{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-taylor-polynomial-01",
  "mko_id": "mko-taylor-polynomial",
  "title": {
    "zh-Hant": "中心平移與 factorial 的 Taylor 多項式",
    "en": "A Taylor polynomial retaining the center and factorials"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-taylor-polynomial-01",
    "source": {
      "mko_id": "mko-taylor-polynomial",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "ee8f291872fc7e5a97cdbe468fffe5f7e4e3857373415ca1a65e5ab2b61a53a1",
      "english_sha256": "5009d7ebab6a826e0e1ca6a349eadfed04ba911cc721a2b103d2e4e14229974f"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "algebra",
      "polynomial",
      "definition_here"
    ],
    "goal": {
      "zh-Hant": "以中心 a=1、order 2 構造指定三次多項式的 T_2。",
      "en": "Construct T_2 at center a=1 for the specified cubic polynomial."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "f(x)=1+2(x-1)+3(x-1)^2-(x-1)^3，a=1，n=2。",
          "en": "f(x)=1+2(x-1)+3(x-1)^2-(x-1)^3, a=1, n=2."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1"
        ],
        "before_tex": "h=x-1",
        "after_tex": "f(x)=1+2h+3h^2-h^3",
        "relation": "definition",
        "rule": "definition_here",
        "conditions": [],
        "explanation": {
          "zh-Hant": "固定中心平移 h，所有 jet 與冪基底都用同一 a。",
          "en": "Fix the shift h so every jet and power basis uses the same center a."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "h 不是 x，不能省略 -1。",
            "en": "h is not x; the shift -1 cannot be omitted."
          }
        },
        "leaf_basis": "definition_here"
      },
      {
        "id": "S02",
        "uses": [
          "S01"
        ],
        "before_tex": "f'(x)=2+6h-3h^2",
        "after_tex": "f(1)=1,\\quad f'(1)=2",
        "relation": "implies",
        "rule": "polynomial",
        "conditions": [],
        "explanation": {
          "zh-Hant": "h 對 x 的導數為 1，代 h=0 得前兩個 jet 值。",
          "en": "The derivative of h with respect to x is 1; setting h=0 gives the first two jet values."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S01"
          ],
          "explanation": {
            "zh-Hant": "導數都在同一中心 1 評估。",
            "en": "Every derivative is evaluated at the same center 1."
          }
        },
        "leaf_basis": "polynomial"
      },
      {
        "id": "S03",
        "uses": [
          "S02"
        ],
        "before_tex": "f''(x)=6-6h",
        "after_tex": "f''(1)=6",
        "relation": "implies",
        "rule": "polynomial",
        "conditions": [],
        "explanation": {
          "zh-Hant": "再次微分後於中心評估，補齊 0≤k≤2 的全部 jet。",
          "en": "Differentiate again and evaluate at the center, completing the entire jet for orders zero through two."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S02"
          ],
          "explanation": {
            "zh-Hant": "沒有只給最高階而漏低階。",
            "en": "No lower order is omitted in favor of only the highest one."
          }
        },
        "leaf_basis": "polynomial"
      },
      {
        "id": "S04",
        "uses": [
          "S02",
          "S03"
        ],
        "before_tex": "T_2(x)=\\sum_{k=0}^{2}\\frac{f^{(k)}(1)}{k!}(x-1)^k",
        "after_tex": "T_2(x)=1+2h+\\frac6{2!}h^2",
        "relation": "definition",
        "rule": "definition_here",
        "conditions": [],
        "explanation": {
          "zh-Hant": "將 jet 依 factorial 縮放後放進中心冪基底。",
          "en": "Scale each jet value by its factorial in the centered power basis."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S02",
            "S03"
          ],
          "explanation": {
            "zh-Hant": "二次係數不是裸導數 6。",
            "en": "The quadratic coefficient is not the unscaled derivative 6."
          }
        },
        "leaf_basis": "definition_here"
      },
      {
        "id": "S05",
        "uses": [
          "S04"
        ],
        "before_tex": "2!=2",
        "after_tex": "T_2(x)=1+2(x-1)+3(x-1)^2",
        "relation": "equivalent",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "精確除 factorial，得到所求有限多項式。",
          "en": "Exact factorial division gives the required finite polynomial."
        },
        "reverse_audit": {
          "kind": "inverse_equivalence",
          "required_ids": [
            "S04"
          ],
          "explanation": {
            "zh-Hant": "可反向由二次係數 3 乘 2! 回讀二階導數 6。",
            "en": "Multiplying coefficient 3 by 2! recovers the second derivative 6."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "S05"
        ],
        "before_tex": "f(x)-T_2(x)",
        "after_tex": "f(x)-T_2(x)=-(x-1)^3",
        "relation": "equivalent",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "完整原函數讓此例可直接相減；一般只有 jet 並無此完整差式。",
          "en": "The full function permits exact subtraction here; a general finite jet would not provide this difference formula."
