{"schema_version":"ocme-site-results-v1","locale":"en","page_type":"derivations","facts":{"questions":36000,"reviewed":36000,"display_candidates":0,"knowledge":302,"guides":299,"paths":59,"batches":360,"families":1441,"atlas":302,"core":80,"evidence":9,"formal_evidence":5,"formal_subjects":4,"lean_files":6,"lean_toolchain":"leanprover/lean4:v4.30.0","question_state":"candidate","english_state":"ai_candidate","question_Lean":"NOT_RUN","canonical_truth_promotion":false},"filters":{"q":"mko-antiderivative"},"kind":"guides","total":1,"pages":1,"page":1,"entries":[{"id":"derivation-mko-antiderivative-01","title":{"zh-Hant":"原函數：多項式代表原函數、常數族與候選驗收","en":"Antiderivatives: the representative polynomial antiderivative, the constant family and checking candidates"},"summary":{"zh-Hant":"對 f=[3,−4,6]（f(x)=3−4x+6x²）：由定義證明四次以下多項式的求導公式；構造代表原函數 F0=[0,3,−2,2] 並驗證 F0'=f；證明 F0+C 對每個實數 C 都是原函數，且在至多五個係數的多項式中每個原函數都是 F0+C；驗收候選 G=[7,3,−2,2]（接受）、H=[0,3,−4,2]（拒絕）、K=[0,3,−3,8/3]（兩個測點相符仍拒絕）；求 F0(2)+C（C=−5），並說明原函數不是反函數。","en":"For f=[3,−4,6] (f(x)=3−4x+6x²): prove from the definition the derivative formula for polynomials of degree at most four; build the representative antiderivative F0=[0,3,−2,2] and verify F0'=f; prove that F0+C is an antiderivative for every real C and that, among polynomials with at most five coefficients, every antiderivative is F0+C; check the candidates G=[7,3,−2,2] (accept), H=[0,3,−4,2] (reject) and K=[0,3,−3,8/3] (reject although it matches at two test points); evaluate F0(2)+C with C=−5; and explain why an antiderivative is not an inverse function."},"mko_ids":["mko-antiderivative"]}],"title":"Concept derivation guides"}