{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261007-advance-20261008-r04-derivative-0028",
  "pilot_number": "R0016-08553",
  "number": 8553,
  "family_id": "qf-derivative-quadratic-point",
  "title": {
    "zh-Hant": "第 8553 題：至多二次式的點導數",
    "en": "Question 8553: Point Derivative of a Degree-at-Most-Two Polynomial"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261007-advance-20261008-r04-derivative-0028",
    "batch_id": "batch-20261007-advance-20261008-r04-derivative-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-derivative"
    ],
    "title_zh": "至多二次式的點導數",
    "stem_zh": "全實數論域，以下函數／點均以明示實數模型解讀。f(x)=3x²+(-10)x+(1)，求 f′(0)，允許 a2=0。",
    "answer": {
      "kind": "short_text",
      "value": "-10",
      "explanation_zh": "導數=-10。產生端以值與導數的雙數 Horner 計算，不依有限點抽樣；常數項不改變導數。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "至多二次式的點導數"
      ]
    },
    "variant": {
      "family_id": "qf-derivative-quadratic-point",
      "seed": 202610084028,
      "parameters": {
        "a2": 3,
        "a1": -10,
        "a0": 1,
        "x": 0,
        "operation": "quadratic_derivative",
        "domain": "reals",
        "semantic_template": "qf-derivative-quadratic-point/{\"inputs\":[\"slope_only\",\"-10\"],\"operation\":\"quadratic_derivative\"}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_integer_rational_arithmetic",
        "producer_finite_enumeration",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-07",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261007-advance-20261008-r04-derivative-0028",
    "batch_id": "batch-20261007-advance-20261008-r04-derivative-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "dc63e477c05268886ec32c26bf083a29c3b9e768032c9d765e75d6f6d81aa64e",
      "family_id": "qf-derivative-quadratic-point",
      "seed": 202610084028,
      "semantic_template": "qf-derivative-quadratic-point/{\"inputs\":[\"slope_only\",\"-10\"],\"operation\":\"quadratic_derivative\"}"
    },
    "fields": {
      "title": "Point Derivative of a Degree-at-Most-Two Polynomial",
      "stem": "On all R, f(t)=(3)t²+(-10)t+(1), allowing a2=0. Find the derivative at x=0.",
      "answer": {
        "kind": "short_text",
        "value": "-10",
        "explanation": "The derivative is 2a2*x+a1=-10. A zero a2 gives a valid lower-degree case, not a falsely declared nonzero-leading-coefficient quadratic."
      },
      "skills": [
        "Point Derivative of a Degree-at-Most-Two Polynomial"
      ],
      "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."
    }
  },
  "givens": [
    {
      "id": "G001",
      "text": {
        "zh-Hant": "全實數論域，以下函數／點均以明示實數模型解讀。f(x)=3x²+(-10)x+(1)，求 f′(0)，允許 a2=0。",
        "en": "On all R, f(t)=(3)t²+(-10)t+(1), allowing a2=0. Find the derivative at x=0."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "declared real formula on all R;derivative means the two-sided h->0 quotient limit",
      "after": "finite-step evaluation and derivative existence are separate",
      "relation": "implies",
      "rule_id": "derivative_limit_definition_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "導數由[f(x+h)−f(x)]/h在h→0、h≠0的雙側極限定義；有限一次代值不是任意函數可微的證據。",
        "en": "A derivative is the two-sided limit of[f(x+h)−f(x)]/h as h→0 with h≠0.One finite evaluation is not evidence that an arbitrary function is differentiable."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "at original x a*x := mul(3, 0)",
      "after": "at original x a*x = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本步的原始操作數與運算順序精確計算；除法前須先核分母非零。",
        "en": "Evaluate exactly with this step's source operands and operation order.Check a denominator is nonzero before division."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 at original x a*x，不以小數近似替代。",
          "en": "Recompute at original x a*x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "0"
        ],
        "output": "0",
        "label": "at original x a*x"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "at original x a*x+b := sum(0, -10)",
      "after": "at original x a*x+b = -10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本步的原始操作數與運算順序精確計算；除法前須先核分母非零。",
        "en": "Evaluate exactly with this step's source operands and operation order.Check a denominator is nonzero before division."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 at original x a*x+b，不以小數近似替代。",
          "en": "Recompute at original x a*x+b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "-10"
        ],
        "output": "-10",
        "label": "at original x a*x+b"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "at original x (a*x+b)*x := mul(-10, 0)",
      "after": "at original x (a*x+b)*x = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本步的原始操作數與運算順序精確計算；除法前須先核分母非零。",
        "en": "Evaluate exactly with this step's source operands and operation order.Check a denominator is nonzero before division."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 at original x (a*x+b)*x，不以小數近似替代。",
          "en": "Recompute at original x (a*x+b)*x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-10",
          "0"
        ],
        "output": "0",
        "label": "at original x (a*x+b)*x"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "at original x f(x) := sum(0, 1)",
      "after": "at original x f(x) = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本步的原始操作數與運算順序精確計算；除法前須先核分母非零。",
        "en": "Evaluate exactly with this step's source operands and operation order.Check a denominator is nonzero before division."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 at original x f(x)，不以小數近似替代。",
