Quiz — سلسلة الاشتقاقية و الاستمرارية 2020.docx

Question 1 sur 10 20:00

[{"id":18609,"question":"Quelle est la dérivée de la fonction f(x) = 3x² + 2x - 5 ?","option_a":"6x + 2","option_b":"3x + 2","option_c":"6x² + 2x","option_d":"3x² + 2","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de x² est 2x, donc 3x² devient 6x. La dérivée de 2x est 2. La constante -5 disparaît.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"6x + 2\", \"b\": \"3x + 2\", \"c\": \"6x² + 2x\", \"d\": \"3x² + 2\"}}","_debug_options_count":4},{"id":18610,"question":"Si une fonction est continue en un point, est-elle toujours 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":18611,"question":"Quelle est la limite de (sin x)\/x lorsque x tend vers 0 ?","option_a":"0","option_b":"1","option_c":"∞","option_d":"n'existe pas","option_e":"","option_f":"","bonne_reponse":"b","explication":"C'est un résultat fondamental en analyse : lim(x→0) (sin x)\/x = 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"0\", \"b\": \"1\", \"c\": \"∞\", \"d\": \"n'existe pas\"}}","_debug_options_count":4},{"id":18612,"question":"La fonction f(x) = x³ - 3x est-elle continue sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Les fonctions polynômes sont continues sur tout ℝ, car elles sont dérivables en tout point.","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":18613,"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":"La dérivée de e^u est u'·e^u. Ici, u = 2x, donc u' = 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\": \"2xe^(2x)\", \"d\": \"e^x\"}}","_debug_options_count":4},{"id":18614,"question":"Si f'(a) = 0, que peut-on dire de la tangente à la courbe de f en x = a ?","option_a":"Elle est horizontale","option_b":"Elle est verticale","option_c":"Elle n'existe pas","option_d":"Elle est parallèle à l'axe des ordonnées","option_e":"","option_f":"","bonne_reponse":"a","explication":"f'(a) = 0 signifie que le coefficient directeur de la tangente est 0, donc elle 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\": \"Elle est horizontale\", \"b\": \"Elle est verticale\", \"c\": \"Elle n'ex","_debug_options_count":4},{"id":18615,"question":"La fonction f(x) = 1\/x est-elle continue en x = 0 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction n'est pas définie en x = 0, donc elle ne peut pas être continue en ce point.","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":18616,"question":"Quelle est la dérivée de f(x) = ln(x) ?","option_a":"1\/x","option_b":"x","option_c":"1","option_d":"e^x","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de ln(x) est 1\/x, pour x \u003E 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\": \"1\/x\", \"b\": \"x\", \"c\": \"1\", \"d\": \"e^x\"}}","_debug_options_count":4},{"id":18617,"question":"Si une fonction est dérivable en un point, est-elle nécessairement continue en ce point ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivabilité implique la continuité. Si f est dérivable en a, alors f est continue en 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":18618,"question":"Quelle est la dérivée de f(x) = (x² + 1)\/(x - 1) ?","option_a":"(2x)\/(x - 1)","option_b":"(2x(x - 1) - (x² + 1))\/(x - 1)²","option_c":"2x - 1","option_d":"(x² + 1)\/(x - 1)²","option_e":"","option_f":"","bonne_reponse":"b","explication":"On utilise la formule de dérivation d'un quotient : (u\/v)' = (u'v - uv')\/v². Ici, u = x² + 1 et v = x - 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"(2x)\/(x - 1)\", \"b\": \"(2x(x - 1) - (x² + 1))\/(x - 1)²\", \"c\": \"2x","_debug_options_count":4}]