{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261013-advance-20261024-r04-partial-derivative-0032", "pilot_number": "R0022-11557", "number": 11557, "family_id": "qf-mod24-partial-partial-y", "title": { "zh-Hant": "第 11557 題：偏導數", "en": "Question 11557: Differentiate in the Second Coordinate while Holding the First Fixed" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261013-advance-20261024-r04-partial-derivative-0032", "batch_id": "batch-20261013-advance-20261024-r04-partial-derivative-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-partial-derivative" ], "title_zh": "偏導數", "stem_zh": "固定另一座標的多項式切片。操作=partial-partial-y；資料={\"coefficients\":[-6,-1,3,1,-1,-1],\"x\":-1,\"y\":3}。請依 [coefficients,x,y,partial_y] 欄序回答。", "answer": { "kind": "expression", "value": "([-6, -1, 3, 1, -1, -1], -1, 3, -2)", "explanation_zh": "依賴結構核對=([-6, -1, 3, 1, -1, -1], -1, 3, -2)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。" }, "difficulty": { "band": "undergraduate_core", "level": 1, "skills": [ "偏導數" ] }, "variant": { "family_id": "qf-mod24-partial-partial-y", "seed": 202610244032, "parameters": { "coefficients": [ -6, -1, 3, 1, -1, -1 ], "x": -1, "y": 3, "domain": "Polynomial_R2_partial_derivatives", "operation": "partial-partial-y", "semantic_template": "qf-mod24-partial-partial-y/{\"coefficients\":[-6,-1,3,1,-1,-1],\"x\":-1,\"y\":3}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-13", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261013-advance-20261024-r04-partial-derivative-0032", "batch_id": "batch-20261013-advance-20261024-r04-partial-derivative-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "3b6c98c139d21512b5224a01bc690f893f51c97821f9a5d7b2ee8fc8628c24ad", "family_id": "qf-mod24-partial-partial-y", "seed": 202610244032, "semantic_template": "qf-mod24-partial-partial-y/{\"coefficients\":[-6,-1,3,1,-1,-1],\"x\":-1,\"y\":3}" }, "fields": { "title": "Differentiate in the Second Coordinate while Holding the First Fixed", "stem": "Use coefficients=[-6, -1, 3, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. Evaluate at x=-1, y=3. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).", "answer": { "kind": "expression", "value": "([-6, -1, 3, 1, -1, -1], -1, 3, -2)", "explanation": "Differentiate only the y-dependent monomials while treating x as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-6, -1, 3, 1, -1, -1], -1, 3, -2)." }, "skills": [ "Differentiate in the Second Coordinate while Holding the First Fixed" ], "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": "固定另一座標的多項式切片。操作=partial-partial-y；資料={\"coefficients\":[-6,-1,3,1,-1,-1],\"x\":-1,\"y\":3}。請依 [coefficients,x,y,partial_y] 欄序回答。", "en": "Use coefficients=[-6, -1, 3, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. Evaluate at x=-1, y=3. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "original coefficients=[-6, -1, 3, 1, -1, -1];basis=[1,x,y,x^2,xy,y^2]", "after": "u(x,y)=(-6)+(-1)*x+(3)*y+(1)*x^2+(-1)*x*y+(-1)*y^2", "relation": "implies", "rule_id": "bivariate_scalar_polynomial_basis_roles", "conditions": [], "explanation": { "zh-Hant": "六個係數按固定單項式順序讀，不與單變量二次式的[a,b,c]降冪次序混用。", "en": "Read the six coefficients in their fixed monomial order,not the descending[a,b,c] order of a univariate quadratic." }, "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": "y partial slice fixed coordinate squared := mul(-1, -1)", "after": "y partial slice fixed coordinate squared = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依本步指定的原始角色與已算中間量精確運算；保留正負號、分母及係數次序。", "en": "Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 y partial slice fixed coordinate squared，不以小數近似替代。", "en": "Recompute y partial slice fixed coordinate squared with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-1", "-1" ], "output": "1", "label": "y partial slice fixed coordinate squared" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "y partial slice fixed linear term := mul(-1, -1)", "after": "y partial slice fixed linear term = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依本步指定的原始角色與已算中間量精確運算；保留正負號、分母及係數次序。", "en": "Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 y partial slice fixed linear term，不以小數近似替代。", "en": "Recompute y partial slice fixed linear term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-1", "-1" ], "output": "1", "label": "y partial slice fixed linear term" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "y partial slice fixed square term := mul(1, 1)", "after": "y partial slice fixed square term = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依本步指定的原始角色與已算中間量精確運算；保留正負號、分母及係數次序。", "en": "Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 y partial slice fixed square term，不以小數近似替代。", "en": "Recompute y partial slice fixed square term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "y