{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0001","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"明示函數兩點的平均變化率","stem_zh":"求平均變化率：同一函數明示 f(-13)=-9、f(-12)=-9，兩個輸入不同。答案為最簡正分母 p/q，整數保留 /1。","answer":{"kind":"expression","value":"0/1","explanation_zh":"差商=0/1。以對應輸出差除以非零輸入差，反轉順序須兩差一起換號；這是平均變化率，不是導數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["明示函數兩點的平均變化率"]},"variant":{"family_id":"qf-rate-endpoint-quotient","seed":202610063001,"parameters":{"x0":-13,"x1":-12,"y0":-9,"y1":-9,"operation":"endpoint_quotient","semantic_template":"qf-rate-endpoint-quotient/{\"inputs\":[\"1\",\"0\"],\"operation\":\"endpoint_quotient\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0002","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"明示函數兩點的平均變化率","stem_zh":"求平均變化率：同一函數明示 f(-12)=-7、f(-14)=-5，兩個輸入不同。答案為最簡正分母 p/q，整數保留 /1。","answer":{"kind":"expression","value":"-1/1","explanation_zh":"差商=-1/1。以對應輸出差除以非零輸入差，反轉順序須兩差一起換號；這是平均變化率，不是導數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["明示函數兩點的平均變化率"]},"variant":{"family_id":"qf-rate-endpoint-quotient","seed":202610063002,"parameters":{"x0":-12,"x1":-14,"y0":-7,"y1":-5,"operation":"endpoint_quotient","semantic_template":"qf-rate-endpoint-quotient/{\"inputs\":[\"2\",\"-2\"],\"operation\":\"endpoint_quotient\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0003","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"明示函數兩點的平均變化率","stem_zh":"求平均變化率：同一函數明示 f(-11)=-5、f(-8)=-2，兩個輸入不同。答案為最簡正分母 p/q，整數保留 /1。","answer":{"kind":"expression","value":"1/1","explanation_zh":"差商=1/1。以對應輸出差除以非零輸入差，反轉順序須兩差一起換號；這是平均變化率，不是導數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["明示函數兩點的平均變化率"]},"variant":{"family_id":"qf-rate-endpoint-quotient","seed":202610063003,"parameters":{"x0":-11,"x1":-8,"y0":-5,"y1":-2,"operation":"endpoint_quotient","semantic_template":"qf-rate-endpoint-quotient/{\"inputs\":[\"3\",\"3\"],\"operation\":\"endpoint_quotient\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0004","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"明示函數兩點的平均變化率","stem_zh":"求平均變化率：同一函數明示 f(-10)=-3、f(-14)=1，兩個輸入不同。答案為最簡正分母 p/q，整數保留 /1。","answer":{"kind":"expression","value":"-1/1","explanation_zh":"差商=-1/1。以對應輸出差除以非零輸入差，反轉順序須兩差一起換號；這是平均變化率，不是導數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["明示函數兩點的平均變化率"]},"variant":{"family_id":"qf-rate-endpoint-quotient","seed":202610063004,"parameters":{"x0":-10,"x1":-14,"y0":-3,"y1":1,"operation":"endpoint_quotient","semantic_template":"qf-rate-endpoint-quotient/{\"inputs\":[\"4\",\"-4\"],\"operation\":\"endpoint_quotient\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0005","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"明示函數兩點的平均變化率","stem_zh":"求平均變化率：同一函數明示 f(-9)=-1、f(-4)=4，兩個輸入不同。答案為最簡正分母 p/q，整數保留 /1。","answer":{"kind":"expression","value":"1/1","explanation_zh":"差商=1/1。以對應輸出差除以非零輸入差，反轉順序須兩差一起換號；這是平均變化率，不是導數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["明示函數兩點的平均變化率"]},"variant":{"family_id":"qf-rate-endpoint-quotient","seed":202610063005,"parameters":{"x0":-9,"x1":-4,"y0":-1,"y1":4,"operation":"endpoint_quotient","semantic_template":"qf-rate-endpoint-quotient/{\"inputs\":[\"5\",\"5\"],\"operation\":\"endpoint_quotient\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0006","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"明示函數兩點的平均變化率","stem_zh":"求平均變化率：同一函數明示 f(-8)=1、f(-14)=7，兩個輸入不同。答案為最簡正分母 p/q，整數保留 /1。","answer":{"kind":"expression","value":"-1/1","explanation_zh":"差商=-1/1。以對應輸出差除以非零輸入差，反轉順序須兩差一起換號；這是平均變化率，不是導數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["明示函數兩點的平均變化率"]},"variant":{"family_id":"qf-rate-endpoint-quotient","seed":202610063006,"parameters":{"x0":-8,"x1":-14,"y0":1,"y1":7,"operation":"endpoint_quotient","semantic_template":"qf-rate-endpoint-quotient/{\"inputs\":[\"6\",\"-6\"],\"operation\":\"endpoint_quotient\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0007","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"明示函數兩點的平均變化率","stem_zh":"求平均變化率：同一函數明示 f(-7)=3、f(0)=10，兩個輸入不同。答案為最簡正分母 p/q，整數保留 /1。","answer":{"kind":"expression","value":"1/1","explanation_zh":"差商=1/1。以對應輸出差除以非零輸入差，反轉順序須兩差一起換號；這是平均變化率，不是導數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["明示函數兩點的平均變化率"]},"variant":{"family_id":"qf-rate-endpoint-quotient","seed":202610063007,"parameters":{"x0":-7,"x1":0,"y0":3,"y1":10,"operation":"endpoint_quotient","semantic_template":"qf-rate-endpoint-quotient/{\"inputs\":[\"7\",\"7\"],\"operation\":\"endpoint_quotient\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0008","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"明示函數兩點的平均變化率","stem_zh":"求平均變化率：同一函數明示 f(-6)=5、f(-7)=13，兩個輸入不同。答案為最簡正分母 p/q，整數保留 /1。","answer":{"kind":"expression","value":"-8/1","explanation_zh":"差商=-8/1。以對應輸出差除以非零輸入差，反轉順序須兩差一起換號；這是平均變化率，不是導數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["明示函數兩點的平均變化率"]},"variant":{"family_id":"qf-rate-endpoint-quotient","seed":202610063008,"parameters":{"x0":-6,"x1":-7,"y0":5,"y1":13,"operation":"endpoint_quotient","semantic_template":"qf-rate-endpoint-quotient/{\"inputs\":[\"1\",\"-8\"],\"operation\":\"endpoint_quotient\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0009","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"明示函數兩點的平均變化率","stem_zh":"求平均變化率：同一函數明示 f(-5)=7、f(-3)=16，兩個輸入不同。答案為最簡正分母 p/q，整數保留 /1。","answer":{"kind":"expression","value":"9/2","explanation_zh":"差商=9/2。以對應輸出差除以非零輸入差，反轉順序須兩差一起換號；這是平均變化率，不是導數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["明示函數兩點的平均變化率"]},"variant":{"family_id":"qf-rate-endpoint-quotient","seed":202610063009,"parameters":{"x0":-5,"x1":-3,"y0":7,"y1":16,"operation":"endpoint_quotient","semantic_template":"qf-rate-endpoint-quotient/{\"inputs\":[\"2\",\"9\"],\"operation\":\"endpoint_quotient\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0010","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"明示函數兩點的平均變化率","stem_zh":"求平均變化率：同一函數明示 f(-4)=9、f(-7)=19，兩個輸入不同。答案為最簡正分母 p/q，整數保留 /1。","answer":{"kind":"expression","value":"-10/3","explanation_zh":"差商=-10/3。以對應輸出差除以非零輸入差，反轉順序須兩差一起換號；這是平均變化率，不是導數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["明示函數兩點的平均變化率"]},"variant":{"family_id":"qf-rate-endpoint-quotient","seed":202610063010,"parameters":{"x0":-4,"x1":-7,"y0":9,"y1":19,"operation":"endpoint_quotient","semantic_template":"qf-rate-endpoint-quotient/{\"inputs\":[\"3\",\"-10\"],\"operation\":\"endpoint_quotient\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0011","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"明示函數兩點的平均變化率","stem_zh":"求平均變化率：同一函數明示 f(-3)=11、f(1)=22，兩個輸入不同。答案為最簡正分母 p/q，整數保留 /1。","answer":{"kind":"expression","value":"11/4","explanation_zh":"差商=11/4。以對應輸出差除以非零輸入差，反轉順序須兩差一起換號；這是平均變化率，不是導數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["明示函數兩點的平均變化率"]},"variant":{"family_id":"qf-rate-endpoint-quotient","seed":202610063011,"parameters":{"x0":-3,"x1":1,"y0":11,"y1":22,"operation":"endpoint_quotient","semantic_template":"qf-rate-endpoint-quotient/{\"inputs\":[\"4\",\"11\"],\"operation\":\"endpoint_quotient\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0012","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"明示函數兩點的平均變化率","stem_zh":"求平均變化率：同一函數明示 f(-2)=13、f(-7)=25，兩個輸入不同。答案為最簡正分母 p/q，整數保留 /1。","answer":{"kind":"expression","value":"-12/5","explanation_zh":"差商=-12/5。以對應輸出差除以非零輸入差，反轉順序須兩差一起換號；這是平均變化率，不是導數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["明示函數兩點的平均變化率"]},"variant":{"family_id":"qf-rate-endpoint-quotient","seed":202610063012,"parameters":{"x0":-2,"x1":-7,"y0":13,"y1":25,"operation":"endpoint_quotient","semantic_template":"qf-rate-endpoint-quotient/{\"inputs\":[\"5\",\"-12\"],\"operation\":\"endpoint_quotient\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0013","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"明示函數兩點的平均變化率","stem_zh":"求平均變化率：同一函數明示 f(-1)=15、f(5)=28，兩個輸入不同。