{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261007-advance-20261009-r01-fundamental-theorem-calculus-0037", "pilot_number": "R0019-10162", "number": 10162, "family_id": "qf-ftc-moving-bounds-derivative", "title": { "zh-Hant": "第 10162 題：移動上下限的有向積分導數", "en": "Question 10162: Derivative of an Oriented Integral with Moving Bounds" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261007-advance-20261009-r01-fundamental-theorem-calculus-0037", "batch_id": "batch-20261007-advance-20261009-r01-fundamental-theorem-calculus-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-fundamental-theorem-calculus" ], "title_zh": "移動上下限的有向積分導數", "stem_zh": "全實數論域；連續多項式被積函數，FTC的明示充分前提。升冪係數 f=[-1, 12, 1, 4]，l(t)=-3t+(-3)、u(t)=0t+(-1)。H(t)=∫_l(t)^u(t) f(s)ds，求 H′(1)。", "answer": { "kind": "expression", "value": "-2703/1", "explanation_zh": "移動端點導數=-2703/1。分別以 f(u)u′ 與 f(l)l′ 計算再相減；某點上下限相等不保證導數零，這是連續模型的條件計算。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "移動上下限的有向積分導數" ] }, "variant": { "family_id": "qf-ftc-moving-bounds-derivative", "seed": 202610091037, "parameters": { "f": [ -1, 12, 1, 4 ], "lower_m": -3, "lower_b": -3, "upper_m": 0, "upper_b": -1, "x": 1, "operation": "ftc_moving_bounds_derivative", "domain": "reals", "representation": "ascending_powers", "semantic_template": "qf-ftc-moving-bounds-derivative/[[-1,12,1,4],-3,-3,0,-1,1]" } }, "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-20261009-r01-fundamental-theorem-calculus-0037", "batch_id": "batch-20261007-advance-20261009-r01-fundamental-theorem-calculus-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "8932d2d79cc19596376a1c09a9d1d61cb032c63ac0d29b71a4b2cdad575cf036", "family_id": "qf-ftc-moving-bounds-derivative", "seed": 202610091037, "semantic_template": "qf-ftc-moving-bounds-derivative/[[-1,12,1,4],-3,-3,0,-1,1]" }, "fields": { "title": "Derivative of an Oriented Integral with Moving Bounds", "stem": "On all R, continuous f has ascending coefficients [-1, 12, 1, 4], constant term first. Let lower l(x)=(-3)x+(-3) and upper u(x)=(0)x+(-1). Find the derivative of ∫_l(x)^u(x) f(t)dt at x=1. Use reduced positive-denominator fractions, retaining /1 for integers.", "answer": { "kind": "expression", "value": "-2703/1", "explanation": "Use f(u(x))*u′(x)−f(l(x))*l′(x) with the declared continuous integrand and affine bounds, obtaining -2703/1. Bounds coinciding at this point do not force equal velocities or a zero derivative." }, "skills": [ "Derivative of an Oriented Integral with Moving Bounds" ], "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；連續多項式被積函數，FTC的明示充分前提。升冪係數 f=[-1, 12, 1, 4]，l(t)=-3t+(-3)、u(t)=0t+(-1)。H(t)=∫_l(t)^u(t) f(s)ds，求 H′(1)。", "en": "On all R, continuous f has ascending coefficients [-1, 12, 1, 4], constant term first. Let lower l(x)=(-3)x+(-3) and upper u(x)=(0)x+(-1). Find the derivative of ∫_l(x)^u(x) f(t)dt at x=1. Use reduced positive-denominator fractions, retaining /1 for integers." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "raw integrand f=[-1, 12, 1, 4];ascending powers on all R", "after": "normalized coefficients=[-1/1, 12/1, 1/1, 4/1];declared continuous real polynomial", "relation": "implies", "rule_id": "declared_FTC_continuity_hypothesis", "conditions": [], "explanation": { "zh-Hant": "本題明示全R連續多項式，積分採有向端點。FTC關係只在這些充分前提下使用，不以有限核對宣稱新增一般FTC形式證明。", "en": "The declared integrand is a continuous real polynomial on all R,and endpoints are oriented.Use FTC only under these supplied sufficient hypotheses;finite checks are not a new general formal FTC proof." }, "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": "zero primitive coefficient1 := div(-1, 1)", "after": "zero primitive coefficient1 = -1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。", "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 zero primitive coefficient1，不以小數近似替代。", "en": "Recompute zero primitive coefficient1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "-1", "1" ], "output": "-1", "label": "zero primitive coefficient1" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "integrand power=0;coefficient=-1/1", "after": "primitive power=1;coefficient=-1/1", "relation": "implies", "rule_id": "polynomial_primitive_coefficient_rule", "conditions": [], "explanation": { "zh-Hant": "c*x^k的零常數原函數把c/(k+1)放在x^(k+1)，分母k+1正；導回即c*x^k。