{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0001","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a7f1fefa609a5d649515a80a3e77185c4f51b2997626804c1bea7c23c17ea9e","family_id":"qf-mod24-partial-partial-x","seed":202610244001,"semantic_template":"qf-mod24-partial-partial-x/{\"coefficients\":[-12,-2,-3,1,-1,-1],\"x\":-2,\"y\":-3}"},"fields":{"title":"Differentiate in the First Coordinate while Holding the Second Fixed","stem":"Use coefficients=[-12, -2, -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=-2, y=-3. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).","answer":{"kind":"expression","value":"([-12, -2, -3, 1, -1, -1], -2, -3, -3)","explanation":"Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-12, -2, -3, 1, -1, -1], -2, -3, -3)."},"skills":["Differentiate in the First Coordinate while Holding the Second 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0002","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e584b64a43e00efca87bbc3d2084617d9e93a989847b4b3f1535a26e45213d9","family_id":"qf-mod24-partial-partial-x","seed":202610244002,"semantic_template":"qf-mod24-partial-partial-x/{\"coefficients\":[-11,-1,-2,1,0,-1],\"x\":-1,\"y\":-2}"},"fields":{"title":"Differentiate in the First Coordinate while Holding the Second Fixed","stem":"Use coefficients=[-11, -1, -2, 1, 0, -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=-2. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).","answer":{"kind":"expression","value":"([-11, -1, -2, 1, 0, -1], -1, -2, -3)","explanation":"Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-11, -1, -2, 1, 0, -1], -1, -2, -3)."},"skills":["Differentiate in the First Coordinate while Holding the Second 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0003","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0cb3f56e5457ceb57f324d10484b8fc150baf446ea7cfe0021fcd63a8a2f33ed","family_id":"qf-mod24-partial-partial-x","seed":202610244003,"semantic_template":"qf-mod24-partial-partial-x/{\"coefficients\":[-10,0,-1,1,1,-1],\"x\":0,\"y\":-1}"},"fields":{"title":"Differentiate in the First Coordinate while Holding the Second Fixed","stem":"Use coefficients=[-10, 0, -1, 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=0, y=-1. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).","answer":{"kind":"expression","value":"([-10, 0, -1, 1, 1, -1], 0, -1, -1)","explanation":"Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-10, 0, -1, 1, 1, -1], 0, -1, -1)."},"skills":["Differentiate in the First Coordinate while Holding the Second 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0004","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f772102df8829f1f5fdd01d873e200e3aded45f965474487f3532c144f9fe538","family_id":"qf-mod24-partial-partial-x","seed":202610244004,"semantic_template":"qf-mod24-partial-partial-x/{\"coefficients\":[-9,1,0,1,-1,-1],\"x\":1,\"y\":0}"},"fields":{"title":"Differentiate in the First Coordinate while Holding the Second Fixed","stem":"Use coefficients=[-9, 1, 0, 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=0. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).","answer":{"kind":"expression","value":"([-9, 1, 0, 1, -1, -1], 1, 0, 3)","explanation":"Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-9, 1, 0, 1, -1, -1], 1, 0, 3)."},"skills":["Differentiate in the First Coordinate while Holding the Second 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0005","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"572aa4575dfa48266cc551d6897704d4ecf64fee1550becbb4410044c1df3810","family_id":"qf-mod24-partial-partial-x","seed":202610244005,"semantic_template":"qf-mod24-partial-partial-x/{\"coefficients\":[-8,2,1,1,0,-1],\"x\":2,\"y\":1}"},"fields":{"title":"Differentiate in the First Coordinate while Holding the Second Fixed","stem":"Use coefficients=[-8, 2, 1, 1, 0, -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=2, y=1. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).","answer":{"kind":"expression","value":"([-8, 2, 1, 1, 0, -1], 2, 1, 6)","explanation":"Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-8, 2, 1, 1, 0, -1], 2, 1, 6)."},"skills":["Differentiate in the First Coordinate while Holding the Second 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0006","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8b84ace6f6bb972bfc694a86797ee2f61f3c6fbf836053902515ccff88cfa98","family_id":"qf-mod24-partial-partial-x","seed":202610244006,"semantic_template":"qf-mod24-partial-partial-x/{\"coefficients\":[-7,-2,2,1,1,-1],\"x\":-2,\"y\":2}"},"fields":{"title":"Differentiate in the First Coordinate while Holding the Second Fixed","stem":"Use coefficients=[-7, -2, 2, 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=-2, y=2. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).","answer":{"kind":"expression","value":"([-7, -2, 2, 1, 1, -1], -2, 2, -4)","explanation":"Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-7, -2, 2, 1, 1, -1], -2, 2, -4)."},"skills":["Differentiate in the First Coordinate while Holding the Second 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0007","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"05f4e7890217842a3850b2c12714f97a72e029a34f7033bbc823d11703d280cf","family_id":"qf-mod24-partial-partial-x","seed":202610244007,"semantic_template":"qf-mod24-partial-partial-x/{\"coefficients\":[-6,-1,3,1,-1,-1],\"x\":-1,\"y\":3}"},"fields":{"title":"Differentiate in the First Coordinate while Holding the Second 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_x+2*c_xx*x+c_xy*y).","answer":{"kind":"expression","value":"([-6, -1, 3, 1, -1, -1], -1, 3, -6)","explanation":"Differentiate only the x-dependent monomials while treating y 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, -6)."},"skills":["Differentiate in the First Coordinate while Holding the Second 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0008","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f5ef7be651d446e9eec81fcad0f618492716db4fb92f3e68b565573c9bc31c9d","family_id":"qf-mod24-partial-partial-x","seed":202610244008,"semantic_template":"qf-mod24-partial-partial-x/{\"coefficients\":[-5,0,-3,1,0,-1],\"x\":0,\"y\":-3}"},"fields":{"title":"Differentiate in the First Coordinate while Holding the Second Fixed","stem":"Use coefficients=[-5, 0, -3, 1, 0, -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=0, y=-3. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).","answer":{"kind":"expression","value":"([-5, 0, -3, 1, 0, -1], 0, -3, 0)","explanation":"Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-5, 0, -3, 1, 0, -1], 0, -3, 0)."},"skills":["Differentiate in the First Coordinate while Holding the Second 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0009","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0fd98da5237866bf0671ba9610db2a12f3e46f7ff2837fb4914f9118ff3c295c","family_id":"qf-mod24-partial-partial-x","seed":202610244009,"semantic_template":"qf-mod24-partial-partial-x/{\"coefficients\":[-4,1,-2,1,1,-1],\"x\":1,\"y\":-2}"},"fields":{"title":"Differentiate in the First Coordinate while Holding the Second Fixed","stem":"Use coefficients=[-4, 1, -2, 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=-2. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).","answer":{"kind":"expression","value":"([-4, 1, -2, 1, 1, -1], 1, -2, 1)","explanation":"Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-4, 1, -2, 1, 1, -1], 1, -2, 1)."},"skills":["Differentiate in the First Coordinate while Holding the Second 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0010","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"611426e443ab4d4681d63911d252bfd00b8d7e5163f143ee093420551026dd02","family_id":"qf-mod24-partial-partial-x","seed":202610244010,"semantic_template":"qf-mod24-partial-partial-x/{\"coefficients\":[-3,2,-1,1,-1,-1],\"x\":2,\"y\":-1}"},"fields":{"title":"Differentiate in the First Coordinate while Holding the Second Fixed","stem":"Use coefficients=[-3, 2, -1, 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=2, y=-1. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).","answer":{"kind":"expression","value":"([-3, 2, -1, 1, -1, -1], 2, -1, 7)","explanation":"Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-3, 2, -1, 1, -1, -1], 2, -1, 7)."},"skills":["Differentiate in the First Coordinate while Holding the Second 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0011","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"47bfaa9ad0202b90cc094ca97f15f8bf76fc92d4805f39132c027bc8a39c42d5","family_id":"qf-mod24-partial-partial-x","seed":202610244011,"semantic_template":"qf-mod24-partial-partial-x/{\"coefficients\":[-2,-2,0,1,0,-1],\"x\":-2,\"y\":0}"},"fields":{"title":"Differentiate in the First Coordinate while Holding the Second Fixed","stem":"Use coefficients=[-2, -2, 0, 1, 0, -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=-2, y=0. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).","answer":{"kind":"expression","value":"([-2, -2, 0, 1, 0, -1], -2, 0, -6)","explanation":"Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-2, -2, 0, 1, 0, -1], -2, 0, -6)."},"skills":["Differentiate in the First Coordinate while Holding the Second 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0012","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5402e224b4d036480d3fce080ed5b3e20363aad685c26e7c6a1fb5a83b91806","family_id":"qf-mod24-partial-partial-x","seed":202610244012,"semantic_template":"qf-mod24-partial-partial-x/{\"coefficients\":[-1,-1,1,1,1,-1],\"x\":-1,\"y\":1}"},"fields":{"title":"Differentiate in the First Coordinate while Holding the Second Fixed","stem":"Use coefficients=[-1, -1, 1, 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=1. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).","answer":{"kind":"expression","value":"([-1, -1, 1, 1, 1, -1], -1, 1, -2)","explanation":"Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-1, -1, 1, 1, 1, -1], -1, 1, -2)."},"skills":["Differentiate in the First Coordinate while Holding the Second 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0013","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"84a7c0fd6b68b41aa61fd89c8207de512a99c0c25e898965e009d0039db08fe4","family_id":"qf-mod24-partial-partial-x","seed":202610244013,"semantic_template":"qf-mod24-partial-partial-x/{\"coefficients\":[0,0,2,1,-1,-1],\"x\":0,\"y\":2}"},"fields":{"title":"Differentiate in the First Coordinate while Holding the Second Fixed","stem":"Use coefficients=[0, 0, 2, 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=0, y=2. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).","answer":{"kind":"expression","value":"([0, 0, 2, 1, -1, -1], 0, 2, -2)","explanation":"Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([0, 0, 2, 1, -1, -1], 0, 2, -2)."},"skills":["Differentiate in the First Coordinate while Holding the Second 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0014","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"717d69366520d9acd3efe67ceec28b2a5d4cebecf57e179a20e0ede29bb00728","family_id":"qf-mod24-partial-partial-x","seed":202610244014,"semantic_template":"qf-mod24-partial-partial-x/{\"coefficients\":[1,1,3,1,0,-1],\"x\":1,\"y\":3}"},"fields":{"title":"Differentiate in the First Coordinate while Holding the Second Fixed","stem":"Use coefficients=[1, 1, 3, 1, 0, -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_x+2*c_xx*x+c_xy*y).","answer":{"kind":"expression","value":"([1, 1, 3, 1, 0, -1], 1, 3, 3)","explanation":"Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1, 3, 1, 0, -1], 1, 3, 3)."},"skills":["Differentiate in the First Coordinate while Holding the Second 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0015","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ed12c9dfef247e5f94f21de78b7b5e33b1263cb4e0933e5a88ed41adb051553","family_id":"qf-mod24-partial-partial-x","seed":202610244015,"semantic_template":"qf-mod24-partial-partial-x/{\"coefficients\":[2,2,-3,1,1,-1],\"x\":2,\"y\":-3}"},"fields":{"title":"Differentiate in the First Coordinate while Holding the Second Fixed","stem":"Use coefficients=[2, 2, -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=2, y=-3. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).","answer":{"kind":"expression","value":"([2, 2, -3, 1, 1, -1], 2, -3, 3)","explanation":"Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([2, 2, -3, 1, 1, -1], 2, -3, 3)."},"skills":["Differentiate in the First Coordinate while Holding the Second 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0016","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24a8e537efdc2a69c37c0f5a5ee39ae603f90b683a69cf79605353ee91366f61","family_id":"qf-mod24-partial-partial-x","seed":202610244016,"semantic_template":"qf-mod24-partial-partial-x/{\"coefficients\":[3,-2,-2,1,-1,-1],\"x\":-2,\"y\":-2}"},"fields":{"title":"Differentiate in the First Coordinate while Holding the Second Fixed","stem":"Use coefficients=[3, -2, -2, 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=-2, y=-2. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).","answer":{"kind":"expression","value":"([3, -2, -2, 1, -1, -1], -2, -2, -4)","explanation":"Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([3, -2, -2, 1, -1, -1], -2, -2, -4)."},"skills":["Differentiate in the First Coordinate while Holding the Second 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0017","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc9684d7543b6000fec6fa3c8f57ca03f307e4f03aacf124ab3ad75800833408","family_id":"qf-mod24-partial-partial-x","seed":202610244017,"semantic_template":"qf-mod24-partial-partial-x/{\"coefficients\":[4,-1,-1,1,0,-1],\"x\":-1,\"y\":-1}"},"fields":{"title":"Differentiate in the First Coordinate while Holding the Second Fixed","stem":"Use coefficients=[4, -1, -1, 1, 0, -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=-1. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).","answer":{"kind":"expression","value":"([4, -1, -1, 1, 0, -1], -1, -1, -3)","explanation":"Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([4, -1, -1, 1, 0, -1], -1, -1, -3)."},"skills":["Differentiate in the First Coordinate while Holding the Second 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0018","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2023b1f9f76b186a13e11b4bae8d6df1a603b025d30d7f4cba2fc4e712680bb9","family_id":"qf-mod24-partial-partial-x","seed":202610244018,"semantic_template":"qf-mod24-partial-partial-x/{\"coefficients\":[5,0,0,1,1,-1],\"x\":0,\"y\":0}"},"fields":{"title":"Differentiate in the First Coordinate while Holding the Second Fixed","stem":"Use coefficients=[5, 0, 0, 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=0, y=0. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).","answer":{"kind":"expression","value":"([5, 0, 0, 1, 1, -1], 0, 0, 0)","explanation":"Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([5, 0, 0, 1, 1, -1], 0, 0, 0)."},"skills":["Differentiate in the First Coordinate while Holding the Second 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0019","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"70dab8baf24ad8c101332bcd03a7bde81663116f6a3afa193bc1b2a4d0577551","family_id":"qf-mod24-partial-partial-x","seed":202610244019,"semantic_template":"qf-mod24-partial-partial-x/{\"coefficients\":[6,1,1,1,-1,-1],\"x\":1,\"y\":1}"},"fields":{"title":"Differentiate in the First Coordinate while Holding the Second Fixed","stem":"Use coefficients=[6, 1, 1, 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=1. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).","answer":{"kind":"expression","value":"([6, 1, 1, 1, -1, -1], 1, 1, 2)","explanation":"Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([6, 1, 1, 1, -1, -1], 1, 1, 2)."},"skills":["Differentiate in the First Coordinate while Holding the Second 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0020","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c23283124c63692dd8a5103573853a28b440d5e1a151e6703f0481d858d38c8","family_id":"qf-mod24-partial-partial-x","seed":202610244020,"semantic_template":"qf-mod24-partial-partial-x/{\"coefficients\":[7,2,2,1,0,-1],\"x\":2,\"y\":2}"},"fields":{"title":"Differentiate in the First Coordinate while Holding the Second Fixed","stem":"Use coefficients=[7, 2, 2, 1, 0, -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=2, y=2. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).","answer":{"kind":"expression","value":"([7, 2, 2, 1, 0, -1], 2, 2, 6)","explanation":"Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([7, 2, 2, 1, 0, -1], 2, 2, 6)."},"skills":["Differentiate in the First Coordinate while Holding the Second 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0021","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63ec12dcde958e3a367b08dc1d8c9743de2f6c16fe2479abe552bf4ed55e6b4b","family_id":"qf-mod24-partial-partial-x","seed":202610244021,"semantic_template":"qf-mod24-partial-partial-x/{\"coefficients\":[8,-2,3,1,1,-1],\"x\":-2,\"y\":3}"},"fields":{"title":"Differentiate in the First Coordinate while Holding the Second Fixed","stem":"Use coefficients=[8, -2, 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=-2, y=3. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).","answer":{"kind":"expression","value":"([8, -2, 3, 1, 1, -1], -2, 3, -3)","explanation":"Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([8, -2, 3, 1, 1, -1], -2, 3, -3)."},"skills":["Differentiate in the First Coordinate while Holding the Second 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0022","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c7c28594b2fbd365b7b2382ca3cf1e1b09ff2b6b2d19fc8a19326f1fbbb38faf","family_id":"qf-mod24-partial-partial-x","seed":202610244022,"semantic_template":"qf-mod24-partial-partial-x/{\"coefficients\":[9,-1,-3,1,-1,-1],\"x\":-1,\"y\":-3}"},"fields":{"title":"Differentiate in the First Coordinate while Holding the Second Fixed","stem":"Use coefficients=[9, -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_x+2*c_xx*x+c_xy*y).","answer":{"kind":"expression","value":"([9, -1, -3, 1, -1, -1], -1, -3, 0)","explanation":"Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([9, -1, -3, 1, -1, -1], -1, -3, 0)."},"skills":["Differentiate in the First Coordinate while Holding the Second 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0023","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b965e5d9c247063c827aa9ed3bb47852af785d37f6f2c5afbad6cd3457728310","family_id":"qf-mod24-partial-partial-x","seed":202610244023,"semantic_template":"qf-mod24-partial-partial-x/{\"coefficients\":[10,0,-2,1,0,-1],\"x\":0,\"y\":-2}"},"fields":{"title":"Differentiate in the First Coordinate while Holding the Second Fixed","stem":"Use coefficients=[10, 0, -2, 1, 0, -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=0, y=-2. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).","answer":{"kind":"expression","value":"([10, 0, -2, 1, 0, -1], 0, -2, 0)","explanation":"Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([10, 0, -2, 1, 0, -1], 0, -2, 0)."},"skills":["Differentiate in the First Coordinate while Holding the Second 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0024","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dbcbb15a134a4e44c4f2e1b77feeb6ce5442a90e839fb39f770709276bae638e","family_id":"qf-mod24-partial-partial-x","seed":202610244024,"semantic_template":"qf-mod24-partial-partial-x/{\"coefficients\":[11,1,-1,1,1,-1],\"x\":1,\"y\":-1}"},"fields":{"title":"Differentiate in the First Coordinate while Holding the Second Fixed","stem":"Use coefficients=[11, 1, -1, 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=-1. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).","answer":{"kind":"expression","value":"([11, 1, -1, 1, 1, -1], 1, -1, 2)","explanation":"Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([11, 1, -1, 1, 1, -1], 1, -1, 2)."},"skills":["Differentiate in the First Coordinate while Holding the Second 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0025","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81a548c799e4e5173d1b56c927ef1a56afdf1815b8f1ee99202841b12ebd29f9","family_id":"qf-mod24-partial-partial-x","seed":202610244025,"semantic_template":"qf-mod24-partial-partial-x/{\"coefficients\":[12,2,0,1,-1,-1],\"x\":2,\"y\":0}"},"fields":{"title":"Differentiate in the First Coordinate while Holding the Second Fixed","stem":"Use coefficients=[12, 2, 0, 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=2, y=0. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).","answer":{"kind":"expression","value":"([12, 2, 0, 1, -1, -1], 2, 0, 6)","explanation":"Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([12, 2, 0, 1, -1, -1], 2, 0, 6)."},"skills":["Differentiate in the First Coordinate while Holding the Second 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0026","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b1460187355772648c42d9b4f43b7526a2581ce36b33da2b784dd1b4edb92ce","family_id":"qf-mod24-partial-partial-y","seed":202610244026,"semantic_template":"qf-mod24-partial-partial-y/{\"coefficients\":[-12,-2,-3,1,-1,-1],\"x\":-2,\"y\":-3}"},"fields":{"title":"Differentiate in the Second Coordinate while Holding the First Fixed","stem":"Use coefficients=[-12, -2, -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=-2, y=-3. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).","answer":{"kind":"expression","value":"([-12, -2, -3, 1, -1, -1], -2, -3, 5)","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 ([-12, -2, -3, 1, -1, -1], -2, -3, 5)."},"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0027","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"678d3975f5c34ae2a0f75b0a2bf0562d92ecc296bf3ab1523150b3702bf6efd0","family_id":"qf-mod24-partial-partial-y","seed":202610244027,"semantic_template":"qf-mod24-partial-partial-y/{\"coefficients\":[-11,-1,-2,1,0,-1],\"x\":-1,\"y\":-2}"},"fields":{"title":"Differentiate in the Second Coordinate while Holding the First Fixed","stem":"Use coefficients=[-11, -1, -2, 1, 0, -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=-2. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).","answer":{"kind":"expression","value":"([-11, -1, -2, 1, 0, -1], -1, -2, 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 ([-11, -1, -2, 1, 0, -1], -1, -2, 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0028","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a152a2d7ba069f9543ed3c99ba7a65a16b9bd6b104bec32e91b857ed57f7b14d","family_id":"qf-mod24-partial-partial-y","seed":202610244028,"semantic_template":"qf-mod24-partial-partial-y/{\"coefficients\":[-10,0,-1,1,1,-1],\"x\":0,\"y\":-1}"},"fields":{"title":"Differentiate in the Second Coordinate while Holding the First Fixed","stem":"Use coefficients=[-10, 0, -1, 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=0, y=-1. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).","answer":{"kind":"expression","value":"([-10, 0, -1, 1, 1, -1], 0, -1, 1)","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 ([-10, 0, -1, 1, 1, -1], 0, -1, 1)."},"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0029","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"77f8d864ab44dc1e4d3794edd84ae57f30a6d74b7bcdaf777a4933564549da3b","family_id":"qf-mod24-partial-partial-y","seed":202610244029,"semantic_template":"qf-mod24-partial-partial-y/{\"coefficients\":[-9,1,0,1,-1,-1],\"x\":1,\"y\":0}"},"fields":{"title":"Differentiate in the Second Coordinate while Holding the First Fixed","stem":"Use coefficients=[-9, 1, 0, 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=0. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).","answer":{"kind":"expression","value":"([-9, 1, 0, 1, -1, -1], 1, 0, -1)","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 ([-9, 1, 0, 1, -1, -1], 1, 0, -1)."},"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0030","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eee16f6568cd7d1e166c4e6c5044930cf4693be251aa978ef87cd65df40dd5aa","family_id":"qf-mod24-partial-partial-y","seed":202610244030,"semantic_template":"qf-mod24-partial-partial-y/{\"coefficients\":[-8,2,1,1,0,-1],\"x\":2,\"y\":1}"},"fields":{"title":"Differentiate in the Second Coordinate while Holding the First Fixed","stem":"Use coefficients=[-8, 2, 1, 1, 0, -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=2, y=1. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).","answer":{"kind":"expression","value":"([-8, 2, 1, 1, 0, -1], 2, 1, -1)","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 ([-8, 2, 1, 1, 0, -1], 2, 1, -1)."},"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0031","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ecfc27d3030c409fe02666d57698ab8a5b098790d10b534e41ecb5ffb8c9feb1","family_id":"qf-mod24-partial-partial-y","seed":202610244031,"semantic_template":"qf-mod24-partial-partial-y/{\"coefficients\":[-7,-2,2,1,1,-1],\"x\":-2,\"y\":2}"},"fields":{"title":"Differentiate in the Second Coordinate while Holding the First Fixed","stem":"Use coefficients=[-7, -2, 2, 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=-2, y=2. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).","answer":{"kind":"expression","value":"([-7, -2, 2, 1, 1, -1], -2, 2, -4)","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 ([-7, -2, 2, 1, 1, -1], -2, 2, -4)."},"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."}}
{"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0033","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81dd17b579934015cddecdce846ebc1a1c4b0a33bc3d112919d1d4801e8f7365","family_id":"qf-mod24-partial-partial-y","seed":202610244033,"semantic_template":"qf-mod24-partial-partial-y/{\"coefficients\":[-5,0,-3,1,0,-1],\"x\":0,\"y\":-3}"},"fields":{"title":"Differentiate in the Second Coordinate while Holding the First Fixed","stem":"Use coefficients=[-5, 0, -3, 1, 0, -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=0, y=-3. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).","answer":{"kind":"expression","value":"([-5, 0, -3, 1, 0, -1], 0, -3, 3)","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 ([-5, 0, -3, 1, 0, -1], 0, -3, 3)."},"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0034","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b916fb964e27d60eb9b0ca654c91011e8f8e8a1e189a80f6e74d08400752baf","family_id":"qf-mod24-partial-partial-y","seed":202610244034,"semantic_template":"qf-mod24-partial-partial-y/{\"coefficients\":[-4,1,-2,1,1,-1],\"x\":1,\"y\":-2}"},"fields":{"title":"Differentiate in the Second Coordinate while Holding the First Fixed","stem":"Use coefficients=[-4, 1, -2, 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=-2. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).","answer":{"kind":"expression","value":"([-4, 1, -2, 1, 1, -1], 1, -2, 3)","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 ([-4, 1, -2, 1, 1, -1], 1, -2, 3)."},"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0035","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d2e13d7971e7fefcaa089093c3ee91d4be14bf4a8e55bd8ace953c1a45f7c3b9","family_id":"qf-mod24-partial-partial-y","seed":202610244035,"semantic_template":"qf-mod24-partial-partial-y/{\"coefficients\":[-3,2,-1,1,-1,-1],\"x\":2,\"y\":-1}"},"fields":{"title":"Differentiate in the Second Coordinate while Holding the First Fixed","stem":"Use coefficients=[-3, 2, -1, 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=2, y=-1. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).","answer":{"kind":"expression","value":"([-3, 2, -1, 1, -1, -1], 2, -1, -1)","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 ([-3, 2, -1, 1, -1, -1], 2, -1, -1)."},"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0036","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f05cdea4bdcb14d3137a229f9082e74a85db2146f0f3219df395ca88a46a508d","family_id":"qf-mod24-partial-partial-y","seed":202610244036,"semantic_template":"qf-mod24-partial-partial-y/{\"coefficients\":[-2,-2,0,1,0,-1],\"x\":-2,\"y\":0}"},"fields":{"title":"Differentiate in the Second Coordinate while Holding the First Fixed","stem":"Use coefficients=[-2, -2, 0, 1, 0, -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=-2, y=0. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).","answer":{"kind":"expression","value":"([-2, -2, 0, 1, 0, -1], -2, 0, 0)","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 ([-2, -2, 0, 1, 0, -1], -2, 0, 0)."},"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0037","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ca7304491335011e8767575069efbd441b3842dd27e1104648df66e603edea6","family_id":"qf-mod24-partial-partial-y","seed":202610244037,"semantic_template":"qf-mod24-partial-partial-y/{\"coefficients\":[-1,-1,1,1,1,-1],\"x\":-1,\"y\":1}"},"fields":{"title":"Differentiate in the Second Coordinate while Holding the First Fixed","stem":"Use coefficients=[-1, -1, 1, 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=1. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).","answer":{"kind":"expression","value":"([-1, -1, 1, 1, 1, -1], -1, 1, -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 ([-1, -1, 1, 1, 1, -1], -1, 1, -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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0038","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dbabd49329796eab3a0cce3b875eee22e89638e1f0e7201a5409247796475eaa","family_id":"qf-mod24-partial-partial-y","seed":202610244038,"semantic_template":"qf-mod24-partial-partial-y/{\"coefficients\":[0,0,2,1,-1,-1],\"x\":0,\"y\":2}"},"fields":{"title":"Differentiate in the Second Coordinate while Holding the First Fixed","stem":"Use coefficients=[0, 0, 2, 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=0, y=2. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).","answer":{"kind":"expression","value":"([0, 0, 2, 1, -1, -1], 0, 2, -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 ([0, 0, 2, 1, -1, -1], 0, 2, -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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0039","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"630803a1b211e2be6e6ba2f1570f20465f6c1b108ad4d13b128c50c8b5976631","family_id":"qf-mod24-partial-partial-y","seed":202610244039,"semantic_template":"qf-mod24-partial-partial-y/{\"coefficients\":[1,1,3,1,0,-1],\"x\":1,\"y\":3}"},"fields":{"title":"Differentiate in the Second Coordinate while Holding the First Fixed","stem":"Use coefficients=[1, 1, 3, 1, 0, -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":"([1, 1, 3, 1, 0, -1], 1, 3, -3)","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 ([1, 1, 3, 1, 0, -1], 1, 3, -3)."},"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0040","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"481417f99e05ca476731015a3709e66da822bbd39a1352e259715dcf4dca0da7","family_id":"qf-mod24-partial-partial-y","seed":202610244040,"semantic_template":"qf-mod24-partial-partial-y/{\"coefficients\":[2,2,-3,1,1,-1],\"x\":2,\"y\":-3}"},"fields":{"title":"Differentiate in the Second Coordinate while Holding the First Fixed","stem":"Use coefficients=[2, 2, -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=2, y=-3. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).","answer":{"kind":"expression","value":"([2, 2, -3, 1, 1, -1], 2, -3, 5)","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 ([2, 2, -3, 1, 1, -1], 2, -3, 5)."},"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0041","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9d77d8a3c86bbb6a616fee2beb73a6065f79f2cfe17380c2605b23eda4a5ab62","family_id":"qf-mod24-partial-partial-y","seed":202610244041,"semantic_template":"qf-mod24-partial-partial-y/{\"coefficients\":[3,-2,-2,1,-1,-1],\"x\":-2,\"y\":-2}"},"fields":{"title":"Differentiate in the Second Coordinate while Holding the First Fixed","stem":"Use coefficients=[3, -2, -2, 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=-2, y=-2. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).","answer":{"kind":"expression","value":"([3, -2, -2, 1, -1, -1], -2, -2, 4)","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 ([3, -2, -2, 1, -1, -1], -2, -2, 4)."},"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0042","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"87f8793fbcf537c60ee6b42a12d494ccb91be8301d017f11fcf633d962423a02","family_id":"qf-mod24-partial-partial-y","seed":202610244042,"semantic_template":"qf-mod24-partial-partial-y/{\"coefficients\":[4,-1,-1,1,0,-1],\"x\":-1,\"y\":-1}"},"fields":{"title":"Differentiate in the Second Coordinate while Holding the First Fixed","stem":"Use coefficients=[4, -1, -1, 1, 0, -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=-1. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).","answer":{"kind":"expression","value":"([4, -1, -1, 1, 0, -1], -1, -1, 1)","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 ([4, -1, -1, 1, 0, -1], -1, -1, 1)."},"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0043","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"edae06c8dcdfc50f3865f3fab29d50a69f1632e7b89db3f017a499a9415290f5","family_id":"qf-mod24-partial-partial-y","seed":202610244043,"semantic_template":"qf-mod24-partial-partial-y/{\"coefficients\":[5,0,0,1,1,-1],\"x\":0,\"y\":0}"},"fields":{"title":"Differentiate in the Second Coordinate while Holding the First Fixed","stem":"Use coefficients=[5, 0, 0, 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=0, y=0. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).","answer":{"kind":"expression","value":"([5, 0, 0, 1, 1, -1], 0, 0, 0)","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 ([5, 0, 0, 1, 1, -1], 0, 0, 0)."},"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0044","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e3e3b7cc643f7ff5f51489ea48c32158ee9d6439105cce04525caec3d4f35605","family_id":"qf-mod24-partial-partial-y","seed":202610244044,"semantic_template":"qf-mod24-partial-partial-y/{\"coefficients\":[6,1,1,1,-1,-1],\"x\":1,\"y\":1}"},"fields":{"title":"Differentiate in the Second Coordinate while Holding the First Fixed","stem":"Use coefficients=[6, 1, 1, 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=1. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).","answer":{"kind":"expression","value":"([6, 1, 1, 1, -1, -1], 1, 1, -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, 1, 1, -1, -1], 1, 1, -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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0045","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b0c82b658f6d268505d5c42dd2ab38c611b8c694af0390dcaf36dd86ad669bc","family_id":"qf-mod24-partial-partial-y","seed":202610244045,"semantic_template":"qf-mod24-partial-partial-y/{\"coefficients\":[7,2,2,1,0,-1],\"x\":2,\"y\":2}"},"fields":{"title":"Differentiate in the Second Coordinate while Holding the First Fixed","stem":"Use coefficients=[7, 2, 2, 1, 0, -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=2, y=2. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).","answer":{"kind":"expression","value":"([7, 2, 2, 1, 0, -1], 2, 2, -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 ([7, 2, 2, 1, 0, -1], 2, 2, -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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0046","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"88c8bffe69345938835f492696e00de092fb97c0a7bf6f62c93086e5c933b26d","family_id":"qf-mod24-partial-partial-y","seed":202610244046,"semantic_template":"qf-mod24-partial-partial-y/{\"coefficients\":[8,-2,3,1,1,-1],\"x\":-2,\"y\":3}"},"fields":{"title":"Differentiate in the Second Coordinate while Holding the First Fixed","stem":"Use coefficients=[8, -2, 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=-2, y=3. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).","answer":{"kind":"expression","value":"([8, -2, 3, 1, 1, -1], -2, 3, -5)","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 ([8, -2, 3, 1, 1, -1], -2, 3, -5)."},"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0047","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a1c26c4f0115d9303fd283b2cb92bf7c95f223e2b58e88bdb0847592a8d8c5b0","family_id":"qf-mod24-partial-partial-y","seed":202610244047,"semantic_template":"qf-mod24-partial-partial-y/{\"coefficients\":[9,-1,-3,1,-1,-1],\"x\":-1,\"y\":-3}"},"fields":{"title":"Differentiate in the Second Coordinate while Holding the First Fixed","stem":"Use coefficients=[9, -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":"([9, -1, -3, 1, -1, -1], -1, -3, 4)","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 ([9, -1, -3, 1, -1, -1], -1, -3, 4)."},"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0048","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b8ef51ace32094a284b7189c5e56cb4570ff1707a67061a829a97f67e04809a","family_id":"qf-mod24-partial-partial-y","seed":202610244048,"semantic_template":"qf-mod24-partial-partial-y/{\"coefficients\":[10,0,-2,1,0,-1],\"x\":0,\"y\":-2}"},"fields":{"title":"Differentiate in the Second Coordinate while Holding the First Fixed","stem":"Use coefficients=[10, 0, -2, 1, 0, -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=0, y=-2. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).","answer":{"kind":"expression","value":"([10, 0, -2, 1, 0, -1], 0, -2, 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 ([10, 0, -2, 1, 0, -1], 0, -2, 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0049","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90183534072851cfcaba8162e965041a16d727e2eca82cf9a75e9d0f1586de5a","family_id":"qf-mod24-partial-partial-y","seed":202610244049,"semantic_template":"qf-mod24-partial-partial-y/{\"coefficients\":[11,1,-1,1,1,-1],\"x\":1,\"y\":-1}"},"fields":{"title":"Differentiate in the Second Coordinate while Holding the First Fixed","stem":"Use coefficients=[11, 1, -1, 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=-1. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).","answer":{"kind":"expression","value":"([11, 1, -1, 1, 1, -1], 1, -1, 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 ([11, 1, -1, 1, 1, -1], 1, -1, 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0050","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d41b6edd6f5fc51a6b14ab425aee750b6a9a8d78fe2edcb72ea822c64db5eaa0","family_id":"qf-mod24-partial-partial-y","seed":202610244050,"semantic_template":"qf-mod24-partial-partial-y/{\"coefficients\":[12,2,0,1,-1,-1],\"x\":2,\"y\":0}"},"fields":{"title":"Differentiate in the Second Coordinate while Holding the First Fixed","stem":"Use coefficients=[12, 2, 0, 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=2, y=0. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).","answer":{"kind":"expression","value":"([12, 2, 0, 1, -1, -1], 2, 0, -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 ([12, 2, 0, 1, -1, -1], 2, 0, -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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0051","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"60ba489033d82212ca1f5cab2c5a02746dd780bea264c8fb2591845a3f9fb522","family_id":"qf-mod24-partial-gradient","seed":202610244051,"semantic_template":"qf-mod24-partial-gradient/{\"coefficients\":[-12,-2,-3,1,-1,-1],\"x\":-2,\"y\":-3}"},"fields":{"title":"Collect the Two Coordinate Partials in Their Declared Order","stem":"Use coefficients=[-12, -2, -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. x=-2, y=-3. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.","answer":{"kind":"expression","value":"([-12, -2, -3, 1, -1, -1], -2, -3, [-3, 5])","explanation":"Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-12, -2, -3, 1, -1, -1], -2, -3, [-3, 5])."},"skills":["Collect the Two Coordinate Partials in Their Declared Order"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0052","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aced5eae3e02992b15df714a69ab0cee4b5ed97fb8ad108ce263afa38568ad1a","family_id":"qf-mod24-partial-gradient","seed":202610244052,"semantic_template":"qf-mod24-partial-gradient/{\"coefficients\":[-11,-1,-2,1,0,-1],\"x\":-1,\"y\":-2}"},"fields":{"title":"Collect the Two Coordinate Partials in Their Declared Order","stem":"Use coefficients=[-11, -1, -2, 1, 0, -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. x=-1, y=-2. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.","answer":{"kind":"expression","value":"([-11, -1, -2, 1, 0, -1], -1, -2, [-3, 2])","explanation":"Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-11, -1, -2, 1, 0, -1], -1, -2, [-3, 2])."},"skills":["Collect the Two Coordinate Partials in Their Declared Order"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0053","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a8d8f78fdb9e3d39db94c71045e9f49d56b5e625cc0d2b6fe8b82542f25e609a","family_id":"qf-mod24-partial-gradient","seed":202610244053,"semantic_template":"qf-mod24-partial-gradient/{\"coefficients\":[-10,0,-1,1,1,-1],\"x\":0,\"y\":-1}"},"fields":{"title":"Collect the Two Coordinate Partials in Their Declared Order","stem":"Use coefficients=[-10, 0, -1, 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. x=0, y=-1. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.","answer":{"kind":"expression","value":"([-10, 0, -1, 1, 1, -1], 0, -1, [-1, 1])","explanation":"Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-10, 0, -1, 1, 1, -1], 0, -1, [-1, 1])."},"skills":["Collect the Two Coordinate Partials in Their Declared Order"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0054","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b4ef0e06ab5687999bea7e125b03cd13a7a7fa3d7a8120606a3f47a5abb3130","family_id":"qf-mod24-partial-gradient","seed":202610244054,"semantic_template":"qf-mod24-partial-gradient/{\"coefficients\":[-9,1,0,1,-1,-1],\"x\":1,\"y\":0}"},"fields":{"title":"Collect the Two Coordinate Partials in Their Declared Order","stem":"Use coefficients=[-9, 1, 0, 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. x=1, y=0. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.","answer":{"kind":"expression","value":"([-9, 1, 0, 1, -1, -1], 1, 0, [3, -1])","explanation":"Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-9, 1, 0, 1, -1, -1], 1, 0, [3, -1])."