答案為最簡正分母 p/q，整數保留 /1。","answer":{"kind":"expression","value":"13/6","explanation_zh":"差商=13/6。以對應輸出差除以非零輸入差，反轉順序須兩差一起換號；這是平均變化率，不是導數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["明示函數兩點的平均變化率"]},"variant":{"family_id":"qf-rate-endpoint-quotient","seed":202610063013,"parameters":{"x0":-1,"x1":5,"y0":15,"y1":28,"operation":"endpoint_quotient","semantic_template":"qf-rate-endpoint-quotient/{\"inputs\":[\"6\",\"13\"],\"operation\":\"endpoint_quotient\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0014","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"明示函數兩點的平均變化率","stem_zh":"求平均變化率：同一函數明示 f(0)=17、f(-7)=31，兩個輸入不同。答案為最簡正分母 p/q，整數保留 /1。","answer":{"kind":"expression","value":"-2/1","explanation_zh":"差商=-2/1。以對應輸出差除以非零輸入差，反轉順序須兩差一起換號；這是平均變化率，不是導數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["明示函數兩點的平均變化率"]},"variant":{"family_id":"qf-rate-endpoint-quotient","seed":202610063014,"parameters":{"x0":0,"x1":-7,"y0":17,"y1":31,"operation":"endpoint_quotient","semantic_template":"qf-rate-endpoint-quotient/{\"inputs\":[\"7\",\"-14\"],\"operation\":\"endpoint_quotient\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0015","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"明示函數兩點的平均變化率","stem_zh":"求平均變化率：同一函數明示 f(1)=19、f(2)=34，兩個輸入不同。答案為最簡正分母 p/q，整數保留 /1。","answer":{"kind":"expression","value":"15/1","explanation_zh":"差商=15/1。以對應輸出差除以非零輸入差，反轉順序須兩差一起換號；這是平均變化率，不是導數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["明示函數兩點的平均變化率"]},"variant":{"family_id":"qf-rate-endpoint-quotient","seed":202610063015,"parameters":{"x0":1,"x1":2,"y0":19,"y1":34,"operation":"endpoint_quotient","semantic_template":"qf-rate-endpoint-quotient/{\"inputs\":[\"1\",\"15\"],\"operation\":\"endpoint_quotient\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0016","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"明示函數兩點的平均變化率","stem_zh":"求平均變化率：同一函數明示 f(2)=21、f(0)=37，兩個輸入不同。答案為最簡正分母 p/q，整數保留 /1。","answer":{"kind":"expression","value":"-8/1","explanation_zh":"差商=-8/1。以對應輸出差除以非零輸入差，反轉順序須兩差一起換號；這是平均變化率，不是導數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["明示函數兩點的平均變化率"]},"variant":{"family_id":"qf-rate-endpoint-quotient","seed":202610063016,"parameters":{"x0":2,"x1":0,"y0":21,"y1":37,"operation":"endpoint_quotient","semantic_template":"qf-rate-endpoint-quotient/{\"inputs\":[\"2\",\"-16\"],\"operation\":\"endpoint_quotient\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0017","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"明示函數兩點的平均變化率","stem_zh":"求平均變化率：同一函數明示 f(3)=23、f(6)=40，兩個輸入不同。答案為最簡正分母 p/q，整數保留 /1。","answer":{"kind":"expression","value":"17/3","explanation_zh":"差商=17/3。以對應輸出差除以非零輸入差，反轉順序須兩差一起換號；這是平均變化率，不是導數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["明示函數兩點的平均變化率"]},"variant":{"family_id":"qf-rate-endpoint-quotient","seed":202610063017,"parameters":{"x0":3,"x1":6,"y0":23,"y1":40,"operation":"endpoint_quotient","semantic_template":"qf-rate-endpoint-quotient/{\"inputs\":[\"3\",\"17\"],\"operation\":\"endpoint_quotient\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0018","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"明示函數兩點的平均變化率","stem_zh":"求平均變化率：同一函數明示 f(4)=25、f(0)=43，兩個輸入不同。答案為最簡正分母 p/q，整數保留 /1。","answer":{"kind":"expression","value":"-9/2","explanation_zh":"差商=-9/2。以對應輸出差除以非零輸入差，反轉順序須兩差一起換號；這是平均變化率，不是導數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["明示函數兩點的平均變化率"]},"variant":{"family_id":"qf-rate-endpoint-quotient","seed":202610063018,"parameters":{"x0":4,"x1":0,"y0":25,"y1":43,"operation":"endpoint_quotient","semantic_template":"qf-rate-endpoint-quotient/{\"inputs\":[\"4\",\"-18\"],\"operation\":\"endpoint_quotient\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0019","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"明示函數兩點的平均變化率","stem_zh":"求平均變化率：同一函數明示 f(5)=27、f(10)=46，兩個輸入不同。答案為最簡正分母 p/q，整數保留 /1。","answer":{"kind":"expression","value":"19/5","explanation_zh":"差商=19/5。以對應輸出差除以非零輸入差，反轉順序須兩差一起換號；這是平均變化率，不是導數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["明示函數兩點的平均變化率"]},"variant":{"family_id":"qf-rate-endpoint-quotient","seed":202610063019,"parameters":{"x0":5,"x1":10,"y0":27,"y1":46,"operation":"endpoint_quotient","semantic_template":"qf-rate-endpoint-quotient/{\"inputs\":[\"5\",\"19\"],\"operation\":\"endpoint_quotient\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0020","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"明示函數兩點的平均變化率","stem_zh":"求平均變化率：同一函數明示 f(6)=29、f(0)=49，兩個輸入不同。答案為最簡正分母 p/q，整數保留 /1。","answer":{"kind":"expression","value":"-10/3","explanation_zh":"差商=-10/3。以對應輸出差除以非零輸入差，反轉順序須兩差一起換號；這是平均變化率，不是導數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["明示函數兩點的平均變化率"]},"variant":{"family_id":"qf-rate-endpoint-quotient","seed":202610063020,"parameters":{"x0":6,"x1":0,"y0":29,"y1":49,"operation":"endpoint_quotient","semantic_template":"qf-rate-endpoint-quotient/{\"inputs\":[\"6\",\"-20\"],\"operation\":\"endpoint_quotient\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0021","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"明示函數兩點的平均變化率","stem_zh":"求平均變化率：同一函數明示 f(7)=31、f(14)=52，兩個輸入不同。答案為最簡正分母 p/q，整數保留 /1。","answer":{"kind":"expression","value":"3/1","explanation_zh":"差商=3/1。以對應輸出差除以非零輸入差，反轉順序須兩差一起換號；這是平均變化率，不是導數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["明示函數兩點的平均變化率"]},"variant":{"family_id":"qf-rate-endpoint-quotient","seed":202610063021,"parameters":{"x0":7,"x1":14,"y0":31,"y1":52,"operation":"endpoint_quotient","semantic_template":"qf-rate-endpoint-quotient/{\"inputs\":[\"7\",\"21\"],\"operation\":\"endpoint_quotient\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0022","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"明示函數兩點的平均變化率","stem_zh":"求平均變化率：同一函數明示 f(8)=33、f(7)=55，兩個輸入不同。答案為最簡正分母 p/q，整數保留 /1。","answer":{"kind":"expression","value":"-22/1","explanation_zh":"差商=-22/1。以對應輸出差除以非零輸入差，反轉順序須兩差一起換號；這是平均變化率，不是導數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["明示函數兩點的平均變化率"]},"variant":{"family_id":"qf-rate-endpoint-quotient","seed":202610063022,"parameters":{"x0":8,"x1":7,"y0":33,"y1":55,"operation":"endpoint_quotient","semantic_template":"qf-rate-endpoint-quotient/{\"inputs\":[\"1\",\"-22\"],\"operation\":\"endpoint_quotient\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0023","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"明示函數兩點的平均變化率","stem_zh":"求平均變化率：同一函數明示 f(9)=35、f(11)=58，兩個輸入不同。答案為最簡正分母 p/q，整數保留 /1。","answer":{"kind":"expression","value":"23/2","explanation_zh":"差商=23/2。以對應輸出差除以非零輸入差，反轉順序須兩差一起換號；這是平均變化率，不是導數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["明示函數兩點的平均變化率"]},"variant":{"family_id":"qf-rate-endpoint-quotient","seed":202610063023,"parameters":{"x0":9,"x1":11,"y0":35,"y1":58,"operation":"endpoint_quotient","semantic_template":"qf-rate-endpoint-quotient/{\"inputs\":[\"2\",\"23\"],\"operation\":\"endpoint_quotient\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0024","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"明示函數兩點的平均變化率","stem_zh":"求平均變化率：同一函數明示 f(10)=37、f(7)=61，兩個輸入不同。答案為最簡正分母 p/q，整數保留 /1。","answer":{"kind":"expression","value":"-8/1","explanation_zh":"差商=-8/1。以對應輸出差除以非零輸入差，反轉順序須兩差一起換號；這是平均變化率，不是導數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["明示函數兩點的平均變化率"]},"variant":{"family_id":"qf-rate-endpoint-quotient","seed":202610063024,"parameters":{"x0":10,"x1":7,"y0":37,"y1":61,"operation":"endpoint_quotient","semantic_template":"qf-rate-endpoint-quotient/{\"inputs\":[\"3\",\"-24\"],\"operation\":\"endpoint_quotient\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0025","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"明示函數兩點的平均變化率","stem_zh":"求平均變化率：同一函數明示 f(11)=39、f(15)=64，兩個輸入不同。答案為最簡正分母 p/q，整數保留 /1。","answer":{"kind":"expression","value":"25/4","explanation_zh":"差商=25/4。