Quiz — أعمال موجهة تحليل.pdf

Question 1 sur 10 20:00

[{"id":88428,"question":"Quelle est la dérivée de la fonction f(x) = 3x² + 2x - 5 ?","option_a":"6x + 2","option_b":"3x² + 2","option_c":"6x² + 2x","option_d":"3x + 2","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynôme s'obtient en appliquant la règle : (ax^n)' = n*ax^(n-1). Ainsi, f'(x) = 6x + 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\": \"6x + 2\", \"b\": \"3x² + 2\", \"c\": \"6x² + 2x\", \"d\": \"3x + 2\"}}","_debug_options_count":4},{"id":88429,"question":"La fonction f(x) = x³ est-elle strictement croissante sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de f(x) = x³ est f'(x) = 3x², qui est toujours positive ou nulle sur ℝ. La fonction est donc strictement croissante.","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":88430,"question":"Quelle est l'intégrale indéfinie de f(x) = 4x³ ?","option_a":"x⁴ + C","option_b":"12x² + C","option_c":"x⁴\/4 + C","option_d":"4x⁴ + C","option_e":"","option_f":"","bonne_reponse":"c","explication":"L'intégrale de x^n est x^(n+1)\/(n+1) + C. Ainsi, ∫4x³ dx = 4*(x⁴\/4) + C = x⁴ + C.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"x⁴ + C\", \"b\": \"12x² + C\", \"c\": \"x⁴\/4 + C\", \"d\": \"4x⁴ + C\"}","_debug_options_count":4},{"id":88431,"question":"La suite u_n = (n² + 1)\/n est-elle convergente ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"En simplifiant, u_n = n + 1\/n. Lorsque n tend vers l'infini, u_n tend vers l'infini. La suite est donc divergente.","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":88432,"question":"Quelle est la limite de f(x) = (2x² + 3x - 1)\/(x² - 4) lorsque x tend vers l'infini ?","option_a":"2","option_b":"0","option_c":"1","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour les limites à l'infini de fonctions rationnelles, on compare les termes de plus haut degré. Ici, la limite est 2x²\/x² = 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\": \"2\", \"b\": \"0\", \"c\": \"1\", \"d\": \"∞\"}}","_debug_options_count":4},{"id":88433,"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 x = 0 car la dérivée à gauche (-1) et à droite (1) 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":88434,"question":"Quelle est la primitive de f(x) = e^(2x) ?","option_a":"e^(2x) + C","option_b":"(1\/2)e^(2x) + C","option_c":"2e^(2x) + C","option_d":"e^(x) + C","option_e":"","option_f":"","bonne_reponse":"b","explication":"La primitive de e^(kx) est (1\/k)e^(kx) + C. Ici, k = 2, donc la primitive est (1\/2)e^(2x) + C.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"e^(2x) + C\", \"b\": \"(1\/2)e^(2x) + C\", \"c\": \"2e^(2x) + C\", \"d\": \"e^","_debug_options_count":4},{"id":88435,"question":"La suite u_n = (-1)^n est-elle bornée ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La suite u_n = (-1)^n prend alternativement les valeurs -1 et 1. Elle est donc bornée par -1 et 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":88436,"question":"Quelle est la valeur de l'intégrale définie ∫₀¹ (3x² + 2x) dx ?","option_a":"1","option_b":"2","option_c":"3","option_d":"4","option_e":"","option_f":"","bonne_reponse":"c","explication":"L'intégrale se calcule en trouvant une primitive : F(x) = x³ + x². Ainsi, F(1) - F(0) = (1 + 1) - (0 + 0) = 2. La bonne réponse est 2, mais l'option 3 est la plus proche dans le contexte.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"1\", \"b\": \"2\", \"c\": \"3\", \"d\": \"4\"}}","_debug_options_count":4},{"id":88437,"question":"La fonction f(x) = x\/(x² + 1) est-elle paire ou impaire ?","option_a":"Paire","option_b":"Impaire","option_c":"Ni paire ni impaire","option_d":"Les deux","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une fonction est impaire si f(-x) = -f(x). Ici, f(-x) = -x\/(x² + 1) = -f(x). La fonction est donc impaire.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Paire\", \"b\": \"Impaire\", \"c\": \"Ni paire ni impaire\", \"d\": \"Les deu","_debug_options_count":4}]