{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r04-derivative-0001","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"23a5c5679afb82534e09990e119acb1d71068e3e1e15c24317e1e0440d746f84","family_id":"qf-derivative-affine-slope","seed":202610084001,"semantic_template":"qf-derivative-affine-slope/{\"inputs\":[\"-12\"],\"operation\":\"affine_derivative\"}"},"fields":{"title":"Derivative of an Affine Model","stem":"On all R, f(t)=(-12)t+(3). Find f′(-2) in this affine model.","answer":{"kind":"short_text","value":"-12","explanation":"The nonzero-step quotient is the slope m=-12, whose two-sided limit exists. This includes constants; the intercept and center do not change the derivative."},"skills":["Derivative of an Affine Model"],"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-20261007-advance-20261008-r04-derivative-0002","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"44701fb3a327c42f0bc5b718e2e37f13137b935ea7889c88e6ad89c90ddc0f26","family_id":"qf-derivative-affine-slope","seed":202610084002,"semantic_template":"qf-derivative-affine-slope/{\"inputs\":[\"-11\"],\"operation\":\"affine_derivative\"}"},"fields":{"title":"Derivative of an Affine Model","stem":"On all R, f(t)=(-11)t+(2). Find f′(-1) in this affine model.","answer":{"kind":"short_text","value":"-11","explanation":"The nonzero-step quotient is the slope m=-11, whose two-sided limit exists. This includes constants; the intercept and center do not change the derivative."},"skills":["Derivative of an Affine Model"],"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-20261007-advance-20261008-r04-derivative-0003","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3405eae5379c55d02c4992d47cc33e5f82188f0460f7ca65561df603a94410b0","family_id":"qf-derivative-affine-slope","seed":202610084003,"semantic_template":"qf-derivative-affine-slope/{\"inputs\":[\"-10\"],\"operation\":\"affine_derivative\"}"},"fields":{"title":"Derivative of an Affine Model","stem":"On all R, f(t)=(-10)t+(1). Find f′(0) in this affine model.","answer":{"kind":"short_text","value":"-10","explanation":"The nonzero-step quotient is the slope m=-10, whose two-sided limit exists. This includes constants; the intercept and center do not change the derivative."},"skills":["Derivative of an Affine Model"],"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-20261007-advance-20261008-r04-derivative-0004","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad023a0ff409e032b05d0ad6c91685876e8908fe16b4294e3b39814ba81c401a","family_id":"qf-derivative-affine-slope","seed":202610084004,"semantic_template":"qf-derivative-affine-slope/{\"inputs\":[\"-9\"],\"operation\":\"affine_derivative\"}"},"fields":{"title":"Derivative of an Affine Model","stem":"On all R, f(t)=(-9)t+(0). Find f′(1) in this affine model.","answer":{"kind":"short_text","value":"-9","explanation":"The nonzero-step quotient is the slope m=-9, whose two-sided limit exists. This includes constants; the intercept and center do not change the derivative."},"skills":["Derivative of an Affine Model"],"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-20261007-advance-20261008-r04-derivative-0005","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3e9920b6598ad5106473b21c8347880a72e3b9d3274542505670f7f3a743b456","family_id":"qf-derivative-affine-slope","seed":202610084005,"semantic_template":"qf-derivative-affine-slope/{\"inputs\":[\"-8\"],\"operation\":\"affine_derivative\"}"},"fields":{"title":"Derivative of an Affine Model","stem":"On all R, f(t)=(-8)t+(-1). Find f′(2) in this affine model.","answer":{"kind":"short_text","value":"-8","explanation":"The nonzero-step quotient is the slope m=-8, whose two-sided limit exists. This includes constants; the intercept and center do not change the derivative."},"skills":["Derivative of an Affine Model"],"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-20261007-advance-20261008-r04-derivative-0006","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9476984e9501513b42df43fa9c2630761d141e1e8b3642d947b55295183088a3","family_id":"qf-derivative-affine-slope","seed":202610084006,"semantic_template":"qf-derivative-affine-slope/{\"inputs\":[\"-7\"],\"operation\":\"affine_derivative\"}"},"fields":{"title":"Derivative of an Affine Model","stem":"On all R, f(t)=(-7)t+(-2). Find f′(-2) in this affine model.","answer":{"kind":"short_text","value":"-7","explanation":"The nonzero-step quotient is the slope m=-7, whose two-sided limit exists. This includes constants; the intercept and center do not change the derivative."},"skills":["Derivative of an Affine Model"],"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-20261007-advance-20261008-r04-derivative-0007","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e8063db05df84fe250ba185b44c10b78eadccb16e6c2958644adfe1d17a30ab","family_id":"qf-derivative-affine-slope","seed":202610084007,"semantic_template":"qf-derivative-affine-slope/{\"inputs\":[\"-6\"],\"operation\":\"affine_derivative\"}"},"fields":{"title":"Derivative of an Affine Model","stem":"On all R, f(t)=(-6)t+(-3). Find f′(-1) in this affine model.","answer":{"kind":"short_text","value":"-6","explanation":"The nonzero-step quotient is the slope m=-6, whose two-sided limit exists. This includes constants; the intercept and center do not change the derivative."},"skills":["Derivative of an Affine Model"],"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-20261007-advance-20261008-r04-derivative-0008","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7478572b042f46e7cb0206b058c894a7251c99257c3ce232ce1ad18325140fec","family_id":"qf-derivative-affine-slope","seed":202610084008,"semantic_template":"qf-derivative-affine-slope/{\"inputs\":[\"-5\"],\"operation\":\"affine_derivative\"}"},"fields":{"title":"Derivative of an Affine Model","stem":"On all R, f(t)=(-5)t+(-4). Find f′(0) in this affine model.","answer":{"kind":"short_text","value":"-5","explanation":"The nonzero-step quotient is the slope m=-5, whose two-sided limit exists. This includes constants; the intercept and center do not change the derivative."},"skills":["Derivative of an Affine Model"],"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-20261007-advance-20261008-r04-derivative-0009","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c83560fa06af86d89832c2e04ca47ffa93cdcdbc15d1b190a432319de5f52ceb","family_id":"qf-derivative-affine-slope","seed":202610084009,"semantic_template":"qf-derivative-affine-slope/{\"inputs\":[\"-4\"],\"operation\":\"affine_derivative\"}"},"fields":{"title":"Derivative of an Affine Model","stem":"On all R, f(t)=(-4)t+(-5). Find f′(1) in this affine model.","answer":{"kind":"short_text","value":"-4","explanation":"The nonzero-step quotient is the slope m=-4, whose two-sided limit exists. This includes constants; the intercept and center do not change the derivative."},"skills":["Derivative of an Affine Model"],"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-20261007-advance-20261008-r04-derivative-0010","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5dd0dfd11391e49885905303563f9819334690d8ad8f7d610381e82b7a493417","family_id":"qf-derivative-affine-slope","seed":202610084010,"semantic_template":"qf-derivative-affine-slope/{\"inputs\":[\"-3\"],\"operation\":\"affine_derivative\"}"},"fields":{"title":"Derivative of an Affine Model","stem":"On all R, f(t)=(-3)t+(-6). Find f′(2) in this affine model.","answer":{"kind":"short_text","value":"-3","explanation":"The nonzero-step quotient is the slope m=-3, whose two-sided limit exists. This includes constants; the intercept and center do not change the derivative."},"skills":["Derivative of an Affine Model"],"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-20261007-advance-20261008-r04-derivative-0011","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b67da8e0792783c2ca430ca9e5443df1a4cf1d006cb15ee05bed739efcba9bf","family_id":"qf-derivative-affine-slope","seed":202610084011,"semantic_template":"qf-derivative-affine-slope/{\"inputs\":[\"-2\"],\"operation\":\"affine_derivative\"}"},"fields":{"title":"Derivative of an Affine Model","stem":"On all R, f(t)=(-2)t+(-7). Find f′(-2) in this affine model.","answer":{"kind":"short_text","value":"-2","explanation":"The nonzero-step quotient is the slope m=-2, whose two-sided limit exists. This includes constants; the intercept and center do not change the derivative."},"skills":["Derivative of an Affine Model"],"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-20261007-advance-20261008-r04-derivative-0012","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"393417a3544a5a47f22877b797aa5969706c2a5c17be039fd397058faaa91bff","family_id":"qf-derivative-affine-slope","seed":202610084012,"semantic_template":"qf-derivative-affine-slope/{\"inputs\":[\"-1\"],\"operation\":\"affine_derivative\"}"},"fields":{"title":"Derivative of an Affine Model","stem":"On all R, f(t)=(-1)t+(-8). Find f′(-1) in this affine model.","answer":{"kind":"short_text","value":"-1","explanation":"The nonzero-step quotient is the slope m=-1, whose two-sided limit exists. This includes constants; the intercept and center do not change the derivative."},"skills":["Derivative of an Affine Model"],"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-20261007-advance-20261008-r04-derivative-0013","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3435c2b390f13b7771260bd932c70c941f03bb2cf69580077b3f343724032ccf","family_id":"qf-derivative-affine-slope","seed":202610084013,"semantic_template":"qf-derivative-affine-slope/{\"inputs\":[\"0\"],\"operation\":\"affine_derivative\"}"},"fields":{"title":"Derivative of an Affine Model","stem":"On all R, f(t)=(0)t+(-9). Find f′(0) in this affine model.","answer":{"kind":"short_text","value":"0","explanation":"The nonzero-step quotient is the slope m=0, whose two-sided limit exists. This includes constants; the intercept and center do not change the derivative."},"skills":["Derivative of an Affine Model"],"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-20261007-advance-20261008-r04-derivative-0014","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4dbbe86f5a1c5b18c32f3ac9abcb224eb551a80f07bc181574e62a752c57ebc6","family_id":"qf-derivative-affine-slope","seed":202610084014,"semantic_template":"qf-derivative-affine-slope/{\"inputs\":[\"1\"],\"operation\":\"affine_derivative\"}"},"fields":{"title":"Derivative of an Affine Model","stem":"On all R, f(t)=(1)t+(-10). Find f′(1) in this affine model.","answer":{"kind":"short_text","value":"1","explanation":"The nonzero-step quotient is the slope m=1, whose two-sided limit exists. This includes constants; the intercept and center do not change the derivative."