_k_over_k_factorial)
Wave6第六層核對=(-3, 3, [10, 20, 15, 20/3], [[0, 10], [1, 20], [2, 15], [3, 20/3]], coefficient_k_equals_derivative_k_over_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-coefficient-construction · candidate
36. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={"derivatives":[11,22,33,44,55],"center":-2,"order":4}。請依 [center, order, shifted_coefficients, indexed_coefficients, certificate] 欄序回答。
查看答案與解釋
(-2, 4, [11, 22, 33/2, 22/3, 55/24], [[0, 11], [1, 22], [2, 33/2], [3, 22/3], [4, 55/24]], coefficient_k_equals_derivative_k_over_k_factorial)
Wave6第六層核對=(-2, 4, [11, 22, 33/2, 22/3, 55/24], [[0, 11], [1, 22], [2, 33/2], [3, 22/3], [4, 55/24]], coefficient_k_equals_derivative_k_over_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-coefficient-construction · candidate
37. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={"derivatives":[12,24,36,48,60,72],"center":-1,"order":5}。請依 [center, order, shifted_coefficients, indexed_coefficients, certificate] 欄序回答。
查看答案與解釋
(-1, 5, [12, 24, 18, 8, 5/2, 3/5], [[0, 12], [1, 24], [2, 18], [3, 8], [4, 5/2], [5, 3/5]], coefficient_k_equals_derivative_k_over_k_factorial)
Wave6第六層核對=(-1, 5, [12, 24, 18, 8, 5/2, 3/5], [[0, 12], [1, 24], [2, 18], [3, 8], [4, 5/2], [5, 3/5]], coefficient_k_equals_derivative_k_over_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-coefficient-construction · candidate
38. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={"derivatives":[13,26,39],"center":0,"order":2}。請依 [center, order, shifted_coefficients, indexed_coefficients, certificate] 欄序回答。
查看答案與解釋
(0, 2, [13, 26, 39/2], [[0, 13], [1, 26], [2, 39/2]], coefficient_k_equals_derivative_k_over_k_factorial)
Wave6第六層核對=(0, 2, [13, 26, 39/2], [[0, 13], [1, 26], [2, 39/2]], coefficient_k_equals_derivative_k_over_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-coefficient-construction · candidate
39. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={"derivatives":[14,28,42,56],"center":1,"order":3}。請依 [center, order, shifted_coefficients, indexed_coefficients, certificate] 欄序回答。
查看答案與解釋
(1, 3, [14, 28, 21, 28/3], [[0, 14], [1, 28], [2, 21], [3, 28/3]], coefficient_k_equals_derivative_k_over_k_factorial)
Wave6第六層核對=(1, 3, [14, 28, 21, 28/3], [[0, 14], [1, 28], [2, 21], [3, 28/3]], coefficient_k_equals_derivative_k_over_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-coefficient-construction · candidate
40. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={"derivatives":[15,30,45,60,75],"center":2,"order":4}。請依 [center, order, shifted_coefficients, indexed_coefficients, certificate] 欄序回答。
查看答案與解釋
(2, 4, [15, 30, 45/2, 10, 25/8], [[0, 15], [1, 30], [2, 45/2], [3, 10], [4, 25/8]], coefficient_k_equals_derivative_k_over_k_factorial)
Wave6第六層核對=(2, 4, [15, 30, 45/2, 10, 25/8], [[0, 15], [1, 30], [2, 45/2], [3, 10], [4, 25/8]], coefficient_k_equals_derivative_k_over_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-coefficient-construction · candidate
41. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={"derivatives":[16,32,48,64,80,96],"center":3,"order":5}。請依 [center, order, shifted_coefficients, indexed_coefficients, certificate] 欄序回答。
查看答案與解釋
(3, 5, [16, 32, 24, 32/3, 10/3, 4/5], [[0, 16], [1, 32], [2, 24], [3, 32/3], [4, 10/3], [5, 4/5]], coefficient_k_equals_derivative_k_over_k_factorial)
Wave6第六層核對=(3, 5, [16, 32, 24, 32/3, 10/3, 4/5], [[0, 16], [1, 32], [2, 24], [3, 32/3], [4, 10/3], [5, 4/5]], coefficient_k_equals_derivative_k_over_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-coefficient-construction · candidate
42. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={"derivatives":[17,34,51],"center":4,"order":2}。請依 [center, order, shifted_coefficients, indexed_coefficients, certificate] 欄序回答。
查看答案與解釋
(4, 2, [17, 34, 51/2], [[0, 17], [1, 34], [2, 51/2]], coefficient_k_equals_derivative_k_over_k_factorial)
Wave6第六層核對=(4, 2, [17, 34, 51/2], [[0, 17], [1, 34], [2, 51/2]], coefficient_k_equals_derivative_k_over_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-coefficient-construction · candidate
43. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={"derivatives":[18,36,54,72],"center":5,"order":3}。請依 [center, order, shifted_coefficients, indexed_coefficients, certificate] 欄序回答。
查看答案與解釋
(5, 3, [18, 36, 27, 12], [[0, 18], [1, 36], [2, 27], [3, 12]], coefficient_k_equals_derivative_k_over_k_factorial)
Wave6第六層核對=(5, 3, [18, 36, 27, 12], [[0, 18], [1, 36], [2, 27], [3, 12]], coefficient_k_equals_derivative_k_over_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-coefficient-construction · candidate
44. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={"derivatives":[19,38,57,76,95],"center":6,"order":4}。請依 [center, order, shifted_coefficients, indexed_coefficients, certificate] 欄序回答。
查看答案與解釋
(6, 4, [19, 38, 57/2, 38/3, 95/24], [[0, 19], [1, 38], [2, 57/2], [3, 38/3], [4, 95/24]], coefficient_k_equals_derivative_k_over_k_factorial)
Wave6第六層核對=(6, 4, [19, 38, 57/2, 38/3, 95/24], [[0, 19], [1, 38], [2, 57/2], [3, 38/3], [4, 95/24]], coefficient_k_equals_derivative_k_over_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-coefficient-construction · candidate
45. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={"derivatives":[20,40,60,80,100,120],"center":7,"order":5}。請依 [center, order, shifted_coefficients, indexed_coefficients, certificate] 欄序回答。
查看答案與解釋
(7, 5, [20, 40, 30, 40/3, 25/6, 1], [[0, 20], [1, 40], [2, 30], [3, 40/3], [4, 25/6], [5, 1]], coefficient_k_equals_derivative_k_over_k_factorial)
Wave6第六層核對=(7, 5, [20, 40, 30, 40/3, 25/6, 1], [[0, 20], [1, 40], [2, 30], [3, 40/3], [4, 25/6], [5, 1]], coefficient_k_equals_derivative_k_over_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-coefficient-construction · candidate
46. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={"derivatives":[21,42,63],"center":8,"order":2}。請依 [center, order, shifted_coefficients, indexed_coefficients, certificate] 欄序回答。
查看答案與解釋
(8, 2, [21, 42, 63/2], [[0, 21], [1, 42], [2, 63/2]], coefficient_k_equals_derivative_k_over_k_factorial)
Wave6第六層核對=(8, 2, [21, 42, 63/2], [[0, 21], [1, 42], [2, 63/2]], coefficient_k_equals_derivative_k_over_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-coefficient-construction · candidate
47. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={"derivatives":[22,44,66,88],"center":9,"order":3}。請依 [center, order, shifted_coefficients, indexed_coefficients, certificate] 欄序回答。
查看答案與解釋
(9, 3, [22, 44, 33, 44/3], [[0, 22], [1, 44], [2, 33], [3, 44/3]], coefficient_k_equals_derivative_k_over_k_factorial)
Wave6第六層核對=(9, 3, [22, 44, 33, 44/3], [[0, 22], [1, 44], [2, 33], [3, 44/3]], coefficient_k_equals_derivative_k_over_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-coefficient-construction · candidate
48. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={"derivatives":[23,46,69,92,115],"center":10,"order":4}。請依 [center, order, shifted_coefficients, indexed_coefficients, certificate] 欄序回答。
查看答案與解釋
(10, 4, [23, 46, 69/2, 46/3, 115/24], [[0, 23], [1, 46], [2, 69/2], [3, 46/3], [4, 115/24]], coefficient_k_equals_derivative_k_over_k_factorial)
Wave6第六層核對=(10, 4, [23, 46, 69/2, 46/3, 115/24], [[0, 23], [1, 46], [2, 69/2], [3, 46/3], [4, 115/24]], coefficient_k_equals_derivative_k_over_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-coefficient-construction · candidate
49. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={"derivatives":[24,48,72,96,120,144],"center":11,"order":5}。請依 [center, order, shifted_coefficients, indexed_coefficients, certificate] 欄序回答。
查看答案與解釋
(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)
Wave6第六層核對=(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)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-coefficient-construction · candidate
50. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={"derivatives":[25,50,75],"center":12,"order":2}。請依 [center, order, shifted_coefficients, indexed_coefficients, certificate] 欄序回答。
查看答案與解釋
(12, 2, [25, 50, 75/2], [[0, 25], [1, 50], [2, 75/2]], coefficient_k_equals_derivative_k_over_k_factorial)
Wave6第六層核對=(12, 2, [25, 50, 75/2], [[0, 25], [1, 50], [2, 75/2]], coefficient_k_equals_derivative_k_over_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-coefficient-construction · candidate
51. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-exact-polynomial-reconstruction；資料={"shifted_coefficients":[1,2,3],"center":-12,"point":-14}。請依 [shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate] 欄序回答。
查看答案與解釋
([1, 2, 3], [1, 2, 6], [1, 2, 3], -14, 9, 9, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)
Wave6第六層核對=([1, 2, 3], [1, 2, 6], [1, 2, 3], -14, 9, 9, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-exact-polynomial-reconstruction · candidate
52. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-exact-polynomial-reconstruction；資料={"shifted_coefficients":[2,4,6,8],"center":-11,"point":-12}。請依 [shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate] 欄序回答。
查看答案與解釋
([2, 4, 6, 8], [2, 4, 12, 48], [2, 4, 6, 8], -12, -4, -4, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)
Wave6第六層核對=([2, 4, 6, 8], [2, 4, 12, 48], [2, 4, 6, 8], -12, -4, -4, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-exact-polynomial-reconstruction · candidate
53. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-exact-polynomial-reconstruction；資料={"shifted_coefficients":[3,6,9,12,15],"center":-10,"point":-10}。請依 [shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate] 欄序回答。
查看答案與解釋
([3, 6, 9, 12, 15], [3, 6, 18, 72, 360], [3, 6, 9, 12, 15], -10, 3, 3, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)
Wave6第六層核對=([3, 6, 9, 12, 15], [3, 6, 18, 72, 360], [3, 6, 9, 12, 15], -10, 3, 3, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-exact-polynomial-reconstruction · candidate
54. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-exact-polynomial-reconstruction；資料={"shifted_coefficients":[4,8,12,16,20,24],"center":-9,"point":-8}。請依 [shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate] 欄序回答。
查看答案與解釋
([4, 8, 12, 16, 20, 24], [4, 8, 24, 96, 480, 2880], [4, 8, 12, 16, 20, 24], -8, 84, 84, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)
Wave6第六層核對=([4, 8, 12, 16, 20, 24], [4, 8, 24, 96, 480, 2880], [4, 8, 12, 16, 20, 24], -8, 84, 84, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-exact-polynomial-reconstruction · candidate
55. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-exact-polynomial-reconstruction；資料={"shifted_coefficients":[5,10,15],"center":-8,"point":-6}。請依 [shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate] 欄序回答。
查看答案與解釋
([5, 10, 15], [5, 10, 30], [5, 10, 15], -6, 85, 85, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)
Wave6第六層核對=([5, 10, 15], [5, 10, 30], [5, 10, 15], -6, 85, 85, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-exact-polynomial-reconstruction · candidate
56. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-exact-polynomial-reconstruction；資料={"shifted_coefficients":[6,12,18,24],"center":-7,"point":-9}。請依 [shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate] 欄序回答。
查看答案與解釋
([6, 12, 18, 24], [6, 12, 36, 144], [6, 12, 18, 24], -9, -138, -138, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)
Wave6第六層核對=([6, 12, 18, 24], [6, 12, 36, 144], [6, 12, 18, 24], -9, -138, -138, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-exact-polynomial-reconstruction · candidate
57. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-exact-polynomial-reconstruction；資料={"shifted_coefficients":[7,14,21,28,35],"center":-6,"point":-7}。請依 [shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate] 欄序回答。
查看答案與解釋
([7, 14, 21, 28, 35], [7, 14, 42, 168, 840], [7, 14, 21, 28, 35], -7, 21, 21, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)
Wave6第六層核對=([7, 14, 21, 28, 35], [7, 14, 42, 168, 840], [7, 14, 21, 28, 35], -7, 21, 21, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-exact-polynomial-reconstruction · candidate
58. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-exact-polynomial-reconstruction；資料={"shifted_coefficients":[8,16,24,32,40,48],"center":-5,"point":-5}。請依 [shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate] 欄序回答。
查看答案與解釋
([8, 16, 24, 32, 40, 48], [8, 16, 48, 192, 960, 5760], [8, 16, 24, 32, 40, 48], -5, 8, 8, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)
Wave6第六層核對=([8, 16, 24, 32, 40, 48], [8, 16, 48, 192, 960, 5760], [8, 16, 24, 32, 40, 48], -5, 8, 8, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-exact-polynomial-reconstruction · candidate
59. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-exact-polynomial-reconstruction；資料={"shifted_coefficients":[9,18,27],"center":-4,"point":-3}。請依 [shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate] 欄序回答。
查看答案與解釋
([9, 18, 27], [9, 18, 54], [9, 18, 27], -3, 54, 54, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)
Wave6第六層核對=([9, 18, 27], [9, 18, 54], [9, 18, 27], -3, 54, 54, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-exact-polynomial-reconstruction · candidate
60. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-exact-polynomial-reconstruction；資料={"shifted_coefficients":[10,20,30,40],"center":-3,"point":-1}。請依 [shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate] 欄序回答。
查看答案與解釋
([10, 20, 30, 40], [10, 20, 60, 240], [10, 20, 30, 40], -1, 490, 490, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)
Wave6第六層核對=([10, 20, 30, 40], [10, 20, 60, 240], [10, 20, 30, 40], -1, 490, 490, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-exact-polynomial-reconstruction · candidate
61. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-exact-polynomial-reconstruction；資料={"shifted_coefficients":[11,22,33,44,55],"center":-2,"point":-4}。請依 [shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate] 欄序回答。
查看答案與解釋
([11, 22, 33, 44, 55], [11, 22, 66, 264, 1320], [11, 22, 33, 44, 55], -4, 627, 627, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)