以對應輸出差除以非零輸入差，反轉順序須兩差一起換號；這是平均變化率，不是導數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["明示函數兩點的平均變化率"]},"variant":{"family_id":"qf-rate-endpoint-quotient","seed":202610063025,"parameters":{"x0":11,"x1":15,"y0":39,"y1":64,"operation":"endpoint_quotient","semantic_template":"qf-rate-endpoint-quotient/{\"inputs\":[\"4\",\"25\"],\"operation\":\"endpoint_quotient\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0026","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"平均變化率的非零分母資格","stem_zh":"平均變化率使用輸入差作分母。給兩個輸入 x0=0、x1=0，是否滿足本題差商的非零分母資格？","answer":{"kind":"short_text","value":"否","explanation_zh":"輸入差=0。只有輸入差非零才具此資格；相同輸入不把商填零或 0/0，且本題不是導數判定。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["平均變化率的非零分母資格"]},"variant":{"family_id":"qf-rate-denominator-validity","seed":202610063026,"parameters":{"x0":0,"x1":0,"operation":"denominator_validity","semantic_template":"qf-rate-denominator-validity/{\"inputs\":[\"0\"],\"operation\":\"denominator_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0027","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"平均變化率的非零分母資格","stem_zh":"平均變化率使用輸入差作分母。給兩個輸入 x0=-1、x1=-2，是否滿足本題差商的非零分母資格？","answer":{"kind":"short_text","value":"是","explanation_zh":"輸入差=-1。只有輸入差非零才具此資格；相同輸入不把商填零或 0/0，且本題不是導數判定。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["平均變化率的非零分母資格"]},"variant":{"family_id":"qf-rate-denominator-validity","seed":202610063027,"parameters":{"x0":-1,"x1":-2,"operation":"denominator_validity","semantic_template":"qf-rate-denominator-validity/{\"inputs\":[\"1\"],\"operation\":\"denominator_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0028","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"平均變化率的非零分母資格","stem_zh":"平均變化率使用輸入差作分母。給兩個輸入 x0=-2、x1=0，是否滿足本題差商的非零分母資格？","answer":{"kind":"short_text","value":"是","explanation_zh":"輸入差=2。只有輸入差非零才具此資格；相同輸入不把商填零或 0/0，且本題不是導數判定。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["平均變化率的非零分母資格"]},"variant":{"family_id":"qf-rate-denominator-validity","seed":202610063028,"parameters":{"x0":-2,"x1":0,"operation":"denominator_validity","semantic_template":"qf-rate-denominator-validity/{\"inputs\":[\"2\"],\"operation\":\"denominator_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0029","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"平均變化率的非零分母資格","stem_zh":"平均變化率使用輸入差作分母。給兩個輸入 x0=-3、x1=-6，是否滿足本題差商的非零分母資格？","answer":{"kind":"short_text","value":"是","explanation_zh":"輸入差=-3。只有輸入差非零才具此資格；相同輸入不把商填零或 0/0，且本題不是導數判定。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["平均變化率的非零分母資格"]},"variant":{"family_id":"qf-rate-denominator-validity","seed":202610063029,"parameters":{"x0":-3,"x1":-6,"operation":"denominator_validity","semantic_template":"qf-rate-denominator-validity/{\"inputs\":[\"3\"],\"operation\":\"denominator_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0030","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"平均變化率的非零分母資格","stem_zh":"平均變化率使用輸入差作分母。給兩個輸入 x0=-4、x1=0，是否滿足本題差商的非零分母資格？","answer":{"kind":"short_text","value":"是","explanation_zh":"輸入差=4。只有輸入差非零才具此資格；相同輸入不把商填零或 0/0，且本題不是導數判定。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["平均變化率的非零分母資格"]},"variant":{"family_id":"qf-rate-denominator-validity","seed":202610063030,"parameters":{"x0":-4,"x1":0,"operation":"denominator_validity","semantic_template":"qf-rate-denominator-validity/{\"inputs\":[\"4\"],\"operation\":\"denominator_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0031","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"平均變化率的非零分母資格","stem_zh":"平均變化率使用輸入差作分母。給兩個輸入 x0=-5、x1=-10，是否滿足本題差商的非零分母資格？","answer":{"kind":"short_text","value":"是","explanation_zh":"輸入差=-5。只有輸入差非零才具此資格；相同輸入不把商填零或 0/0，且本題不是導數判定。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["平均變化率的非零分母資格"]},"variant":{"family_id":"qf-rate-denominator-validity","seed":202610063031,"parameters":{"x0":-5,"x1":-10,"operation":"denominator_validity","semantic_template":"qf-rate-denominator-validity/{\"inputs\":[\"5\"],\"operation\":\"denominator_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0032","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"平均變化率的非零分母資格","stem_zh":"平均變化率使用輸入差作分母。給兩個輸入 x0=-6、x1=0，是否滿足本題差商的非零分母資格？","answer":{"kind":"short_text","value":"是","explanation_zh":"輸入差=6。只有輸入差非零才具此資格；相同輸入不把商填零或 0/0，且本題不是導數判定。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["平均變化率的非零分母資格"]},"variant":{"family_id":"qf-rate-denominator-validity","seed":202610063032,"parameters":{"x0":-6,"x1":0,"operation":"denominator_validity","semantic_template":"qf-rate-denominator-validity/{\"inputs\":[\"6\"],\"operation\":\"denominator_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0033","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"平均變化率的非零分母資格","stem_zh":"平均變化率使用輸入差作分母。給兩個輸入 x0=-7、x1=-14，是否滿足本題差商的非零分母資格？","answer":{"kind":"short_text","value":"是","explanation_zh":"輸入差=-7。只有輸入差非零才具此資格；相同輸入不把商填零或 0/0，且本題不是導數判定。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["平均變化率的非零分母資格"]},"variant":{"family_id":"qf-rate-denominator-validity","seed":202610063033,"parameters":{"x0":-7,"x1":-14,"operation":"denominator_validity","semantic_template":"qf-rate-denominator-validity/{\"inputs\":[\"7\"],\"operation\":\"denominator_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0034","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"平均變化率的非零分母資格","stem_zh":"平均變化率使用輸入差作分母。給兩個輸入 x0=-8、x1=0，是否滿足本題差商的非零分母資格？","answer":{"kind":"short_text","value":"是","explanation_zh":"輸入差=8。只有輸入差非零才具此資格；相同輸入不把商填零或 0/0，且本題不是導數判定。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["平均變化率的非零分母資格"]},"variant":{"family_id":"qf-rate-denominator-validity","seed":202610063034,"parameters":{"x0":-8,"x1":0,"operation":"denominator_validity","semantic_template":"qf-rate-denominator-validity/{\"inputs\":[\"8\"],\"operation\":\"denominator_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0035","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"平均變化率的非零分母資格","stem_zh":"平均變化率使用輸入差作分母。給兩個輸入 x0=-9、x1=-18，是否滿足本題差商的非零分母資格？","answer":{"kind":"short_text","value":"是","explanation_zh":"輸入差=-9。只有輸入差非零才具此資格；相同輸入不把商填零或 0/0，且本題不是導數判定。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["平均變化率的非零分母資格"]},"variant":{"family_id":"qf-rate-denominator-validity","seed":202610063035,"parameters":{"x0":-9,"x1":-18,"operation":"denominator_validity","semantic_template":"qf-rate-denominator-validity/{\"inputs\":[\"9\"],\"operation\":\"denominator_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0036","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"平均變化率的非零分母資格","stem_zh":"平均變化率使用輸入差作分母。給兩個輸入 x0=-10、x1=0，是否滿足本題差商的非零分母資格？","answer":{"kind":"short_text","value":"是","explanation_zh":"輸入差=10。只有輸入差非零才具此資格；相同輸入不把商填零或 0/0，且本題不是導數判定。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["平均變化率的非零分母資格"]},"variant":{"family_id":"qf-rate-denominator-validity","seed":202610063036,"parameters":{"x0":-10,"x1":0,"operation":"denominator_validity","semantic_template":"qf-rate-denominator-validity/{\"inputs\":[\"10\"],\"operation\":\"denominator_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0037","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"平均變化率的非零分母資格","stem_zh":"平均變化率使用輸入差作分母。給兩個輸入 x0=-11、x1=-22，是否滿足本題差商的非零分母資格？","answer":{"kind":"short_text","value":"是","explanation_zh":"輸入差=-11。只有輸入差非零才具此資格；相同輸入不把商填零或 0/0，且本題不是導數判定。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["平均變化率的非零分母資格"]},"variant":{"family_id":"qf-rate-denominator-validity","seed":202610063037,"parameters":{"x0":-11,"x1":-22,"operation":"denominator_validity","semantic_template":"qf-rate-denominator-validity/{\"inputs\":[\"11\"],\"operation\":\"denominator_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0038","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"平均變化率的非零分母資格","stem_zh":"平均變化率使用輸入差作分母。給兩個輸入 x0=-12、x1=0，是否滿足本題差商的非零分母資格？","answer":{"kind":"short_text","value":"是","explanation_zh":"輸入差=12。只有輸入差非零才具此資格；相同輸入不把商填零或 0/0，且本題不是導數判定。