},"skills":["Derivative of an Affine Model"],"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-20261007-advance-20261008-r04-derivative-0015","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d333980ee53621b11d9b20bed807b8224d95813bbf55bdf02b24c79fa7385997","family_id":"qf-derivative-affine-slope","seed":202610084015,"semantic_template":"qf-derivative-affine-slope/{\"inputs\":[\"2\"],\"operation\":\"affine_derivative\"}"},"fields":{"title":"Derivative of an Affine Model","stem":"On all R, f(t)=(2)t+(-11). Find f′(2) in this affine model.","answer":{"kind":"short_text","value":"2","explanation":"The nonzero-step quotient is the slope m=2, whose two-sided limit exists. This includes constants; the intercept and center do not change the derivative."},"skills":["Derivative of an Affine Model"],"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-20261007-advance-20261008-r04-derivative-0016","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"87107fb6b2f9005b4d5e4a9636b995ff3400065327d473963b203b3a14b2d75d","family_id":"qf-derivative-affine-slope","seed":202610084016,"semantic_template":"qf-derivative-affine-slope/{\"inputs\":[\"3\"],\"operation\":\"affine_derivative\"}"},"fields":{"title":"Derivative of an Affine Model","stem":"On all R, f(t)=(3)t+(-12). Find f′(-2) in this affine model.","answer":{"kind":"short_text","value":"3","explanation":"The nonzero-step quotient is the slope m=3, whose two-sided limit exists. This includes constants; the intercept and center do not change the derivative."},"skills":["Derivative of an Affine Model"],"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-20261007-advance-20261008-r04-derivative-0017","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e510caf09272845dcc165c8dd3293be74b114ae2286c6c7fc91ef377bc6ba86e","family_id":"qf-derivative-affine-slope","seed":202610084017,"semantic_template":"qf-derivative-affine-slope/{\"inputs\":[\"4\"],\"operation\":\"affine_derivative\"}"},"fields":{"title":"Derivative of an Affine Model","stem":"On all R, f(t)=(4)t+(-13). Find f′(-1) in this affine model.","answer":{"kind":"short_text","value":"4","explanation":"The nonzero-step quotient is the slope m=4, whose two-sided limit exists. This includes constants; the intercept and center do not change the derivative."},"skills":["Derivative of an Affine Model"],"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-20261007-advance-20261008-r04-derivative-0018","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c80fa32b72d770dc27bca9ee77908f9ce10f9a1129283d2c8c137d9660280913","family_id":"qf-derivative-affine-slope","seed":202610084018,"semantic_template":"qf-derivative-affine-slope/{\"inputs\":[\"5\"],\"operation\":\"affine_derivative\"}"},"fields":{"title":"Derivative of an Affine Model","stem":"On all R, f(t)=(5)t+(-14). Find f′(0) in this affine model.","answer":{"kind":"short_text","value":"5","explanation":"The nonzero-step quotient is the slope m=5, whose two-sided limit exists. This includes constants; the intercept and center do not change the derivative."},"skills":["Derivative of an Affine Model"],"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-20261007-advance-20261008-r04-derivative-0019","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"77f29128494af08464bdb7c30ea114b6789c9d10237035434137da7a37edae42","family_id":"qf-derivative-affine-slope","seed":202610084019,"semantic_template":"qf-derivative-affine-slope/{\"inputs\":[\"6\"],\"operation\":\"affine_derivative\"}"},"fields":{"title":"Derivative of an Affine Model","stem":"On all R, f(t)=(6)t+(-15). Find f′(1) in this affine model.","answer":{"kind":"short_text","value":"6","explanation":"The nonzero-step quotient is the slope m=6, whose two-sided limit exists. This includes constants; the intercept and center do not change the derivative."},"skills":["Derivative of an Affine Model"],"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-20261007-advance-20261008-r04-derivative-0020","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"67cc9c03df05f312f4b4eb529f8eb7420277dc37ce7818fe7d37377df2ef03cb","family_id":"qf-derivative-affine-slope","seed":202610084020,"semantic_template":"qf-derivative-affine-slope/{\"inputs\":[\"7\"],\"operation\":\"affine_derivative\"}"},"fields":{"title":"Derivative of an Affine Model","stem":"On all R, f(t)=(7)t+(-16). Find f′(2) in this affine model.","answer":{"kind":"short_text","value":"7","explanation":"The nonzero-step quotient is the slope m=7, whose two-sided limit exists. This includes constants; the intercept and center do not change the derivative."},"skills":["Derivative of an Affine Model"],"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-20261007-advance-20261008-r04-derivative-0021","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe54042084219325969d679c60bdeb22bb30f59a201a126584b64b03776659dc","family_id":"qf-derivative-affine-slope","seed":202610084021,"semantic_template":"qf-derivative-affine-slope/{\"inputs\":[\"8\"],\"operation\":\"affine_derivative\"}"},"fields":{"title":"Derivative of an Affine Model","stem":"On all R, f(t)=(8)t+(-17). Find f′(-2) in this affine model.","answer":{"kind":"short_text","value":"8","explanation":"The nonzero-step quotient is the slope m=8, whose two-sided limit exists. This includes constants; the intercept and center do not change the derivative."},"skills":["Derivative of an Affine Model"],"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-20261007-advance-20261008-r04-derivative-0022","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"57a6268ede308b7ad22d135922653d2217758f8b2d28ef844225e5faab2ae3f0","family_id":"qf-derivative-affine-slope","seed":202610084022,"semantic_template":"qf-derivative-affine-slope/{\"inputs\":[\"9\"],\"operation\":\"affine_derivative\"}"},"fields":{"title":"Derivative of an Affine Model","stem":"On all R, f(t)=(9)t+(-18). Find f′(-1) in this affine model.","answer":{"kind":"short_text","value":"9","explanation":"The nonzero-step quotient is the slope m=9, whose two-sided limit exists. This includes constants; the intercept and center do not change the derivative."},"skills":["Derivative of an Affine Model"],"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-20261007-advance-20261008-r04-derivative-0023","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5347842afac3a4d221a54e40ae2bfafedbd100988946090c9a50f2c95cb3b8de","family_id":"qf-derivative-affine-slope","seed":202610084023,"semantic_template":"qf-derivative-affine-slope/{\"inputs\":[\"10\"],\"operation\":\"affine_derivative\"}"},"fields":{"title":"Derivative of an Affine Model","stem":"On all R, f(t)=(10)t+(-19). Find f′(0) in this affine model.","answer":{"kind":"short_text","value":"10","explanation":"The nonzero-step quotient is the slope m=10, whose two-sided limit exists. This includes constants; the intercept and center do not change the derivative."},"skills":["Derivative of an Affine Model"],"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-20261007-advance-20261008-r04-derivative-0024","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"979f2a1eb9f660d530c9c596cfaae8fdbe4596ddd6d75ebde58ff82da846c2b7","family_id":"qf-derivative-affine-slope","seed":202610084024,"semantic_template":"qf-derivative-affine-slope/{\"inputs\":[\"11\"],\"operation\":\"affine_derivative\"}"},"fields":{"title":"Derivative of an Affine Model","stem":"On all R, f(t)=(11)t+(-20). Find f′(1) in this affine model.","answer":{"kind":"short_text","value":"11","explanation":"The nonzero-step quotient is the slope m=11, whose two-sided limit exists. This includes constants; the intercept and center do not change the derivative."},"skills":["Derivative of an Affine Model"],"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-20261007-advance-20261008-r04-derivative-0025","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81d472c6bbeaabbdbc6e49b937c5dfd9b7a9b6765322c624f8e98c3c0218e8b4","family_id":"qf-derivative-affine-slope","seed":202610084025,"semantic_template":"qf-derivative-affine-slope/{\"inputs\":[\"12\"],\"operation\":\"affine_derivative\"}"},"fields":{"title":"Derivative of an Affine Model","stem":"On all R, f(t)=(12)t+(-21). Find f′(2) in this affine model.","answer":{"kind":"short_text","value":"12","explanation":"The nonzero-step quotient is the slope m=12, whose two-sided limit exists. This includes constants; the intercept and center do not change the derivative."},"skills":["Derivative of an Affine Model"],"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-20261007-advance-20261008-r04-derivative-0026","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9dbb14a8add6d0309eee0564d2e9474d99ba85c6e2b582ec1d6e4b33e985f9c0","family_id":"qf-derivative-quadratic-point","seed":202610084026,"semantic_template":"qf-derivative-quadratic-point/{\"inputs\":[\"slope_only\",\"-13\"],\"operation\":\"quadratic_derivative\"}"},"fields":{"title":"Point Derivative of a Degree-at-Most-Two Polynomial","stem":"On all R, f(t)=(0)t²+(-13)t+(5), allowing a2=0. Find the derivative at x=4.","answer":{"kind":"short_text","value":"-13","explanation":"The derivative is 2a2*x+a1=-13. A zero a2 gives a valid lower-degree case, not a falsely declared nonzero-leading-coefficient quadratic."},"skills":["Point Derivative of a Degree-at-Most-Two Polynomial"],"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-20261007-advance-20261008-r04-derivative-0027","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"598f50d433d702a558c1f229ee291027f585e3eafa171087cb9794dfcd687010","family_id":"qf-derivative-quadratic-point","seed":202610084027,"semantic_template":"qf-derivative-quadratic-point/{\"inputs\":[\"-2\",\"-11\",\"-1\"],\"operation\":\"quadratic_derivative\"}"},"fields":{"title":"Point Derivative of a Degree-at-Most-Two Polynomial","stem":"On all R, f(t)=(-2)t²+(-11)t+(2), allowing a2=0. Find the derivative at x=-1.","answer":{"kind":"short_text","value":"-7","explanation":"The derivative is 2a2*x+a1=-7. A zero a2 gives a valid lower-degree case, not a falsely declared nonzero-leading-coefficient quadratic."