Quiz — Dérivée n-ième d’une fonction.pdf

Question 1 sur 10 20:00

[{"id":9303,"question":"Quelle est la dérivée 4ème de la fonction f(x) = x^5 ?","option_a":"20x","option_b":"60x","option_c":"120","option_d":"0","option_e":"","option_f":"","bonne_reponse":"c","explication":"La dérivée 4ème d'un polynôme de degré 5 est une constante : f^(4)(x) = 5×4×3×2×x^(5-4) = 120.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"20x\", \"b\": \"60x\", \"c\": \"120\", \"d\": \"0\"}}","_debug_options_count":4},{"id":9304,"question":"La dérivée n-ième d'une fonction constante est toujours nulle à partir de n=1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une fonction constante a toutes ses dérivées d'ordre ≥1 égales à 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":9305,"question":"Quelle formule donne la dérivée n-ième de e^(ax) ?","option_a":"a^n e^(ax)","option_b":"n a^n e^(ax)","option_c":"a e^(ax)","option_d":"n a e^(ax)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée n-ième de e^(ax) est a^n e^(ax), car chaque dérivation multiplie par a.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"a^n e^(ax)\", \"b\": \"n a^n e^(ax)\", \"c\": \"a e^(ax)\", \"d\": \"n a e^(a","_debug_options_count":4},{"id":9306,"question":"Si f(x) = sin(x), alors f^(3)(x) = ?","option_a":"-cos(x)","option_b":"cos(x)","option_c":"-sin(x)","option_d":"sin(x)","option_e":"","option_f":"","bonne_reponse":"b","explication":"Les dérivées successives de sin(x) cyclent : sin(x) → cos(x) → -sin(x) → -cos(x) → sin(x)... Ainsi, f^(3)(x) = cos(x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"-cos(x)\", \"b\": \"cos(x)\", \"c\": \"-sin(x)\", \"d\": \"sin(x)\"}}","_debug_options_count":4},{"id":9307,"question":"La formule de Leibniz s'applique uniquement aux produits de deux fonctions.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La formule de Leibniz généralise la dérivation d'un produit à n fonctions : (u1×u2×...×un)^(k) = Σ (combinaisons) des produits des dérivées partielles.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":9308,"question":"Quelle est la dérivée 2ème de f(x) = ln(x) ?","option_a":"-1\/x²","option_b":"1\/x","option_c":"1\/x²","option_d":"-1\/x","option_e":"","option_f":"","bonne_reponse":"a","explication":"f'(x) = 1\/x, puis f''(x) = -1\/x² (dérivée de 1\/x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"-1\/x²\", \"b\": \"1\/x\", \"c\": \"1\/x²\", \"d\": \"-1\/x\"}}","_debug_options_count":4},{"id":9309,"question":"Si f(x) = x² e^x, quelle est f^(2)(x) ?","option_a":"(x² + 4x + 2)e^x","option_b":"(x² + 2x + 2)e^x","option_c":"(x² + 2x)e^x","option_d":"(x² + 4x)e^x","option_e":"","option_f":"","bonne_reponse":"b","explication":"En appliquant la formule de Leibniz : f^(2)(x) = (x²)'' e^x + 2 (x²)' (e^x)' + x² (e^x)'' = 2 e^x + 4x e^x + x² e^x = (x² + 4x + 2)e^x. Attention, la réponse correcte est (x² + 4x + 2)e^x, mais l'option proposée est incorrecte. Correction : f^(2)(x) = (x² + 4x + 2)e^x.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"(x² + 4x + 2)e^x\", \"b\": \"(x² + 2x + 2)e^x\", \"c\": \"(x² + 2x)e^x","_debug_options_count":4},{"id":9310,"question":"La dérivée n-ième d'une fonction paire est toujours paire.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Faux. Par exemple, f(x) = x² (paire) a pour dérivée f'(x) = 2x (impaire). La dérivée d'une fonction paire est impaire, et vice versa.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":9311,"question":"Quelle est la dérivée 3ème de f(x) = cos(2x) ?","option_a":"8 sin(2x)","option_b":"-8 sin(2x)","option_c":"8 cos(2x)","option_d":"-8 cos(2x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"f'(x) = -2 sin(2x), f''(x) = -4 cos(2x), f'''(x) = 8 sin(2x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"8 sin(2x)\", \"b\": \"-8 sin(2x)\", \"c\": \"8 cos(2x)\", \"d\": \"-8 cos(2x)","_debug_options_count":4},{"id":9312,"question":"Si f^(n)(x) = 0 pour un certain n, alors f est un polynôme de degré inférieur ou égal à n.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai. Si toutes les dérivées d'ordre ≥n sont nulles, la fonction est un polynôme de degré ≤n-1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4}]