Wave6第六層核對=([11, 22, 33, 44, 55], [11, 22, 66, 264, 1320], [11, 22, 33, 44, 55], -4, 627, 627, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-exact-polynomial-reconstruction · candidate
62. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-exact-polynomial-reconstruction；資料={"shifted_coefficients":[12,24,36,48,60,72],"center":-1,"point":-2}。請依 [shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate] 欄序回答。
查看答案與解釋
([12, 24, 36, 48, 60, 72], [12, 24, 72, 288, 1440, 8640], [12, 24, 36, 48, 60, 72], -2, -36, -36, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)
Wave6第六層核對=([12, 24, 36, 48, 60, 72], [12, 24, 72, 288, 1440, 8640], [12, 24, 36, 48, 60, 72], -2, -36, -36, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-exact-polynomial-reconstruction · candidate
63. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-exact-polynomial-reconstruction；資料={"shifted_coefficients":[13,26,39],"center":0,"point":0}。請依 [shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate] 欄序回答。
查看答案與解釋
([13, 26, 39], [13, 26, 78], [13, 26, 39], 0, 13, 13, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)
Wave6第六層核對=([13, 26, 39], [13, 26, 78], [13, 26, 39], 0, 13, 13, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-exact-polynomial-reconstruction · candidate
64. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-exact-polynomial-reconstruction；資料={"shifted_coefficients":[14,28,42,56],"center":1,"point":2}。請依 [shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate] 欄序回答。
查看答案與解釋
([14, 28, 42, 56], [14, 28, 84, 336], [14, 28, 42, 56], 2, 140, 140, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)
Wave6第六層核對=([14, 28, 42, 56], [14, 28, 84, 336], [14, 28, 42, 56], 2, 140, 140, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-exact-polynomial-reconstruction · candidate
65. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-exact-polynomial-reconstruction；資料={"shifted_coefficients":[15,30,45,60,75],"center":2,"point":4}。請依 [shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate] 欄序回答。
查看答案與解釋
([15, 30, 45, 60, 75], [15, 30, 90, 360, 1800], [15, 30, 45, 60, 75], 4, 1935, 1935, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)
Wave6第六層核對=([15, 30, 45, 60, 75], [15, 30, 90, 360, 1800], [15, 30, 45, 60, 75], 4, 1935, 1935, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-exact-polynomial-reconstruction · candidate
66. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-exact-polynomial-reconstruction；資料={"shifted_coefficients":[16,32,48,64,80,96],"center":3,"point":1}。請依 [shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate] 欄序回答。
查看答案與解釋
([16, 32, 48, 64, 80, 96], [16, 32, 96, 384, 1920, 11520], [16, 32, 48, 64, 80, 96], 1, -2160, -2160, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)
Wave6第六層核對=([16, 32, 48, 64, 80, 96], [16, 32, 96, 384, 1920, 11520], [16, 32, 48, 64, 80, 96], 1, -2160, -2160, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-exact-polynomial-reconstruction · candidate
67. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-exact-polynomial-reconstruction；資料={"shifted_coefficients":[17,34,51],"center":4,"point":3}。請依 [shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate] 欄序回答。
查看答案與解釋
([17, 34, 51], [17, 34, 102], [17, 34, 51], 3, 34, 34, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)
Wave6第六層核對=([17, 34, 51], [17, 34, 102], [17, 34, 51], 3, 34, 34, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-exact-polynomial-reconstruction · candidate
68. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-exact-polynomial-reconstruction；資料={"shifted_coefficients":[18,36,54,72],"center":5,"point":5}。請依 [shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate] 欄序回答。
查看答案與解釋
([18, 36, 54, 72], [18, 36, 108, 432], [18, 36, 54, 72], 5, 18, 18, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)
Wave6第六層核對=([18, 36, 54, 72], [18, 36, 108, 432], [18, 36, 54, 72], 5, 18, 18, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-exact-polynomial-reconstruction · candidate
69. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-exact-polynomial-reconstruction；資料={"shifted_coefficients":[19,38,57,76,95],"center":6,"point":7}。請依 [shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate] 欄序回答。
查看答案與解釋
([19, 38, 57, 76, 95], [19, 38, 114, 456, 2280], [19, 38, 57, 76, 95], 7, 285, 285, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)
Wave6第六層核對=([19, 38, 57, 76, 95], [19, 38, 114, 456, 2280], [19, 38, 57, 76, 95], 7, 285, 285, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-exact-polynomial-reconstruction · candidate
70. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-exact-polynomial-reconstruction；資料={"shifted_coefficients":[20,40,60,80,100,120],"center":7,"point":9}。請依 [shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate] 欄序回答。
查看答案與解釋
([20, 40, 60, 80, 100, 120], [20, 40, 120, 480, 2400, 14400], [20, 40, 60, 80, 100, 120], 9, 6420, 6420, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)
Wave6第六層核對=([20, 40, 60, 80, 100, 120], [20, 40, 120, 480, 2400, 14400], [20, 40, 60, 80, 100, 120], 9, 6420, 6420, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-exact-polynomial-reconstruction · candidate
71. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-exact-polynomial-reconstruction；資料={"shifted_coefficients":[21,42,63],"center":8,"point":6}。請依 [shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate] 欄序回答。
查看答案與解釋
([21, 42, 63], [21, 42, 126], [21, 42, 63], 6, 189, 189, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)
Wave6第六層核對=([21, 42, 63], [21, 42, 126], [21, 42, 63], 6, 189, 189, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-exact-polynomial-reconstruction · candidate
72. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-exact-polynomial-reconstruction；資料={"shifted_coefficients":[22,44,66,88],"center":9,"point":8}。請依 [shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate] 欄序回答。
查看答案與解釋
([22, 44, 66, 88], [22, 44, 132, 528], [22, 44, 66, 88], 8, -44, -44, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)
Wave6第六層核對=([22, 44, 66, 88], [22, 44, 132, 528], [22, 44, 66, 88], 8, -44, -44, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-exact-polynomial-reconstruction · candidate
73. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-exact-polynomial-reconstruction；資料={"shifted_coefficients":[23,46,69,92,115],"center":10,"point":10}。請依 [shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate] 欄序回答。
查看答案與解釋
([23, 46, 69, 92, 115], [23, 46, 138, 552, 2760], [23, 46, 69, 92, 115], 10, 23, 23, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)
Wave6第六層核對=([23, 46, 69, 92, 115], [23, 46, 138, 552, 2760], [23, 46, 69, 92, 115], 10, 23, 23, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-exact-polynomial-reconstruction · candidate
74. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-exact-polynomial-reconstruction；資料={"shifted_coefficients":[24,48,72,96,120,144],"center":11,"point":12}。請依 [shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate] 欄序回答。
查看答案與解釋
([24, 48, 72, 96, 120, 144], [24, 48, 144, 576, 2880, 17280], [24, 48, 72, 96, 120, 144], 12, 504, 504, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)
Wave6第六層核對=([24, 48, 72, 96, 120, 144], [24, 48, 144, 576, 2880, 17280], [24, 48, 72, 96, 120, 144], 12, 504, 504, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-exact-polynomial-reconstruction · candidate
75. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-exact-polynomial-reconstruction；資料={"shifted_coefficients":[25,50,75],"center":12,"point":14}。請依 [shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate] 欄序回答。
查看答案與解釋
([25, 50, 75], [25, 50, 150], [25, 50, 75], 14, 425, 425, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)
Wave6第六層核對=([25, 50, 75], [25, 50, 150], [25, 50, 75], 14, 425, 425, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-exact-polynomial-reconstruction · candidate
76. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-definition-boundary；資料={"mode":"missing_derivative","derivatives":[1,2],"center":-12,"order":2,"alternate_center":-11}。請依 [mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary] 欄序回答。
zero through n是導數階數0至n，不是所有導數數值都必須為0。固定的否拒絕漏階、錯中心、漏factorial或無依據餘項／級數結論。
查看答案與解釋
(missing_derivative, 2, [1, 2], -12, -11, 否, all_derivatives_zero_through_n_are_required)
Wave6第六層核對=(missing_derivative, 2, [1, 2], -12, -11, 否, all_derivatives_zero_through_n_are_required)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-definition-boundary · candidate
77. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-definition-boundary；資料={"mode":"wrong_center","derivatives":[2,4,6],"center":-11,"order":2,"alternate_center":-10}。請依 [mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary] 欄序回答。
zero through n是導數階數0至n，不是所有導數數值都必須為0。固定的否拒絕漏階、錯中心、漏factorial或無依據餘項／級數結論。
查看答案與解釋
(wrong_center, 2, [2, 4, 6], -11, -10, 否, a_jet_is_bound_to_its_declared_center)
Wave6第六層核對=(wrong_center, 2, [2, 4, 6], -11, -10, 否, a_jet_is_bound_to_its_declared_center)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-definition-boundary · candidate
78. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-definition-boundary；資料={"mode":"missing_factorial","derivatives":[3,6,9],"center":-10,"order":2,"alternate_center":-9}。請依 [mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary] 欄序回答。
zero through n是導數階數0至n，不是所有導數數值都必須為0。固定的否拒絕漏階、錯中心、漏factorial或無依據餘項／級數結論。
查看答案與解釋
(missing_factorial, 2, [3, 6, 9], -10, -9, 否, coefficients_require_division_by_k_factorial)
Wave6第六層核對=(missing_factorial, 2, [3, 6, 9], -10, -9, 否, coefficients_require_division_by_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-definition-boundary · candidate
79. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-definition-boundary；資料={"mode":"remainder_claim","derivatives":[4,8,12],"center":-9,"order":2,"alternate_center":-8}。請依 [mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary] 欄序回答。
zero through n是導數階數0至n，不是所有導數數值都必須為0。固定的否拒絕漏階、錯中心、漏factorial或無依據餘項／級數結論。
查看答案與解釋
(remainder_claim, 2, [4, 8, 12], -9, -8, 否, a_Taylor_polynomial_alone_gives_no_remainder_or_series_equality)
Wave6第六層核對=(remainder_claim, 2, [4, 8, 12], -9, -8, 否, a_Taylor_polynomial_alone_gives_no_remainder_or_series_equality)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-definition-boundary · candidate
80. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-definition-boundary；資料={"mode":"missing_derivative","derivatives":[5,10],"center":-8,"order":2,"alternate_center":-7}。請依 [mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary] 欄序回答。
zero through n是導數階數0至n，不是所有導數數值都必須為0。固定的否拒絕漏階、錯中心、漏factorial或無依據餘項／級數結論。
查看答案與解釋
(missing_derivative, 2, [5, 10], -8, -7, 否, all_derivatives_zero_through_n_are_required)
Wave6第六層核對=(missing_derivative, 2, [5, 10], -8, -7, 否, all_derivatives_zero_through_n_are_required)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-definition-boundary · candidate
81. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-definition-boundary；資料={"mode":"wrong_center","derivatives":[6,12,18],"center":-7,"order":2,"alternate_center":-6}。請依 [mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary] 欄序回答。
zero through n是導數階數0至n，不是所有導數數值都必須為0。固定的否拒絕漏階、錯中心、漏factorial或無依據餘項／級數結論。
查看答案與解釋
(wrong_center, 2, [6, 12, 18], -7, -6, 否, a_jet_is_bound_to_its_declared_center)
Wave6第六層核對=(wrong_center, 2, [6, 12, 18], -7, -6, 否, a_jet_is_bound_to_its_declared_center)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-definition-boundary · candidate
82. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-definition-boundary；資料={"mode":"missing_factorial","derivatives":[7,14,21],"center":-6,"order":2,"alternate_center":-5}。請依 [mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary] 欄序回答。
zero through n是導數階數0至n，不是所有導數數值都必須為0。固定的否拒絕漏階、錯中心、漏factorial或無依據餘項／級數結論。
查看答案與解釋
(missing_factorial, 2, [7, 14, 21], -6, -5, 否, coefficients_require_division_by_k_factorial)
Wave6第六層核對=(missing_factorial, 2, [7, 14, 21], -6, -5, 否, coefficients_require_division_by_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-definition-boundary · candidate
83. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-definition-boundary；資料={"mode":"remainder_claim","derivatives":[8,16,24],"center":-5,"order":2,"alternate_center":-4}。請依 [mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary] 欄序回答。
zero through n是導數階數0至n，不是所有導數數值都必須為0。固定的否拒絕漏階、錯中心、漏factorial或無依據餘項／級數結論。
查看答案與解釋
(remainder_claim, 2, [8, 16, 24], -5, -4, 否, a_Taylor_polynomial_alone_gives_no_remainder_or_series_equality)
Wave6第六層核對=(remainder_claim, 2, [8, 16, 24], -5, -4, 否, a_Taylor_polynomial_alone_gives_no_remainder_or_series_equality)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-definition-boundary · candidate
84. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-definition-boundary；資料={"mode":"missing_derivative","derivatives":[9,18],"center":-4,"order":2,"alternate_center":-3}。請依 [mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary] 欄序回答。
zero through n是導數階數0至n，不是所有導數數值都必須為0。固定的否拒絕漏階、錯中心、漏factorial或無依據餘項／級數結論。
查看答案與解釋
(missing_derivative, 2, [9, 18], -4, -3, 否, all_derivatives_zero_through_n_are_required)
Wave6第六層核對=(missing_derivative, 2, [9, 18], -4, -3, 否, all_derivatives_zero_through_n_are_required)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-definition-boundary · candidate
85. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-definition-boundary；資料={"mode":"wrong_center","derivatives":[10,20,30],"center":-3,"order":2,"alternate_center":-2}。請依 [mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary] 欄序回答。
zero through n是導數階數0至n，不是所有導數數值都必須為0。固定的否拒絕漏階、錯中心、漏factorial或無依據餘項／級數結論。
查看答案與解釋
(wrong_center, 2, [10, 20, 30], -3, -2, 否, a_jet_is_bound_to_its_declared_center)