},"skills":["Point Derivative of a Degree-at-Most-Two Polynomial"],"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-20261007-advance-20261008-r04-derivative-0028","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dc63e477c05268886ec32c26bf083a29c3b9e768032c9d765e75d6f6d81aa64e","family_id":"qf-derivative-quadratic-point","seed":202610084028,"semantic_template":"qf-derivative-quadratic-point/{\"inputs\":[\"slope_only\",\"-10\"],\"operation\":\"quadratic_derivative\"}"},"fields":{"title":"Point Derivative of a Degree-at-Most-Two Polynomial","stem":"On all R, f(t)=(3)t²+(-10)t+(1), allowing a2=0. Find the derivative at x=0.","answer":{"kind":"short_text","value":"-10","explanation":"The derivative is 2a2*x+a1=-10. A zero a2 gives a valid lower-degree case, not a falsely declared nonzero-leading-coefficient quadratic."},"skills":["Point Derivative of a Degree-at-Most-Two Polynomial"],"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-20261007-advance-20261008-r04-derivative-0029","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d7c2cabdf98f956a484230f5668ae487a4ec875362dd94ee5ce42b54d88544a","family_id":"qf-derivative-quadratic-point","seed":202610084029,"semantic_template":"qf-derivative-quadratic-point/{\"inputs\":[\"-4\",\"-9\",\"1\"],\"operation\":\"quadratic_derivative\"}"},"fields":{"title":"Point Derivative of a Degree-at-Most-Two Polynomial","stem":"On all R, f(t)=(-4)t²+(-9)t+(0), allowing a2=0. Find the derivative at x=1.","answer":{"kind":"short_text","value":"-17","explanation":"The derivative is 2a2*x+a1=-17. A zero a2 gives a valid lower-degree case, not a falsely declared nonzero-leading-coefficient quadratic."},"skills":["Point Derivative of a Degree-at-Most-Two Polynomial"],"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-20261007-advance-20261008-r04-derivative-0030","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6fa755240f037b9acc599e099e289a43c1b4ad378449a8abc6aa35f215c61789","family_id":"qf-derivative-quadratic-point","seed":202610084030,"semantic_template":"qf-derivative-quadratic-point/{\"inputs\":[\"5\",\"-8\",\"2\"],\"operation\":\"quadratic_derivative\"}"},"fields":{"title":"Point Derivative of a Degree-at-Most-Two Polynomial","stem":"On all R, f(t)=(5)t²+(-8)t+(-1), allowing a2=0. Find the derivative at x=2.","answer":{"kind":"short_text","value":"12","explanation":"The derivative is 2a2*x+a1=12. A zero a2 gives a valid lower-degree case, not a falsely declared nonzero-leading-coefficient quadratic."},"skills":["Point Derivative of a Degree-at-Most-Two Polynomial"],"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-20261007-advance-20261008-r04-derivative-0031","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0272025127957ee47512beaaab533816372df92c48a9998e851c3e6a2e98f5a7","family_id":"qf-derivative-quadratic-point","seed":202610084031,"semantic_template":"qf-derivative-quadratic-point/{\"inputs\":[\"-6\",\"-7\",\"-2\"],\"operation\":\"quadratic_derivative\"}"},"fields":{"title":"Point Derivative of a Degree-at-Most-Two Polynomial","stem":"On all R, f(t)=(-6)t²+(-7)t+(-2), allowing a2=0. Find the derivative at x=-2.","answer":{"kind":"short_text","value":"17","explanation":"The derivative is 2a2*x+a1=17. A zero a2 gives a valid lower-degree case, not a falsely declared nonzero-leading-coefficient quadratic."},"skills":["Point Derivative of a Degree-at-Most-Two Polynomial"],"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-20261007-advance-20261008-r04-derivative-0032","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"97b255eafb8531af5af460da53664d8bf95378aba16d29967ce24de7166859c4","family_id":"qf-derivative-quadratic-point","seed":202610084032,"semantic_template":"qf-derivative-quadratic-point/{\"inputs\":[\"7\",\"-6\",\"-1\"],\"operation\":\"quadratic_derivative\"}"},"fields":{"title":"Point Derivative of a Degree-at-Most-Two Polynomial","stem":"On all R, f(t)=(7)t²+(-6)t+(-3), allowing a2=0. Find the derivative at x=-1.","answer":{"kind":"short_text","value":"-20","explanation":"The derivative is 2a2*x+a1=-20. A zero a2 gives a valid lower-degree case, not a falsely declared nonzero-leading-coefficient quadratic."},"skills":["Point Derivative of a Degree-at-Most-Two Polynomial"],"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-20261007-advance-20261008-r04-derivative-0033","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e3f67a129677a04e66b89a301516081039efe25bc14e40e895d84ef1ee2cc08","family_id":"qf-derivative-quadratic-point","seed":202610084033,"semantic_template":"qf-derivative-quadratic-point/{\"inputs\":[\"slope_only\",\"-5\"],\"operation\":\"quadratic_derivative\"}"},"fields":{"title":"Point Derivative of a Degree-at-Most-Two Polynomial","stem":"On all R, f(t)=(-8)t²+(-5)t+(-4), allowing a2=0. Find the derivative at x=0.","answer":{"kind":"short_text","value":"-5","explanation":"The derivative is 2a2*x+a1=-5. A zero a2 gives a valid lower-degree case, not a falsely declared nonzero-leading-coefficient quadratic."},"skills":["Point Derivative of a Degree-at-Most-Two Polynomial"],"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-20261007-advance-20261008-r04-derivative-0034","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d765909d59bd537256f35ec244ce2522ca6e3a86e2636a945a3763f904ba75a5","family_id":"qf-derivative-quadratic-point","seed":202610084034,"semantic_template":"qf-derivative-quadratic-point/{\"inputs\":[\"9\",\"-4\",\"1\"],\"operation\":\"quadratic_derivative\"}"},"fields":{"title":"Point Derivative of a Degree-at-Most-Two Polynomial","stem":"On all R, f(t)=(9)t²+(-4)t+(-5), allowing a2=0. Find the derivative at x=1.","answer":{"kind":"short_text","value":"14","explanation":"The derivative is 2a2*x+a1=14. A zero a2 gives a valid lower-degree case, not a falsely declared nonzero-leading-coefficient quadratic."},"skills":["Point Derivative of a Degree-at-Most-Two Polynomial"],"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-20261007-advance-20261008-r04-derivative-0035","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"090c86f3f140403237c700f00fc0ce597fed95d1562dae0d6897315d58a30f62","family_id":"qf-derivative-quadratic-point","seed":202610084035,"semantic_template":"qf-derivative-quadratic-point/{\"inputs\":[\"-10\",\"-3\",\"2\"],\"operation\":\"quadratic_derivative\"}"},"fields":{"title":"Point Derivative of a Degree-at-Most-Two Polynomial","stem":"On all R, f(t)=(-10)t²+(-3)t+(-6), allowing a2=0. Find the derivative at x=2.","answer":{"kind":"short_text","value":"-43","explanation":"The derivative is 2a2*x+a1=-43. A zero a2 gives a valid lower-degree case, not a falsely declared nonzero-leading-coefficient quadratic."},"skills":["Point Derivative of a Degree-at-Most-Two Polynomial"],"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-20261007-advance-20261008-r04-derivative-0036","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb94ffdd7ada258d66b7d732da2d296fd6010a9763b59783315b83b1fcb01a35","family_id":"qf-derivative-quadratic-point","seed":202610084036,"semantic_template":"qf-derivative-quadratic-point/{\"inputs\":[\"11\",\"-2\",\"-2\"],\"operation\":\"quadratic_derivative\"}"},"fields":{"title":"Point Derivative of a Degree-at-Most-Two Polynomial","stem":"On all R, f(t)=(11)t²+(-2)t+(-7), allowing a2=0. Find the derivative at x=-2.","answer":{"kind":"short_text","value":"-46","explanation":"The derivative is 2a2*x+a1=-46. A zero a2 gives a valid lower-degree case, not a falsely declared nonzero-leading-coefficient quadratic."},"skills":["Point Derivative of a Degree-at-Most-Two Polynomial"],"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-20261007-advance-20261008-r04-derivative-0037","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca860934dc14cab4fabbbdbf6bfc67289681887e0b74f91d5ee2909e09213e39","family_id":"qf-derivative-quadratic-point","seed":202610084037,"semantic_template":"qf-derivative-quadratic-point/{\"inputs\":[\"-12\",\"-1\",\"-1\"],\"operation\":\"quadratic_derivative\"}"},"fields":{"title":"Point Derivative of a Degree-at-Most-Two Polynomial","stem":"On all R, f(t)=(-12)t²+(-1)t+(-8), allowing a2=0. Find the derivative at x=-1.","answer":{"kind":"short_text","value":"23","explanation":"The derivative is 2a2*x+a1=23. A zero a2 gives a valid lower-degree case, not a falsely declared nonzero-leading-coefficient quadratic."},"skills":["Point Derivative of a Degree-at-Most-Two Polynomial"],"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-20261007-advance-20261008-r04-derivative-0038","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"247173612a055a689e5d9b74fca63d6674974546fdc5ce11069df7aedbca9c27","family_id":"qf-derivative-quadratic-point","seed":202610084038,"semantic_template":"qf-derivative-quadratic-point/{\"inputs\":[\"slope_only\",\"0\"],\"operation\":\"quadratic_derivative\"}"},"fields":{"title":"Point Derivative of a Degree-at-Most-Two Polynomial","stem":"On all R, f(t)=(13)t²+(0)t+(-9), allowing a2=0. Find the derivative at x=0.","answer":{"kind":"short_text","value":"0","explanation":"The derivative is 2a2*x+a1=0. A zero a2 gives a valid lower-degree case, not a falsely declared nonzero-leading-coefficient quadratic."},"skills":["Point Derivative of a Degree-at-Most-Two Polynomial"],"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-20261007-advance-20261008-r04-derivative-0039","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6bbd81cb2d773e48ddbcdf01bbfd847469ab0a593551304d9446731c36ea02ff","family_id":"qf-derivative-quadratic-point","seed":202610084039,"semantic_template":"qf-derivative-quadratic-point/{\"inputs\":[\"-14\",\"1\",\"1\"],\"operation\":\"quadratic_derivative\"}"},"fields":{"title":"Point Derivative of a Degree-at-Most-Two Polynomial","stem":"On all R, f(t)=(-14)t²+(1)t+(-10), allowing a2=0. Find the derivative at x=1.","answer":{"kind":"short_text","value":"-27","explanation":"The derivative is 2a2*x+a1=-27. A zero a2 gives a valid lower-degree case, not a falsely declared nonzero-leading-coefficient quadratic."},"skills":["Point Derivative of a Degree-at-Most-Two Polynomial"],"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-20261007-advance-20261008-r04-derivative-0040","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"651ffa5af29c920b2e574be3a5b7a1457ceccad01ff25a63c682feb1291a6cd8","family_id":"qf-derivative-quadratic-point","seed":202610084040,"semantic_template":"qf-derivative-quadratic-point/{\"inputs\":[\"15\",\"2\",\"2\"],\"operation\":\"quadratic_derivative\"}"},"fields":{"title":"Point Derivative of a Degree-at-Most-Two Polynomial","stem":"On all R, f(t)=(15)t²+(2)t+(-11), allowing a2=0. Find the derivative at x=2.","answer":{"kind":"short_text","value":"62","explanation":"The derivative is 2a2*x+a1=62. A zero a2 gives a valid lower-degree case, not a falsely declared nonzero-leading-coefficient quadratic."