Quiz — Lycée Pilote Sfax - Dérivabilité 1.pdf

Question 1 sur 10 20:00

[{"id":35600,"question":"Soit f(x) = x². Quel est le nombre dérivé de f en x = 3 ?","option_a":"6","option_b":"9","option_c":"3","option_d":"12","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le nombre dérivé de f(x) = x² en x = 3 est f'(3) = 2*3 = 6, d'après la formule de dérivation des fonctions polynômes.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"6\", \"b\": \"9\", \"c\": \"3\", \"d\": \"12\"}}","_debug_options_count":4},{"id":35601,"question":"La fonction f(x) = |x| est-elle dérivable en x = 0 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction valeur absolue n'est pas dérivable en 0 car la limite du taux d'accroissement à gauche et à droite n'est pas la même.","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":35602,"question":"Quelle est la dérivée de f(x) = 3x³ + 2x² - 5x + 1 ?","option_a":"9x² + 4x - 5","option_b":"6x² + 4x - 5","option_c":"3x² + 2x - 5","option_d":"9x² + 4x + 1","option_e":"","option_f":"","bonne_reponse":"a","explication":"En appliquant les règles de dérivation, f'(x) = 9x² + 4x - 5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"9x² + 4x - 5\", \"b\": \"6x² + 4x - 5\", \"c\": \"3x² + 2x - 5\", \"d\": ","_debug_options_count":4},{"id":35603,"question":"Si une fonction est continue en un point, est-elle nécessairement dérivable en ce point ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La continuité est une condition nécessaire mais pas suffisante pour la dérivabilité. Exemple : f(x) = |x| est continue en 0 mais pas dérivable.","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":35604,"question":"Soit f(x) = 1\/x. Quel est le domaine de dérivabilité de f ?","option_a":"ℝ*","option_b":"ℝ","option_c":"ℝ+*","option_d":"ℝ-{0}","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction f(x) = 1\/x est dérivable sur ℝ* (tous les réels sauf 0), car elle n'est pas définie en 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\": \"ℝ*\", \"b\": \"ℝ\", \"c\": \"ℝ+*\", \"d\": \"ℝ-{0}\"}}","_debug_options_count":4},{"id":35605,"question":"La dérivée de f(x) = sin(x) est :","option_a":"cos(x)","option_b":"-cos(x)","option_c":"sin(x)","option_d":"tan(x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de la fonction sinus est la fonction cosinus : f'(x) = cos(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\": \"cos(x)\", \"b\": \"-cos(x)\", \"c\": \"sin(x)\", \"d\": \"tan(x)\"}}","_debug_options_count":4},{"id":35606,"question":"Soit f(x) = x^(1\/3). Cette fonction est-elle dérivable en x = 0 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction f(x) = x^(1\/3) n'est pas dérivable en 0 car la tangente verticale rend le nombre dérivé infini.","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":35607,"question":"Quelle est la dérivée de f(x) = e^(2x) ?","option_a":"2e^(2x)","option_b":"e^(2x)","option_c":"2xe^(2x)","option_d":"e^x","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant la règle de dérivation des fonctions exponentielles composées, f'(x) = 2e^(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\": \"2e^(2x)\", \"b\": \"e^(2x)\", \"c\": \"2xe^(2x)\", \"d\": \"e^x\"}}","_debug_options_count":4},{"id":35608,"question":"La fonction f(x) = x² + 3x - 2 est-elle dérivable sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Toute fonction polynôme est dérivable sur ℝ, donc f(x) = x² + 3x - 2 est dérivable sur tout son domaine de définition.","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":35609,"question":"Soit f(x) = ln(x). Quel est le nombre dérivé de f en x = 1 ?","option_a":"1","option_b":"0","option_c":"-1","option_d":"e","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de la fonction logarithme népérien est f'(x) = 1\/x. Donc f'(1) = 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\": \"1\", \"b\": \"0\", \"c\": \"-1\", \"d\": \"e\"}}","_debug_options_count":4}]