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["平均變化率的非零分母資格"]},"variant":{"family_id":"qf-rate-denominator-validity","seed":202610063038,"parameters":{"x0":-12,"x1":0,"operation":"denominator_validity","semantic_template":"qf-rate-denominator-validity/{\"inputs\":[\"12\"],\"operation\":\"denominator_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0039","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"平均變化率的非零分母資格","stem_zh":"平均變化率使用輸入差作分母。給兩個輸入 x0=-13、x1=-26，是否滿足本題差商的非零分母資格？","answer":{"kind":"short_text","value":"是","explanation_zh":"輸入差=-13。只有輸入差非零才具此資格；相同輸入不把商填零或 0/0，且本題不是導數判定。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["平均變化率的非零分母資格"]},"variant":{"family_id":"qf-rate-denominator-validity","seed":202610063039,"parameters":{"x0":-13,"x1":-26,"operation":"denominator_validity","semantic_template":"qf-rate-denominator-validity/{\"inputs\":[\"13\"],\"operation\":\"denominator_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0040","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"平均變化率的非零分母資格","stem_zh":"平均變化率使用輸入差作分母。給兩個輸入 x0=-14、x1=0，是否滿足本題差商的非零分母資格？","answer":{"kind":"short_text","value":"是","explanation_zh":"輸入差=14。只有輸入差非零才具此資格；相同輸入不把商填零或 0/0，且本題不是導數判定。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["平均變化率的非零分母資格"]},"variant":{"family_id":"qf-rate-denominator-validity","seed":202610063040,"parameters":{"x0":-14,"x1":0,"operation":"denominator_validity","semantic_template":"qf-rate-denominator-validity/{\"inputs\":[\"14\"],\"operation\":\"denominator_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
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{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0045","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"平均變化率的非零分母資格","stem_zh":"平均變化率使用輸入差作分母。給兩個輸入 x0=-19、x1=-38，是否滿足本題差商的非零分母資格？","answer":{"kind":"short_text","value":"是","explanation_zh":"輸入差=-19。只有輸入差非零才具此資格；相同輸入不把商填零或 0/0，且本題不是導數判定。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["平均變化率的非零分母資格"]},"variant":{"family_id":"qf-rate-denominator-validity","seed":202610063045,"parameters":{"x0":-19,"x1":-38,"operation":"denominator_validity","semantic_template":"qf-rate-denominator-validity/{\"inputs\":[\"19\"],\"operation\":\"denominator_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0046","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"平均變化率的非零分母資格","stem_zh":"平均變化率使用輸入差作分母。給兩個輸入 x0=-20、x1=0，是否滿足本題差商的非零分母資格？","answer":{"kind":"short_text","value":"是","explanation_zh":"輸入差=20。只有輸入差非零才具此資格；相同輸入不把商填零或 0/0，且本題不是導數判定。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["平均變化率的非零分母資格"]},"variant":{"family_id":"qf-rate-denominator-validity","seed":202610063046,"parameters":{"x0":-20,"x1":0,"operation":"denominator_validity","semantic_template":"qf-rate-denominator-validity/{\"inputs\":[\"20\"],\"operation\":\"denominator_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0047","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"平均變化率的非零分母資格","stem_zh":"平均變化率使用輸入差作分母。給兩個輸入 x0=-21、x1=-42，是否滿足本題差商的非零分母資格？","answer":{"kind":"short_text","value":"是","explanation_zh":"輸入差=-21。只有輸入差非零才具此資格；相同輸入不把商填零或 0/0，且本題不是導數判定。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["平均變化率的非零分母資格"]},"variant":{"family_id":"qf-rate-denominator-validity","seed":202610063047,"parameters":{"x0":-21,"x1":-42,"operation":"denominator_validity","semantic_template":"qf-rate-denominator-validity/{\"inputs\":[\"21\"],\"operation\":\"denominator_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0048","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"平均變化率的非零分母資格","stem_zh":"平均變化率使用輸入差作分母。給兩個輸入 x0=-22、x1=0，是否滿足本題差商的非零分母資格？","answer":{"kind":"short_text","value":"是","explanation_zh":"輸入差=22。只有輸入差非零才具此資格；相同輸入不把商填零或 0/0，且本題不是導數判定。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["平均變化率的非零分母資格"]},"variant":{"family_id":"qf-rate-denominator-validity","seed":202610063048,"parameters":{"x0":-22,"x1":0,"operation":"denominator_validity","semantic_template":"qf-rate-denominator-validity/{\"inputs\":[\"22\"],\"operation\":\"denominator_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0049","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"平均變化率的非零分母資格","stem_zh":"平均變化率使用輸入差作分母。給兩個輸入 x0=-23、x1=-46，是否滿足本題差商的非零分母資格？","answer":{"kind":"short_text","value":"是","explanation_zh":"輸入差=-23。只有輸入差非零才具此資格；相同輸入不把商填零或 0/0，且本題不是導數判定。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["平均變化率的非零分母資格"]},"variant":{"family_id":"qf-rate-denominator-validity","seed":202610063049,"parameters":{"x0":-23,"x1":-46,"operation":"denominator_validity","semantic_template":"qf-rate-denominator-validity/{\"inputs\":[\"23\"],\"operation\":\"denominator_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0050","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"平均變化率的非零分母資格","stem_zh":"平均變化率使用輸入差作分母。給兩個輸入 x0=-24、x1=0，是否滿足本題差商的非零分母資格？","answer":{"kind":"short_text","value":"是","explanation_zh":"輸入差=24。只有輸入差非零才具此資格；相同輸入不把商填零或 0/0，且本題不是導數判定。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["平均變化率的非零分母資格"]},"variant":{"family_id":"qf-rate-denominator-validity","seed":202610063050,"parameters":{"x0":-24,"x1":0,"operation":"denominator_validity","semantic_template":"qf-rate-denominator-validity/{\"inputs\":[\"24\"],\"operation\":\"denominator_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0051","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"實仿射函數的平均變化率","stem_zh":"在實數論域指定 f(x)=-12x+(-7)，求 x0=0 到 x1=1 的平均變化率，兩個輸入不同；答案採最簡正分母 p/q。","answer":{"kind":"expression","value":"-12/1","explanation_zh":"平均變化率=-12/1。仿射函數的輸出差為 m 倍輸入差，非零差相消後為 m，常數式 m=0 也合法；這不是導數題。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["實仿射函數的平均變化率"]},"variant":{"family_id":"qf-rate-affine-average","seed":202610063051,"parameters":{"m":-12,"b":-7,"x0":0,"x1":1,"domain":"reals","operation":"affine_average","semantic_template":"qf-rate-affine-average/{\"inputs\":[\"-12\"],\"operation\":\"affine_average\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0052","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"實仿射函數的平均變化率","stem_zh":"在實數論域指定 f(x)=-11x+(-4)，求 x0=0 到 x1=-2 的平均變化率，兩個輸入不同；答案採最簡正分母 p/q。","answer":{"kind":"expression","value":"-11/1","explanation_zh":"平均變化率=-11/1。仿射函數的輸出差為 m 倍輸入差，非零差相消後為 m，常數式 m=0 也合法；這不是導數題。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["實仿射函數的平均變化率"]},"variant":{"family_id":"qf-rate-affine-average","seed":202610063052,"parameters":{"m":-11,"b":-4,"x0":0,"x1":-2,"domain":"reals","operation":"affine_average","semantic_template":"qf-rate-affine-average/{\"inputs\":[\"-11\"],\"operation\":\"affine_average\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0053","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"實仿射函數的平均變化率","stem_zh":"在實數論域指定 f(x)=-10x+(-1)，求 x0=0 到 x1=3 的平均變化率，兩個輸入不同；答案採最簡正分母 p/q。","answer":{"kind":"expression","value":"-10/1","explanation_zh":"平均變化率=-10/1。仿射函數的輸出差為 m 倍輸入差，非零差相消後為 m，常數式 m=0 也合法；這不是導數題。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["實仿射函數的平均變化率"]},"variant":{"family_id":"qf-rate-affine-average","seed":202610063053,"parameters":{"m":-10,"b":-1,"x0":0,"x1":3,"domain":"reals","operation":"affine_average","semantic_template":"qf-rate-affine-average/{\"inputs\":[\"-10\"],\"operation\":\"affine_average\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0054","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"實仿射函數的平均變化率","stem_zh":"在實數論域指定 f(x)=-9x+(2)，求 x0=0 到 x1=-4 的平均變化率，兩個輸入不同；答案採最簡正分母 p/q。","answer":{"kind":"expression","value":"-9/1","explanation_zh":"平均變化率=-9/1。仿射函數的輸出差為 m 倍輸入差，非零差相消後為 m，常數式 m=0 也合法；這不是導數題。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["實仿射函數的平均變化率"]},"variant":{"family_id":"qf-rate-affine-average","seed":202610063054,"parameters":{"m":-9,"b":2,"x0":0,"x1":-4,"domain":"reals","operation":"affine_average","semantic_template":"qf-rate-affine-average/{\"inputs\":[\"-9\"],\"operation\":\"affine_average\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0055","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"實仿射函數的平均變化率","stem_zh":"在實數論域指定 f(x)=-8x+(5)，求 x0=0 到 x1=5 的平均變化率，兩個輸入不同；答案採最簡正分母 p/q。","answer":{"kind":"expression","value":"-8/1","explanation_zh":"平均變化率=-8/1。