另加任意常數仍是原函數。", "en": "For c*x^k place c/(k+1) at x^(k+1),with positive denominator k+1.Differentiating recovers c*x^k.Any additive constant gives another primitive." }, "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": "S004", "uses": [ "G001", "S003" ], "before": "zero primitive coefficient2 := div(12, 2)", "after": "zero primitive coefficient2 = 6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。", "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 zero primitive coefficient2，不以小數近似替代。", "en": "Recompute zero primitive coefficient2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "12", "2" ], "output": "6", "label": "zero primitive coefficient2" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "integrand power=1;coefficient=12/1", "after": "primitive power=2;coefficient=6/1", "relation": "implies", "rule_id": "polynomial_primitive_coefficient_rule", "conditions": [], "explanation": { "zh-Hant": "c*x^k的零常數原函數把c/(k+1)放在x^(k+1)，分母k+1正；導回即c*x^k。另加任意常數仍是原函數。", "en": "For c*x^k place c/(k+1) at x^(k+1),with positive denominator k+1.Differentiating recovers c*x^k.Any additive constant gives another primitive." }, "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": "S006", "uses": [ "G001", "S005" ], "before": "zero primitive coefficient3 := div(1, 3)", "after": "zero primitive coefficient3 = 1/3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。", "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 zero primitive coefficient3，不以小數近似替代。", "en": "Recompute zero primitive coefficient3 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1", "3" ], "output": "1/3", "label": "zero primitive coefficient3" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "integrand power=2;coefficient=1/1", "after": "primitive power=3;coefficient=1/3", "relation": "implies", "rule_id": "polynomial_primitive_coefficient_rule", "conditions": [], "explanation": { "zh-Hant": "c*x^k的零常數原函數把c/(k+1)放在x^(k+1)，分母k+1正；導回即c*x^k。另加任意常數仍是原函數。", "en": "For c*x^k place c/(k+1) at x^(k+1),with positive denominator k+1.Differentiating recovers c*x^k.Any additive constant gives another primitive." }, "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": "S008", "uses": [ "G001", "S007" ], "before": "zero primitive coefficient4 := div(4, 4)", "after": "zero primitive coefficient4 = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。", "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 zero primitive coefficient4，不以小數近似替代。", "en": "Recompute zero primitive coefficient4 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "4", "4" ], "output": "1", "label": "zero primitive coefficient4" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "integrand power=3;coefficient=4/1", "after": "primitive power=4;coefficient=1/1", "relation": "implies", "rule_id": "polynomial_primitive_coefficient_rule", "conditions": [], "explanation": { "zh-Hant": "c*x^k的零常數原函數把c/(k+1)放在x^(k+1)，分母k+1正；導回即c*x^k。另加任意常數仍是原函數。", "en": "For c*x^k place c/(k+1) at x^(k+1),with positive denominator k+1.Differentiating recovers c*x^k.Any additive constant gives another primitive." }, "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": "S010", "uses": [ "G001", "S009" ], "before": "original integrand coefficients=[-1/1, 12/1, 1/1, 4/1]", "after": "zero-constant primitive coefficients=[0/1, -1/1, 6/1, 1/3, 1/1]", "relation": "implies", "rule_id": "zero_constant_polynomial_primitive", "conditions": [], "explanation": { "zh-Hant": "這是方便計算的一個原函數代表，不把常數族丟掉當成唯一原函數。", "en": "This is one convenient representative,not a claim that the primitive is unique without its constant family." }, "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": "S011", "uses": [ "G001", "S010" ], "before": "constructed P0 derivative coefficient0 := mul(1, -1)", "after": "constructed P0 derivative coefficient0 = -1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。", "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 constructed P0 derivative coefficient0，不以小數近似替代。", "en": "Recompute constructed P0 derivative coefficient0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "-1" ], "output": "-1", "label": "constructed P0 derivative coefficient0" } }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "constructed P0:source power=1;coefficient=-1/1", "after": "derivative power=0;coefficient=-1/1", "relation": "implies", "rule_id": "polynomial_derivative_coefficient_rule", "conditions": [], "explanation": { "zh-Hant": "次數k乘原係數並把次數降1，常數項不參與；完整保留所有非零高次項。", "en": "Multiply the coefficient by k and lower its power