},"skills":["Collect the Two Coordinate Partials in Their Declared Order"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0055","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa2b0b471e292892a5fa3a6fb6ea7d187c27b6c313351779eb0de7fc918e3b7a","family_id":"qf-mod24-partial-gradient","seed":202610244055,"semantic_template":"qf-mod24-partial-gradient/{\"coefficients\":[-8,2,1,1,0,-1],\"x\":2,\"y\":1}"},"fields":{"title":"Collect the Two Coordinate Partials in Their Declared Order","stem":"Use coefficients=[-8, 2, 1, 1, 0, -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. x=2, y=1. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.","answer":{"kind":"expression","value":"([-8, 2, 1, 1, 0, -1], 2, 1, [6, -1])","explanation":"Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-8, 2, 1, 1, 0, -1], 2, 1, [6, -1])."},"skills":["Collect the Two Coordinate Partials in Their Declared Order"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0056","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2591dcfbde944430c2f8670c1737407008329e291e0ed3eae82da6fd5f92b28","family_id":"qf-mod24-partial-gradient","seed":202610244056,"semantic_template":"qf-mod24-partial-gradient/{\"coefficients\":[-7,-2,2,1,1,-1],\"x\":-2,\"y\":2}"},"fields":{"title":"Collect the Two Coordinate Partials in Their Declared Order","stem":"Use coefficients=[-7, -2, 2, 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. x=-2, y=2. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.","answer":{"kind":"expression","value":"([-7, -2, 2, 1, 1, -1], -2, 2, [-4, -4])","explanation":"Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-7, -2, 2, 1, 1, -1], -2, 2, [-4, -4])."},"skills":["Collect the Two Coordinate Partials in Their Declared Order"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0057","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f27d614b66afed80141de9d5987885f018bb7d27562d2a4774071f40452f787","family_id":"qf-mod24-partial-gradient","seed":202610244057,"semantic_template":"qf-mod24-partial-gradient/{\"coefficients\":[-6,-1,3,1,-1,-1],\"x\":-1,\"y\":3}"},"fields":{"title":"Collect the Two Coordinate Partials in Their Declared Order","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. x=-1, y=3. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.","answer":{"kind":"expression","value":"([-6, -1, 3, 1, -1, -1], -1, 3, [-6, -2])","explanation":"Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. 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, [-6, -2])."},"skills":["Collect the Two Coordinate Partials in Their Declared Order"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0058","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e96ed94002b0eeba5444471780be61e86db1f3fe3310c6259a9f6ae0c9a5ca3","family_id":"qf-mod24-partial-gradient","seed":202610244058,"semantic_template":"qf-mod24-partial-gradient/{\"coefficients\":[-5,0,-3,1,0,-1],\"x\":0,\"y\":-3}"},"fields":{"title":"Collect the Two Coordinate Partials in Their Declared Order","stem":"Use coefficients=[-5, 0, -3, 1, 0, -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. x=0, y=-3. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.","answer":{"kind":"expression","value":"([-5, 0, -3, 1, 0, -1], 0, -3, [0, 3])","explanation":"Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-5, 0, -3, 1, 0, -1], 0, -3, [0, 3])."},"skills":["Collect the Two Coordinate Partials in Their Declared Order"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0059","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90217e027fddfab998a5a9b347e7ea0bd357059bdf067095231edb6795e83778","family_id":"qf-mod24-partial-gradient","seed":202610244059,"semantic_template":"qf-mod24-partial-gradient/{\"coefficients\":[-4,1,-2,1,1,-1],\"x\":1,\"y\":-2}"},"fields":{"title":"Collect the Two Coordinate Partials in Their Declared Order","stem":"Use coefficients=[-4, 1, -2, 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. x=1, y=-2. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.","answer":{"kind":"expression","value":"([-4, 1, -2, 1, 1, -1], 1, -2, [1, 3])","explanation":"Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-4, 1, -2, 1, 1, -1], 1, -2, [1, 3])."},"skills":["Collect the Two Coordinate Partials in Their Declared Order"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0060","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1f854aac46faecfabc5d7480bbf0d63e78b4a8d7152ea23748142a8fc133d984","family_id":"qf-mod24-partial-gradient","seed":202610244060,"semantic_template":"qf-mod24-partial-gradient/{\"coefficients\":[-3,2,-1,1,-1,-1],\"x\":2,\"y\":-1}"},"fields":{"title":"Collect the Two Coordinate Partials in Their Declared Order","stem":"Use coefficients=[-3, 2, -1, 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. x=2, y=-1. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.","answer":{"kind":"expression","value":"([-3, 2, -1, 1, -1, -1], 2, -1, [7, -1])","explanation":"Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-3, 2, -1, 1, -1, -1], 2, -1, [7, -1])."},"skills":["Collect the Two Coordinate Partials in Their Declared Order"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0061","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d118652f7ac85f1182f7fde67e7d0f6d12da27d58cac3affb315342522122c6","family_id":"qf-mod24-partial-gradient","seed":202610244061,"semantic_template":"qf-mod24-partial-gradient/{\"coefficients\":[-2,-2,0,1,0,-1],\"x\":-2,\"y\":0}"},"fields":{"title":"Collect the Two Coordinate Partials in Their Declared Order","stem":"Use coefficients=[-2, -2, 0, 1, 0, -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. x=-2, y=0. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.","answer":{"kind":"expression","value":"([-2, -2, 0, 1, 0, -1], -2, 0, [-6, 0])","explanation":"Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-2, -2, 0, 1, 0, -1], -2, 0, [-6, 0])."},"skills":["Collect the Two Coordinate Partials in Their Declared Order"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0062","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf2ab6f5e2c94462c0fa143780b3beba867a14b55f2ed647fe5bfdda50af3297","family_id":"qf-mod24-partial-gradient","seed":202610244062,"semantic_template":"qf-mod24-partial-gradient/{\"coefficients\":[-1,-1,1,1,1,-1],\"x\":-1,\"y\":1}"},"fields":{"title":"Collect the Two Coordinate Partials in Their Declared Order","stem":"Use coefficients=[-1, -1, 1, 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. x=-1, y=1. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.","answer":{"kind":"expression","value":"([-1, -1, 1, 1, 1, -1], -1, 1, [-2, -2])","explanation":"Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-1, -1, 1, 1, 1, -1], -1, 1, [-2, -2])."},"skills":["Collect the Two Coordinate Partials in Their Declared Order"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0063","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"804a0450e815c554240d3a99673129c626e830ee4a222c2502cc2ea707bb7e09","family_id":"qf-mod24-partial-gradient","seed":202610244063,"semantic_template":"qf-mod24-partial-gradient/{\"coefficients\":[0,0,2,1,-1,-1],\"x\":0,\"y\":2}"},"fields":{"title":"Collect the Two Coordinate Partials in Their Declared Order","stem":"Use coefficients=[0, 0, 2, 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. x=0, y=2. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.","answer":{"kind":"expression","value":"([0, 0, 2, 1, -1, -1], 0, 2, [-2, -2])","explanation":"Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([0, 0, 2, 1, -1, -1], 0, 2, [-2, -2])."},"skills":["Collect the Two Coordinate Partials in Their Declared Order"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0064","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30b8536340662d54376fa26fdfd9e8a32864ded48d8af1f56f623ffbebbdeaf7","family_id":"qf-mod24-partial-gradient","seed":202610244064,"semantic_template":"qf-mod24-partial-gradient/{\"coefficients\":[1,1,3,1,0,-1],\"x\":1,\"y\":3}"},"fields":{"title":"Collect the Two Coordinate Partials in Their Declared Order","stem":"Use coefficients=[1, 1, 3, 1, 0, -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. x=1, y=3. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.","answer":{"kind":"expression","value":"([1, 1, 3, 1, 0, -1], 1, 3, [3, -3])","explanation":"Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1, 3, 1, 0, -1], 1, 3, [3, -3])."},"skills":["Collect the Two Coordinate Partials in Their Declared Order"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0065","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a95239706b3cbe6122c4ccd98bdcf95f04261af9f29075733eeb387889b7a195","family_id":"qf-mod24-partial-gradient","seed":202610244065,"semantic_template":"qf-mod24-partial-gradient/{\"coefficients\":[2,2,-3,1,1,-1],\"x\":2,\"y\":-3}"},"fields":{"title":"Collect the Two Coordinate Partials in Their Declared Order","stem":"Use coefficients=[2, 2, -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. x=2, y=-3. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.","answer":{"kind":"expression","value":"([2, 2, -3, 1, 1, -1], 2, -3, [3, 5])","explanation":"Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([2, 2, -3, 1, 1, -1], 2, -3, [3, 5])."},"skills":["Collect the Two Coordinate Partials in Their Declared Order"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0066","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba3fb0f4c8774bc586cfd1a18b93c84fdac95d8933c07b61f4acc5ced13d00f8","family_id":"qf-mod24-partial-gradient","seed":202610244066,"semantic_template":"qf-mod24-partial-gradient/{\"coefficients\":[3,-2,-2,1,-1,-1],\"x\":-2,\"y\":-2}"},"fields":{"title":"Collect the Two Coordinate Partials in Their Declared Order","stem":"Use coefficients=[3, -2, -2, 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. x=-2, y=-2. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.","answer":{"kind":"expression","value":"([3, -2, -2, 1, -1, -1], -2, -2, [-4, 4])","explanation":"Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([3, -2, -2, 1, -1, -1], -2, -2, [-4, 4])."},"skills":["Collect the Two Coordinate Partials in Their Declared Order"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0067","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d9de86030dd8ec992d31e349c0e7a5b793d09604aef455859bfde3789a67507c","family_id":"qf-mod24-partial-gradient","seed":202610244067,"semantic_template":"qf-mod24-partial-gradient/{\"coefficients\":[4,-1,-1,1,0,-1],\"x\":-1,\"y\":-1}"},"fields":{"title":"Collect the Two Coordinate Partials in Their Declared Order","stem":"Use coefficients=[4, -1, -1, 1, 0, -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. x=-1, y=-1. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.","answer":{"kind":"expression","value":"([4, -1, -1, 1, 0, -1], -1, -1, [-3, 1])","explanation":"Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([4, -1, -1, 1, 0, -1], -1, -1, [-3, 1])."},"skills":["Collect the Two Coordinate Partials in Their Declared Order"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0068","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"31fe432fb751a0dcc1517fdb10b542a5e53be229ae8db0eb8e30e666985cdde3","family_id":"qf-mod24-partial-gradient","seed":202610244068,"semantic_template":"qf-mod24-partial-gradient/{\"coefficients\":[5,0,0,1,1,-1],\"x\":0,\"y\":0}"},"fields":{"title":"Collect the Two Coordinate Partials in Their Declared Order","stem":"Use coefficients=[5, 0, 0, 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. x=0, y=0. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.","answer":{"kind":"expression","value":"([5, 0, 0, 1, 1, -1], 0, 0, [0, 0])","explanation":"Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([5, 0, 0, 1, 1, -1], 0, 0, [0, 0])."},"skills":["Collect the Two Coordinate Partials in Their Declared Order"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0069","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c634c8b3df5e88ec9c889b472d25dc9a16c59b54ee4526151ac441c17c94c0f","family_id":"qf-mod24-partial-gradient","seed":202610244069,"semantic_template":"qf-mod24-partial-gradient/{\"coefficients\":[6,1,1,1,-1,-1],\"x\":1,\"y\":1}"},"fields":{"title":"Collect the Two Coordinate Partials in Their Declared Order","stem":"Use coefficients=[6, 1, 1, 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. x=1, y=1. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.","answer":{"kind":"expression","value":"([6, 1, 1, 1, -1, -1], 1, 1, [2, -2])","explanation":"Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([6, 1, 1, 1, -1, -1], 1, 1, [2, -2])."},"skills":["Collect the Two Coordinate Partials in Their Declared Order"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0070","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"243df1ffc2625ea110f10065f629c0a0e25f2384cad608646c42ed0f97b6ddd9","family_id":"qf-mod24-partial-gradient","seed":202610244070,"semantic_template":"qf-mod24-partial-gradient/{\"coefficients\":[7,2,2,1,0,-1],\"x\":2,\"y\":2}"},"fields":{"title":"Collect the Two Coordinate Partials in Their Declared Order","stem":"Use coefficients=[7, 2, 2, 1, 0, -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. x=2, y=2. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.","answer":{"kind":"expression","value":"([7, 2, 2, 1, 0, -1], 2, 2, [6, -2])","explanation":"Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([7, 2, 2, 1, 0, -1], 2, 2, [6, -2])."},"skills":["Collect the Two Coordinate Partials in Their Declared Order"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0071","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d8b2c606d1c6e7cfc1d4a6c1bf3b0cf0007c96f7f3a8e38ed7332071327eba6b","family_id":"qf-mod24-partial-gradient","seed":202610244071,"semantic_template":"qf-mod24-partial-gradient/{\"coefficients\":[8,-2,3,1,1,-1],\"x\":-2,\"y\":3}"},"fields":{"title":"Collect the Two Coordinate Partials in Their Declared Order","stem":"Use coefficients=[8, -2, 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. x=-2, y=3. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.","answer":{"kind":"expression","value":"([8, -2, 3, 1, 1, -1], -2, 3, [-3, -5])","explanation":"Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([8, -2, 3, 1, 1, -1], -2, 3, [-3, -5])."},"skills":["Collect the Two Coordinate Partials in Their Declared Order"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0072","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf90dd3c3f6a9a20d414e048c5218be5013757608e4e9dc9421392aad9f1a88d","family_id":"qf-mod24-partial-gradient","seed":202610244072,"semantic_template":"qf-mod24-partial-gradient/{\"coefficients\":[9,-1,-3,1,-1,-1],\"x\":-1,\"y\":-3}"},"fields":{"title":"Collect the Two Coordinate Partials in Their Declared Order","stem":"Use coefficients=[9, -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. x=-1, y=-3. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.","answer":{"kind":"expression","value":"([9, -1, -3, 1, -1, -1], -1, -3, [0, 4])","explanation":"Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([9, -1, -3, 1, -1, -1], -1, -3, [0, 4])."},"skills":["Collect the Two Coordinate Partials in Their Declared Order"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0073","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5e64432cda15333fd3b96d45a146a72825cc0f3f5d0553f4e29d5e184dc725c4","family_id":"qf-mod24-partial-gradient","seed":202610244073,"semantic_template":"qf-mod24-partial-gradient/{\"coefficients\":[10,0,-2,1,0,-1],\"x\":0,\"y\":-2}"},"fields":{"title":"Collect the Two Coordinate Partials in Their Declared Order","stem":"Use coefficients=[10, 0, -2, 1, 0, -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. x=0, y=-2. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.","answer":{"kind":"expression","value":"([10, 0, -2, 1, 0, -1], 0, -2, [0, 2])","explanation":"Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([10, 0, -2, 1, 0, -1], 0, -2, [0, 2])."},"skills":["Collect the Two Coordinate Partials in Their Declared Order"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0074","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0c7576700a71431ea75a3d286bed5f200a2606e34d313d14e709233d876b46e","family_id":"qf-mod24-partial-gradient","seed":202610244074,"semantic_template":"qf-mod24-partial-gradient/{\"coefficients\":[11,1,-1,1,1,-1],\"x\":1,\"y\":-1}"},"fields":{"title":"Collect the Two Coordinate Partials in Their Declared Order","stem":"Use coefficients=[11, 1, -1, 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. x=1, y=-1. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.","answer":{"kind":"expression","value":"([11, 1, -1, 1, 1, -1], 1, -1, [2, 2])","explanation":"Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([11, 1, -1, 1, 1, -1], 1, -1, [2, 2])."},"skills":["Collect the Two Coordinate Partials in Their Declared Order"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0075","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c9cd775bdd1bd7a11f6965ab9c4ea86233c0492a041e51c6ab1d2f056b70d8e8","family_id":"qf-mod24-partial-gradient","seed":202610244075,"semantic_template":"qf-mod24-partial-gradient/{\"coefficients\":[12,2,0,1,-1,-1],\"x\":2,\"y\":0}"},"fields":{"title":"Collect the Two Coordinate Partials in Their Declared Order","stem":"Use coefficients=[12, 2, 0, 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. x=2, y=0. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.","answer":{"kind":"expression","value":"([12, 2, 0, 1, -1, -1], 2, 0, [6, -2])","explanation":"Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([12, 2, 0, 1, -1, -1], 2, 0, [6, -2])."},"skills":["Collect the Two Coordinate Partials in Their Declared Order"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0076","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f3bda2e1cf9cad864d62cb23e55e64024c7fb6e60296a5fb6e6d2d76561be5d","family_id":"qf-mod24-partial-x-slice-derivative","seed":202610244076,"semantic_template":"qf-mod24-partial-x-slice-derivative/{\"coefficients\":[-12,-2,-3,1,-1,-1],\"fixed_y\":-2,\"x\":-3}"},"fields":{"title":"A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative","stem":"Use coefficients=[-12, -2, -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. fixed y=-2, evaluation x=-3. Give (original coefficients,fixed y,ascending x-slice coefficients,ascending derivative coefficients,x,derivative value). The slice has degrees up to 2 and its derivative degrees up to 1.","answer":{"kind":"expression","value":"([-12, -2, -3, 1, -1, -1], -2, [-10, 0, 1], [0, 2], -3, -6)","explanation":"Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-12, -2, -3, 1, -1, -1], -2, [-10, 0, 1], [0, 2], -3, -6)."},"skills":["A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0077","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"efe401e5afeedb599f3b9433d205e4504ee111f5cd4e08b8cabf88de8a4557e8","family_id":"qf-mod24-partial-x-slice-derivative","seed":202610244077,"semantic_template":"qf-mod24-partial-x-slice-derivative/{\"coefficients\":[-11,-1,-2,1,0,-1],\"fixed_y\":-1,\"x\":-2}"},"fields":{"title":"A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative","stem":"Use coefficients=[-11, -1, -2, 1, 0, -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. fixed y=-1, evaluation x=-2. Give (original coefficients,fixed y,ascending x-slice coefficients,ascending derivative coefficients,x,derivative value). The slice has degrees up to 2 and its derivative degrees up to 1.","answer":{"kind":"expression","value":"([-11, -1, -2, 1, 0, -1], -1, [-10, -1, 1], [-1, 2], -2, -5)","explanation":"Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-11, -1, -2, 1, 0, -1], -1, [-10, -1, 1], [-1, 2], -2, -5)."},"skills":["A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0078","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"47400f6d41149a8d09684bfc0843f7ac08f03b429773459cccfd22d05a3dc0c4","family_id":"qf-mod24-partial-x-slice-derivative","seed":202610244078,"semantic_template":"qf-mod24-partial-x-slice-derivative/{\"coefficients\":[-10,0,-1,1,1,-1],\"fixed_y\":0,\"x\":-1}"},"fields":{"title":"A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative","stem":"Use coefficients=[-10, 0, -1, 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. fixed y=0, evaluation x=-1. Give (original coefficients,fixed y,ascending x-slice coefficients,ascending derivative coefficients,x,derivative value). The slice has degrees up to 2 and its derivative degrees up to 1.","answer":{"kind":"expression","value":"([-10, 0, -1, 1, 1, -1], 0, [-10, 0, 1], [0, 2], -1, -2)","explanation":"Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-10, 0, -1, 1, 1, -1], 0, [-10, 0, 1], [0, 2], -1, -2)."},"skills":["A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0079","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b5038b8df971b0108b4e9b629efaeb776db96e26d66be3b87f7db712cf2d9990","family_id":"qf-mod24-partial-x-slice-derivative","seed":202610244079,"semantic_template":"qf-mod24-partial-x-slice-derivative/{\"coefficients\":[-9,1,0,1,-1,-1],\"fixed_y\":1,\"x\":0}"},"fields":{"title":"A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative","stem":"Use coefficients=[-9, 1, 0, 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. fixed y=1, evaluation x=0. Give (original coefficients,fixed y,ascending x-slice coefficients,ascending derivative coefficients,x,derivative value). The slice has degrees up to 2 and its derivative degrees up to 1.","answer":{"kind":"expression","value":"([-9, 1, 0, 1, -1, -1], 1, [-10, 0, 1], [0, 2], 0, 0)","explanation":"Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-9, 1, 0, 1, -1, -1], 1, [-10, 0, 1], [0, 2], 0, 0)."