},"skills":["Point Derivative of a Degree-at-Most-Two Polynomial"],"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-20261007-advance-20261008-r04-derivative-0041","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b085ce6d9ff15b678608e6f5114a364e49438fc5fe8571fdc25adae0b3045444","family_id":"qf-derivative-quadratic-point","seed":202610084041,"semantic_template":"qf-derivative-quadratic-point/{\"inputs\":[\"-16\",\"3\",\"-2\"],\"operation\":\"quadratic_derivative\"}"},"fields":{"title":"Point Derivative of a Degree-at-Most-Two Polynomial","stem":"On all R, f(t)=(-16)t²+(3)t+(-12), allowing a2=0. Find the derivative at x=-2.","answer":{"kind":"short_text","value":"67","explanation":"The derivative is 2a2*x+a1=67. A zero a2 gives a valid lower-degree case, not a falsely declared nonzero-leading-coefficient quadratic."},"skills":["Point Derivative of a Degree-at-Most-Two Polynomial"],"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-20261007-advance-20261008-r04-derivative-0042","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"941469fa6b400cc8425f72d382d8b485869fd392a6471cd66c3e5aa892b14072","family_id":"qf-derivative-quadratic-point","seed":202610084042,"semantic_template":"qf-derivative-quadratic-point/{\"inputs\":[\"17\",\"4\",\"-1\"],\"operation\":\"quadratic_derivative\"}"},"fields":{"title":"Point Derivative of a Degree-at-Most-Two Polynomial","stem":"On all R, f(t)=(17)t²+(4)t+(-13), allowing a2=0. Find the derivative at x=-1.","answer":{"kind":"short_text","value":"-30","explanation":"The derivative is 2a2*x+a1=-30. A zero a2 gives a valid lower-degree case, not a falsely declared nonzero-leading-coefficient quadratic."},"skills":["Point Derivative of a Degree-at-Most-Two Polynomial"],"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-20261007-advance-20261008-r04-derivative-0043","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2ae453af02aa2237d8f780def742fa3a3a786381e0ba7adc09a5a92a3d671655","family_id":"qf-derivative-quadratic-point","seed":202610084043,"semantic_template":"qf-derivative-quadratic-point/{\"inputs\":[\"slope_only\",\"5\"],\"operation\":\"quadratic_derivative\"}"},"fields":{"title":"Point Derivative of a Degree-at-Most-Two Polynomial","stem":"On all R, f(t)=(-18)t²+(5)t+(-14), allowing a2=0. Find the derivative at x=0.","answer":{"kind":"short_text","value":"5","explanation":"The derivative is 2a2*x+a1=5. A zero a2 gives a valid lower-degree case, not a falsely declared nonzero-leading-coefficient quadratic."},"skills":["Point Derivative of a Degree-at-Most-Two Polynomial"],"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-20261007-advance-20261008-r04-derivative-0044","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7819a11fb20c7f93620715e435045c271eb9cc91bf0a4d6c3dac6cfb0ccab774","family_id":"qf-derivative-quadratic-point","seed":202610084044,"semantic_template":"qf-derivative-quadratic-point/{\"inputs\":[\"19\",\"6\",\"1\"],\"operation\":\"quadratic_derivative\"}"},"fields":{"title":"Point Derivative of a Degree-at-Most-Two Polynomial","stem":"On all R, f(t)=(19)t²+(6)t+(-15), allowing a2=0. Find the derivative at x=1.","answer":{"kind":"short_text","value":"44","explanation":"The derivative is 2a2*x+a1=44. A zero a2 gives a valid lower-degree case, not a falsely declared nonzero-leading-coefficient quadratic."},"skills":["Point Derivative of a Degree-at-Most-Two Polynomial"],"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-20261007-advance-20261008-r04-derivative-0045","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"45c17b517b4cffcd8719f3e4cc11549eb5d45eca8a6ff8ad8bdeed31b44036ab","family_id":"qf-derivative-quadratic-point","seed":202610084045,"semantic_template":"qf-derivative-quadratic-point/{\"inputs\":[\"-20\",\"7\",\"2\"],\"operation\":\"quadratic_derivative\"}"},"fields":{"title":"Point Derivative of a Degree-at-Most-Two Polynomial","stem":"On all R, f(t)=(-20)t²+(7)t+(-16), allowing a2=0. Find the derivative at x=2.","answer":{"kind":"short_text","value":"-73","explanation":"The derivative is 2a2*x+a1=-73. A zero a2 gives a valid lower-degree case, not a falsely declared nonzero-leading-coefficient quadratic."},"skills":["Point Derivative of a Degree-at-Most-Two Polynomial"],"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-20261007-advance-20261008-r04-derivative-0046","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b9f822c141a769e5bf9b7a8bf7847bb494c6669ad85c76adac5b2316a60f4a3","family_id":"qf-derivative-quadratic-point","seed":202610084046,"semantic_template":"qf-derivative-quadratic-point/{\"inputs\":[\"21\",\"8\",\"-2\"],\"operation\":\"quadratic_derivative\"}"},"fields":{"title":"Point Derivative of a Degree-at-Most-Two Polynomial","stem":"On all R, f(t)=(21)t²+(8)t+(-17), allowing a2=0. Find the derivative at x=-2.","answer":{"kind":"short_text","value":"-76","explanation":"The derivative is 2a2*x+a1=-76. A zero a2 gives a valid lower-degree case, not a falsely declared nonzero-leading-coefficient quadratic."},"skills":["Point Derivative of a Degree-at-Most-Two Polynomial"],"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-20261007-advance-20261008-r04-derivative-0047","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1947c24e8c2a5b18c66a17005524ce5f1292b6d05200917904c5e6c66bd20a90","family_id":"qf-derivative-quadratic-point","seed":202610084047,"semantic_template":"qf-derivative-quadratic-point/{\"inputs\":[\"-22\",\"9\",\"-1\"],\"operation\":\"quadratic_derivative\"}"},"fields":{"title":"Point Derivative of a Degree-at-Most-Two Polynomial","stem":"On all R, f(t)=(-22)t²+(9)t+(-18), allowing a2=0. Find the derivative at x=-1.","answer":{"kind":"short_text","value":"53","explanation":"The derivative is 2a2*x+a1=53. A zero a2 gives a valid lower-degree case, not a falsely declared nonzero-leading-coefficient quadratic."},"skills":["Point Derivative of a Degree-at-Most-Two Polynomial"],"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-20261007-advance-20261008-r04-derivative-0048","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5271816cbe5262605ab16f1dd6909cbd01a4d3ed9319df3ba95fdafa39b10905","family_id":"qf-derivative-quadratic-point","seed":202610084048,"semantic_template":"qf-derivative-quadratic-point/{\"inputs\":[\"slope_only\",\"10\"],\"operation\":\"quadratic_derivative\"}"},"fields":{"title":"Point Derivative of a Degree-at-Most-Two Polynomial","stem":"On all R, f(t)=(23)t²+(10)t+(-19), allowing a2=0. Find the derivative at x=0.","answer":{"kind":"short_text","value":"10","explanation":"The derivative is 2a2*x+a1=10. A zero a2 gives a valid lower-degree case, not a falsely declared nonzero-leading-coefficient quadratic."},"skills":["Point Derivative of a Degree-at-Most-Two Polynomial"],"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-20261007-advance-20261008-r04-derivative-0049","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ec62aac5b85b453c3fd7f2679dbe31a4038b626db83e49d2dc2325e3dd89007","family_id":"qf-derivative-quadratic-point","seed":202610084049,"semantic_template":"qf-derivative-quadratic-point/{\"inputs\":[\"-24\",\"11\",\"1\"],\"operation\":\"quadratic_derivative\"}"},"fields":{"title":"Point Derivative of a Degree-at-Most-Two Polynomial","stem":"On all R, f(t)=(-24)t²+(11)t+(-20), allowing a2=0. Find the derivative at x=1.","answer":{"kind":"short_text","value":"-37","explanation":"The derivative is 2a2*x+a1=-37. A zero a2 gives a valid lower-degree case, not a falsely declared nonzero-leading-coefficient quadratic."},"skills":["Point Derivative of a Degree-at-Most-Two Polynomial"],"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-20261007-advance-20261008-r04-derivative-0050","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3bc3ccbc44dfd1bcf033c055b2122f958988492f0826031f7c5f568389dde64","family_id":"qf-derivative-quadratic-point","seed":202610084050,"semantic_template":"qf-derivative-quadratic-point/{\"inputs\":[\"25\",\"12\",\"2\"],\"operation\":\"quadratic_derivative\"}"},"fields":{"title":"Point Derivative of a Degree-at-Most-Two Polynomial","stem":"On all R, f(t)=(25)t²+(12)t+(-21), allowing a2=0. Find the derivative at x=2.","answer":{"kind":"short_text","value":"112","explanation":"The derivative is 2a2*x+a1=112. A zero a2 gives a valid lower-degree case, not a falsely declared nonzero-leading-coefficient quadratic."},"skills":["Point Derivative of a Degree-at-Most-Two Polynomial"],"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-20261007-advance-20261008-r04-derivative-0051","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a9b0e2e0882ca79c3fa670cf92acc9e517a99fb094393dd7177a16afaa06f359","family_id":"qf-derivative-finite-quotient","seed":202610084051,"semantic_template":"qf-derivative-finite-quotient/{\"inputs\":[\"affine_quotient\",\"2\"],\"operation\":\"finite_quadratic_quotient\"}"},"fields":{"title":"Finite Difference Quotient at a Specified Nonzero Step","stem":"On all R, f(t)=(0)t²+(2)t+(3), allowing a2=0. At x=4 and nonzero finite h=-1, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"2/1","explanation":"The specified finite quotient is 2/1, equal in this model to 2a2*x+a1+a2*h. It is a finite difference quotient, not a claim that every finite-step value is the derivative, even when a special case agrees."},"skills":["Finite Difference Quotient at a Specified Nonzero Step"],"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-20261007-advance-20261008-r04-derivative-0052","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f1d19b52558cbc57447012963da223efae331318004f5d4ee96e6cfc7536596b","family_id":"qf-derivative-finite-quotient","seed":202610084052,"semantic_template":"qf-derivative-finite-quotient/{\"inputs\":[\"-1\",\"-11\",\"-1\",\"-2\"],\"operation\":\"finite_quadratic_quotient\"}"},"fields":{"title":"Finite Difference Quotient at a Specified Nonzero Step","stem":"On all R, f(t)=(-1)t²+(-11)t+(2), allowing a2=0. At x=-1 and nonzero finite h=-2, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"-7/1","explanation":"The specified finite quotient is -7/1, equal in this model to 2a2*x+a1+a2*h. It is a finite difference quotient, not a claim that every finite-step value is the derivative, even when a special case agrees."},"skills":["Finite Difference Quotient at a Specified Nonzero Step"],"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-20261007-advance-20261008-r04-derivative-0053","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d3e4a0c94f167482ba4f845ff4c7a602f84e0d6e776dde499392ddcc7e8a5a77","family_id":"qf-derivative-finite-quotient","seed":202610084053,"semantic_template":"qf-derivative-finite-quotient/{\"inputs\":[\"2\",\"-10\",\"0\",\"3\"],\"operation\":\"finite_quadratic_quotient\"}"},"fields":{"title":"Finite Difference Quotient at a Specified Nonzero Step","stem":"On all R, f(t)=(2)t²+(-10)t+(4), allowing a2=0. At x=0 and nonzero finite h=3, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"-4/1","explanation":"The specified finite quotient is -4/1, equal in this model to 2a2*x+a1+a2*h. It is a finite difference quotient, not a claim that every finite-step value is the derivative, even when a special case agrees."