by1.The constant contributes nothing;retain every higher nonzero term." }, "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": "S013", "uses": [ "G001", "S012" ], "before": "constructed P0 derivative coefficient1 := mul(2, 6)", "after": "constructed P0 derivative coefficient1 = 12", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。", "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 constructed P0 derivative coefficient1，不以小數近似替代。", "en": "Recompute constructed P0 derivative coefficient1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "6" ], "output": "12", "label": "constructed P0 derivative coefficient1" } }, { "id": "S014", "uses": [ "G001", "S013" ], "before": "constructed P0:source power=2;coefficient=6/1", "after": "derivative power=1;coefficient=12/1", "relation": "implies", "rule_id": "polynomial_derivative_coefficient_rule", "conditions": [], "explanation": { "zh-Hant": "次數k乘原係數並把次數降1，常數項不參與；完整保留所有非零高次項。", "en": "Multiply the coefficient by k and lower its power by1.The constant contributes nothing;retain every higher nonzero term." }, "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": "S015", "uses": [ "G001", "S014" ], "before": "constructed P0 derivative coefficient2 := mul(3, 1/3)", "after": "constructed P0 derivative coefficient2 = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。", "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 constructed P0 derivative coefficient2，不以小數近似替代。", "en": "Recompute constructed P0 derivative coefficient2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "3", "1/3" ], "output": "1", "label": "constructed P0 derivative coefficient2" } }, { "id": "S016", "uses": [ "G001", "S015" ], "before": "constructed P0:source power=3;coefficient=1/3", "after": "derivative power=2;coefficient=1/1", "relation": "implies", "rule_id": "polynomial_derivative_coefficient_rule", "conditions": [], "explanation": { "zh-Hant": "次數k乘原係數並把次數降1，常數項不參與；完整保留所有非零高次項。", "en": "Multiply the coefficient by k and lower its power by1.The constant contributes nothing;retain every higher nonzero term." }, "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": "S017", "uses": [ "G001", "S016" ], "before": "constructed P0 derivative coefficient3 := mul(4, 1)", "after": "constructed P0 derivative coefficient3 = 4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。", "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 constructed P0 derivative coefficient3，不以小數近似替代。", "en": "Recompute constructed P0 derivative coefficient3 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "4", "1" ], "output": "4", "label": "constructed P0 derivative coefficient3" } }, { "id": "S018", "uses": [ "G001", "S017" ], "before": "constructed P0:source power=4;coefficient=1/1", "after": "derivative power=3;coefficient=4/1", "relation": "implies", "rule_id": "polynomial_derivative_coefficient_rule", "conditions": [], "explanation": { "zh-Hant": "次數k乘原係數並把次數降1，常數項不參與；完整保留所有非零高次項。", "en": "Multiply the coefficient by k and lower its power by1.The constant contributes nothing;retain every higher nonzero term." }, "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": "S019", "uses": [ "G001", "S018" ], "before": "P0 derivative coefficients=[-1/1, 12/1, 1/1, 4/1];f=[-1/1, 12/1, 1/1, 4/1]", "after": "P0'=f on all R in this declared polynomial model", "relation": "implies", "rule_id": "constructed_primitive_coefficient_identity", "conditions": [], "explanation": { "zh-Hant": "完整係數身份給原函數關係，搭配明示連續前提可用FTC端點差；不是觀察有限點猜函數。", "en": "The full coefficient identity establishes the primitive relation in the declared model.With the supplied continuity premise use the FTC endpoint difference,not a function guessed from sample points." }, "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": "S020", "uses": [ "G001", "S019" ], "before": "lower bound m*x := mul(-3, 1)", "after": "lower bound m*x = -3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。", "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 lower bound m*x，不以小數近似替代。", "en": "Recompute lower bound m*x with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-3", "1" ], "output": "-3", "label": "lower bound m*x" } }, { "id": "S021", "uses": [ "G001", "S020" ], "before": "lower bound m*x+b := sum(-3, -3)", "after": "lower bound m*x+b = -6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。", "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 lower bound