},"skills":["A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0080","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8a07fd68eeaf033077bd7887a50ebc9a6d3d07990b308ffdd44f74ac00e4598c","family_id":"qf-mod24-partial-x-slice-derivative","seed":202610244080,"semantic_template":"qf-mod24-partial-x-slice-derivative/{\"coefficients\":[-8,2,1,1,0,-1],\"fixed_y\":2,\"x\":1}"},"fields":{"title":"A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative","stem":"Use coefficients=[-8, 2, 1, 1, 0, -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. fixed y=2, evaluation x=1. Give (original coefficients,fixed y,ascending x-slice coefficients,ascending derivative coefficients,x,derivative value). The slice has degrees up to 2 and its derivative degrees up to 1.","answer":{"kind":"expression","value":"([-8, 2, 1, 1, 0, -1], 2, [-10, 2, 1], [2, 2], 1, 4)","explanation":"Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-8, 2, 1, 1, 0, -1], 2, [-10, 2, 1], [2, 2], 1, 4)."},"skills":["A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0081","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ed0ef903a3714bbb48a3540359df468b05e542cd6e42f13b4364ab98b80e6765","family_id":"qf-mod24-partial-x-slice-derivative","seed":202610244081,"semantic_template":"qf-mod24-partial-x-slice-derivative/{\"coefficients\":[-7,-2,2,1,1,-1],\"fixed_y\":-2,\"x\":2}"},"fields":{"title":"A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative","stem":"Use coefficients=[-7, -2, 2, 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. fixed y=-2, evaluation x=2. Give (original coefficients,fixed y,ascending x-slice coefficients,ascending derivative coefficients,x,derivative value). The slice has degrees up to 2 and its derivative degrees up to 1.","answer":{"kind":"expression","value":"([-7, -2, 2, 1, 1, -1], -2, [-15, -4, 1], [-4, 2], 2, 0)","explanation":"Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-7, -2, 2, 1, 1, -1], -2, [-15, -4, 1], [-4, 2], 2, 0)."},"skills":["A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0082","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f3177111de392c80390a326de50ce48b782b6c332919efb986205a5c52af355b","family_id":"qf-mod24-partial-x-slice-derivative","seed":202610244082,"semantic_template":"qf-mod24-partial-x-slice-derivative/{\"coefficients\":[-6,-1,3,1,-1,-1],\"fixed_y\":-1,\"x\":3}"},"fields":{"title":"A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative","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. fixed y=-1, evaluation x=3. Give (original coefficients,fixed y,ascending x-slice coefficients,ascending derivative coefficients,x,derivative value). The slice has degrees up to 2 and its derivative degrees up to 1.","answer":{"kind":"expression","value":"([-6, -1, 3, 1, -1, -1], -1, [-10, 0, 1], [0, 2], 3, 6)","explanation":"Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. 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, [-10, 0, 1], [0, 2], 3, 6)."},"skills":["A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0083","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b8aee6a678f3946da6dfff67ca6c82d95e78c1ec68628c62e265475c7529326","family_id":"qf-mod24-partial-x-slice-derivative","seed":202610244083,"semantic_template":"qf-mod24-partial-x-slice-derivative/{\"coefficients\":[-5,0,-3,1,0,-1],\"fixed_y\":0,\"x\":-3}"},"fields":{"title":"A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative","stem":"Use coefficients=[-5, 0, -3, 1, 0, -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. fixed y=0, evaluation x=-3. Give (original coefficients,fixed y,ascending x-slice coefficients,ascending derivative coefficients,x,derivative value). The slice has degrees up to 2 and its derivative degrees up to 1.","answer":{"kind":"expression","value":"([-5, 0, -3, 1, 0, -1], 0, [-5, 0, 1], [0, 2], -3, -6)","explanation":"Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-5, 0, -3, 1, 0, -1], 0, [-5, 0, 1], [0, 2], -3, -6)."},"skills":["A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0084","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e473d3db215845c79333d54d249a6248828083a83c941d3ed8d8c13b92e9084","family_id":"qf-mod24-partial-x-slice-derivative","seed":202610244084,"semantic_template":"qf-mod24-partial-x-slice-derivative/{\"coefficients\":[-4,1,-2,1,1,-1],\"fixed_y\":1,\"x\":-2}"},"fields":{"title":"A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative","stem":"Use coefficients=[-4, 1, -2, 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. fixed y=1, evaluation x=-2. Give (original coefficients,fixed y,ascending x-slice coefficients,ascending derivative coefficients,x,derivative value). The slice has degrees up to 2 and its derivative degrees up to 1.","answer":{"kind":"expression","value":"([-4, 1, -2, 1, 1, -1], 1, [-7, 2, 1], [2, 2], -2, -2)","explanation":"Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-4, 1, -2, 1, 1, -1], 1, [-7, 2, 1], [2, 2], -2, -2)."},"skills":["A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0085","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a9b20d70f143a9cb7275fd2ceecf73269e4c8c5f892af12fd5132c71a916837c","family_id":"qf-mod24-partial-x-slice-derivative","seed":202610244085,"semantic_template":"qf-mod24-partial-x-slice-derivative/{\"coefficients\":[-3,2,-1,1,-1,-1],\"fixed_y\":2,\"x\":-1}"},"fields":{"title":"A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative","stem":"Use coefficients=[-3, 2, -1, 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. fixed y=2, evaluation x=-1. Give (original coefficients,fixed y,ascending x-slice coefficients,ascending derivative coefficients,x,derivative value). The slice has degrees up to 2 and its derivative degrees up to 1.","answer":{"kind":"expression","value":"([-3, 2, -1, 1, -1, -1], 2, [-9, 0, 1], [0, 2], -1, -2)","explanation":"Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-3, 2, -1, 1, -1, -1], 2, [-9, 0, 1], [0, 2], -1, -2)."},"skills":["A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0086","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7451aa46a07dbd87d6646c938af1a9d49802da167b5574bb4f42cf94cbac2c82","family_id":"qf-mod24-partial-x-slice-derivative","seed":202610244086,"semantic_template":"qf-mod24-partial-x-slice-derivative/{\"coefficients\":[-2,-2,0,1,0,-1],\"fixed_y\":-2,\"x\":0}"},"fields":{"title":"A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative","stem":"Use coefficients=[-2, -2, 0, 1, 0, -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. fixed y=-2, evaluation x=0. Give (original coefficients,fixed y,ascending x-slice coefficients,ascending derivative coefficients,x,derivative value). The slice has degrees up to 2 and its derivative degrees up to 1.","answer":{"kind":"expression","value":"([-2, -2, 0, 1, 0, -1], -2, [-6, -2, 1], [-2, 2], 0, -2)","explanation":"Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-2, -2, 0, 1, 0, -1], -2, [-6, -2, 1], [-2, 2], 0, -2)."},"skills":["A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0087","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c75a1b335fdc946d04b331d62a73c63569599363ea72af752a0eef6a469062b","family_id":"qf-mod24-partial-x-slice-derivative","seed":202610244087,"semantic_template":"qf-mod24-partial-x-slice-derivative/{\"coefficients\":[-1,-1,1,1,1,-1],\"fixed_y\":-1,\"x\":1}"},"fields":{"title":"A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative","stem":"Use coefficients=[-1, -1, 1, 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. fixed y=-1, evaluation x=1. Give (original coefficients,fixed y,ascending x-slice coefficients,ascending derivative coefficients,x,derivative value). The slice has degrees up to 2 and its derivative degrees up to 1.","answer":{"kind":"expression","value":"([-1, -1, 1, 1, 1, -1], -1, [-3, -2, 1], [-2, 2], 1, 0)","explanation":"Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-1, -1, 1, 1, 1, -1], -1, [-3, -2, 1], [-2, 2], 1, 0)."},"skills":["A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0088","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f75a7915a1ac87a9c56fea3615caea098d7fe6b7192f9b4bc8b470923966fa71","family_id":"qf-mod24-partial-x-slice-derivative","seed":202610244088,"semantic_template":"qf-mod24-partial-x-slice-derivative/{\"coefficients\":[0,0,2,1,-1,-1],\"fixed_y\":0,\"x\":2}"},"fields":{"title":"A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative","stem":"Use coefficients=[0, 0, 2, 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. fixed y=0, evaluation x=2. Give (original coefficients,fixed y,ascending x-slice coefficients,ascending derivative coefficients,x,derivative value). The slice has degrees up to 2 and its derivative degrees up to 1.","answer":{"kind":"expression","value":"([0, 0, 2, 1, -1, -1], 0, [0, 0, 1], [0, 2], 2, 4)","explanation":"Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([0, 0, 2, 1, -1, -1], 0, [0, 0, 1], [0, 2], 2, 4)."},"skills":["A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0089","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1a4f4ce265ff4f46f22c8a941ee407483da2e43c9d5c02b2219366e123c256d5","family_id":"qf-mod24-partial-x-slice-derivative","seed":202610244089,"semantic_template":"qf-mod24-partial-x-slice-derivative/{\"coefficients\":[1,1,3,1,0,-1],\"fixed_y\":1,\"x\":3}"},"fields":{"title":"A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative","stem":"Use coefficients=[1, 1, 3, 1, 0, -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. fixed y=1, evaluation x=3. Give (original coefficients,fixed y,ascending x-slice coefficients,ascending derivative coefficients,x,derivative value). The slice has degrees up to 2 and its derivative degrees up to 1.","answer":{"kind":"expression","value":"([1, 1, 3, 1, 0, -1], 1, [3, 1, 1], [1, 2], 3, 7)","explanation":"Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1, 3, 1, 0, -1], 1, [3, 1, 1], [1, 2], 3, 7)."