仿射函數的輸出差為 m 倍輸入差，非零差相消後為 m，常數式 m=0 也合法；這不是導數題。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["實仿射函數的平均變化率"]},"variant":{"family_id":"qf-rate-affine-average","seed":202610063055,"parameters":{"m":-8,"b":5,"x0":0,"x1":5,"domain":"reals","operation":"affine_average","semantic_template":"qf-rate-affine-average/{\"inputs\":[\"-8\"],\"operation\":\"affine_average\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0056","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"實仿射函數的平均變化率","stem_zh":"在實數論域指定 f(x)=-7x+(8)，求 x0=0 到 x1=-6 的平均變化率，兩個輸入不同；答案採最簡正分母 p/q。","answer":{"kind":"expression","value":"-7/1","explanation_zh":"平均變化率=-7/1。仿射函數的輸出差為 m 倍輸入差，非零差相消後為 m，常數式 m=0 也合法；這不是導數題。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["實仿射函數的平均變化率"]},"variant":{"family_id":"qf-rate-affine-average","seed":202610063056,"parameters":{"m":-7,"b":8,"x0":0,"x1":-6,"domain":"reals","operation":"affine_average","semantic_template":"qf-rate-affine-average/{\"inputs\":[\"-7\"],\"operation\":\"affine_average\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0057","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"實仿射函數的平均變化率","stem_zh":"在實數論域指定 f(x)=-6x+(11)，求 x0=0 到 x1=7 的平均變化率，兩個輸入不同；答案採最簡正分母 p/q。","answer":{"kind":"expression","value":"-6/1","explanation_zh":"平均變化率=-6/1。仿射函數的輸出差為 m 倍輸入差，非零差相消後為 m，常數式 m=0 也合法；這不是導數題。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["實仿射函數的平均變化率"]},"variant":{"family_id":"qf-rate-affine-average","seed":202610063057,"parameters":{"m":-6,"b":11,"x0":0,"x1":7,"domain":"reals","operation":"affine_average","semantic_template":"qf-rate-affine-average/{\"inputs\":[\"-6\"],\"operation\":\"affine_average\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0058","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"實仿射函數的平均變化率","stem_zh":"在實數論域指定 f(x)=-5x+(14)，求 x0=0 到 x1=-8 的平均變化率，兩個輸入不同；答案採最簡正分母 p/q。","answer":{"kind":"expression","value":"-5/1","explanation_zh":"平均變化率=-5/1。仿射函數的輸出差為 m 倍輸入差，非零差相消後為 m，常數式 m=0 也合法；這不是導數題。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["實仿射函數的平均變化率"]},"variant":{"family_id":"qf-rate-affine-average","seed":202610063058,"parameters":{"m":-5,"b":14,"x0":0,"x1":-8,"domain":"reals","operation":"affine_average","semantic_template":"qf-rate-affine-average/{\"inputs\":[\"-5\"],\"operation\":\"affine_average\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0059","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"實仿射函數的平均變化率","stem_zh":"在實數論域指定 f(x)=-4x+(17)，求 x0=0 到 x1=9 的平均變化率，兩個輸入不同；答案採最簡正分母 p/q。","answer":{"kind":"expression","value":"-4/1","explanation_zh":"平均變化率=-4/1。仿射函數的輸出差為 m 倍輸入差，非零差相消後為 m，常數式 m=0 也合法；這不是導數題。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["實仿射函數的平均變化率"]},"variant":{"family_id":"qf-rate-affine-average","seed":202610063059,"parameters":{"m":-4,"b":17,"x0":0,"x1":9,"domain":"reals","operation":"affine_average","semantic_template":"qf-rate-affine-average/{\"inputs\":[\"-4\"],\"operation\":\"affine_average\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0060","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"實仿射函數的平均變化率","stem_zh":"在實數論域指定 f(x)=-3x+(20)，求 x0=0 到 x1=-10 的平均變化率，兩個輸入不同；答案採最簡正分母 p/q。","answer":{"kind":"expression","value":"-3/1","explanation_zh":"平均變化率=-3/1。仿射函數的輸出差為 m 倍輸入差，非零差相消後為 m，常數式 m=0 也合法；這不是導數題。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["實仿射函數的平均變化率"]},"variant":{"family_id":"qf-rate-affine-average","seed":202610063060,"parameters":{"m":-3,"b":20,"x0":0,"x1":-10,"domain":"reals","operation":"affine_average","semantic_template":"qf-rate-affine-average/{\"inputs\":[\"-3\"],\"operation\":\"affine_average\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0061","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"實仿射函數的平均變化率","stem_zh":"在實數論域指定 f(x)=-2x+(23)，求 x0=0 到 x1=11 的平均變化率，兩個輸入不同；答案採最簡正分母 p/q。","answer":{"kind":"expression","value":"-2/1","explanation_zh":"平均變化率=-2/1。仿射函數的輸出差為 m 倍輸入差，非零差相消後為 m，常數式 m=0 也合法；這不是導數題。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["實仿射函數的平均變化率"]},"variant":{"family_id":"qf-rate-affine-average","seed":202610063061,"parameters":{"m":-2,"b":23,"x0":0,"x1":11,"domain":"reals","operation":"affine_average","semantic_template":"qf-rate-affine-average/{\"inputs\":[\"-2\"],\"operation\":\"affine_average\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0062","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"實仿射函數的平均變化率","stem_zh":"在實數論域指定 f(x)=-1x+(26)，求 x0=0 到 x1=-12 的平均變化率，兩個輸入不同；答案採最簡正分母 p/q。","answer":{"kind":"expression","value":"-1/1","explanation_zh":"平均變化率=-1/1。仿射函數的輸出差為 m 倍輸入差，非零差相消後為 m，常數式 m=0 也合法；這不是導數題。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["實仿射函數的平均變化率"]},"variant":{"family_id":"qf-rate-affine-average","seed":202610063062,"parameters":{"m":-1,"b":26,"x0":0,"x1":-12,"domain":"reals","operation":"affine_average","semantic_template":"qf-rate-affine-average/{\"inputs\":[\"-1\"],\"operation\":\"affine_average\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0063","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"實仿射函數的平均變化率","stem_zh":"在實數論域指定 f(x)=0x+(29)，求 x0=0 到 x1=13 的平均變化率，兩個輸入不同；答案採最簡正分母 p/q。","answer":{"kind":"expression","value":"0/1","explanation_zh":"平均變化率=0/1。仿射函數的輸出差為 m 倍輸入差，非零差相消後為 m，常數式 m=0 也合法；這不是導數題。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["實仿射函數的平均變化率"]},"variant":{"family_id":"qf-rate-affine-average","seed":202610063063,"parameters":{"m":0,"b":29,"x0":0,"x1":13,"domain":"reals","operation":"affine_average","semantic_template":"qf-rate-affine-average/{\"inputs\":[\"0\"],\"operation\":\"affine_average\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0064","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"實仿射函數的平均變化率","stem_zh":"在實數論域指定 f(x)=1x+(32)，求 x0=0 到 x1=-14 的平均變化率，兩個輸入不同；答案採最簡正分母 p/q。","answer":{"kind":"expression","value":"1/1","explanation_zh":"平均變化率=1/1。仿射函數的輸出差為 m 倍輸入差，非零差相消後為 m，常數式 m=0 也合法；這不是導數題。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["實仿射函數的平均變化率"]},"variant":{"family_id":"qf-rate-affine-average","seed":202610063064,"parameters":{"m":1,"b":32,"x0":0,"x1":-14,"domain":"reals","operation":"affine_average","semantic_template":"qf-rate-affine-average/{\"inputs\":[\"1\"],\"operation\":\"affine_average\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0065","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"實仿射函數的平均變化率","stem_zh":"在實數論域指定 f(x)=2x+(35)，求 x0=0 到 x1=15 的平均變化率，兩個輸入不同；答案採最簡正分母 p/q。","answer":{"kind":"expression","value":"2/1","explanation_zh":"平均變化率=2/1。仿射函數的輸出差為 m 倍輸入差，非零差相消後為 m，常數式 m=0 也合法；這不是導數題。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["實仿射函數的平均變化率"]},"variant":{"family_id":"qf-rate-affine-average","seed":202610063065,"parameters":{"m":2,"b":35,"x0":0,"x1":15,"domain":"reals","operation":"affine_average","semantic_template":"qf-rate-affine-average/{\"inputs\":[\"2\"],\"operation\":\"affine_average\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0066","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"實仿射函數的平均變化率","stem_zh":"在實數論域指定 f(x)=3x+(38)，求 x0=0 到 x1=-16 的平均變化率，兩個輸入不同；答案採最簡正分母 p/q。","answer":{"kind":"expression","value":"3/1","explanation_zh":"平均變化率=3/1。仿射函數的輸出差為 m 倍輸入差，非零差相消後為 m，常數式 m=0 也合法；這不是導數題。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["實仿射函數的平均變化率"]},"variant":{"family_id":"qf-rate-affine-average","seed":202610063066,"parameters":{"m":3,"b":38,"x0":0,"x1":-16,"domain":"reals","operation":"affine_average","semantic_template":"qf-rate-affine-average/{\"inputs\":[\"3\"],\"operation\":\"affine_average\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0067","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"實仿射函數的平均變化率","stem_zh":"在實數論域指定 f(x)=4x+(41)，求 x0=0 到 x1=17 的平均變化率，兩個輸入不同；答案採最簡正分母 p/q。","answer":{"kind":"expression","value":"4/1","explanation_zh":"平均變化率=4/1。