m*x+b，不以小數近似替代。", "en": "Recompute lower bound m*x+b with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-3", "-3" ], "output": "-6", "label": "lower bound m*x+b" } }, { "id": "S022", "uses": [ "G001", "S021" ], "before": "upper bound m*x := mul(0, 1)", "after": "upper bound m*x = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。", "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 upper bound m*x，不以小數近似替代。", "en": "Recompute upper bound m*x with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "1" ], "output": "0", "label": "upper bound m*x" } }, { "id": "S023", "uses": [ "G001", "S022" ], "before": "upper bound m*x+b := sum(0, -1)", "after": "upper bound m*x+b = -1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。", "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 upper bound m*x+b，不以小數近似替代。", "en": "Recompute upper bound m*x+b with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "-1" ], "output": "-1", "label": "upper bound m*x+b" } }, { "id": "S024", "uses": [ "G001", "S023" ], "before": "H(x)=P0(u(x))-P0(l(x));P0'=f;lower slope=-3;upper slope=0", "after": "H'(x)=f(u(x))*upper_slope-f(l(x))*lower_slope", "relation": "implies", "rule_id": "FTC_moving_bounds_two_chain_factors", "conditions": [], "explanation": { "zh-Hant": "先由有向端點差，再對兩個合成各用鏈鎖律；下限整項要減，常數端點斜率0。某點上下限相等不代表速度相同，也不強迫導數0。", "en": "Use the oriented endpoint difference,then the chain rule for both compositions.Subtract the whole lower-bound term;constant bounds have slope0.Equal bound values at one point do not mean equal velocities or force a zero derivative." }, "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": "S025", "uses": [ "G001", "S024" ], "before": "f at upper Horner multiply3 := mul(0, -1)", "after": "f at upper Horner multiply3 = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。", "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 f at upper Horner multiply3，不以小數近似替代。", "en": "Recompute f at upper Horner multiply3 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "-1" ], "output": "0", "label": "f at upper Horner multiply3" } }, { "id": "S026", "uses": [ "G001", "S025" ], "before": "f at upper Horner add3 := sum(0, 4)", "after": "f at upper Horner add3 = 4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。", "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 f at upper Horner add3，不以小數近似替代。", "en": "Recompute f at upper Horner add3 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "4" ], "output": "4", "label": "f at upper Horner add3" } }, { "id": "S027", "uses": [ "G001", "S026" ], "before": "f at upper Horner multiply2 := mul(4, -1)", "after": "f at upper Horner multiply2 = -4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。", "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 f at upper Horner multiply2，不以小數近似替代。", "en": "Recompute f at upper Horner multiply2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "4", "-1" ], "output": "-4", "label": "f at upper Horner multiply2" } }, { "id": "S028", "uses": [ "G001", "S027" ], "before": "f at upper Horner add2 := sum(-4, 1)", "after": "f at upper Horner add2 = -3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。", "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 f at upper Horner add2，不以小數近似替代。", "en": "Recompute f at upper Horner add2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-4", "1" ], "output": "-3", "label": "f at upper Horner add2" } }, { "id": "S029", "uses": [ "G001", "S028" ], "before": "f at upper Horner multiply1 := mul(-3, -1)", "after": "f at upper Horner multiply1 = 3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。", "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 f at upper Horner multiply1，不以小數近似替代。", "en": "Recompute f at upper Horner multiply1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-3", "-1" ], "output": "3", "label": "f at upper Horner multiply1" } }, { "id": "S030", "uses": [ "G001", "S029" ], "before": "f at upper Horner add1 := sum(3, 12)", "after": "f at upper Horner add1 = 15", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。", "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 f at upper Horner add1，不以小數近似替代。", "en": "Recompute f at upper Horner add1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "3", "12" ], "output": "15", "label": "f at upper Horner add1" } }, { "id": "S031", "uses": [ "G001", "S030" ], "before": "f at upper Horner multiply0 := mul(15, -1)", "after": "f