Quiz — 659d1af00305c_-_Série N°1-Dérivabilité.pdf

Question 1 sur 10 20:00

[{"id":30650,"question":"Soit la fonction f définie par f(x) = x². Quel est le nombre dérivé de f en x = 3 ?","option_a":"6","option_b":"9","option_c":"3","option_d":"0","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le nombre dérivé de f en x = a est donné par f'(a) = lim(h→0) [f(a+h) - f(a)]\/h. Pour f(x) = x², f'(x) = 2x, donc f'(3) = 6.","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\": \"0\"}}","_debug_options_count":4},{"id":30651,"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 a un 'coin' en x = 0, donc elle n'est pas dérivable en ce point (la limite du taux d'accroissement n'existe pas).","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":30652,"question":"Quelle est la dérivée de la fonction 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(x)) est u'(x)e^(u(x)). Ici, u(x) = 2x, donc u'(x) = 2. Ainsi, 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":30653,"question":"Si f est dérivable en a, alors f est-elle nécessairement continue en a ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Oui, si une fonction est dérivable en un point, elle y est nécessairement continue (théorème fondamental).","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":30654,"question":"Quelle est la dérivée de f(x) = ln(x² + 1) ?","option_a":"2x \/ (x² + 1)","option_b":"2x","option_c":"(x² + 1) \/ 2x","option_d":"1 \/ (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\": \"2x \/ (x² + 1)\", \"b\": \"2x\", \"c\": \"(x² + 1) \/ 2x\", \"d\": \"1 \/ (x²","_debug_options_count":4},{"id":30655,"question":"La fonction f(x) = x^(1\/3) 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) a une tangente verticale en x = 0, donc elle n'est pas dérivable en ce point (la limite du taux d'accroissement est infinie).","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":30656,"question":"Quelle est la dérivée de f(x) = sin(3x) ?","option_a":"3cos(3x)","option_b":"cos(3x)","option_c":"3sin(3x)","option_d":"-3cos(3x)","option_e":"","option_f":"","bonne_reponse":"a","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\": \"a\", \"options\": {\"a\": \"3cos(3x)\", \"b\": \"cos(3x)\", \"c\": \"3sin(3x)\", \"d\": \"-3cos(3x)\"}}","_debug_options_count":4},{"id":30657,"question":"Si f est continue en a mais pas dérivable en a, peut-on appliquer le théorème des valeurs intermédiaires en a ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Oui, le théorème des valeurs intermédiaires s'applique aux fonctions continues, même si elles ne sont pas dérivables.","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":30658,"question":"Quelle est la dérivée de f(x) = (x² + 1)\/(x - 1) ?","option_a":"(2x(x - 1) - (x² + 1)) \/ (x - 1)²","option_b":"(2x(x - 1) + (x² + 1)) \/ (x - 1)²","option_c":"2x \/ (x - 1)","option_d":"(x² + 1) \/ (x - 1)²","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'un quotient u\/v est (u'v - uv')\/v². Ici, u = x² + 1 (u' = 2x) et v = x - 1 (v' = 1). Ainsi, f'(x) = (2x(x - 1) - (x² + 1)) \/ (x - 1)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\": \"(2x(x - 1) - (x² + 1)) \/ (x - 1)²\", \"b\": \"(2x(x - 1) + (x² + 1","_debug_options_count":4},{"id":30659,"question":"La fonction f(x) = x^3 est-elle dérivable sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Oui, la fonction f(x) = x^3 est dérivable sur ℝ car elle est continue et admet une dérivée en tout point (f'(x) = 3x²).","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}]