仿射函數的輸出差為 m 倍輸入差，非零差相消後為 m，常數式 m=0 也合法；這不是導數題。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["實仿射函數的平均變化率"]},"variant":{"family_id":"qf-rate-affine-average","seed":202610063067,"parameters":{"m":4,"b":41,"x0":0,"x1":17,"domain":"reals","operation":"affine_average","semantic_template":"qf-rate-affine-average/{\"inputs\":[\"4\"],\"operation\":\"affine_average\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0068","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"實仿射函數的平均變化率","stem_zh":"在實數論域指定 f(x)=5x+(44)，求 x0=0 到 x1=-18 的平均變化率，兩個輸入不同；答案採最簡正分母 p/q。","answer":{"kind":"expression","value":"5/1","explanation_zh":"平均變化率=5/1。仿射函數的輸出差為 m 倍輸入差，非零差相消後為 m，常數式 m=0 也合法；這不是導數題。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["實仿射函數的平均變化率"]},"variant":{"family_id":"qf-rate-affine-average","seed":202610063068,"parameters":{"m":5,"b":44,"x0":0,"x1":-18,"domain":"reals","operation":"affine_average","semantic_template":"qf-rate-affine-average/{\"inputs\":[\"5\"],\"operation\":\"affine_average\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0069","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"實仿射函數的平均變化率","stem_zh":"在實數論域指定 f(x)=6x+(47)，求 x0=0 到 x1=19 的平均變化率，兩個輸入不同；答案採最簡正分母 p/q。","answer":{"kind":"expression","value":"6/1","explanation_zh":"平均變化率=6/1。仿射函數的輸出差為 m 倍輸入差，非零差相消後為 m，常數式 m=0 也合法；這不是導數題。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["實仿射函數的平均變化率"]},"variant":{"family_id":"qf-rate-affine-average","seed":202610063069,"parameters":{"m":6,"b":47,"x0":0,"x1":19,"domain":"reals","operation":"affine_average","semantic_template":"qf-rate-affine-average/{\"inputs\":[\"6\"],\"operation\":\"affine_average\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0070","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"實仿射函數的平均變化率","stem_zh":"在實數論域指定 f(x)=7x+(50)，求 x0=0 到 x1=-20 的平均變化率，兩個輸入不同；答案採最簡正分母 p/q。","answer":{"kind":"expression","value":"7/1","explanation_zh":"平均變化率=7/1。仿射函數的輸出差為 m 倍輸入差，非零差相消後為 m，常數式 m=0 也合法；這不是導數題。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["實仿射函數的平均變化率"]},"variant":{"family_id":"qf-rate-affine-average","seed":202610063070,"parameters":{"m":7,"b":50,"x0":0,"x1":-20,"domain":"reals","operation":"affine_average","semantic_template":"qf-rate-affine-average/{\"inputs\":[\"7\"],\"operation\":\"affine_average\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0071","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"實仿射函數的平均變化率","stem_zh":"在實數論域指定 f(x)=8x+(53)，求 x0=0 到 x1=21 的平均變化率，兩個輸入不同；答案採最簡正分母 p/q。","answer":{"kind":"expression","value":"8/1","explanation_zh":"平均變化率=8/1。仿射函數的輸出差為 m 倍輸入差，非零差相消後為 m，常數式 m=0 也合法；這不是導數題。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["實仿射函數的平均變化率"]},"variant":{"family_id":"qf-rate-affine-average","seed":202610063071,"parameters":{"m":8,"b":53,"x0":0,"x1":21,"domain":"reals","operation":"affine_average","semantic_template":"qf-rate-affine-average/{\"inputs\":[\"8\"],\"operation\":\"affine_average\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0072","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"實仿射函數的平均變化率","stem_zh":"在實數論域指定 f(x)=9x+(56)，求 x0=0 到 x1=-22 的平均變化率，兩個輸入不同；答案採最簡正分母 p/q。","answer":{"kind":"expression","value":"9/1","explanation_zh":"平均變化率=9/1。仿射函數的輸出差為 m 倍輸入差，非零差相消後為 m，常數式 m=0 也合法；這不是導數題。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["實仿射函數的平均變化率"]},"variant":{"family_id":"qf-rate-affine-average","seed":202610063072,"parameters":{"m":9,"b":56,"x0":0,"x1":-22,"domain":"reals","operation":"affine_average","semantic_template":"qf-rate-affine-average/{\"inputs\":[\"9\"],\"operation\":\"affine_average\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0073","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"實仿射函數的平均變化率","stem_zh":"在實數論域指定 f(x)=10x+(59)，求 x0=0 到 x1=23 的平均變化率，兩個輸入不同；答案採最簡正分母 p/q。","answer":{"kind":"expression","value":"10/1","explanation_zh":"平均變化率=10/1。仿射函數的輸出差為 m 倍輸入差，非零差相消後為 m，常數式 m=0 也合法；這不是導數題。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["實仿射函數的平均變化率"]},"variant":{"family_id":"qf-rate-affine-average","seed":202610063073,"parameters":{"m":10,"b":59,"x0":0,"x1":23,"domain":"reals","operation":"affine_average","semantic_template":"qf-rate-affine-average/{\"inputs\":[\"10\"],\"operation\":\"affine_average\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0074","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"實仿射函數的平均變化率","stem_zh":"在實數論域指定 f(x)=11x+(62)，求 x0=0 到 x1=-24 的平均變化率，兩個輸入不同；答案採最簡正分母 p/q。","answer":{"kind":"expression","value":"11/1","explanation_zh":"平均變化率=11/1。仿射函數的輸出差為 m 倍輸入差，非零差相消後為 m，常數式 m=0 也合法；這不是導數題。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["實仿射函數的平均變化率"]},"variant":{"family_id":"qf-rate-affine-average","seed":202610063074,"parameters":{"m":11,"b":62,"x0":0,"x1":-24,"domain":"reals","operation":"affine_average","semantic_template":"qf-rate-affine-average/{\"inputs\":[\"11\"],\"operation\":\"affine_average\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0075","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"實仿射函數的平均變化率","stem_zh":"在實數論域指定 f(x)=12x+(65)，求 x0=0 到 x1=25 的平均變化率，兩個輸入不同；答案採最簡正分母 p/q。","answer":{"kind":"expression","value":"12/1","explanation_zh":"平均變化率=12/1。仿射函數的輸出差為 m 倍輸入差，非零差相消後為 m，常數式 m=0 也合法；這不是導數題。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["實仿射函數的平均變化率"]},"variant":{"family_id":"qf-rate-affine-average","seed":202610063075,"parameters":{"m":12,"b":65,"x0":0,"x1":25,"domain":"reals","operation":"affine_average","semantic_template":"qf-rate-affine-average/{\"inputs\":[\"12\"],\"operation\":\"affine_average\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0076","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"二次頂點式的平均變化率","stem_zh":"在全實數論域，指定 q:R→R，q(x)=1(x−(0))²+(0)，a=1≠0。求 x0=-1 到 x1=1 的平均變化率，明示兩個輸入不同；以最簡正分母 p/q 回答。","answer":{"kind":"expression","value":"0/1","explanation_zh":"平均變化率=0/1。平方差除以輸入差化為 a(x0+x1−2h)，k 抵消；對稱端點可得零差商，但不是導數或常數函數聲明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["二次頂點式的平均變化率"]},"variant":{"family_id":"qf-rate-quadratic-average","seed":202610063076,"parameters":{"a":1,"h":0,"k":0,"x0":-1,"x1":1,"domain":"reals","operation":"quadratic_average","semantic_template":"qf-rate-quadratic-average/{\"inputs\":[\"symmetric_endpoints\"],\"operation\":\"quadratic_average\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0077","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"二次頂點式的平均變化率","stem_zh":"在全實數論域，指定 q:R→R，q(x)=-2(x−(-11))²+(-11)，a=-2≠0。求 x0=-9 到 x1=-12 的平均變化率，明示兩個輸入不同；以最簡正分母 p/q 回答。","answer":{"kind":"expression","value":"-2/1","explanation_zh":"平均變化率=-2/1。平方差除以輸入差化為 a(x0+x1−2h)，k 抵消；對稱端點可得零差商，但不是導數或常數函數聲明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["二次頂點式的平均變化率"]},"variant":{"family_id":"qf-rate-quadratic-average","seed":202610063077,"parameters":{"a":-2,"h":-11,"k":-11,"x0":-9,"x1":-12,"domain":"reals","operation":"quadratic_average","semantic_template":"qf-rate-quadratic-average/{\"inputs\":[\"-2\",\"1\"],\"operation\":\"quadratic_average\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0078","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"二次頂點式的平均變化率","stem_zh":"在全實數論域，指定 q:R→R，q(x)=3(x−(-10))²+(-10)，a=3≠0。求 x0=-11 到 x1=-7 的平均變化率，明示兩個輸入不同；以最簡正分母 p/q 回答。","answer":{"kind":"expression","value":"6/1","explanation_zh":"平均變化率=6/1。平方差除以輸入差化為 a(x0+x1−2h)，k 抵消；對稱端點可得零差商，但不是導數或常數函數聲明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["二次頂點式的平均變化率"]},"variant":{"family_id":"qf-rate-quadratic-average","seed":202610063078,"parameters":{"a":3,"h":-10,"k":-10,"x0":-11,"x1":-7,"domain":"reals","operation":"quadratic_average","semantic_template":"qf-rate-quadratic-average/{\"inputs\":[\"3\",\"2\"],\"operation\":\"quadratic_average\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0079","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"二次頂點式的平均變化率","stem_zh":"在全實數論域，指定 q:R→R，q(x)=-4(x−(-9))²+(-9)，a=-4≠0。