{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-rate-of-change",
  "semantic_family_id": "sf-rate-of-change",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "兩點間的平均變化率"
  },
  "summary": {
    "zh-Hant": "平均變化率是輸出差除以非零輸入差，不是瞬時導數。"
  },
  "statement": {
    "zh-Hant": "對已知兩個函數值 f(x0)=y0、f(x1)=y1 且 x0≠x1，兩點間平均變化率為 (y1−y0)/(x1−x0)。若 x0=x1，此商不具本條目資格，不以零或 0/0 代替。對 f(x)=mx+b，該商為 m；對 q(x)=a(x−h)²+k，該商為 a(x0+x1−2h)。"
  },
  "formula": {
    "tex": "\\frac{b-a}{d-c}",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mfrac><mrow><mi>b</mi><mo>−</mo><mi>a</mi></mrow><mrow><mi>d</mi><mo>−</mo><mi>c</mi></mrow></mfrac></math>",
    "semantic_ast": {
      "type": "fraction",
      "numerator": {
        "type": "subtraction",
        "left": {
          "type": "symbol",
          "name": "b",
          "glyph": "b"
        },
        "right": {
          "type": "symbol",
          "name": "a",
          "glyph": "a"
        }
      },
      "denominator": {
        "type": "subtraction",
        "left": {
          "type": "symbol",
          "name": "d",
          "glyph": "d"
        },
        "right": {
          "type": "symbol",
          "name": "c",
          "glyph": "c"
        }
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "a5a2f5ab3756a5fde866e849202dbeb569f9d11eb9a36e90bc3966f2b3748362"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "端點須為同一指定函數的合法輸入；直接兩點題只使用明示輸出值。"
    },
    {
      "id": "a2",
      "zh-Hant": "非資格題外，輸入差必須非零；題庫採整數端點與係數，答案規約為最簡正分母分數。"
    },
    {
      "id": "a3",
      "zh-Hant": "二次切片 a≠0；代表分式 a、b 是兩個輸出，c、d 是對應輸入，不與頂點式係數混用。"
    }
  ],
  "symbols": [
    {
      "token": "a",
      "role": "first_output",
      "role_zh": "第一輸出 y0",
      "scope": "representative_formula"
    },
    {
      "token": "b",
      "role": "second_output",
      "role_zh": "第二輸出 y1",
      "scope": "representative_formula"
    },
    {
      "token": "c",
      "role": "first_input",
      "role_zh": "第一輸入 x0",
      "scope": "representative_formula"
    },
    {
      "token": "d",
      "role": "second_input",
      "role_zh": "第二輸入 x1",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-ratio-proportion",
      "title_zh": "比與比例",
      "reason_zh": "非零分母比值的語義。"
    },
    {
      "id": "mko-function-graph",
      "title_zh": "函數圖像",
      "reason_zh": "兩點來自同一指定函數。"
    },
    {
      "id": "mko-linear-function",
      "title_zh": "仿射函數與一次函數",
      "reason_zh": "仿射式的平均變化率切片。"
    },
    {
      "id": "mko-quadratic-function",
      "title_zh": "頂點式二次函數",
      "reason_zh": "二次切片先固定頂點式。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "端點順序同時反轉時，分子分母同換號，商不變；不能只反轉其中一對。負的輸入差仍可規約到正分母。",
      "平均變化率描述兩個指定輸入之間的差比，不聲稱處處導數存在；即使二次式的商可化簡，也不把它當作某一端點的瞬時導數。",
      "常數仿射式 m=0 在不同輸入上平均變化率為零。二次式中 k 平移輸出，於差中抵消；對稱端點可有零平均變化率，而函數並不是常數。"
    ],
    "common_misconception_zh": "相同輸入也把差商填零，或把平均變化率直接叫導數。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "非零輸入差的有限精確差商可重算；導數、極限及一般變化率定理未新增形式 Evidence。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "綁定固定版 slope API 並保留非零分母；Mathlib 對除零的全定義不能替代本教學資格前提。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-rate-of-change.md 整理；2026-10-05 脈衝 R03（使用者手動接續 2026-10-06 工作序列；實際日期仍為 2026-10-05），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-05"
  }
}