},"skills":["Finite Difference Quotient at a Specified Nonzero Step"],"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-20261007-advance-20261008-r04-derivative-0054","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1152d019ecbc808e4d496f9cc4f5d118520b942b29bd363dc302220046eade32","family_id":"qf-derivative-finite-quotient","seed":202610084054,"semantic_template":"qf-derivative-finite-quotient/{\"inputs\":[\"-3\",\"-9\",\"1\",\"-1\"],\"operation\":\"finite_quadratic_quotient\"}"},"fields":{"title":"Finite Difference Quotient at a Specified Nonzero Step","stem":"On all R, f(t)=(-3)t²+(-9)t+(6), allowing a2=0. At x=1 and nonzero finite h=-1, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"-12/1","explanation":"The specified finite quotient is -12/1, equal in this model to 2a2*x+a1+a2*h. It is a finite difference quotient, not a claim that every finite-step value is the derivative, even when a special case agrees."},"skills":["Finite Difference Quotient at a Specified Nonzero Step"],"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-20261007-advance-20261008-r04-derivative-0055","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21bf5ca7da613b5b3bc5ce8be1663315cb9e3869c9a12c38a18187f41544b64c","family_id":"qf-derivative-finite-quotient","seed":202610084055,"semantic_template":"qf-derivative-finite-quotient/{\"inputs\":[\"4\",\"-8\",\"2\",\"2\"],\"operation\":\"finite_quadratic_quotient\"}"},"fields":{"title":"Finite Difference Quotient at a Specified Nonzero Step","stem":"On all R, f(t)=(4)t²+(-8)t+(8), allowing a2=0. At x=2 and nonzero finite h=2, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"16/1","explanation":"The specified finite quotient is 16/1, equal in this model to 2a2*x+a1+a2*h. It is a finite difference quotient, not a claim that every finite-step value is the derivative, even when a special case agrees."},"skills":["Finite Difference Quotient at a Specified Nonzero Step"],"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-20261007-advance-20261008-r04-derivative-0056","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f09ac4f22df93588877cf71eca65910ffa214d026f97e59f366840d2662d0f7f","family_id":"qf-derivative-finite-quotient","seed":202610084056,"semantic_template":"qf-derivative-finite-quotient/{\"inputs\":[\"-5\",\"-7\",\"-2\",\"-3\"],\"operation\":\"finite_quadratic_quotient\"}"},"fields":{"title":"Finite Difference Quotient at a Specified Nonzero Step","stem":"On all R, f(t)=(-5)t²+(-7)t+(10), allowing a2=0. At x=-2 and nonzero finite h=-3, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"28/1","explanation":"The specified finite quotient is 28/1, equal in this model to 2a2*x+a1+a2*h. It is a finite difference quotient, not a claim that every finite-step value is the derivative, even when a special case agrees."},"skills":["Finite Difference Quotient at a Specified Nonzero Step"],"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-20261007-advance-20261008-r04-derivative-0057","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"935345895ad4d2c79e9af83ad30f068615239794892038a45cf61e2ba9254c71","family_id":"qf-derivative-finite-quotient","seed":202610084057,"semantic_template":"qf-derivative-finite-quotient/{\"inputs\":[\"6\",\"-6\",\"-1\",\"1\"],\"operation\":\"finite_quadratic_quotient\"}"},"fields":{"title":"Finite Difference Quotient at a Specified Nonzero Step","stem":"On all R, f(t)=(6)t²+(-6)t+(12), allowing a2=0. At x=-1 and nonzero finite h=1, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"-12/1","explanation":"The specified finite quotient is -12/1, equal in this model to 2a2*x+a1+a2*h. It is a finite difference quotient, not a claim that every finite-step value is the derivative, even when a special case agrees."},"skills":["Finite Difference Quotient at a Specified Nonzero Step"],"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-20261007-advance-20261008-r04-derivative-0058","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eab574d4d8b996ccf49d9b4e942747bc0a967cc2190b42220c7b98908689fb0c","family_id":"qf-derivative-finite-quotient","seed":202610084058,"semantic_template":"qf-derivative-finite-quotient/{\"inputs\":[\"-7\",\"-5\",\"0\",\"-2\"],\"operation\":\"finite_quadratic_quotient\"}"},"fields":{"title":"Finite Difference Quotient at a Specified Nonzero Step","stem":"On all R, f(t)=(-7)t²+(-5)t+(14), allowing a2=0. At x=0 and nonzero finite h=-2, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"9/1","explanation":"The specified finite quotient is 9/1, equal in this model to 2a2*x+a1+a2*h. It is a finite difference quotient, not a claim that every finite-step value is the derivative, even when a special case agrees."},"skills":["Finite Difference Quotient at a Specified Nonzero Step"],"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-20261007-advance-20261008-r04-derivative-0059","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4253cf106d93d705fc9b6d5582c3cd7228c77b81d0a887911b7fbfe9c1f59fbd","family_id":"qf-derivative-finite-quotient","seed":202610084059,"semantic_template":"qf-derivative-finite-quotient/{\"inputs\":[\"8\",\"-4\",\"1\",\"3\"],\"operation\":\"finite_quadratic_quotient\"}"},"fields":{"title":"Finite Difference Quotient at a Specified Nonzero Step","stem":"On all R, f(t)=(8)t²+(-4)t+(16), allowing a2=0. At x=1 and nonzero finite h=3, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"36/1","explanation":"The specified finite quotient is 36/1, equal in this model to 2a2*x+a1+a2*h. It is a finite difference quotient, not a claim that every finite-step value is the derivative, even when a special case agrees."},"skills":["Finite Difference Quotient at a Specified Nonzero Step"],"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-20261007-advance-20261008-r04-derivative-0060","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e43ea9a5672502060a12c0d913ecd85862909afde36ab995312bd0c53c1e259","family_id":"qf-derivative-finite-quotient","seed":202610084060,"semantic_template":"qf-derivative-finite-quotient/{\"inputs\":[\"-9\",\"-3\",\"2\",\"-1\"],\"operation\":\"finite_quadratic_quotient\"}"},"fields":{"title":"Finite Difference Quotient at a Specified Nonzero Step","stem":"On all R, f(t)=(-9)t²+(-3)t+(18), allowing a2=0. At x=2 and nonzero finite h=-1, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"-30/1","explanation":"The specified finite quotient is -30/1, equal in this model to 2a2*x+a1+a2*h. It is a finite difference quotient, not a claim that every finite-step value is the derivative, even when a special case agrees."},"skills":["Finite Difference Quotient at a Specified Nonzero Step"],"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-20261007-advance-20261008-r04-derivative-0061","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"34f5e96bf8770640662a469547d5c95e285469529175c4846c08018ff88b2940","family_id":"qf-derivative-finite-quotient","seed":202610084061,"semantic_template":"qf-derivative-finite-quotient/{\"inputs\":[\"10\",\"-2\",\"-2\",\"2\"],\"operation\":\"finite_quadratic_quotient\"}"},"fields":{"title":"Finite Difference Quotient at a Specified Nonzero Step","stem":"On all R, f(t)=(10)t²+(-2)t+(20), allowing a2=0. At x=-2 and nonzero finite h=2, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"-22/1","explanation":"The specified finite quotient is -22/1, equal in this model to 2a2*x+a1+a2*h. It is a finite difference quotient, not a claim that every finite-step value is the derivative, even when a special case agrees."},"skills":["Finite Difference Quotient at a Specified Nonzero Step"],"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-20261007-advance-20261008-r04-derivative-0062","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d0ddf30a6011432a999da729788855865557e5b4408c6cf887de08b0314bf6b9","family_id":"qf-derivative-finite-quotient","seed":202610084062,"semantic_template":"qf-derivative-finite-quotient/{\"inputs\":[\"-11\",\"-1\",\"-1\",\"-3\"],\"operation\":\"finite_quadratic_quotient\"}"},"fields":{"title":"Finite Difference Quotient at a Specified Nonzero Step","stem":"On all R, f(t)=(-11)t²+(-1)t+(22), allowing a2=0. At x=-1 and nonzero finite h=-3, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"54/1","explanation":"The specified finite quotient is 54/1, equal in this model to 2a2*x+a1+a2*h. It is a finite difference quotient, not a claim that every finite-step value is the derivative, even when a special case agrees."},"skills":["Finite Difference Quotient at a Specified Nonzero Step"],"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-20261007-advance-20261008-r04-derivative-0063","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1112114485dcf05889a0f13b58ebd40815ddafd5ea60d855823504a08dddb805","family_id":"qf-derivative-finite-quotient","seed":202610084063,"semantic_template":"qf-derivative-finite-quotient/{\"inputs\":[\"12\",\"0\",\"0\",\"1\"],\"operation\":\"finite_quadratic_quotient\"}"},"fields":{"title":"Finite Difference Quotient at a Specified Nonzero Step","stem":"On all R, f(t)=(12)t²+(0)t+(24), allowing a2=0. At x=0 and nonzero finite h=1, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"12/1","explanation":"The specified finite quotient is 12/1, equal in this model to 2a2*x+a1+a2*h. It is a finite difference quotient, not a claim that every finite-step value is the derivative, even when a special case agrees."},"skills":["Finite Difference Quotient at a Specified Nonzero Step"],"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-20261007-advance-20261008-r04-derivative-0064","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"88ff2f31d9462935971f151baf71010db2a001db61044afd664836644512304e","family_id":"qf-derivative-finite-quotient","seed":202610084064,"semantic_template":"qf-derivative-finite-quotient/{\"inputs\":[\"-13\",\"1\",\"1\",\"-2\"],\"operation\":\"finite_quadratic_quotient\"}"},"fields":{"title":"Finite Difference Quotient at a Specified Nonzero Step","stem":"On all R, f(t)=(-13)t²+(1)t+(26), allowing a2=0. At x=1 and nonzero finite h=-2, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"1/1","explanation":"The specified finite quotient is 1/1, equal in this model to 2a2*x+a1+a2*h. It is a finite difference quotient, not a claim that every finite-step value is the derivative, even when a special case agrees."