Wave6第六層核對=(wrong_center, 2, [10, 20, 30], -3, -2, 否, a_jet_is_bound_to_its_declared_center)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-definition-boundary · candidate
86. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-definition-boundary；資料={"mode":"missing_factorial","derivatives":[11,22,33],"center":-2,"order":2,"alternate_center":-1}。請依 [mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary] 欄序回答。
zero through n是導數階數0至n，不是所有導數數值都必須為0。固定的否拒絕漏階、錯中心、漏factorial或無依據餘項／級數結論。
查看答案與解釋
(missing_factorial, 2, [11, 22, 33], -2, -1, 否, coefficients_require_division_by_k_factorial)
Wave6第六層核對=(missing_factorial, 2, [11, 22, 33], -2, -1, 否, coefficients_require_division_by_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-definition-boundary · candidate
87. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-definition-boundary；資料={"mode":"remainder_claim","derivatives":[12,24,36],"center":-1,"order":2,"alternate_center":0}。請依 [mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary] 欄序回答。
zero through n是導數階數0至n，不是所有導數數值都必須為0。固定的否拒絕漏階、錯中心、漏factorial或無依據餘項／級數結論。
查看答案與解釋
(remainder_claim, 2, [12, 24, 36], -1, 0, 否, a_Taylor_polynomial_alone_gives_no_remainder_or_series_equality)
Wave6第六層核對=(remainder_claim, 2, [12, 24, 36], -1, 0, 否, a_Taylor_polynomial_alone_gives_no_remainder_or_series_equality)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-definition-boundary · candidate
88. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-definition-boundary；資料={"mode":"missing_derivative","derivatives":[13,26],"center":0,"order":2,"alternate_center":1}。請依 [mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary] 欄序回答。
zero through n是導數階數0至n，不是所有導數數值都必須為0。固定的否拒絕漏階、錯中心、漏factorial或無依據餘項／級數結論。
查看答案與解釋
(missing_derivative, 2, [13, 26], 0, 1, 否, all_derivatives_zero_through_n_are_required)
Wave6第六層核對=(missing_derivative, 2, [13, 26], 0, 1, 否, all_derivatives_zero_through_n_are_required)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-definition-boundary · candidate
89. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-definition-boundary；資料={"mode":"wrong_center","derivatives":[14,28,42],"center":1,"order":2,"alternate_center":2}。請依 [mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary] 欄序回答。
zero through n是導數階數0至n，不是所有導數數值都必須為0。固定的否拒絕漏階、錯中心、漏factorial或無依據餘項／級數結論。
查看答案與解釋
(wrong_center, 2, [14, 28, 42], 1, 2, 否, a_jet_is_bound_to_its_declared_center)
Wave6第六層核對=(wrong_center, 2, [14, 28, 42], 1, 2, 否, a_jet_is_bound_to_its_declared_center)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-definition-boundary · candidate
90. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-definition-boundary；資料={"mode":"missing_factorial","derivatives":[15,30,45],"center":2,"order":2,"alternate_center":3}。請依 [mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary] 欄序回答。
zero through n是導數階數0至n，不是所有導數數值都必須為0。固定的否拒絕漏階、錯中心、漏factorial或無依據餘項／級數結論。
查看答案與解釋
(missing_factorial, 2, [15, 30, 45], 2, 3, 否, coefficients_require_division_by_k_factorial)
Wave6第六層核對=(missing_factorial, 2, [15, 30, 45], 2, 3, 否, coefficients_require_division_by_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-definition-boundary · candidate
91. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-definition-boundary；資料={"mode":"remainder_claim","derivatives":[16,32,48],"center":3,"order":2,"alternate_center":4}。請依 [mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary] 欄序回答。
zero through n是導數階數0至n，不是所有導數數值都必須為0。固定的否拒絕漏階、錯中心、漏factorial或無依據餘項／級數結論。
查看答案與解釋
(remainder_claim, 2, [16, 32, 48], 3, 4, 否, a_Taylor_polynomial_alone_gives_no_remainder_or_series_equality)
Wave6第六層核對=(remainder_claim, 2, [16, 32, 48], 3, 4, 否, a_Taylor_polynomial_alone_gives_no_remainder_or_series_equality)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-definition-boundary · candidate
92. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-definition-boundary；資料={"mode":"missing_derivative","derivatives":[17,34],"center":4,"order":2,"alternate_center":5}。請依 [mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary] 欄序回答。
zero through n是導數階數0至n，不是所有導數數值都必須為0。固定的否拒絕漏階、錯中心、漏factorial或無依據餘項／級數結論。
查看答案與解釋
(missing_derivative, 2, [17, 34], 4, 5, 否, all_derivatives_zero_through_n_are_required)
Wave6第六層核對=(missing_derivative, 2, [17, 34], 4, 5, 否, all_derivatives_zero_through_n_are_required)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-definition-boundary · candidate
93. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-definition-boundary；資料={"mode":"wrong_center","derivatives":[18,36,54],"center":5,"order":2,"alternate_center":6}。請依 [mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary] 欄序回答。
zero through n是導數階數0至n，不是所有導數數值都必須為0。固定的否拒絕漏階、錯中心、漏factorial或無依據餘項／級數結論。
