Quiz — Série n°2 - Dérivabilité.pdf

Question 1 sur 10 20:00

[{"id":4796,"question":"Quelle est la dérivée de la fonction f(x) = x² + 3x - 5 ?","option_a":"2x + 3","option_b":"x² + 3x","option_c":"2x - 5","option_d":"x + 3","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une somme est la somme des dérivées. La dérivée de x² est 2x, celle de 3x est 3, et celle de -5 est 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\": \"2x + 3\", \"b\": \"x² + 3x\", \"c\": \"2x - 5\", \"d\": \"x + 3\"}}","_debug_options_count":4},{"id":4797,"question":"Si une fonction est dérivable en un point, alors elle est nécessairement continue en ce point.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est un théorème fondamental : la dérivabilité implique la continuité.","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":4798,"question":"Quelle est la dérivée de la fonction f(x) = 1\/x ?","option_a":"-1\/x²","option_b":"1\/x²","option_c":"x","option_d":"-x","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de 1\/x est -1\/x², car f(x) = x^(-1) et on applique la règle de dérivation des puissances.","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\/x²\", \"c\": \"x\", \"d\": \"-x\"}}","_debug_options_count":4},{"id":4799,"question":"Le théorème de Rolle s'applique à une fonction f sur [a, b] si :","option_a":"f est continue sur [a, b], dérivable sur ]a, b[, et f(a) = f(b)","option_b":"f est dérivable sur [a, b] et f(a) ≠ f(b)","option_c":"f est continue sur ]a, b[ et dérivable sur [a, b]","option_d":"f est continue sur [a, b] et discontinue sur ]a, b[","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le théorème de Rolle nécessite que la fonction soit continue sur [a, b], dérivable sur ]a, b[, et que f(a) = f(b).","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 est continue sur [a, b], dérivable sur ]a, b[, et f(a) = f(b)\"","_debug_options_count":4},{"id":4800,"question":"La dérivée d'une fonction constante est toujours égale à 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une constante est toujours 0, car la pente de la tangente est horizontale.","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":4801,"question":"Quelle est la dérivée de la fonction f(x) = e^(2x) ?","option_a":"2e^(2x)","option_b":"e^(2x)","option_c":"2x e^(2x)","option_d":"e^x","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de e^(u(x)) est u'(x) * e^(u(x)). Ici, u(x) = 2x, donc u'(x) = 2.","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\": \"2x e^(2x)\", \"d\": \"e^x\"}}","_debug_options_count":4},{"id":4802,"question":"Si f'(x) = 0 pour tout x dans un intervalle, alors f est constante sur cet intervalle.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est le théorème des fonctions constantes : si la dérivée est nulle sur un intervalle, la fonction est constante sur cet intervalle.","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":4803,"question":"Quelle est la dérivée de la fonction f(x) = sin(x) ?","option_a":"cos(x)","option_b":"-cos(x)","option_c":"sin(x)","option_d":"-sin(x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de sin(x) est cos(x), c'est une formule fondamentale à connaître.","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\": \"-sin(x)\"}}","_debug_options_count":4},{"id":4804,"question":"Le théorème des accroissements finis stipule qu'il existe c dans ]a, b[ tel que :","option_a":"f'(c) = (f(b) - f(a)) \/ (b - a)","option_b":"f'(c) = f(b) - f(a)","option_c":"f(c) = (f(b) - f(a)) \/ (b - a)","option_d":"f'(c) = 0","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le théorème des accroissements finis affirme qu'il existe un point c où la dérivée est égale au taux d'accroissement moyen.","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'(c) = (f(b) - f(a)) \/ (b - a)\", \"b\": \"f'(c) = f(b) - f(a)\", \"c\"","_debug_options_count":4},{"id":4805,"question":"La dérivée d'une fonction paire est toujours impaire.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Si f est paire (f(-x) = f(x)), alors sa dérivée f' est impaire (f'(-x) = -f'(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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4}]