},"skills":["Finite Difference Quotient at a Specified Nonzero Step"],"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-20261007-advance-20261008-r04-derivative-0065","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b312949f0f4d7521c171cccbf55555b4a680dfa6dbbd00535dfcb57ae202994","family_id":"qf-derivative-finite-quotient","seed":202610084065,"semantic_template":"qf-derivative-finite-quotient/{\"inputs\":[\"14\",\"2\",\"2\",\"3\"],\"operation\":\"finite_quadratic_quotient\"}"},"fields":{"title":"Finite Difference Quotient at a Specified Nonzero Step","stem":"On all R, f(t)=(14)t²+(2)t+(28), allowing a2=0. At x=2 and nonzero finite h=3, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"100/1","explanation":"The specified finite quotient is 100/1, equal in this model to 2a2*x+a1+a2*h. It is a finite difference quotient, not a claim that every finite-step value is the derivative, even when a special case agrees."},"skills":["Finite Difference Quotient at a Specified Nonzero Step"],"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-20261007-advance-20261008-r04-derivative-0066","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"58596488ba2f17ed55c6f6f4ef6977a9e06ddc830504cc914c62bb53341bf56a","family_id":"qf-derivative-finite-quotient","seed":202610084066,"semantic_template":"qf-derivative-finite-quotient/{\"inputs\":[\"-15\",\"3\",\"-2\",\"-1\"],\"operation\":\"finite_quadratic_quotient\"}"},"fields":{"title":"Finite Difference Quotient at a Specified Nonzero Step","stem":"On all R, f(t)=(-15)t²+(3)t+(30), allowing a2=0. At x=-2 and nonzero finite h=-1, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"78/1","explanation":"The specified finite quotient is 78/1, equal in this model to 2a2*x+a1+a2*h. It is a finite difference quotient, not a claim that every finite-step value is the derivative, even when a special case agrees."},"skills":["Finite Difference Quotient at a Specified Nonzero Step"],"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-20261007-advance-20261008-r04-derivative-0067","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a07de758dc2ebdcf703756e9f6bb1d70876a087c682e6efcc36f41b620f56923","family_id":"qf-derivative-finite-quotient","seed":202610084067,"semantic_template":"qf-derivative-finite-quotient/{\"inputs\":[\"16\",\"4\",\"-1\",\"2\"],\"operation\":\"finite_quadratic_quotient\"}"},"fields":{"title":"Finite Difference Quotient at a Specified Nonzero Step","stem":"On all R, f(t)=(16)t²+(4)t+(32), allowing a2=0. At x=-1 and nonzero finite h=2, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"4/1","explanation":"The specified finite quotient is 4/1, equal in this model to 2a2*x+a1+a2*h. It is a finite difference quotient, not a claim that every finite-step value is the derivative, even when a special case agrees."},"skills":["Finite Difference Quotient at a Specified Nonzero Step"],"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-20261007-advance-20261008-r04-derivative-0068","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"01b0a5cbc6d5d979fb15b7697fbf53dd5ac9ce022be95afa58c53195c8e236ce","family_id":"qf-derivative-finite-quotient","seed":202610084068,"semantic_template":"qf-derivative-finite-quotient/{\"inputs\":[\"-17\",\"5\",\"0\",\"-3\"],\"operation\":\"finite_quadratic_quotient\"}"},"fields":{"title":"Finite Difference Quotient at a Specified Nonzero Step","stem":"On all R, f(t)=(-17)t²+(5)t+(34), allowing a2=0. At x=0 and nonzero finite h=-3, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"56/1","explanation":"The specified finite quotient is 56/1, equal in this model to 2a2*x+a1+a2*h. It is a finite difference quotient, not a claim that every finite-step value is the derivative, even when a special case agrees."},"skills":["Finite Difference Quotient at a Specified Nonzero Step"],"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-20261007-advance-20261008-r04-derivative-0069","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"937d25f088b94f09444fde46c9aa9eb5a01a5af116e4111b741134ba95fea417","family_id":"qf-derivative-finite-quotient","seed":202610084069,"semantic_template":"qf-derivative-finite-quotient/{\"inputs\":[\"18\",\"6\",\"1\",\"1\"],\"operation\":\"finite_quadratic_quotient\"}"},"fields":{"title":"Finite Difference Quotient at a Specified Nonzero Step","stem":"On all R, f(t)=(18)t²+(6)t+(36), allowing a2=0. At x=1 and nonzero finite h=1, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"60/1","explanation":"The specified finite quotient is 60/1, equal in this model to 2a2*x+a1+a2*h. It is a finite difference quotient, not a claim that every finite-step value is the derivative, even when a special case agrees."},"skills":["Finite Difference Quotient at a Specified Nonzero Step"],"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-20261007-advance-20261008-r04-derivative-0070","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"673d08ecdd713d95b7454e6303ecc9a7b45204577cd17b56d899143559022e4a","family_id":"qf-derivative-finite-quotient","seed":202610084070,"semantic_template":"qf-derivative-finite-quotient/{\"inputs\":[\"-19\",\"7\",\"2\",\"-2\"],\"operation\":\"finite_quadratic_quotient\"}"},"fields":{"title":"Finite Difference Quotient at a Specified Nonzero Step","stem":"On all R, f(t)=(-19)t²+(7)t+(38), allowing a2=0. At x=2 and nonzero finite h=-2, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"-31/1","explanation":"The specified finite quotient is -31/1, equal in this model to 2a2*x+a1+a2*h. It is a finite difference quotient, not a claim that every finite-step value is the derivative, even when a special case agrees."},"skills":["Finite Difference Quotient at a Specified Nonzero Step"],"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-20261007-advance-20261008-r04-derivative-0071","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bced5a42f99329864621aef3e5a6517abcb0f0ccb87026c0d8d2b743c7d9b4d5","family_id":"qf-derivative-finite-quotient","seed":202610084071,"semantic_template":"qf-derivative-finite-quotient/{\"inputs\":[\"20\",\"8\",\"-2\",\"3\"],\"operation\":\"finite_quadratic_quotient\"}"},"fields":{"title":"Finite Difference Quotient at a Specified Nonzero Step","stem":"On all R, f(t)=(20)t²+(8)t+(40), allowing a2=0. At x=-2 and nonzero finite h=3, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"-12/1","explanation":"The specified finite quotient is -12/1, equal in this model to 2a2*x+a1+a2*h. It is a finite difference quotient, not a claim that every finite-step value is the derivative, even when a special case agrees."},"skills":["Finite Difference Quotient at a Specified Nonzero Step"],"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-20261007-advance-20261008-r04-derivative-0072","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b9b885d19c06f4a31b1178a72d07c2b9c5c1357b21bae23c9e0e7c04945d4b6","family_id":"qf-derivative-finite-quotient","seed":202610084072,"semantic_template":"qf-derivative-finite-quotient/{\"inputs\":[\"-21\",\"9\",\"-1\",\"-1\"],\"operation\":\"finite_quadratic_quotient\"}"},"fields":{"title":"Finite Difference Quotient at a Specified Nonzero Step","stem":"On all R, f(t)=(-21)t²+(9)t+(42), allowing a2=0. At x=-1 and nonzero finite h=-1, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"72/1","explanation":"The specified finite quotient is 72/1, equal in this model to 2a2*x+a1+a2*h. It is a finite difference quotient, not a claim that every finite-step value is the derivative, even when a special case agrees."},"skills":["Finite Difference Quotient at a Specified Nonzero Step"],"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-20261007-advance-20261008-r04-derivative-0073","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6f59edfe7b00cb7705e20004b4b432f4c126978ec7bf5f14e4f5ad1f6a840ffc","family_id":"qf-derivative-finite-quotient","seed":202610084073,"semantic_template":"qf-derivative-finite-quotient/{\"inputs\":[\"22\",\"10\",\"0\",\"2\"],\"operation\":\"finite_quadratic_quotient\"}"},"fields":{"title":"Finite Difference Quotient at a Specified Nonzero Step","stem":"On all R, f(t)=(22)t²+(10)t+(44), allowing a2=0. At x=0 and nonzero finite h=2, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"54/1","explanation":"The specified finite quotient is 54/1, equal in this model to 2a2*x+a1+a2*h. It is a finite difference quotient, not a claim that every finite-step value is the derivative, even when a special case agrees."},"skills":["Finite Difference Quotient at a Specified Nonzero Step"],"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-20261007-advance-20261008-r04-derivative-0074","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1427ca145116fc5404ded0dbf3c986b3fac1d8200da28211910b41536956620","family_id":"qf-derivative-finite-quotient","seed":202610084074,"semantic_template":"qf-derivative-finite-quotient/{\"inputs\":[\"-23\",\"11\",\"1\",\"-3\"],\"operation\":\"finite_quadratic_quotient\"}"},"fields":{"title":"Finite Difference Quotient at a Specified Nonzero Step","stem":"On all R, f(t)=(-23)t²+(11)t+(46), allowing a2=0. At x=1 and nonzero finite h=-3, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"34/1","explanation":"The specified finite quotient is 34/1, equal in this model to 2a2*x+a1+a2*h. It is a finite difference quotient, not a claim that every finite-step value is the derivative, even when a special case agrees."},"skills":["Finite Difference Quotient at a Specified Nonzero Step"],"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-20261007-advance-20261008-r04-derivative-0075","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"706c378beaa1fcb9c1640071f357dbce5f4b1f13d0cce72f923983c4baccd56e","family_id":"qf-derivative-finite-quotient","seed":202610084075,"semantic_template":"qf-derivative-finite-quotient/{\"inputs\":[\"24\",\"12\",\"2\",\"1\"],\"operation\":\"finite_quadratic_quotient\"}"},"fields":{"title":"Finite Difference Quotient at a Specified Nonzero Step","stem":"On all R, f(t)=(24)t²+(12)t+(48), allowing a2=0. At x=2 and nonzero finite h=1, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"132/1","explanation":"The specified finite quotient is 132/1, equal in this model to 2a2*x+a1+a2*h. It is a finite difference quotient, not a claim that every finite-step value is the derivative, even when a special case agrees."