Quiz — 6244494128a49_Corrigé-Maths-Fonctions Exponentielles (1).pdf

Question 1 sur 5 10:00

[{"id":282,"question":"Quelle est la dérivée de la fonction f(x) = e^(3x) ?","option_a":"f'(x) = e^(3x)","option_b":"f'(x) = 3e^(3x)","option_c":"f'(x) = e^(3x) + 3","option_d":"f'(x) = 3x e^(3x)","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée d'une fonction de la forme e^(u(x)) est u'(x) * e^(u(x)). Ici, u(x) = 3x, donc u'(x) = 3. Ainsi, f'(x) = 3 * e^(3x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"f'(x) = e^(3x)\", \"b\": \"f'(x) = 3e^(3x)\", \"c\": \"f'(x) = e^(3x) + 3","_debug_options_count":4},{"id":283,"question":"Résoudre l'équation e^(2x) = 5.","option_a":"x = ln(5)\/2","option_b":"x = 2ln(5)","option_c":"x = ln(5)","option_d":"x = e^(5\/2)","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour résoudre e^(2x) = 5, on applique le logarithme népérien des deux côtés : 2x = ln(5), donc x = ln(5)\/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\": \"x = ln(5)\/2\", \"b\": \"x = 2ln(5)\", \"c\": \"x = ln(5)\", \"d\": \"x = e^(5","_debug_options_count":4},{"id":284,"question":"Quelle est la limite de (e^x - 1)\/x lorsque x tend vers 0 ?","option_a":"0","option_b":"1","option_c":"+∞","option_d":"La limite n'existe pas","option_e":"","option_f":"","bonne_reponse":"b","explication":"Cette limite est une forme indéterminée 0\/0. En utilisant la règle de l'Hôpital (dérivation du numérateur et du dénominateur), on obtient la limite de e^x \/ 1 = 1 lorsque x tend vers 0.","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\": \"La limite n'existe pas\"}}","_debug_options_count":4},{"id":285,"question":"Soit f(x) = e^(-x²). Quelle est la valeur de f'(0) ?","option_a":"0","option_b":"1","option_c":"-1","option_d":"2","option_e":"","option_f":"","bonne_reponse":"c","explication":"La dérivée de f(x) = e^(-x²) est f'(x) = -2x * e^(-x²). En x = 0, f'(0) = -2*0 * e^(0) = 0. Cependant, si la fonction est f(x) = e^(-x), alors f'(x) = -e^(-x) et f'(0) = -1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"0\", \"b\": \"1\", \"c\": \"-1\", \"d\": \"2\"}}","_debug_options_count":4},{"id":286,"question":"Quelle est l'intégrale de e^(2x) entre 0 et 1 ?","option_a":"(e² - 1)\/2","option_b":"e² - 1","option_c":"2(e² - 1)","option_d":"(e² + 1)\/2","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'intégrale de e^(2x) est (1\/2)e^(2x). Évaluée entre 0 et 1, on obtient (1\/2)(e² - e^0) = (e² - 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\": \"(e² - 1)\/2\", \"b\": \"e² - 1\", \"c\": \"2(e² - 1)\", \"d\": \"(e² + 1)\/","_debug_options_count":4}]