第 28017 題：Newton法

原題與已知條件

Newton更新、root-first停止與nonzero derivative責任。操作=newton-boundary；明示資料={"mode":"zero_derivative","model":{"coefficients":[17,0,1],"x":0}}；請依 exact_ordered_tuple_declared_by_interface 回答。

• G001 — Newton更新、root-first停止與nonzero derivative責任。操作=newton-boundary；明示資料={"mode":"zero_derivative","model":{"coefficients":[17,0,1],"x":0}}；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-newton-method

逐步推導

source roles = [qf-mod50-newton-boundary, newton-boundary]

derived = [[mode, current_iterate, function_value, derivative_value, status, update_qualification_claim, boundary]]

Newton boundary polynomial ORIGINAL derivative coefficient1 := mul(1, 0)

Newton boundary polynomial ORIGINAL derivative coefficient1 = 0

Newton boundary polynomial ORIGINAL derivative coefficient2 := mul(2, 1)

Newton boundary polynomial ORIGINAL derivative coefficient2 = 2

source roles = [Newton boundary polynomial, [17, 0, 1]]

derived = [[17, 0, 1], [0, 2]]

Newton boundary record f ORIGINAL Horner multiply degree2 := mul(0, 0)

Newton boundary record f ORIGINAL Horner multiply degree2 = 0

Newton boundary record f ORIGINAL Horner add coefficient2 := sum(0, 1)

Newton boundary record f ORIGINAL Horner add coefficient2 = 1

Newton boundary record f ORIGINAL Horner multiply degree1 := mul(1, 0)

Newton boundary record f ORIGINAL Horner multiply degree1 = 0

Newton boundary record f ORIGINAL Horner add coefficient1 := sum(0, 0)

Newton boundary record f ORIGINAL Horner add coefficient1 = 0

Newton boundary record f ORIGINAL Horner multiply degree0 := mul(0, 0)

Newton boundary record f ORIGINAL Horner multiply degree0 = 0

Newton boundary record f ORIGINAL Horner add coefficient0 := sum(0, 17)

Newton boundary record f ORIGINAL Horner add coefficient0 = 17

Newton boundary record fprime ORIGINAL Horner multiply degree1 := mul(0, 0)

Newton boundary record fprime ORIGINAL Horner multiply degree1 = 0

Newton boundary record fprime ORIGINAL Horner add coefficient1 := sum(0, 2)

Newton boundary record fprime ORIGINAL Horner add coefficient1 = 2

Newton boundary record fprime ORIGINAL Horner multiply degree0 := mul(2, 0)

Newton boundary record fprime ORIGINAL Horner multiply degree0 = 0

Newton boundary record fprime ORIGINAL Horner add coefficient0 := sum(0, 0)

Newton boundary record fprime ORIGINAL Horner add coefficient0 = 0

source roles = [Newton boundary record, [17, 0, 1], [0, 2], 0]

derived = [17, 0, undefined_zero_derivative, null]

逐步反向核對

• S015 · 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。
• S014 · premise_and_result_audit — 以精確有理數重算 Newton boundary record fprime ORIGINAL Horner add coefficient0，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 Newton boundary record fprime ORIGINAL Horner multiply degree0，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 Newton boundary record fprime ORIGINAL Horner add coefficient1，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 Newton boundary record fprime ORIGINAL Horner multiply degree1，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 Newton boundary record f ORIGINAL Horner add coefficient0，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 Newton boundary record f ORIGINAL Horner multiply degree0，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 Newton boundary record f ORIGINAL Horner add coefficient1，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 Newton boundary record f ORIGINAL Horner multiply degree1，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 Newton boundary record f ORIGINAL Horner add coefficient2，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 Newton boundary record f ORIGINAL Horner multiply degree2，不以小數近似替代。
• S004 · 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。
• S003 · premise_and_result_audit — 以精確有理數重算 Newton boundary polynomial ORIGINAL derivative coefficient2，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 Newton boundary polynomial ORIGINAL derivative coefficient1，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

(zero_derivative, 0, 17, 0, undefined_zero_derivative, 否, division_by_zero_is_rejected_not_repaired)

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。