at upper Horner multiply0 = -15", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。", "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 f at upper Horner multiply0，不以小數近似替代。", "en": "Recompute f at upper Horner multiply0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "15", "-1" ], "output": "-15", "label": "f at upper Horner multiply0" } }, { "id": "S032", "uses": [ "G001", "S031" ], "before": "f at upper Horner add0 := sum(-15, -1)", "after": "f at upper Horner add0 = -16", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。", "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 f at upper Horner add0，不以小數近似替代。", "en": "Recompute f at upper Horner add0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-15", "-1" ], "output": "-16", "label": "f at upper Horner add0" } }, { "id": "S033", "uses": [ "G001", "S032" ], "before": "f at lower Horner multiply3 := mul(0, -6)", "after": "f at lower Horner multiply3 = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。", "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 f at lower Horner multiply3，不以小數近似替代。", "en": "Recompute f at lower Horner multiply3 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "-6" ], "output": "0", "label": "f at lower Horner multiply3" } }, { "id": "S034", "uses": [ "G001", "S033" ], "before": "f at lower Horner add3 := sum(0, 4)", "after": "f at lower Horner add3 = 4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。", "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 f at lower Horner add3，不以小數近似替代。", "en": "Recompute f at lower Horner add3 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "4" ], "output": "4", "label": "f at lower Horner add3" } }, { "id": "S035", "uses": [ "G001", "S034" ], "before": "f at lower Horner multiply2 := mul(4, -6)", "after": "f at lower Horner multiply2 = -24", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。", "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 f at lower Horner multiply2，不以小數近似替代。", "en": "Recompute f at lower Horner multiply2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "4", "-6" ], "output": "-24", "label": "f at lower Horner multiply2" } }, { "id": "S036", "uses": [ "G001", "S035" ], "before": "f at lower Horner add2 := sum(-24, 1)", "after": "f at lower Horner add2 = -23", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。", "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 f at lower Horner add2，不以小數近似替代。", "en": "Recompute f at lower Horner add2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-24", "1" ], "output": "-23", "label": "f at lower Horner add2" } }, { "id": "S037", "uses": [ "G001", "S036" ], "before": "f at lower Horner multiply1 := mul(-23, -6)", "after": "f at lower Horner multiply1 = 138", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。", "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 f at lower Horner multiply1，不以小數近似替代。", "en": "Recompute f at lower Horner multiply1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-23", "-6" ], "output": "138", "label": "f at lower Horner multiply1" } }, { "id": "S038", "uses": [ "G001", "S037" ], "before": "f at lower Horner add1 := sum(138, 12)", "after": "f at lower Horner add1 = 150", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。", "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 f at lower Horner add1，不以小數近似替代。", "en": "Recompute f at lower Horner add1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "138", "12" ], "output": "150", "label": "f at lower Horner add1" } }, { "id": "S039", "uses": [ "G001", "S038" ], "before": "f at lower Horner multiply0 := mul(150, -6)", "after": "f at lower Horner multiply0 = -900", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。", "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 f at lower Horner multiply0，不以小數近似替代。", "en": "Recompute f at lower Horner multiply0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "150", "-6" ], "output": "-900", "label": "f at lower Horner multiply0" } }, { "id": "S040", "uses": [ "G001", "S039" ], "before": "f at lower Horner add0 := sum(-900, -1)", "after": "f at lower Horner add0 = -901", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。", "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 f at lower Horner add0，不以小數近似替代。", "en": "Recompute f at lower Horner add0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-900", "-1" ], "output": "-901", "label": "f at