{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261111-r02-higher-order-derivative-0077", "pilot_number": "R0045-23002", "number": 23002, "family_id": "qf-mod42-higher-missing-next-order-boundary", "title": { "zh-Hant": "第 23002 題：高階導數", "en": "Question 23002: A First Derivative May Exist Where the Second Does Not" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261111-r02-higher-order-derivative-0077", "batch_id": "batch-20261014-advance-20261111-r02-higher-order-derivative-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-higher-order-derivative" ], "title_zh": "高階導數", "stem_zh": "f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-missing-next-order-boundary；資料={\"scale\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(2, f(x)=scale*x*abs(x), 0, 是, 0, 否, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)", "explanation_zh": "Wave6第五層核對=(2, f(x)=scale*x*abs(x), 0, 是, 0, 否, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。" }, "difficulty": { "band": "undergraduate_core", "level": 1, "skills": [ "高階導數" ] }, "variant": { "family_id": "qf-mod42-higher-missing-next-order-boundary", "seed": 202611112077, "parameters": { "scale": 2, "domain": "Exact_polynomial_higher_derivative", "operation": "higher-missing-next-order-boundary", "semantic_template": "qf-mod42-higher-missing-next-order-boundary/{\"scale\":2}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_or_symbolic_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-14", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261014-advance-20261111-r02-higher-order-derivative-0077", "batch_id": "batch-20261014-advance-20261111-r02-higher-order-derivative-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "4a0d1f15aafa1ef2e7333ba8ccb30f3d16d01f6a33cc5625a39fe43de69a4537", "family_id": "qf-mod42-higher-missing-next-order-boundary", "seed": 202611112077, "semantic_template": "qf-mod42-higher-missing-next-order-boundary/{\"scale\":2}" }, "fields": { "title": "A First Derivative May Exist Where the Second Does Not", "stem": "Use nonzero scale=2 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).", "answer": { "kind": "expression", "value": "(2, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)", "explanation": "The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (2, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)." }, "skills": [ "A First Derivative May Exist Where the Second Does Not" ], "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review." } }, "givens": [ { "id": "G001", "text": { "zh-Hant": "f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-missing-next-order-boundary；資料={\"scale\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use nonzero scale=2 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source operation=higher-missing-next-order-boundary;original snapshot and mathematics retained", "after": "explicit ordered source output fields=[scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary]", "relation": "implies", "rule_id": "source_ordered_interface_explicit_without_javascript", "conditions": [], "explanation": { "zh-Hant": "原題快照保留歷史介面；此處直接列來源欄序，不依賴JavaScript。tag與No保聲稱範圍，固定來源排序不冒充數學證明。", "en": "Retain the historical interface marker while listing source fields without JavaScript.Tags and No retain their claim scope;source-protocol ordering is not a mathematical proof." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "ORIGINAL nonzero scale=2;WHOLE f(x)=scale*x*abs(x),f(0)=0", "after": "for h!=0,(f(h)-f(0))/h=scale*abs(h) tends0;first derivative at0 exists and equals0", "relation": "implies", "rule_id": "higher_first_derivative_at_zero_from_difference_quotient", "conditions": [], "explanation": { "zh-Hant": "直接用所有h→0的差商，不由有限樣本猜導數。正負h皆有同一趨零絕對值界。", "en": "Use the difference quotient for all h approaching0,not finite samples;both signs are bounded by the same vanishing absolute value." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "ORIGINAL f(x)=(2)x*abs(x);for x!=0,f_prime(x)=2*(2)*abs(x);f_prime(0)=0", "after": "next difference quotient at0 has right limit=4,left limit=-4;nonzero scale makes them unequal,so second derivative at0 does NOT exist", "relation": "implies", "rule_id": "higher_second_derivative_opposite_one_sided_limits", "conditions": [], "explanation": { "zh-Hant": "先有f的一階導數，再核其導數的左右差商；一階存在不保下一階，也不推analytic或Taylor級數收斂。", "en": "After establishing the first derivative,check its two one-sided difference quotients.First-order existence guarantees neither the next derivative nor analyticity or Taylor-series convergence." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null } ], "math_model": { "result": [ 2, "f(x)=scale*x*abs(x)", 0, true, 0, false, "first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits" ] }, "conclusion": { "canonical_value": "(2, f(x)=scale*x*abs(x), 0, 是, 0, 否, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)", "explanation": { "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。", "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged." } }, "definition_note": { "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。", "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate." }, "source": { "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db", "question_binding": { "batch_id": "batch-20261014-advance-20261111-r02-higher-order-derivative-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261111-r02-higher-order-derivative-foundations/questions.jsonl", "batch_sha256": "53a54bdcff64574bbf5f352d4badfe20bbb98f5da23d3abdfdc2f3b20d16f3b1", "original_line": 77, "original_line_sha256": "f0efd2d97b50c94866239e35064824d3dd0c2ed8de700d011e8da3e158578084", "question_compact_sha256": "4a0d1f15aafa1ef2e7333ba8ccb30f3d16d01f6a33cc5625a39fe43de69a4537", "english_row_compact_sha256": "1dc878148f87f69e497727e254c2315ed69374f149462d3afb652ac88a483c96", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261111-r02-higher-order-derivative-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261111-r02-higher-order-derivative-foundations.en.jsonl" }, "mko_bindings": { "mko-higher-order-derivative": { "sha256": "78b36c7cb74b3af2a4e2f3cbb7c42f790660d13aa1347fdd50826cc718717741", "english_sha256": "c1886e7e00a610edfc2e3a3fb75552d4afb78b49f57c24e7e1abd7d5ce5775d6", "original_path": "public/data/mko/higher-order-derivative.json", "english_path": "public/data/i18n/editions/mko-higher-order-derivative.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "9732ef02cb296d2e89f9b6cade5a76a83e9d37bdcf6a3de7f8613ebce02311e8", "original_candidate_sha256": "c7d2b5d434ebba544cd658088f0415d0be87d119e24d1b51f30f875215b195e1", "original_candidate_line_sha256": "c7d2b5d434ebba544cd658088f0415d0be87d119e24d1b51f30f875215b195e1" }, "historical_source_state": { "review_state": { "derivation": "candidate_authored_and_locally_checked", "source_question_state": "candidate", "independent_AI_or_human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN" }, "scope_note": { "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。", "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged." }, "label": "Frozen producer metadata; current per-question review is separate below." }, "revision": { "id": "review-20261026-009-p46", "review_date": "2026-10-26", "owner_authorized_reader_publication": true, "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.", "review_receipts": { "semantic_review_sha256": "357d7ad7a6726b2a757558d3b9586a408ade06526cc023da093719c292704438", "question_first_witness_sha256": "99cefe17e98c7a74321801cf72df6d02fc95b6a6b78aa335accb7e11ed467b2a" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 3, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "0242f93209ce17e04ee4bf30fd47cea4838ef08fa0f8dc88d8b9ba3a294b8429", "arithmetic_steps": 0 }, "uses_answer_as_premise": false, "human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN", "Canonical_truth_promotion": false, "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged." } }