Quiz — Lycée Pilote Sfax - Dérivabilité 3.pdf

Question 1 sur 10 20:00

[{"id":69531,"question":"Si une fonction f est dérivable en a, alors :","option_a":"f est continue en a","option_b":"f est discontinue en a","option_c":"f n'est pas définie en a","option_d":"f est constante en a","option_e":"","option_f":"","bonne_reponse":"a","explication":"Toute fonction dérivable en un point est nécessairement continue en ce point. C'est une propriété fondamentale de la dérivabilité.","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 en a\", \"b\": \"f est discontinue en a\", \"c\": \"f n'es","_debug_options_count":4},{"id":69532,"question":"Le nombre dérivé de f en a est égal à :","option_a":"lim (h→0) (f(a+h)-f(a))\/h","option_b":"lim (x→a) (f(x)-f(a))\/(x-a)","option_c":"f'(a)","option_d":"Toutes les réponses sont correctes","option_e":"","option_f":"","bonne_reponse":"d","explication":"Les trois expressions sont équivalentes et représentent la définition du nombre dérivé de f en a.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"d\", \"options\": {\"a\": \"lim (h→0) (f(a+h)-f(a))\/h\", \"b\": \"lim (x→a) (f(x)-f(a))\/(x-a)","_debug_options_count":4},{"id":69533,"question":"La fonction f(x) = |x| est 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 elle présente un point anguleux (la dérivée à gauche et à droite ne sont pas égales).","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":69534,"question":"Si f(x) = x², alors f'(2) est égal à :","option_a":"2","option_b":"4","option_c":"0","option_d":"8","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de f(x) = x² est f'(x) = 2x. Donc f'(2) = 2*2 = 4.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"2\", \"b\": \"4\", \"c\": \"0\", \"d\": \"8\"}}","_debug_options_count":4},{"id":69535,"question":"La dérivée de f(x) = 1\/x est :","option_a":"-1\/x²","option_b":"1\/x²","option_c":"x²","option_d":"0","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant la formule de dérivation des fonctions puissances, f'(x) = -1*x^(-2) = -1\/x².","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\": \"x²\", \"d\": \"0\"}}","_debug_options_count":4},{"id":69536,"question":"Une fonction peut être continue sans être dérivable.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est vrai. Par exemple, la fonction f(x) = |x| est continue en 0 mais n'y est pas dérivable.","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":69537,"question":"Si f est dérivable sur un intervalle I, alors f est :","option_a":"constante sur I","option_b":"continue sur I","option_c":"discontinue sur I","option_d":"non définie sur I","option_e":"","option_f":"","bonne_reponse":"b","explication":"Toute fonction dérivable sur un intervalle est nécessairement continue sur cet intervalle.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"constante sur I\", \"b\": \"continue sur I\", \"c\": \"discontinue sur I\"","_debug_options_count":4},{"id":69538,"question":"La dérivée de f(x) = √x est :","option_a":"1\/(2√x)","option_b":"1\/√x","option_c":"2√x","option_d":"1\/(2x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant la formule de dérivation des fonctions racines, f'(x) = 1\/(2√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\/(2√x)\", \"b\": \"1\/√x\", \"c\": \"2√x\", \"d\": \"1\/(2x)\"}}","_debug_options_count":4},{"id":69539,"question":"Si f'(a) = 0, alors la tangente à la courbe de f en a est :","option_a":"horizontale","option_b":"verticale","option_c":"oblique","option_d":"parallèle à l'axe des ordonnées","option_e":"","option_f":"","bonne_reponse":"a","explication":"Si la dérivée est nulle en un point, la tangente à la courbe en ce point 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\": \"horizontale\", \"b\": \"verticale\", \"c\": \"oblique\", \"d\": \"parallèle ","_debug_options_count":4},{"id":69540,"question":"La fonction f(x) = x³ est dérivable sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction f(x) = x³ est dérivable sur tout ℝ car sa dérivée f'(x) = 3x² existe pour tout x réel.","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}]