},"skills":["A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0090","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"31095ea079c2a280ca836c35362a55165e672399021b689946e7db00aff02a67","family_id":"qf-mod24-partial-x-slice-derivative","seed":202610244090,"semantic_template":"qf-mod24-partial-x-slice-derivative/{\"coefficients\":[2,2,-3,1,1,-1],\"fixed_y\":2,\"x\":-3}"},"fields":{"title":"A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative","stem":"Use coefficients=[2, 2, -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. fixed y=2, evaluation x=-3. Give (original coefficients,fixed y,ascending x-slice coefficients,ascending derivative coefficients,x,derivative value). The slice has degrees up to 2 and its derivative degrees up to 1.","answer":{"kind":"expression","value":"([2, 2, -3, 1, 1, -1], 2, [-8, 4, 1], [4, 2], -3, -2)","explanation":"Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([2, 2, -3, 1, 1, -1], 2, [-8, 4, 1], [4, 2], -3, -2)."},"skills":["A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0091","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eae8d5cfa52254d80091826f46d758427fc658c5c365f0ee6f8bd28a571c260c","family_id":"qf-mod24-partial-x-slice-derivative","seed":202610244091,"semantic_template":"qf-mod24-partial-x-slice-derivative/{\"coefficients\":[3,-2,-2,1,-1,-1],\"fixed_y\":-2,\"x\":-2}"},"fields":{"title":"A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative","stem":"Use coefficients=[3, -2, -2, 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. fixed y=-2, evaluation x=-2. Give (original coefficients,fixed y,ascending x-slice coefficients,ascending derivative coefficients,x,derivative value). The slice has degrees up to 2 and its derivative degrees up to 1.","answer":{"kind":"expression","value":"([3, -2, -2, 1, -1, -1], -2, [3, 0, 1], [0, 2], -2, -4)","explanation":"Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([3, -2, -2, 1, -1, -1], -2, [3, 0, 1], [0, 2], -2, -4)."},"skills":["A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0092","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"28c50c30d969acb03485e4d54574a9dc9d7050460fbdac72b653d37988a3e701","family_id":"qf-mod24-partial-x-slice-derivative","seed":202610244092,"semantic_template":"qf-mod24-partial-x-slice-derivative/{\"coefficients\":[4,-1,-1,1,0,-1],\"fixed_y\":-1,\"x\":-1}"},"fields":{"title":"A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative","stem":"Use coefficients=[4, -1, -1, 1, 0, -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. fixed y=-1, evaluation x=-1. Give (original coefficients,fixed y,ascending x-slice coefficients,ascending derivative coefficients,x,derivative value). The slice has degrees up to 2 and its derivative degrees up to 1.","answer":{"kind":"expression","value":"([4, -1, -1, 1, 0, -1], -1, [4, -1, 1], [-1, 2], -1, -3)","explanation":"Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([4, -1, -1, 1, 0, -1], -1, [4, -1, 1], [-1, 2], -1, -3)."},"skills":["A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0093","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"218165e2cf535c5e4f8725c0f9b2293f726a7baddbb715e4844bab334716ceac","family_id":"qf-mod24-partial-x-slice-derivative","seed":202610244093,"semantic_template":"qf-mod24-partial-x-slice-derivative/{\"coefficients\":[5,0,0,1,1,-1],\"fixed_y\":0,\"x\":0}"},"fields":{"title":"A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative","stem":"Use coefficients=[5, 0, 0, 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. fixed y=0, evaluation x=0. Give (original coefficients,fixed y,ascending x-slice coefficients,ascending derivative coefficients,x,derivative value). The slice has degrees up to 2 and its derivative degrees up to 1.","answer":{"kind":"expression","value":"([5, 0, 0, 1, 1, -1], 0, [5, 0, 1], [0, 2], 0, 0)","explanation":"Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([5, 0, 0, 1, 1, -1], 0, [5, 0, 1], [0, 2], 0, 0)."},"skills":["A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0094","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"99a0255f6158041c1466e56aa4d6676c23e341f1d8cb2172224d5115feca825b","family_id":"qf-mod24-partial-x-slice-derivative","seed":202610244094,"semantic_template":"qf-mod24-partial-x-slice-derivative/{\"coefficients\":[6,1,1,1,-1,-1],\"fixed_y\":1,\"x\":1}"},"fields":{"title":"A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative","stem":"Use coefficients=[6, 1, 1, 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. fixed y=1, evaluation x=1. Give (original coefficients,fixed y,ascending x-slice coefficients,ascending derivative coefficients,x,derivative value). The slice has degrees up to 2 and its derivative degrees up to 1.","answer":{"kind":"expression","value":"([6, 1, 1, 1, -1, -1], 1, [6, 0, 1], [0, 2], 1, 2)","explanation":"Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([6, 1, 1, 1, -1, -1], 1, [6, 0, 1], [0, 2], 1, 2)."},"skills":["A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0095","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"395b5bc47a96f8cd78fd580ca9c03f58e49677ec02ab2cc9f733db76ce0b1136","family_id":"qf-mod24-partial-x-slice-derivative","seed":202610244095,"semantic_template":"qf-mod24-partial-x-slice-derivative/{\"coefficients\":[7,2,2,1,0,-1],\"fixed_y\":2,\"x\":2}"},"fields":{"title":"A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative","stem":"Use coefficients=[7, 2, 2, 1, 0, -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. fixed y=2, evaluation x=2. Give (original coefficients,fixed y,ascending x-slice coefficients,ascending derivative coefficients,x,derivative value). The slice has degrees up to 2 and its derivative degrees up to 1.","answer":{"kind":"expression","value":"([7, 2, 2, 1, 0, -1], 2, [7, 2, 1], [2, 2], 2, 6)","explanation":"Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([7, 2, 2, 1, 0, -1], 2, [7, 2, 1], [2, 2], 2, 6)."},"skills":["A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0096","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"60336952085a9a42a3b15f652b7a53754f4ba062788a620de9023d17f86ae6dd","family_id":"qf-mod24-partial-x-slice-derivative","seed":202610244096,"semantic_template":"qf-mod24-partial-x-slice-derivative/{\"coefficients\":[8,-2,3,1,1,-1],\"fixed_y\":-2,\"x\":3}"},"fields":{"title":"A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative","stem":"Use coefficients=[8, -2, 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. fixed y=-2, evaluation x=3. Give (original coefficients,fixed y,ascending x-slice coefficients,ascending derivative coefficients,x,derivative value). The slice has degrees up to 2 and its derivative degrees up to 1.","answer":{"kind":"expression","value":"([8, -2, 3, 1, 1, -1], -2, [-2, -4, 1], [-4, 2], 3, 2)","explanation":"Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([8, -2, 3, 1, 1, -1], -2, [-2, -4, 1], [-4, 2], 3, 2)."},"skills":["A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0097","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"232480b9b35311e04c8831c13f0b6fbeb434394dbeb24106789fa850d2d628c9","family_id":"qf-mod24-partial-x-slice-derivative","seed":202610244097,"semantic_template":"qf-mod24-partial-x-slice-derivative/{\"coefficients\":[9,-1,-3,1,-1,-1],\"fixed_y\":-1,\"x\":-3}"},"fields":{"title":"A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative","stem":"Use coefficients=[9, -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. fixed y=-1, evaluation x=-3. Give (original coefficients,fixed y,ascending x-slice coefficients,ascending derivative coefficients,x,derivative value). The slice has degrees up to 2 and its derivative degrees up to 1.","answer":{"kind":"expression","value":"([9, -1, -3, 1, -1, -1], -1, [11, 0, 1], [0, 2], -3, -6)","explanation":"Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([9, -1, -3, 1, -1, -1], -1, [11, 0, 1], [0, 2], -3, -6)."},"skills":["A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0098","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"808198e394615fc4c91bd61629e87ed2fc9762c0217eeaab8cd4548913b8168d","family_id":"qf-mod24-partial-x-slice-derivative","seed":202610244098,"semantic_template":"qf-mod24-partial-x-slice-derivative/{\"coefficients\":[10,0,-2,1,0,-1],\"fixed_y\":0,\"x\":-2}"},"fields":{"title":"A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative","stem":"Use coefficients=[10, 0, -2, 1, 0, -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. fixed y=0, evaluation x=-2. Give (original coefficients,fixed y,ascending x-slice coefficients,ascending derivative coefficients,x,derivative value). The slice has degrees up to 2 and its derivative degrees up to 1.","answer":{"kind":"expression","value":"([10, 0, -2, 1, 0, -1], 0, [10, 0, 1], [0, 2], -2, -4)","explanation":"Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([10, 0, -2, 1, 0, -1], 0, [10, 0, 1], [0, 2], -2, -4)."},"skills":["A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0099","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff41053f63942b208e71e5c3030cdcddd8597dd0691f6f522fce6aa50b769183","family_id":"qf-mod24-partial-x-slice-derivative","seed":202610244099,"semantic_template":"qf-mod24-partial-x-slice-derivative/{\"coefficients\":[11,1,-1,1,1,-1],\"fixed_y\":1,\"x\":-1}"},"fields":{"title":"A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative","stem":"Use coefficients=[11, 1, -1, 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. fixed y=1, evaluation x=-1. Give (original coefficients,fixed y,ascending x-slice coefficients,ascending derivative coefficients,x,derivative value). The slice has degrees up to 2 and its derivative degrees up to 1.","answer":{"kind":"expression","value":"([11, 1, -1, 1, 1, -1], 1, [9, 2, 1], [2, 2], -1, 0)","explanation":"Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([11, 1, -1, 1, 1, -1], 1, [9, 2, 1], [2, 2], -1, 0)."},"skills":["A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r04-partial-derivative-0100","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6085aa0c29a7e94bd05429e589d9218d683fdd20d9275f3766910374ce7509ef","family_id":"qf-mod24-partial-x-slice-derivative","seed":202610244100,"semantic_template":"qf-mod24-partial-x-slice-derivative/{\"coefficients\":[12,2,0,1,-1,-1],\"fixed_y\":2,\"x\":0}"},"fields":{"title":"A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative","stem":"Use coefficients=[12, 2, 0, 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. fixed y=2, evaluation x=0. Give (original coefficients,fixed y,ascending x-slice coefficients,ascending derivative coefficients,x,derivative value). The slice has degrees up to 2 and its derivative degrees up to 1.","answer":{"kind":"expression","value":"([12, 2, 0, 1, -1, -1], 2, [8, 0, 1], [0, 2], 0, 0)","explanation":"Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([12, 2, 0, 1, -1, -1], 2, [8, 0, 1], [0, 2], 0, 0)."},"skills":["A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative"],"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."}}