        },
        "reverse_audit": {
          "kind": "inverse_equivalence",
          "required_ids": [
            "G1",
            "S05"
          ],
          "explanation": {
            "zh-Hant": "此等式不等於宣稱一般 Taylor 餘項定理已證。",
            "en": "This identity does not claim a general Taylor remainder theorem."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S07",
        "uses": [
          "G1"
        ],
        "before_tex": "f^{(3)}(1)=-6",
        "after_tex": "T_3(x)=T_2(x)-h^3=f(x)",
        "relation": "implies",
        "rule": "polynomial",
        "conditions": [],
        "explanation": {
          "zh-Hant": "此 f degree=3，加入三階係數 -6/3!=-1 才精確重建。",
          "en": "Since this f has degree three, adding the cubic coefficient -6/3!=-1 reconstructs it exactly."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "degree≤n 的特殊重建不外推任意 smooth 函數。",
            "en": "Exact reconstruction for degree≤n is not extrapolated to arbitrary smooth functions."
          }
        },
        "leaf_basis": "polynomial"
      },
      {
        "id": "S08",
        "uses": [
          "S02",
          "S05"
        ],
        "before_tex": "n=0\\text{ as a separate order boundary}",
        "after_tex": "T_0(x)=f(1)=1",
        "relation": "definition",
        "rule": "definition_here",
        "conditions": [],
        "explanation": {
          "zh-Hant": "零階沒有 x 的冪項，仍是定義良好的常數多項式。",
          "en": "At order zero there are no positive-power terms; the constant polynomial remains well-defined."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S02",
            "S05"
          ],
          "explanation": {
            "zh-Hant": "不能把 n=0 誤作沒有任何項。",
            "en": "Order zero does not mean there are no terms."
          }
        },
        "leaf_basis": "definition_here"
      }
    ],
    "result": {
      "zh-Hant": "T_2=1+2(x-1)+3(x-1)^2，差為 -(x-1)^3；T_3 才等於此 f。",
      "en": "T_2=1+2(x-1)+3(x-1)^2, with difference -(x-1)^3; T_3 equals this f."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "二次係數是 6。",
          "en": "The quadratic coefficient is 6."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "必除 2! 得 3。",
          "en": "Division by 2! gives 3."
        }
      },
      {
        "claim": {
          "zh-Hant": "相同 jet 可直接放到中心 0。",
          "en": "The same jet can be moved directly to center zero."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "jet 綁中心 1，中心改變需重新評估。",
          "en": "The jet is bound to center 1; a new center requires reevaluation."
        }
      },
      {
        "claim": {
          "zh-Hant": "T_2 必等於完整 f。",
          "en": "T_2 necessarily equals the full f."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "本例差為非零三次項。",
          "en": "The difference here is a nonzero cubic term."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "固定center a及order n≥0，並要求f^(k)(a)對0≤k≤n全部存在。T_n(x)=sum_{k=0}^n f^(k)(a)(x-a)^k/k!；n=0時為常數f(a)。Taylor polynomial本身不含remainder bound，也不保證無窮Taylor series收斂或等於f。若f本身是degree≤n的多項式則可exact重建；一般smooth函數不得由此外推analytic。",
          "en": "Fix center a and nonnegative order n and require every f^(k)(a) for0≤k≤n to exist. T_n(x)=sum over k=0..n of f^(k)(a)*(x−a)^k/k!; at n=0 this is constant f(a). The polynomial itself gives no remainder bound or guarantee that an infinite Taylor series converges or equals f. A polynomial f of degree at most n reconstructs exactly; this does not make an arbitrary smooth function analytic."
        },
        "source": {
          "mko_id": "mko-taylor-polynomial",
          "sha256": "ee8f291872fc7e5a97cdbe468fffe5f7e4e3857373415ca1a65e5ab2b61a53a1",
          "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f"
        }
      },
      {
        "id": "algebra",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "實數域的等式運算；除法前分母必非零，不等式乘正數保持方向。",
          "en": "Use real-field identities; division requires a nonzero denominator and multiplication by a positive number preserves inequalities."
        }
      },
      {
        "id": "polynomial",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "多項式在實數上連續且可逐項微分；d(x^k)/dx=k x^(k-1)，常數導數為零。",
          "en": "Real polynomials are continuous and differentiate termwise: the derivative of x^k is k*x^(k-1), and constants differentiate to zero."
        }
      },
      {
        "id": "definition_here",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "中心 a、階數 n 的 Taylor 多項式為從 k=0 到 n 的 f^(k)(a)*(x-a)^k/k! 之和，須全部中心 jet 存在；n=0 時為 f(a)。",
          "en": "The Taylor polynomial of order n at a sums f^(k)(a)*(x-a)^k/k! from k=0 through n and requires the complete center jet; at n=0 it is f(a)."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "quadratic-coefficient",
        "lhs": "6/factorial(2)",
        "rhs": "3",
        "step": "S05"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "只建有限 Taylor 多項式，不把有限 jet 當一般餘項或解析性證書。",
      "en": "Construct a finite Taylor polynomial only; a finite jet is no general remainder or analyticity certificate."
    },
    "md_symbols": [
      [
        "h=x-1",
        "中心位移",
        "center shift"
      ],
      [
        "T_n",
        "有限 Taylor 多項式",
        "finite Taylor polynomial"
      ]
    ],
    "md_error_controls": [
      {
        "title": "factorial / Factorials",
        "zh": "每個 k 階導數除 k!，含 0!=1。",
        "en": "Divide each k-th derivative by k!, including 0!=1."
      },
      {
        "title": "不帶餘項承諾 / No implicit remainder promise",
        "zh": "有限多項式本身不保證任意區間誤差。",
        "en": "The finite polynomial alone promises no interval error bound."
      }
    ],
    "md_open_items": []
  },
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    "bilingual_sync": "complete_source_bound",
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}