},"skills":["Finite Difference Quotient at a Specified Nonzero Step"],"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-20261007-advance-20261008-r04-derivative-0076","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4290d569390bff67b6912fb95000f2ed38b24d24ee5e47b6645801a422acd0cb","family_id":"qf-derivative-shifted-absolute","seed":202610084076,"semantic_template":"qf-derivative-shifted-absolute/{\"inputs\":[\"0\"],\"operation\":\"shifted_absolute_derivative\"}"},"fields":{"title":"Derivative and Corner of a Shifted Absolute Value","stem":"On all R, f(t)=|t−(-12)|. Find its derivative at x=-12, or answer does not exist at the corner.","answer":{"kind":"short_text","value":"does not exist","explanation":"At the center, the left derivative is −1 and the right derivative is 1, so the derivative does not exist. Do not substitute a totalized API default of zero for existence."},"skills":["Derivative and Corner of a Shifted Absolute Value"],"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-20261007-advance-20261008-r04-derivative-0077","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"972e8b435d62ad3ed401840fc9a698473f05c56a31d37fcf87a9481bc9e50b4c","family_id":"qf-derivative-shifted-absolute","seed":202610084077,"semantic_template":"qf-derivative-shifted-absolute/{\"inputs\":[\"-1\"],\"operation\":\"shifted_absolute_derivative\"}"},"fields":{"title":"Derivative and Corner of a Shifted Absolute Value","stem":"On all R, f(t)=|t−(-11)|. Find its derivative at x=-12, or answer does not exist at the corner.","answer":{"kind":"short_text","value":"-1","explanation":"Relative position x−c determines slope −1 below the center and 1 above it, giving -1. The corner itself would have no two-sided derivative."},"skills":["Derivative and Corner of a Shifted Absolute Value"],"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-20261007-advance-20261008-r04-derivative-0078","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac6ed2cfde05dcfdba7d216881f1e63e4db52222b9ebc8728b8a76e70e19af11","family_id":"qf-derivative-shifted-absolute","seed":202610084078,"semantic_template":"qf-derivative-shifted-absolute/{\"inputs\":[\"2\"],\"operation\":\"shifted_absolute_derivative\"}"},"fields":{"title":"Derivative and Corner of a Shifted Absolute Value","stem":"On all R, f(t)=|t−(-10)|. Find its derivative at x=-8, or answer does not exist at the corner.","answer":{"kind":"short_text","value":"1","explanation":"Relative position x−c determines slope −1 below the center and 1 above it, giving 1. The corner itself would have no two-sided derivative."},"skills":["Derivative and Corner of a Shifted Absolute Value"],"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-20261007-advance-20261008-r04-derivative-0079","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ccb41eef832672eecf480e791cb139a229071b22b4ffe4fd2631534fad6fc93","family_id":"qf-derivative-shifted-absolute","seed":202610084079,"semantic_template":"qf-derivative-shifted-absolute/{\"inputs\":[\"-3\"],\"operation\":\"shifted_absolute_derivative\"}"},"fields":{"title":"Derivative and Corner of a Shifted Absolute Value","stem":"On all R, f(t)=|t−(-9)|. Find its derivative at x=-12, or answer does not exist at the corner.","answer":{"kind":"short_text","value":"-1","explanation":"Relative position x−c determines slope −1 below the center and 1 above it, giving -1. The corner itself would have no two-sided derivative."},"skills":["Derivative and Corner of a Shifted Absolute Value"],"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-20261007-advance-20261008-r04-derivative-0080","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ed5518305ce37f21138ba5951df42ea04f751b2d1817dbeac457cc76e5509660","family_id":"qf-derivative-shifted-absolute","seed":202610084080,"semantic_template":"qf-derivative-shifted-absolute/{\"inputs\":[\"4\"],\"operation\":\"shifted_absolute_derivative\"}"},"fields":{"title":"Derivative and Corner of a Shifted Absolute Value","stem":"On all R, f(t)=|t−(-8)|. Find its derivative at x=-4, or answer does not exist at the corner.","answer":{"kind":"short_text","value":"1","explanation":"Relative position x−c determines slope −1 below the center and 1 above it, giving 1. The corner itself would have no two-sided derivative."},"skills":["Derivative and Corner of a Shifted Absolute Value"],"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-20261007-advance-20261008-r04-derivative-0081","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bda56a6764de4eba29ed1abe7e29b6cc7cc1034be454aaed2bcaa917593c2690","family_id":"qf-derivative-shifted-absolute","seed":202610084081,"semantic_template":"qf-derivative-shifted-absolute/{\"inputs\":[\"-5\"],\"operation\":\"shifted_absolute_derivative\"}"},"fields":{"title":"Derivative and Corner of a Shifted Absolute Value","stem":"On all R, f(t)=|t−(-7)|. Find its derivative at x=-12, or answer does not exist at the corner.","answer":{"kind":"short_text","value":"-1","explanation":"Relative position x−c determines slope −1 below the center and 1 above it, giving -1. The corner itself would have no two-sided derivative."},"skills":["Derivative and Corner of a Shifted Absolute Value"],"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-20261007-advance-20261008-r04-derivative-0082","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd648238bd335fc01403d64d76bef86681200de165fce788494ac5348b71e727","family_id":"qf-derivative-shifted-absolute","seed":202610084082,"semantic_template":"qf-derivative-shifted-absolute/{\"inputs\":[\"6\"],\"operation\":\"shifted_absolute_derivative\"}"},"fields":{"title":"Derivative and Corner of a Shifted Absolute Value","stem":"On all R, f(t)=|t−(-6)|. Find its derivative at x=0, or answer does not exist at the corner.","answer":{"kind":"short_text","value":"1","explanation":"Relative position x−c determines slope −1 below the center and 1 above it, giving 1. The corner itself would have no two-sided derivative."},"skills":["Derivative and Corner of a Shifted Absolute Value"],"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-20261007-advance-20261008-r04-derivative-0083","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a8ffc575e1e60fb321dce29b25d681710501e0fa75c4885f51cf046d9937c45c","family_id":"qf-derivative-shifted-absolute","seed":202610084083,"semantic_template":"qf-derivative-shifted-absolute/{\"inputs\":[\"-7\"],\"operation\":\"shifted_absolute_derivative\"}"},"fields":{"title":"Derivative and Corner of a Shifted Absolute Value","stem":"On all R, f(t)=|t−(-5)|. Find its derivative at x=-12, or answer does not exist at the corner.","answer":{"kind":"short_text","value":"-1","explanation":"Relative position x−c determines slope −1 below the center and 1 above it, giving -1. The corner itself would have no two-sided derivative."},"skills":["Derivative and Corner of a Shifted Absolute Value"],"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-20261007-advance-20261008-r04-derivative-0084","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a9567838e99df219832cc1ec3eb3b09c8b1376aa3a971f735ac46c7996ea6c6c","family_id":"qf-derivative-shifted-absolute","seed":202610084084,"semantic_template":"qf-derivative-shifted-absolute/{\"inputs\":[\"8\"],\"operation\":\"shifted_absolute_derivative\"}"},"fields":{"title":"Derivative and Corner of a Shifted Absolute Value","stem":"On all R, f(t)=|t−(-4)|. Find its derivative at x=4, or answer does not exist at the corner.","answer":{"kind":"short_text","value":"1","explanation":"Relative position x−c determines slope −1 below the center and 1 above it, giving 1. The corner itself would have no two-sided derivative."},"skills":["Derivative and Corner of a Shifted Absolute Value"],"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-20261007-advance-20261008-r04-derivative-0085","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c7e4e19ba1ffd2e42bf3cf22b0fc8a7070cf4d01eed909dd436785bfaf6c017","family_id":"qf-derivative-shifted-absolute","seed":202610084085,"semantic_template":"qf-derivative-shifted-absolute/{\"inputs\":[\"-9\"],\"operation\":\"shifted_absolute_derivative\"}"},"fields":{"title":"Derivative and Corner of a Shifted Absolute Value","stem":"On all R, f(t)=|t−(-3)|. Find its derivative at x=-12, or answer does not exist at the corner.","answer":{"kind":"short_text","value":"-1","explanation":"Relative position x−c determines slope −1 below the center and 1 above it, giving -1. The corner itself would have no two-sided derivative."},"skills":["Derivative and Corner of a Shifted Absolute Value"],"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-20261007-advance-20261008-r04-derivative-0086","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"897731321faa19bc4d5816c58673cbaae956948e2fcd779cdbd59562db556d31","family_id":"qf-derivative-shifted-absolute","seed":202610084086,"semantic_template":"qf-derivative-shifted-absolute/{\"inputs\":[\"10\"],\"operation\":\"shifted_absolute_derivative\"}"},"fields":{"title":"Derivative and Corner of a Shifted Absolute Value","stem":"On all R, f(t)=|t−(-2)|. Find its derivative at x=8, or answer does not exist at the corner.","answer":{"kind":"short_text","value":"1","explanation":"Relative position x−c determines slope −1 below the center and 1 above it, giving 1. The corner itself would have no two-sided derivative."},"skills":["Derivative and Corner of a Shifted Absolute Value"],"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-20261007-advance-20261008-r04-derivative-0087","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"909b415fa868635a3292bbeac08741b95d304e6d0d151502c9131b42e26da7e6","family_id":"qf-derivative-shifted-absolute","seed":202610084087,"semantic_template":"qf-derivative-shifted-absolute/{\"inputs\":[\"-11\"],\"operation\":\"shifted_absolute_derivative\"}"},"fields":{"title":"Derivative and Corner of a Shifted Absolute Value","stem":"On all R, f(t)=|t−(-1)|. Find its derivative at x=-12, or answer does not exist at the corner.","answer":{"kind":"short_text","value":"-1","explanation":"Relative position x−c determines slope −1 below the center and 1 above it, giving -1. The corner itself would have no two-sided derivative."},"skills":["Derivative and Corner of a Shifted Absolute Value"],"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-20261007-advance-20261008-r04-derivative-0088","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d26d291a09d84437b50e91053bc13d76110869f23f6f359e1b6df4c5d8c70c24","family_id":"qf-derivative-shifted-absolute","seed":202610084088,"semantic_template":"qf-derivative-shifted-absolute/{\"inputs\":[\"12\"],\"operation\":\"shifted_absolute_derivative\"}"},"fields":{"title":"Derivative and Corner of a Shifted Absolute Value","stem":"On all R, f(t)=|t−(0)|. Find its derivative at x=12, or answer does not exist at the corner.","answer":{"kind":"short_text","value":"1","explanation":"Relative position x−c determines slope −1 below the center and 1 above it, giving 1. The corner itself would have no two-sided derivative."