Quiz — Lycée Pilote Sfax - Dérivabilité 2.pdf

Question 1 sur 5 10:00

[{"id":2298,"question":"Quelle est la dérivée de la fonction f(x) = x³ + 2x² - 5x + 1 ?","option_a":"f’(x) = 3x² + 4x - 5","option_b":"f’(x) = 3x² + 4x + 1","option_c":"f’(x) = x² + 2x - 5","option_d":"f’(x) = 3x² + 4x - 1","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d’un polynôme s’obtient en appliquant la règle de dérivation terme à terme : la dérivée de x^n est n·x^(n-1). Ainsi, (x³)’ = 3x², (2x²)’ = 4x, (-5x)’ = -5 et (1)’ = 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\": \"f’(x) = 3x² + 4x - 5\", \"b\": \"f’(x) = 3x² + 4x + 1\", \"c\": \"f","_debug_options_count":4},{"id":2299,"question":"Si f(x) = e^(2x), quelle est sa dérivée ?","option_a":"f’(x) = e^(2x)","option_b":"f’(x) = 2e^(2x)","option_c":"f’(x) = e^x","option_d":"f’(x) = 2e^x","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de e^(u(x)) est u’(x)·e^(u(x)). Ici, u(x) = 2x, donc u’(x) = 2. Ainsi, f’(x) = 2·e^(2x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"f’(x) = e^(2x)\", \"b\": \"f’(x) = 2e^(2x)\", \"c\": \"f’(x) = e^x\"","_debug_options_count":4},{"id":2300,"question":"Quel est le théorème qui affirme que si une fonction est continue sur [a,b] et dérivable sur ]a,b[, alors il existe c dans ]a,b[ tel que f’(c) = (f(b)-f(a))\/(b-a) ?","option_a":"Théorème de Rolle","option_b":"Théorème des valeurs intermédiaires","option_c":"Théorème des accroissements finis","option_d":"Théorème de la bijection","option_e":"","option_f":"","bonne_reponse":"c","explication":"Le théorème des accroissements finis généralise le théorème de Rolle en reliant la dérivée moyenne à la pente de la corde reliant deux points de la courbe.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"Théorème de Rolle\", \"b\": \"Théorème des valeurs intermédiaire","_debug_options_count":4},{"id":2301,"question":"Soit f(x) = ln(x² + 1). Quelle est sa dérivée ?","option_a":"f’(x) = 2x\/(x² + 1)","option_b":"f’(x) = 1\/(x² + 1)","option_c":"f’(x) = 2x\/(x² - 1)","option_d":"f’(x) = x\/(x² + 1)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de ln(u(x)) est u’(x)\/u(x). Ici, u(x) = x² + 1, donc u’(x) = 2x. Ainsi, f’(x) = 2x\/(x² + 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\": \"f’(x) = 2x\/(x² + 1)\", \"b\": \"f’(x) = 1\/(x² + 1)\", \"c\": \"f’","_debug_options_count":4},{"id":2302,"question":"Quelle est la dérivée de la fonction f(x) = sin(3x) ?","option_a":"f’(x) = cos(3x)","option_b":"f’(x) = 3cos(3x)","option_c":"f’(x) = -3cos(3x)","option_d":"f’(x) = sin(3x)","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de sin(u(x)) est u’(x)·cos(u(x)). Ici, u(x) = 3x, donc u’(x) = 3. Ainsi, f’(x) = 3cos(3x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"f’(x) = cos(3x)\", \"b\": \"f’(x) = 3cos(3x)\", \"c\": \"f’(x) = -3","_debug_options_count":4}]