lower Horner add0" } }, { "id": "S041", "uses": [ "G001", "S040" ], "before": "upper chain contribution := mul(-16, 0)", "after": "upper chain contribution = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。", "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 upper chain contribution，不以小數近似替代。", "en": "Recompute upper chain contribution with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-16", "0" ], "output": "0", "label": "upper chain contribution" } }, { "id": "S042", "uses": [ "G001", "S041" ], "before": "lower chain contribution := mul(-901, -3)", "after": "lower chain contribution = 2703", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。", "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 lower chain contribution，不以小數近似替代。", "en": "Recompute lower chain contribution with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-901", "-3" ], "output": "2703", "label": "lower chain contribution" } }, { "id": "S043", "uses": [ "G001", "S042" ], "before": "moving bound derivative upper-minus-lower := sub(0, 2703)", "after": "moving bound derivative upper-minus-lower = -2703", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。", "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 moving bound derivative upper-minus-lower，不以小數近似替代。", "en": "Recompute moving bound derivative upper-minus-lower with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "0", "2703" ], "output": "-2703", "label": "moving bound derivative upper-minus-lower" } }, { "id": "S044", "uses": [ "G001", "S043" ], "before": "point=1;lower=-6;upper=-1;f(upper)=-16;f(lower)=-901", "after": "upper contribution=0;lower contribution=2703;derivative=-2703/1", "relation": "implies", "rule_id": "FTC_moving_bounds_actual_values", "conditions": [], "explanation": { "zh-Hant": "保留兩端實際位置、兩個被積函數值與兩個斜率，最後作有號相減；不取絕對面積或省掉零／負斜率。", "en": "Retain both actual endpoints,integrand values and slopes,then subtract with sign.Do not take an absolute area or discard zero/negative velocities." }, "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": "-2703" }, "conclusion": { "canonical_value": "-2703/1", "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": { "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db", "question_binding": { "batch_id": "batch-20261007-advance-20261009-r01-fundamental-theorem-calculus-foundations", "batch_path": "public/data/questions/batches/batch-20261007-advance-20261009-r01-fundamental-theorem-calculus-foundations/questions.jsonl", "batch_sha256": "d2d6fe2173f2779b3f2778eb8354bae1c05ca3ef96e61fac19f950257f89c643", "original_line": 37, "original_line_sha256": "401fb0a5814499d1c5edda4e85a8be29ce2fe15e83cb3d3cba78ad7615eddabc", "question_compact_sha256": "8932d2d79cc19596376a1c09a9d1d61cb032c63ac0d29b71a4b2cdad575cf036", "english_row_compact_sha256": "040fcd75d70f212fdab37e1364843edc872e05e74eea0ed1946a1ff7900ed383", "english_manifest_path": "public/data/i18n/editions/batch-20261007-advance-20261009-r01-fundamental-theorem-calculus-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261007-advance-20261009-r01-fundamental-theorem-calculus-foundations.en.jsonl" }, "mko_bindings": { "mko-fundamental-theorem-calculus": { "sha256": "6b22aa2d4edd9541d84e10a77e3f555318aaef766fcb9ac34f8008c0e2b43b73", "english_sha256": "1b79750a283c6f448f4b0896689d9c71ff9688828879b8c62e6829d099913492", "original_path": "public/data/mko/fundamental-theorem-calculus.json", "english_path": "public/data/i18n/editions/mko-fundamental-theorem-calculus.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "7359cb2346bdb6bf38b7ca4a5fd5b5a99c3007a711f34346754d57b77b2dfcc8", "original_candidate_sha256": "86547b2e1b3c273ac3a3ec4ec09228fc2b9babcb00bd42236f657d263d36f475", "original_candidate_line_sha256": "86547b2e1b3c273ac3a3ec4ec09228fc2b9babcb00bd42236f657d263d36f475" }, "historical_source_state": { "review_state": { "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-20261026-006-p20", "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": "eeb35dc5daab92234bf72827627392b47f38ead4cfb574d4373bebc43c09fbf5", "question_first_witness_sha256": "3cf8e0d3a00b0df1994db2bfde81f4a31b382e170bdda83ea3d0b68da64e8419" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 44, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "53b43c2bddee90a1aa4c8d32fe0f39ff30b71afe888b05a777f85739515b66f6", "arithmetic_steps": 31 }, "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." } }