求 x0=-5 到 x1=-10 的平均變化率，明示兩個輸入不同；以最簡正分母 p/q 回答。","answer":{"kind":"expression","value":"-12/1","explanation_zh":"平均變化率=-12/1。平方差除以輸入差化為 a(x0+x1−2h)，k 抵消；對稱端點可得零差商，但不是導數或常數函數聲明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["二次頂點式的平均變化率"]},"variant":{"family_id":"qf-rate-quadratic-average","seed":202610063079,"parameters":{"a":-4,"h":-9,"k":-9,"x0":-5,"x1":-10,"domain":"reals","operation":"quadratic_average","semantic_template":"qf-rate-quadratic-average/{\"inputs\":[\"-4\",\"3\"],\"operation\":\"quadratic_average\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0080","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"二次頂點式的平均變化率","stem_zh":"在全實數論域，指定 q:R→R，q(x)=5(x−(-8))²+(-8)，a=5≠0。求 x0=-9 到 x1=-3 的平均變化率，明示兩個輸入不同；以最簡正分母 p/q 回答。","answer":{"kind":"expression","value":"20/1","explanation_zh":"平均變化率=20/1。平方差除以輸入差化為 a(x0+x1−2h)，k 抵消；對稱端點可得零差商，但不是導數或常數函數聲明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["二次頂點式的平均變化率"]},"variant":{"family_id":"qf-rate-quadratic-average","seed":202610063080,"parameters":{"a":5,"h":-8,"k":-8,"x0":-9,"x1":-3,"domain":"reals","operation":"quadratic_average","semantic_template":"qf-rate-quadratic-average/{\"inputs\":[\"5\",\"4\"],\"operation\":\"quadratic_average\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0081","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"二次頂點式的平均變化率","stem_zh":"在全實數論域，指定 q:R→R，q(x)=-6(x−(-7))²+(-7)，a=-6≠0。求 x0=-1 到 x1=-8 的平均變化率，明示兩個輸入不同；以最簡正分母 p/q 回答。","answer":{"kind":"expression","value":"-30/1","explanation_zh":"平均變化率=-30/1。平方差除以輸入差化為 a(x0+x1−2h)，k 抵消；對稱端點可得零差商，但不是導數或常數函數聲明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["二次頂點式的平均變化率"]},"variant":{"family_id":"qf-rate-quadratic-average","seed":202610063081,"parameters":{"a":-6,"h":-7,"k":-7,"x0":-1,"x1":-8,"domain":"reals","operation":"quadratic_average","semantic_template":"qf-rate-quadratic-average/{\"inputs\":[\"-6\",\"5\"],\"operation\":\"quadratic_average\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0082","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"二次頂點式的平均變化率","stem_zh":"在全實數論域，指定 q:R→R，q(x)=7(x−(-6))²+(-6)，a=7≠0。求 x0=-7 到 x1=1 的平均變化率，明示兩個輸入不同；以最簡正分母 p/q 回答。","answer":{"kind":"expression","value":"42/1","explanation_zh":"平均變化率=42/1。平方差除以輸入差化為 a(x0+x1−2h)，k 抵消；對稱端點可得零差商，但不是導數或常數函數聲明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["二次頂點式的平均變化率"]},"variant":{"family_id":"qf-rate-quadratic-average","seed":202610063082,"parameters":{"a":7,"h":-6,"k":-6,"x0":-7,"x1":1,"domain":"reals","operation":"quadratic_average","semantic_template":"qf-rate-quadratic-average/{\"inputs\":[\"7\",\"6\"],\"operation\":\"quadratic_average\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0083","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"二次頂點式的平均變化率","stem_zh":"在全實數論域，指定 q:R→R，q(x)=-8(x−(-5))²+(-5)，a=-8≠0。求 x0=3 到 x1=-6 的平均變化率，明示兩個輸入不同；以最簡正分母 p/q 回答。","answer":{"kind":"expression","value":"-56/1","explanation_zh":"平均變化率=-56/1。平方差除以輸入差化為 a(x0+x1−2h)，k 抵消；對稱端點可得零差商，但不是導數或常數函數聲明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["二次頂點式的平均變化率"]},"variant":{"family_id":"qf-rate-quadratic-average","seed":202610063083,"parameters":{"a":-8,"h":-5,"k":-5,"x0":3,"x1":-6,"domain":"reals","operation":"quadratic_average","semantic_template":"qf-rate-quadratic-average/{\"inputs\":[\"-8\",\"7\"],\"operation\":\"quadratic_average\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0084","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"二次頂點式的平均變化率","stem_zh":"在全實數論域，指定 q:R→R，q(x)=9(x−(-4))²+(-4)，a=9≠0。求 x0=-5 到 x1=5 的平均變化率，明示兩個輸入不同；以最簡正分母 p/q 回答。","answer":{"kind":"expression","value":"72/1","explanation_zh":"平均變化率=72/1。平方差除以輸入差化為 a(x0+x1−2h)，k 抵消；對稱端點可得零差商，但不是導數或常數函數聲明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["二次頂點式的平均變化率"]},"variant":{"family_id":"qf-rate-quadratic-average","seed":202610063084,"parameters":{"a":9,"h":-4,"k":-4,"x0":-5,"x1":5,"domain":"reals","operation":"quadratic_average","semantic_template":"qf-rate-quadratic-average/{\"inputs\":[\"9\",\"8\"],\"operation\":\"quadratic_average\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0085","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"二次頂點式的平均變化率","stem_zh":"在全實數論域，指定 q:R→R，q(x)=-10(x−(-3))²+(-3)，a=-10≠0。求 x0=7 到 x1=-4 的平均變化率，明示兩個輸入不同；以最簡正分母 p/q 回答。","answer":{"kind":"expression","value":"-90/1","explanation_zh":"平均變化率=-90/1。平方差除以輸入差化為 a(x0+x1−2h)，k 抵消；對稱端點可得零差商，但不是導數或常數函數聲明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["二次頂點式的平均變化率"]},"variant":{"family_id":"qf-rate-quadratic-average","seed":202610063085,"parameters":{"a":-10,"h":-3,"k":-3,"x0":7,"x1":-4,"domain":"reals","operation":"quadratic_average","semantic_template":"qf-rate-quadratic-average/{\"inputs\":[\"-10\",\"9\"],\"operation\":\"quadratic_average\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0086","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"二次頂點式的平均變化率","stem_zh":"在全實數論域，指定 q:R→R，q(x)=11(x−(-2))²+(-2)，a=11≠0。求 x0=-3 到 x1=9 的平均變化率，明示兩個輸入不同；以最簡正分母 p/q 回答。","answer":{"kind":"expression","value":"110/1","explanation_zh":"平均變化率=110/1。平方差除以輸入差化為 a(x0+x1−2h)，k 抵消；對稱端點可得零差商，但不是導數或常數函數聲明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["二次頂點式的平均變化率"]},"variant":{"family_id":"qf-rate-quadratic-average","seed":202610063086,"parameters":{"a":11,"h":-2,"k":-2,"x0":-3,"x1":9,"domain":"reals","operation":"quadratic_average","semantic_template":"qf-rate-quadratic-average/{\"inputs\":[\"11\",\"10\"],\"operation\":\"quadratic_average\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0087","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"二次頂點式的平均變化率","stem_zh":"在全實數論域，指定 q:R→R，q(x)=-12(x−(-1))²+(-1)，a=-12≠0。求 x0=11 到 x1=-2 的平均變化率，明示兩個輸入不同；以最簡正分母 p/q 回答。","answer":{"kind":"expression","value":"-132/1","explanation_zh":"平均變化率=-132/1。平方差除以輸入差化為 a(x0+x1−2h)，k 抵消；對稱端點可得零差商，但不是導數或常數函數聲明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["二次頂點式的平均變化率"]},"variant":{"family_id":"qf-rate-quadratic-average","seed":202610063087,"parameters":{"a":-12,"h":-1,"k":-1,"x0":11,"x1":-2,"domain":"reals","operation":"quadratic_average","semantic_template":"qf-rate-quadratic-average/{\"inputs\":[\"-12\",\"11\"],\"operation\":\"quadratic_average\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0088","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"二次頂點式的平均變化率","stem_zh":"在全實數論域，指定 q:R→R，q(x)=13(x−(0))²+(0)，a=13≠0。求 x0=-1 到 x1=13 的平均變化率，明示兩個輸入不同；以最簡正分母 p/q 回答。","answer":{"kind":"expression","value":"156/1","explanation_zh":"平均變化率=156/1。平方差除以輸入差化為 a(x0+x1−2h)，k 抵消；對稱端點可得零差商，但不是導數或常數函數聲明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["二次頂點式的平均變化率"]},"variant":{"family_id":"qf-rate-quadratic-average","seed":202610063088,"parameters":{"a":13,"h":0,"k":0,"x0":-1,"x1":13,"domain":"reals","operation":"quadratic_average","semantic_template":"qf-rate-quadratic-average/{\"inputs\":[\"13\",\"12\"],\"operation\":\"quadratic_average\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0089","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"二次頂點式的平均變化率","stem_zh":"在全實數論域，指定 q:R→R，q(x)=-14(x−(1))²+(1)，a=-14≠0。求 x0=15 到 x1=0 的平均變化率，明示兩個輸入不同；以最簡正分母 p/q 回答。","answer":{"kind":"expression","value":"-182/1","explanation_zh":"平均變化率=-182/1。平方差除以輸入差化為 a(x0+x1−2h)，k 抵消；對稱端點可得零差商，但不是導數或常數函數聲明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["二次頂點式的平均變化率"]},"variant":{"family_id":"qf-rate-quadratic-average","seed":202610063089,"parameters":{"a":-14,"h":1,"k":1,"x0":15,"x1":0,"domain":"reals","operation":"quadratic_average","semantic_template":"qf-rate-quadratic-average/{\"inputs\":[\"-14\",\"13\"],\"operation\":\"quadratic_average\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0090","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"二次頂點式的平均變化率","stem_zh":"在全實數論域，指定 q:R→R，q(x)=15(x−(2))²+(2)，a=15≠0。