查看答案與解釋
(wrong_center, 2, [18, 36, 54], 5, 6, 否, a_jet_is_bound_to_its_declared_center)
Wave6第六層核對=(wrong_center, 2, [18, 36, 54], 5, 6, 否, a_jet_is_bound_to_its_declared_center)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-definition-boundary · candidate
94. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-definition-boundary；資料={"mode":"missing_factorial","derivatives":[19,38,57],"center":6,"order":2,"alternate_center":7}。請依 [mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary] 欄序回答。
zero through n是導數階數0至n，不是所有導數數值都必須為0。固定的否拒絕漏階、錯中心、漏factorial或無依據餘項／級數結論。
查看答案與解釋
(missing_factorial, 2, [19, 38, 57], 6, 7, 否, coefficients_require_division_by_k_factorial)
Wave6第六層核對=(missing_factorial, 2, [19, 38, 57], 6, 7, 否, coefficients_require_division_by_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-definition-boundary · candidate
95. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-definition-boundary；資料={"mode":"remainder_claim","derivatives":[20,40,60],"center":7,"order":2,"alternate_center":8}。請依 [mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary] 欄序回答。
zero through n是導數階數0至n，不是所有導數數值都必須為0。固定的否拒絕漏階、錯中心、漏factorial或無依據餘項／級數結論。
查看答案與解釋
(remainder_claim, 2, [20, 40, 60], 7, 8, 否, a_Taylor_polynomial_alone_gives_no_remainder_or_series_equality)
Wave6第六層核對=(remainder_claim, 2, [20, 40, 60], 7, 8, 否, a_Taylor_polynomial_alone_gives_no_remainder_or_series_equality)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-definition-boundary · candidate
96. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-definition-boundary；資料={"mode":"missing_derivative","derivatives":[21,42],"center":8,"order":2,"alternate_center":9}。請依 [mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary] 欄序回答。
zero through n是導數階數0至n，不是所有導數數值都必須為0。固定的否拒絕漏階、錯中心、漏factorial或無依據餘項／級數結論。
查看答案與解釋
(missing_derivative, 2, [21, 42], 8, 9, 否, all_derivatives_zero_through_n_are_required)
Wave6第六層核對=(missing_derivative, 2, [21, 42], 8, 9, 否, all_derivatives_zero_through_n_are_required)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-definition-boundary · candidate
97. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-definition-boundary；資料={"mode":"wrong_center","derivatives":[22,44,66],"center":9,"order":2,"alternate_center":10}。請依 [mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary] 欄序回答。
zero through n是導數階數0至n，不是所有導數數值都必須為0。固定的否拒絕漏階、錯中心、漏factorial或無依據餘項／級數結論。
查看答案與解釋
(wrong_center, 2, [22, 44, 66], 9, 10, 否, a_jet_is_bound_to_its_declared_center)
Wave6第六層核對=(wrong_center, 2, [22, 44, 66], 9, 10, 否, a_jet_is_bound_to_its_declared_center)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-definition-boundary · candidate
98. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-definition-boundary；資料={"mode":"missing_factorial","derivatives":[23,46,69],"center":10,"order":2,"alternate_center":11}。請依 [mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary] 欄序回答。
zero through n是導數階數0至n，不是所有導數數值都必須為0。固定的否拒絕漏階、錯中心、漏factorial或無依據餘項／級數結論。
查看答案與解釋
(missing_factorial, 2, [23, 46, 69], 10, 11, 否, coefficients_require_division_by_k_factorial)
Wave6第六層核對=(missing_factorial, 2, [23, 46, 69], 10, 11, 否, coefficients_require_division_by_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-definition-boundary · candidate
99. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-definition-boundary；資料={"mode":"remainder_claim","derivatives":[24,48,72],"center":11,"order":2,"alternate_center":12}。請依 [mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary] 欄序回答。
zero through n是導數階數0至n，不是所有導數數值都必須為0。固定的否拒絕漏階、錯中心、漏factorial或無依據餘項／級數結論。
查看答案與解釋
(remainder_claim, 2, [24, 48, 72], 11, 12, 否, a_Taylor_polynomial_alone_gives_no_remainder_or_series_equality)
Wave6第六層核對=(remainder_claim, 2, [24, 48, 72], 11, 12, 否, a_Taylor_polynomial_alone_gives_no_remainder_or_series_equality)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-definition-boundary · candidate
100. Taylor多項式
固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-definition-boundary；資料={"mode":"missing_derivative","derivatives":[25,50],"center":12,"order":2,"alternate_center":13}。請依 [mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary] 欄序回答。
zero through n是導數階數0至n，不是所有導數數值都必須為0。固定的否拒絕漏階、錯中心、漏factorial或無依據餘項／級數結論。
查看答案與解釋
(missing_derivative, 2, [25, 50], 12, 13, 否, all_derivatives_zero_through_n_are_required)
Wave6第六層核對=(missing_derivative, 2, [25, 50], 12, 13, 否, all_derivatives_zero_through_n_are_required)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。
本題逐步詳解（AI 二次覆核）
Taylor多項式 · qf-mod43-taylor-definition-boundary · candidate
原始題目 JSONL · 驗收報告