partial slice fixed square term" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "y partial slice constant coefficient := sum(-6, 1, 1)", "after": "y partial slice constant coefficient = -4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依本步指定的原始角色與已算中間量精確運算；保留正負號、分母及係數次序。", "en": "Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 y partial slice constant coefficient，不以小數近似替代。", "en": "Recompute y partial slice constant coefficient with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-6", "1", "1" ], "output": "-4", "label": "y partial slice constant coefficient" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "y partial slice cross contribution := mul(-1, -1)", "after": "y partial slice cross contribution = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依本步指定的原始角色與已算中間量精確運算；保留正負號、分母及係數次序。", "en": "Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 y partial slice cross contribution，不以小數近似替代。", "en": "Recompute y partial slice cross contribution with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-1", "-1" ], "output": "1", "label": "y partial slice cross contribution" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "y partial slice linear coefficient := sum(3, 1)", "after": "y partial slice linear coefficient = 4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依本步指定的原始角色與已算中間量精確運算；保留正負號、分母及係數次序。", "en": "Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 y partial slice linear coefficient，不以小數近似替代。", "en": "Recompute y partial slice linear coefficient with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "3", "1" ], "output": "4", "label": "y partial slice linear coefficient" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "y partial slice:vary y;fix x=-1;original coefficients=[-6, -1, 3, 1, -1, -1]", "after": "ascending y slice coefficients=[-4, 4, -1]", "relation": "implies", "rule_id": "fixed_other_coordinate_before_derivative", "conditions": [], "explanation": { "zh-Hant": "先代固定的另一座標，常數、一次與二次項分開收集；沒有讓兩座標一起變動。", "en": "Substitute the fixed other coordinate first,collecting constant,linear and quadratic terms separately.The two coordinates do not vary together." }, "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": "S009", "uses": [ "G001", "S008" ], "before": "y partial:g(t)=(-4)+(4)*t+(-1)*t^2;point=3;h!=0", "after": "g(3+h)-g(3)=(4)*h+(-1)*(2*(3)*h+h^2)", "relation": "implies", "rule_id": "slice_increment_exact_expansion", "conditions": [], "explanation": { "zh-Hant": "常數項消去，平方項用(z+h)²−z²=2zh+h²展開；此時還沒有代h=0。", "en": "Cancel constants and expand(z+h)^2−z^2=2zh+h^2.Do not substitute h=0 at this stage." }, "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": "y partial twice quadratic coefficient := mul(2, -1)", "after": "y partial twice quadratic coefficient = -2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依本步指定的原始角色與已算中間量精確運算；保留正負號、分母及係數次序。", "en": "Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 y partial twice quadratic coefficient，不以小數近似替代。", "en": "Recompute y partial twice quadratic coefficient with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "-1" ], "output": "-2", "label": "y partial twice quadratic coefficient" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "y partial derivative point term := mul(-2, 3)", "after": "y partial derivative point term = -6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依本步指定的原始角色與已算中間量精確運算；保留正負號、分母及係數次序。", "en": "Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 y partial derivative point term，不以小數近似替代。", "en": "Recompute y partial derivative point term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-2", "3" ], "output": "-6", "label": "y partial derivative point term" } }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "y partial derivative value := sum(4, -6)", "after": "y partial derivative value = -2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依本步指定的原始角色與已算中間量精確運算；保留正負號、分母及係數次序。", "en": "Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 y partial derivative value，不以小數近似替代。", "en": "Recompute y partial derivative value with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "4", "-6" ], "output": "-2", "label": "y partial derivative value" } }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "y partial:h!=0;increment=h*((4)+2*(-1)*(3)+(-1)*h)", "after": "difference quotient=-2+(-1)*h", "relation": "implies", "rule_id": "cancel_only_nonzero_increment", "conditions": [], "explanation": { "zh-Hant": "提出完整因子h後才在h非零條件下約去；差商仍有D*h，不把任意有限差分直接等同導數。", "en": "Factor out the whole h and cancel only under h≠0.The quotient still has D*h;a finite difference is not automatically the 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": "S014", "uses": [ "G001", "S013" ], "before": "y partial:quotient=-2+(-1)*h;epsilon>0;delta=epsilon/1", "after": "0