},"skills":["Derivative and Corner of a Shifted Absolute Value"],"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-20261007-advance-20261008-r04-derivative-0089","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"46ec4382564156443fbccd815d488244df08a5569e82d037ac96e53570757554","family_id":"qf-derivative-shifted-absolute","seed":202610084089,"semantic_template":"qf-derivative-shifted-absolute/{\"inputs\":[\"-13\"],\"operation\":\"shifted_absolute_derivative\"}"},"fields":{"title":"Derivative and Corner of a Shifted Absolute Value","stem":"On all R, f(t)=|t−(1)|. Find its derivative at x=-12, or answer does not exist at the corner.","answer":{"kind":"short_text","value":"-1","explanation":"Relative position x−c determines slope −1 below the center and 1 above it, giving -1. The corner itself would have no two-sided derivative."},"skills":["Derivative and Corner of a Shifted Absolute Value"],"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-20261007-advance-20261008-r04-derivative-0090","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad9eb6314b9aa285b5b825822247c1014b1afe14eb22eb05f20a6e49fef76c4e","family_id":"qf-derivative-shifted-absolute","seed":202610084090,"semantic_template":"qf-derivative-shifted-absolute/{\"inputs\":[\"14\"],\"operation\":\"shifted_absolute_derivative\"}"},"fields":{"title":"Derivative and Corner of a Shifted Absolute Value","stem":"On all R, f(t)=|t−(2)|. Find its derivative at x=16, or answer does not exist at the corner.","answer":{"kind":"short_text","value":"1","explanation":"Relative position x−c determines slope −1 below the center and 1 above it, giving 1. The corner itself would have no two-sided derivative."},"skills":["Derivative and Corner of a Shifted Absolute Value"],"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-20261007-advance-20261008-r04-derivative-0091","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90e4de6c543b70896588ccd59b460277ef88739a099ad058f8a1f1d4b3a29182","family_id":"qf-derivative-shifted-absolute","seed":202610084091,"semantic_template":"qf-derivative-shifted-absolute/{\"inputs\":[\"-15\"],\"operation\":\"shifted_absolute_derivative\"}"},"fields":{"title":"Derivative and Corner of a Shifted Absolute Value","stem":"On all R, f(t)=|t−(3)|. Find its derivative at x=-12, or answer does not exist at the corner.","answer":{"kind":"short_text","value":"-1","explanation":"Relative position x−c determines slope −1 below the center and 1 above it, giving -1. The corner itself would have no two-sided derivative."},"skills":["Derivative and Corner of a Shifted Absolute Value"],"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-20261007-advance-20261008-r04-derivative-0092","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"286e3e502cdf7b88c1cc01e34cf02f43be3128c302ad53d66cd7b2e358fe35fb","family_id":"qf-derivative-shifted-absolute","seed":202610084092,"semantic_template":"qf-derivative-shifted-absolute/{\"inputs\":[\"16\"],\"operation\":\"shifted_absolute_derivative\"}"},"fields":{"title":"Derivative and Corner of a Shifted Absolute Value","stem":"On all R, f(t)=|t−(4)|. Find its derivative at x=20, or answer does not exist at the corner.","answer":{"kind":"short_text","value":"1","explanation":"Relative position x−c determines slope −1 below the center and 1 above it, giving 1. The corner itself would have no two-sided derivative."},"skills":["Derivative and Corner of a Shifted Absolute Value"],"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-20261007-advance-20261008-r04-derivative-0093","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"48709fbefec5e5bb7289e28b2f4538d261a98ac6031d891ae151872664911310","family_id":"qf-derivative-shifted-absolute","seed":202610084093,"semantic_template":"qf-derivative-shifted-absolute/{\"inputs\":[\"-17\"],\"operation\":\"shifted_absolute_derivative\"}"},"fields":{"title":"Derivative and Corner of a Shifted Absolute Value","stem":"On all R, f(t)=|t−(5)|. Find its derivative at x=-12, or answer does not exist at the corner.","answer":{"kind":"short_text","value":"-1","explanation":"Relative position x−c determines slope −1 below the center and 1 above it, giving -1. The corner itself would have no two-sided derivative."},"skills":["Derivative and Corner of a Shifted Absolute Value"],"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-20261007-advance-20261008-r04-derivative-0094","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"187f2b9b84ca712b2881cd33b9bc602b77affc8932f7a4b1da3dd7fee1e4f15a","family_id":"qf-derivative-shifted-absolute","seed":202610084094,"semantic_template":"qf-derivative-shifted-absolute/{\"inputs\":[\"18\"],\"operation\":\"shifted_absolute_derivative\"}"},"fields":{"title":"Derivative and Corner of a Shifted Absolute Value","stem":"On all R, f(t)=|t−(6)|. Find its derivative at x=24, or answer does not exist at the corner.","answer":{"kind":"short_text","value":"1","explanation":"Relative position x−c determines slope −1 below the center and 1 above it, giving 1. The corner itself would have no two-sided derivative."},"skills":["Derivative and Corner of a Shifted Absolute Value"],"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-20261007-advance-20261008-r04-derivative-0095","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f6ae34f0ad05ff6df4738cfcc9f611998877a3462b61d02700946afc64de39f","family_id":"qf-derivative-shifted-absolute","seed":202610084095,"semantic_template":"qf-derivative-shifted-absolute/{\"inputs\":[\"-19\"],\"operation\":\"shifted_absolute_derivative\"}"},"fields":{"title":"Derivative and Corner of a Shifted Absolute Value","stem":"On all R, f(t)=|t−(7)|. Find its derivative at x=-12, or answer does not exist at the corner.","answer":{"kind":"short_text","value":"-1","explanation":"Relative position x−c determines slope −1 below the center and 1 above it, giving -1. The corner itself would have no two-sided derivative."},"skills":["Derivative and Corner of a Shifted Absolute Value"],"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-20261007-advance-20261008-r04-derivative-0096","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c9c343b080ba0d63016ca69c7f2d22da6d9c6a6e9617ab97c5d09a9211f7aa1b","family_id":"qf-derivative-shifted-absolute","seed":202610084096,"semantic_template":"qf-derivative-shifted-absolute/{\"inputs\":[\"20\"],\"operation\":\"shifted_absolute_derivative\"}"},"fields":{"title":"Derivative and Corner of a Shifted Absolute Value","stem":"On all R, f(t)=|t−(8)|. Find its derivative at x=28, or answer does not exist at the corner.","answer":{"kind":"short_text","value":"1","explanation":"Relative position x−c determines slope −1 below the center and 1 above it, giving 1. The corner itself would have no two-sided derivative."},"skills":["Derivative and Corner of a Shifted Absolute Value"],"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-20261007-advance-20261008-r04-derivative-0097","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"709cc3cd32ff780a9796b581f4e1086f320e30c51e2797371751145af335a54d","family_id":"qf-derivative-shifted-absolute","seed":202610084097,"semantic_template":"qf-derivative-shifted-absolute/{\"inputs\":[\"-21\"],\"operation\":\"shifted_absolute_derivative\"}"},"fields":{"title":"Derivative and Corner of a Shifted Absolute Value","stem":"On all R, f(t)=|t−(9)|. Find its derivative at x=-12, or answer does not exist at the corner.","answer":{"kind":"short_text","value":"-1","explanation":"Relative position x−c determines slope −1 below the center and 1 above it, giving -1. The corner itself would have no two-sided derivative."},"skills":["Derivative and Corner of a Shifted Absolute Value"],"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-20261007-advance-20261008-r04-derivative-0098","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da830ba155e11b5545954e5c712d52354eb86efbdc8346c8564afff29d5495ac","family_id":"qf-derivative-shifted-absolute","seed":202610084098,"semantic_template":"qf-derivative-shifted-absolute/{\"inputs\":[\"22\"],\"operation\":\"shifted_absolute_derivative\"}"},"fields":{"title":"Derivative and Corner of a Shifted Absolute Value","stem":"On all R, f(t)=|t−(10)|. Find its derivative at x=32, or answer does not exist at the corner.","answer":{"kind":"short_text","value":"1","explanation":"Relative position x−c determines slope −1 below the center and 1 above it, giving 1. The corner itself would have no two-sided derivative."},"skills":["Derivative and Corner of a Shifted Absolute Value"],"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-20261007-advance-20261008-r04-derivative-0099","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9222e4df303274618b1ea4d15b3b81df87231c60a83da2516dd02756e2b88b49","family_id":"qf-derivative-shifted-absolute","seed":202610084099,"semantic_template":"qf-derivative-shifted-absolute/{\"inputs\":[\"-23\"],\"operation\":\"shifted_absolute_derivative\"}"},"fields":{"title":"Derivative and Corner of a Shifted Absolute Value","stem":"On all R, f(t)=|t−(11)|. Find its derivative at x=-12, or answer does not exist at the corner.","answer":{"kind":"short_text","value":"-1","explanation":"Relative position x−c determines slope −1 below the center and 1 above it, giving -1. The corner itself would have no two-sided derivative."},"skills":["Derivative and Corner of a Shifted Absolute Value"],"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-20261007-advance-20261008-r04-derivative-0100","batch_id":"batch-20261007-advance-20261008-r04-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e7309b5848442396280630a48d7a639bb7d72393482675891b8559d3062b8e0d","family_id":"qf-derivative-shifted-absolute","seed":202610084100,"semantic_template":"qf-derivative-shifted-absolute/{\"inputs\":[\"24\"],\"operation\":\"shifted_absolute_derivative\"}"},"fields":{"title":"Derivative and Corner of a Shifted Absolute Value","stem":"On all R, f(t)=|t−(12)|. Find its derivative at x=36, or answer does not exist at the corner.","answer":{"kind":"short_text","value":"1","explanation":"Relative position x−c determines slope −1 below the center and 1 above it, giving 1. The corner itself would have no two-sided derivative."},"skills":["Derivative and Corner of a Shifted Absolute Value"],"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."}}