求 x0=1 到 x1=17 的平均變化率，明示兩個輸入不同；以最簡正分母 p/q 回答。","answer":{"kind":"expression","value":"210/1","explanation_zh":"平均變化率=210/1。平方差除以輸入差化為 a(x0+x1−2h)，k 抵消；對稱端點可得零差商，但不是導數或常數函數聲明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["二次頂點式的平均變化率"]},"variant":{"family_id":"qf-rate-quadratic-average","seed":202610063090,"parameters":{"a":15,"h":2,"k":2,"x0":1,"x1":17,"domain":"reals","operation":"quadratic_average","semantic_template":"qf-rate-quadratic-average/{\"inputs\":[\"15\",\"14\"],\"operation\":\"quadratic_average\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0091","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"二次頂點式的平均變化率","stem_zh":"在全實數論域，指定 q:R→R，q(x)=-16(x−(3))²+(3)，a=-16≠0。求 x0=19 到 x1=2 的平均變化率，明示兩個輸入不同；以最簡正分母 p/q 回答。","answer":{"kind":"expression","value":"-240/1","explanation_zh":"平均變化率=-240/1。平方差除以輸入差化為 a(x0+x1−2h)，k 抵消；對稱端點可得零差商，但不是導數或常數函數聲明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["二次頂點式的平均變化率"]},"variant":{"family_id":"qf-rate-quadratic-average","seed":202610063091,"parameters":{"a":-16,"h":3,"k":3,"x0":19,"x1":2,"domain":"reals","operation":"quadratic_average","semantic_template":"qf-rate-quadratic-average/{\"inputs\":[\"-16\",\"15\"],\"operation\":\"quadratic_average\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0092","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"二次頂點式的平均變化率","stem_zh":"在全實數論域，指定 q:R→R，q(x)=17(x−(4))²+(4)，a=17≠0。求 x0=3 到 x1=21 的平均變化率，明示兩個輸入不同；以最簡正分母 p/q 回答。","answer":{"kind":"expression","value":"272/1","explanation_zh":"平均變化率=272/1。平方差除以輸入差化為 a(x0+x1−2h)，k 抵消；對稱端點可得零差商，但不是導數或常數函數聲明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["二次頂點式的平均變化率"]},"variant":{"family_id":"qf-rate-quadratic-average","seed":202610063092,"parameters":{"a":17,"h":4,"k":4,"x0":3,"x1":21,"domain":"reals","operation":"quadratic_average","semantic_template":"qf-rate-quadratic-average/{\"inputs\":[\"17\",\"16\"],\"operation\":\"quadratic_average\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0093","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"二次頂點式的平均變化率","stem_zh":"在全實數論域，指定 q:R→R，q(x)=-18(x−(5))²+(5)，a=-18≠0。求 x0=23 到 x1=4 的平均變化率，明示兩個輸入不同；以最簡正分母 p/q 回答。","answer":{"kind":"expression","value":"-306/1","explanation_zh":"平均變化率=-306/1。平方差除以輸入差化為 a(x0+x1−2h)，k 抵消；對稱端點可得零差商，但不是導數或常數函數聲明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["二次頂點式的平均變化率"]},"variant":{"family_id":"qf-rate-quadratic-average","seed":202610063093,"parameters":{"a":-18,"h":5,"k":5,"x0":23,"x1":4,"domain":"reals","operation":"quadratic_average","semantic_template":"qf-rate-quadratic-average/{\"inputs\":[\"-18\",\"17\"],\"operation\":\"quadratic_average\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0094","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"二次頂點式的平均變化率","stem_zh":"在全實數論域，指定 q:R→R，q(x)=19(x−(6))²+(6)，a=19≠0。求 x0=5 到 x1=25 的平均變化率，明示兩個輸入不同；以最簡正分母 p/q 回答。","answer":{"kind":"expression","value":"342/1","explanation_zh":"平均變化率=342/1。平方差除以輸入差化為 a(x0+x1−2h)，k 抵消；對稱端點可得零差商，但不是導數或常數函數聲明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["二次頂點式的平均變化率"]},"variant":{"family_id":"qf-rate-quadratic-average","seed":202610063094,"parameters":{"a":19,"h":6,"k":6,"x0":5,"x1":25,"domain":"reals","operation":"quadratic_average","semantic_template":"qf-rate-quadratic-average/{\"inputs\":[\"19\",\"18\"],\"operation\":\"quadratic_average\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0095","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"二次頂點式的平均變化率","stem_zh":"在全實數論域，指定 q:R→R，q(x)=-20(x−(7))²+(7)，a=-20≠0。求 x0=27 到 x1=6 的平均變化率，明示兩個輸入不同；以最簡正分母 p/q 回答。","answer":{"kind":"expression","value":"-380/1","explanation_zh":"平均變化率=-380/1。平方差除以輸入差化為 a(x0+x1−2h)，k 抵消；對稱端點可得零差商，但不是導數或常數函數聲明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["二次頂點式的平均變化率"]},"variant":{"family_id":"qf-rate-quadratic-average","seed":202610063095,"parameters":{"a":-20,"h":7,"k":7,"x0":27,"x1":6,"domain":"reals","operation":"quadratic_average","semantic_template":"qf-rate-quadratic-average/{\"inputs\":[\"-20\",\"19\"],\"operation\":\"quadratic_average\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0096","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"二次頂點式的平均變化率","stem_zh":"在全實數論域，指定 q:R→R，q(x)=21(x−(8))²+(8)，a=21≠0。求 x0=7 到 x1=29 的平均變化率，明示兩個輸入不同；以最簡正分母 p/q 回答。","answer":{"kind":"expression","value":"420/1","explanation_zh":"平均變化率=420/1。平方差除以輸入差化為 a(x0+x1−2h)，k 抵消；對稱端點可得零差商，但不是導數或常數函數聲明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["二次頂點式的平均變化率"]},"variant":{"family_id":"qf-rate-quadratic-average","seed":202610063096,"parameters":{"a":21,"h":8,"k":8,"x0":7,"x1":29,"domain":"reals","operation":"quadratic_average","semantic_template":"qf-rate-quadratic-average/{\"inputs\":[\"21\",\"20\"],\"operation\":\"quadratic_average\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0097","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"二次頂點式的平均變化率","stem_zh":"在全實數論域，指定 q:R→R，q(x)=-22(x−(9))²+(9)，a=-22≠0。求 x0=31 到 x1=8 的平均變化率，明示兩個輸入不同；以最簡正分母 p/q 回答。","answer":{"kind":"expression","value":"-462/1","explanation_zh":"平均變化率=-462/1。平方差除以輸入差化為 a(x0+x1−2h)，k 抵消；對稱端點可得零差商，但不是導數或常數函數聲明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["二次頂點式的平均變化率"]},"variant":{"family_id":"qf-rate-quadratic-average","seed":202610063097,"parameters":{"a":-22,"h":9,"k":9,"x0":31,"x1":8,"domain":"reals","operation":"quadratic_average","semantic_template":"qf-rate-quadratic-average/{\"inputs\":[\"-22\",\"21\"],\"operation\":\"quadratic_average\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0098","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"二次頂點式的平均變化率","stem_zh":"在全實數論域，指定 q:R→R，q(x)=23(x−(10))²+(10)，a=23≠0。求 x0=9 到 x1=33 的平均變化率，明示兩個輸入不同；以最簡正分母 p/q 回答。","answer":{"kind":"expression","value":"506/1","explanation_zh":"平均變化率=506/1。平方差除以輸入差化為 a(x0+x1−2h)，k 抵消；對稱端點可得零差商，但不是導數或常數函數聲明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["二次頂點式的平均變化率"]},"variant":{"family_id":"qf-rate-quadratic-average","seed":202610063098,"parameters":{"a":23,"h":10,"k":10,"x0":9,"x1":33,"domain":"reals","operation":"quadratic_average","semantic_template":"qf-rate-quadratic-average/{\"inputs\":[\"23\",\"22\"],\"operation\":\"quadratic_average\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0099","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"二次頂點式的平均變化率","stem_zh":"在全實數論域，指定 q:R→R，q(x)=-24(x−(11))²+(11)，a=-24≠0。求 x0=35 到 x1=10 的平均變化率，明示兩個輸入不同；以最簡正分母 p/q 回答。","answer":{"kind":"expression","value":"-552/1","explanation_zh":"平均變化率=-552/1。平方差除以輸入差化為 a(x0+x1−2h)，k 抵消；對稱端點可得零差商，但不是導數或常數函數聲明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["二次頂點式的平均變化率"]},"variant":{"family_id":"qf-rate-quadratic-average","seed":202610063099,"parameters":{"a":-24,"h":11,"k":11,"x0":35,"x1":10,"domain":"reals","operation":"quadratic_average","semantic_template":"qf-rate-quadratic-average/{\"inputs\":[\"-24\",\"23\"],\"operation\":\"quadratic_average\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r03-rate-of-change-0100","batch_id":"batch-20261005-advance-20261006-r03-rate-of-change-foundations","language":"zh-Hant","source_mko_ids":["mko-rate-of-change"],"title_zh":"二次頂點式的平均變化率","stem_zh":"在全實數論域，指定 q:R→R，q(x)=25(x−(12))²+(12)，a=25≠0。求 x0=11 到 x1=37 的平均變化率，明示兩個輸入不同；以最簡正分母 p/q 回答。","answer":{"kind":"expression","value":"600/1","explanation_zh":"平均變化率=600/1。平方差除以輸入差化為 a(x0+x1−2h)，k 抵消；對稱端點可得零差商，但不是導數或常數函數聲明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["二次頂點式的平均變化率"]},"variant":{"family_id":"qf-rate-quadratic-average","seed":202610063100,"parameters":{"a":25,"h":12,"k":12,"x0":11,"x1":37,"domain":"reals","operation":"quadratic_average","semantic_template":"qf-rate-quadratic-average/{\"inputs\":[\"25\",\"24\"],\"operation\":\"quadratic_average\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}