          "en": "Recompute at original x f(x) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1"
        ],
        "output": "1",
        "label": "at original x f(x)"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "f(t)=(3)*t^2+(-10)*t+(1);x=0;f(x)=1;h!=0",
      "after": "f(x+h)-f(x)=h*(2*(3)*(0)+(-10)+(3)*h)",
      "relation": "implies",
      "rule_id": "quadratic_increment_factorization",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用(x+h)²=x²+2xh+h²展開，常數項與原x項相消，再提出h。a2=0時仍是合法的低次模型，不另要求首項非零。",
        "en": "Expand(x+h)²=x²+2xh+h²,cancel the original terms,and factor out h.The a2=0 lower-degree case is valid;no nonzero-leading-coefficient assumption is added."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "twice quadratic coefficient := mul(2, 3)",
      "after": "twice quadratic coefficient = 6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本步的原始操作數與運算順序精確計算；除法前須先核分母非零。",
        "en": "Evaluate exactly with this step's source operands and operation order.Check a denominator is nonzero before division."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 twice quadratic coefficient，不以小數近似替代。",
          "en": "Recompute twice quadratic coefficient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "3"
        ],
        "output": "6",
        "label": "twice quadratic coefficient"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "twice a times x := mul(6, 0)",
      "after": "twice a times x = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本步的原始操作數與運算順序精確計算；除法前須先核分母非零。",
        "en": "Evaluate exactly with this step's source operands and operation order.Check a denominator is nonzero before division."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 twice a times x，不以小數近似替代。",
          "en": "Recompute twice a times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "6",
          "0"
        ],
        "output": "0",
        "label": "twice a times x"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "linear part of quotient := sum(0, -10)",
      "after": "linear part of quotient = -10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本步的原始操作數與運算順序精確計算；除法前須先核分母非零。",
        "en": "Evaluate exactly with this step's source operands and operation order.Check a denominator is nonzero before division."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 linear part of quotient，不以小數近似替代。",
          "en": "Recompute linear part of quotient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "-10"
        ],
        "output": "-10",
        "label": "linear part of quotient"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "for h!=0,quotient=-10+(3)*h",
      "after": "h->0 derivative=-10",
      "relation": "implies",
      "rule_id": "quadratic_two_sided_limit",
      "conditions": [],
      "explanation": {
        "zh-Hant": "非零h可消去後，差商與候選導數的差是a2*h。若a2=0誤差恆0；否則對任意ε>0取δ=ε/(1+|a2|)，0<|h|<δ即誤差<ε，故雙側極限存在。",
        "en": "After cancelling nonzero h,the quotient error is a2*h.If a2=0 the error is zero;otherwise for any ε>0 choose δ=ε/(1+|a2|).Then0<|h|<δ makes the error<ε,establishing the two-sided limit."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": "-10"
  },
  "conclusion": {
    "canonical_value": "-10",
    "explanation": {
      "zh-Hant": "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。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "definition_note": {
    "zh-Hant": "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。",
    "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate."
  },
  "source": {
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      "batch_sha256": "12c2ea336c3fcf77c87dbf06558fe211c51f3f7dd951145c5f88c200dae55752",
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      "english_rows_path": "public/data/i18n/questions/batch-20261007-advance-20261008-r04-derivative-foundations.en.jsonl"
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        "original_path": "public/data/mko/derivative.json",
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    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "8e8954cf690c44da1ec1efa08994a485c73ca8706b7865d854e3f2cdfd4ca89f",
    "original_candidate_sha256": "f38712343e9aebc64f9b2d3b8c600b626361e524b3ccaf91736f09be40c3b809",
    "original_candidate_line_sha256": "f38712343e9aebc64f9b2d3b8c600b626361e524b3ccaf91736f09be40c3b809"
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  "historical_source_state": {
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      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261026-005-p17",
    "review_date": "2026-10-26",
    "owner_authorized_reader_publication": true,
    "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.",
    "review_receipts": {
      "semantic_review_sha256": "00fcddb10a9fb0f56697308976a52e68e2537472bd8e68ecbe96d3c4553f4ad2",
      "question_first_witness_sha256": "198e04380b2d32aef2375b07ca0df1f8afa454d1184993173fad8a3c9187a3d1"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 10,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "6833945d2ccb3fba451309a3b259cd864cbd51dc3eff9242dc8c3036e0f56d85",
      "arithmetic_steps": 7
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    "uses_answer_as_premise": false,
    "human_review": "NOT_PERFORMED",
    "formalization": "not_formalized",
    "Lean": "NOT_RUN",
    "Canonical_truth_promotion": false,
    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
  }
}